diff --git a/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/2301.02172v1.pdf.txt b/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/2301.02172v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2a61a4d15e82c241b02bff34731538de99e04994 --- /dev/null +++ b/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/2301.02172v1.pdf.txt @@ -0,0 +1,1533 @@ +OPTIMISED MORSE TRANSFORM OF A GAUSSIAN PROCESS +FEATURE SPACE +Fabio E. A. Albertani∗† , Alex J. W. Thom∗ +January 5, 2023 +ABSTRACT +Morse projections are well-known in chemistry and allow one, within a Morse potential approximation, +to redefine the potential in a simple quadratic form. The latter, being a non-linear transform, is also +very helpful for machine learning methods as they improve the performance of models by projecting +the feature space onto more well-suited coordinates. Usually, the Morse projection parameters are +taken from numerical benchmarks. We investigate the effect of changing these parameters latter +on the model learning, as well as using the machine learning method itself to make the parameters +decision. We find that learning is not necessarily improved by the latter and that general Morse +projections are extremely susceptible to changes in the training data. +1 +Introduction +Machine learning, as in many fields of science, has rev- +olutionised the way theoretical chemists approach the +interpolation of molecular properties. The many methods +encompassed by the machine learning framework pro- +vide tools to construct models of the former with great +accuracy1–3. A particular method that has seen success is +the Gaussian process (GP) framework which has seen ex- +tensive publications in machine learning potential energy +surface applications4–12. +The representation of the molecular geometry is an essen- +tial part of the ML building process and has seen many +“solutions” spring up through the years13. When using +global or local descriptors of the atomic configuration +of the system to build a “feature space”, one often uses +the internuclear distances as an underlying coordinates. +The latter are often transformed to improve the accu- +racy of models since ML models do not perform equally +when the training data is projected onto different feature +spaces. One known projection of the feature space is +the Morse transform of the internuclear distances which +often improves one’s ability to learn the surface14. +Given the ability of a GPs to learn the underlying pattern +of the target function15,16, it is interesting to consider a +GP which can change the underlying function in its opti- +misation. This is done by making the distance fed to the +kernel (see next section) transform with GP hyperparam- +eters within the kernel itself. +Many more feature space transformations could be con- +sidered (these are also not restricted to transformations +based on internuclear distances) but we will here discuss +the effect of the added “transformation hyperparameters” +on the GP optimisation process. +2 +Gaussian Processes +A Gaussian process is a machine learning regression +method and is defined as a collection of random vari- +ables, any finite number of which have a joint Gaussian +distribution16. An essential part of a GP model is its +kernel function which defines, over a feature space (the +input space of the GP), a measure of similarity. +There are many possible kernel functions one can defined, +as they only need to adhere to a few simple rules16. We +use here the Matérn class kernel multiplied by a constant +kernel (CK) and summed with a White Kernel (WK) to +model noise. The covariance between two vectors over +the feature space, X and X′ here, is given by +K(X, X′) = σ2 21−ν +Γ(ν) +� +√ +2ν d +ρ +�ν +Kν +� +√ +2ν d +ρ +� ++ λ2 +(1) +∗ Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, Lensfield Road, CB2 1EW +† fa381@cam.ac.uk +arXiv:2301.02172v1 [physics.chem-ph] 5 Jan 2023 + +Optimised Morse transform of a Gaussian process feature space +where Γ is the gamma function, Kν is the modified +Bessel function of the second kind of degree ν, ρ are +length scales and d is the Euclidean distance in feature +space |X − X′|. The ν parameter is not optimised and +defines the smoothness of the kernel: a GP with a Matérn +kernel of parameter ν = n + 0.5 is n-times differen- +tiable↓. We also explore an infinitely smooth version of +the Matérn kernel with ν → ∞, commonly known as the +radial basis function (RBF) kernel. +At a set of query points, forming a matrix Xp of size +Np × Nfeatures, a GP model predicts a Gaussian distri- +bution with a mean (sometimes called the latent func- +tion), here denoted y(Xp), and a variance, here denoted +∆(Xp), which is associated to the model confidence. For +a set of prediction points, Xp, the predicted distribution +are given by16: +y(Xp) = Kpt K−1 +tt y +∆(Xp) = Kpp − Kpt K−1 +tt Ktp +(2) +where the kernel matrices are subscripted with the ma- +trices they evaluate (p for query points and t for train- +ing) and the ijth element of the matrix Knm is given +by K(Xn,i, Xm,j). A common metric, used by the ML +community, to define the confidence in predictions is the +∆95% confidence interval which is given as y ± 2∆ for +GPs. +GPs are optimised by finding the most suited hyperpa- +rameters for its kernel. Using a Bayesian approach, one +finds the latter by maximising the log-marginal likelihood +(LML) defined as16 +LML = −1 +2yTK−1 +tt y − 1 +2log|Ktt| − n +2 log(2π) +(3) +where Ktt, as before, is the covariance matrix of the +training set to itself. The terms on the LHS of equa- +tion 3 can be understood as a fit, a regularisation and a +normalisation term respectively. +Practically, the maximisation is done by minimising +−LML but we will use the term LML as the surface +we minimise and the term “minimum” as a set of hy- +perparameters corresponding to a model selected by the +GP. +The LML exploration is done with the GMIN suite17–19 +which allows to give a full description of the minima, +both global and local, of the surface as well as their +connectivity. In order to visualise the surfaces, we use +disconnectivity graphs20–22 which represent, on a −LML +vertical scale, minima as vertical lines connected by tran- +sition states shown by connecting those lines. +3 +Methodology +If one takes the coordinates to be specified as a vector, X, +of N(N − 1)/2 internuclear distances, then the Morse +transformed coordinates form a vector defined as +T (X; M = {α, X0}) : +RN(N−1)/2 +→ RN(N−1)/2 +Xi +�→ exp +� +− (Xi − X0)/α +� +(4) +where α is the Morse parameter and X0 is the Morse +shift parameter. In order to simplify the notation, we will +write the Morse transformed vector XJ as T XJ. +The reasoning behind this transform is that an analytical +Morse potential becomes quadratic when projected onto +the coordinates T X. If one considers non-analytical po- +tentials, one expects the potentials to closer to quadratic +in T X than X. The simpler PES is better described by a +GP since the length scale of the problem becomes more +“unique”. Despite some specific bonds having chemically +derived optimal Morse parameters, there is not always a +straight-forward way to select those parameters. There +are two ways of optimising those parameters: a numeri- +cal optimisation to reduce the error on a testing set (the +traditional “best-fit” approach) and a Bayesian approach +with a Morse hyperparameter. As one does not want the +number of hyperparameters to be too large and optimise +in a very large space, we will set Morse hyparparameters +to be equal for all feature dimensions. +Taking a basic RBF kernel16, on a Morse transformed +feature space, the kernel is evaluated as +K( ˜XA = T XA, ˜XB = T XB; ρ) = +exp +� +− 1 +2( ˜XA − ˜XB)TP( ˜XA − ˜XB) +� +where P = In +� +����� +ρ−2 +1 +ρ−2 +2 +... +ρ−2 +n +� +����� +(5) +where we now used the matrix notation of the RBF ker- +nel and where ρi are the length scales along each feature +dimension. We use here the ˜XJ notation to differentiate +the fixed Morse projection of the kernel input (the Morse +parameters do not appear in the evaluation of the ker- +nel in the LHS of equation 5) from the projection taken +within the kernel itself, like in equation 6. +Instead of equation 5, one can use the internuclear dis- +tances, X as an input and optimise the Morse parameters +↓ For example, with ν = 2.5, one ensures that the GP latent function is physical since both the atomic forces (first derivative) and +atomic Hessians (second derivative) are smooth w.r.t. geometrical changes +2 + +Optimised Morse transform of a Gaussian process feature space +inside a “MorseRBF” kernel which is evaluated as +K(XA, XB; M, ρ) = +exp +� +− 1 +2(T XA − T XB)T P (T XA − T XB) +� +̸≡ exp +� +− 1 +2( ˜XA − ˜XB)T P ( ˜XA − ˜XB) +� +(6) +where P is the same matrix as the one in equation 5. In +the MorseRBF approach, the Morse parameters are hy- +perparameters of the kernel alongside the length scales. +The interesting approach of the kernel is that it does not, +as is common practice, optimise to the Morse parameters +that minimises the error on the testing set, the “best-fit” +approach, but instead uses a Bayesian approach and a +“statistically relevant” set of Morse parameters. +Regarding the X0 parameter, one can see in figure 1-2 +that the covariance function drops very quickly for data +points in the region X < X0 which could potentially lead +to a loss of information. With very compressed kernel +length scales around X < X0, data will not affect the +latent function. Learning was also performed for those +surfaces but were not considered further in this study. +The derivatives of the kernel with respect to each hyper- +parameter can be obtained analytically. The derivatives +with respect to the length scales are not affected by the +Morse transform and are equivalent to simply changing +the feature space in the derivatives of the standard RBF +kernel. The derivative with respect to the Morse parame- +ter can also be analytical obtained and is given by +∂αK(XA, XB) = +� +− 1 +2∂α +��T XA − T XB +�T� +P +�T XA − T XB +�� +K(XA, XB)+ +� +− 1 +2 +�T XA − T XB +�T P +∂α +��T XA − T XB +��� +K(XA, XB) += +�� XA +2α2 +T XA − XB +2α2 +T XB +�T P +�T XA − T XB +�� +K(XA, XB)+ +��T XA − T XB +�T P +� XA +2α2 +T XA − XB +2α2 +T XB +�� +K(XA, XB) +(7) +where T XI are Morse transformed vectors of the original +XI data points (to simplify the notation we did not write +the dependency on α and ρ). +In a very similar manner to the MorseRBF kernel, one +can define a MorseMatérn kernel starting from equation +1 and, despite analytical definitions of the gradient of the +kernel with respect to the Morse hyperparameter being +quite complicated, one can use numerical gradients and +optimise the Morse transformed kernel. +To understand the effect of the Morse kernels, we +compare the shape of the kernel functions in the non- +transformed space. Figure 1-2 shows a Matérn and a +MorseMatérn, projected back to the non-transformed di- +mension, to give a better insight. +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 0 +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 1 +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 2 +Figure 1: Covariance of the Matérn (ν = 2.5) kernel +(black lines) compared to the MorseMatérn kernel (red +lines), projected back onto X, for different X0 and with +α = 2.0. The covariance is quite unsymmetrical an the +forward influence is greater than the backward influence +since the transform expands the dataset at large X values. +The X0 parameter dampens the strong “elongation” of the +covariance at small X values and also strongly contracts +the covariance extent at X< X0 where the exponent in +equation 4 becomes positive. +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 0 +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 1 +−1 +0 +1 +2 +3 +X +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +K(XA, XB) +X0 = 2 +Figure 2: Covariance of the Matérn (ν = 2.5) kernel +(black lines) compared to the MorseMatérn kernel (red +lines), projected back onto X, for a larger α = 5.0. As +opposed to figure 1, the effect of the X0 does not affect +the covariance as much. The widening seen from the +previous figure is just a consequence of the length scale +hyperparameter being equal since one cannot associate +the Morse transformed length scale to the linear space +one. +Morse kernels allow the covariance to be unsymmetrical +in the internuclear space, which affects the correlation of +data in a very particular way over the feature space. As +shown on figure 1, the forward and backward correlation +differ and the extent of that effect depends greatly on the +α parameter. This increased flexibility of the kernel in +the model optimisation, allows to control the long range +effect of the training data for PES modelling. +3 + +Optimised Morse transform of a Gaussian process feature space +4 +Results +We use a training set of 48 water geometries calculated +UHF/aug-cc-pVDZ energies, sampled from a Boltzmann +distribution using the Metropolis–Hastings algorithm +with data up to 0.3 Ha above the equilibrium energy, +using the Q-Chem software23. Firstly, the training data is +projected on the 3 internuclear distances (ID) and Morse +transformed according to equation 4 to create the feature +space of the GPs. Secondly, the optimisable transforma- +tion using GPs with both the MorseRBF kernel and the +MorseMatérn class of kernel, we use the twice differen- +tiable kernel with ν = 2.5. Finally, in order to assess +the performance of each latent function, we define the +MAE of predictions on a testing set also sampled from a +Boltzmann distribution with data up to 0.2 Ha above the +equilibrium energy. +The Bayesian approach optimises the LML(θ) which +only includes the training data. This is very different +from optimising the MAE(θ) which only includes the +testing data (we do not explore this surface here). Since +we use GMIN and explore the whole LML landscape, one +can combine those approaches and rank local minima of +the LML surface with their respective MAEs. One then +selects the minimum which has the lowest error. This +gives an hybrid approach which optimises the MAE(θ | +∂ LML(θ) = 0). +The “best-fit” approach, given we use a single Morse +parameter, is a 1D minimisation of the MAE(α). One +also selects, for each GP trained with a different Morse +parameter, the LML minimum with the lowest MAE. We +first look at the results of the Matérn (ν = 2.5) kernel. +As mentioned before, the better performing minimum +of the LML is not always the global minimum. For the +Matérn (ν = 2.5) kernel, this is seen in figure 3: from +α = 2.2 it is a worse performing model that is lower on +the LML surface while the best performing minimum can +be followed. Even though there is no guarantee that fol- +lowing the better performing minimum on the LML as α +changes ensures selection of the best model, it also seem +unlikely that an eventually better performing minimum +at a different and larger α could not be followed back to +smaller Morse parameters where it disappears (with the +exception of α → 0). +2 +5 +10 +α +2.5 +5.0 +7.5 +MAE [mHa] +5 +10 +α +0 +1 +log(ρ0) = log(ρ1) +0 +1 +log(ρ2) +1 +5 +10 +α +Figure 3: Optimised hyperparameters of the Matérn +(ν = 2.5) kernel along the lengths scales representing +the Morse transformed O-H distances (ρ0 = ρ1) and +the Morse transformed H-H distance (ρ2) for different +minima as well as the MAE (of each respective minima) +on a test set. The blue-green dots represent the lower of +the minima on the LML while the red-yellow dots are +the second lowest minimum. The trajectory highlighted +in black represents the models with the lowest MAE in +both panels. Around α = 2.0, the two modes of selec- +tion yield different models (the grey area is plotted to aid +clarity of the switch between the two regimes). +The overall behaviour of the trajectories in hyperparam- +eters space of both LML minima represented in panel +(b) of figure 3 is expected. As α increases, the length +scales shorten. A larger Morse parameter compresses the +training data, shortening the distance between data. As +a consequence, a constant length scale would flatten the +GP latent function. This is prevented by the data term +of the LML, which causes the minima to move towards +shorter length scales. Moreover, the minima trajectory +can be observed to be almost linear towards larger Morse +parameters as the transform of equation 4 becomes itself +more linear since, in the limit of infinite α, the transform +is linear: +Xi �→ lim +α→∞ exp(−Xi/α) = +lim +α→∞ +� +1 − Xi +α + X2 +i +2α2 + . . . +� +≃ 1 − Xi +α +(8) +This opens the question of redefining length scales to re- +flect the change in α (for example as ρi → ρi/α) hyper- +parameter. This does not seem to affect the optimisation +process and will not be considered further. +Despite figure 3 showing only two minima, the GP has +multiple minima on the LML surface. However, only two +minima provided PES models with low MAEs. Figure +4 shows the disconnectivity graphs of the LML to show +the complexity of the surface for the Matérn (ν = 2.5) +kernel. It is surprising that the variations are so large +despite the training data being unchanged. +4 + +Optimised Morse transform of a Gaussian process feature space +Disconnectivity graphs show some surprising variations +that are sometimes a simple consequence of TSs being +very flat and hard to capture. This leads to the latter disap- +pearing after small changes to the LML space which has +strong consequences on the network of minima that can +be shown. This is the case of the graphs for α = 0.2 and +α = 0.4 in figure 4 where the latter finds TSs between +LML minima more easily. +epsilon +11 +12 +19 +0.2 +epsilon +2 +3 +4 +6 +7 +8 +9 +0.4 +epsilon +4 +5 +6 +7 +1.2 +epsilon +5 +7 +1.4 +epsilon +4 +6 +2.2 +epsilon +5 +7 +2.6 +epsilon +10 +11 +2.8 +epsilon +6 +7 +15 +3.0 +epsilon +7 +8 +9 +3.2 +epsilon +6 +7 +3.6 +epsilon +1 +8 +9 +4.0 +epsilon +1 +6 +4.4 +epsilon +1 +6 +7 +11 +4.6 +epsilon +2 +10 +4.8 +epsilon +7 +8 +5.0 +epsilon +9 +10 +11 +5.8 +epsilon +3 +4 +7.4 +epsilon +2 +3 +7.8 +epsilon +10 +11 +12 +14 +8.2 +epsilon +1 +7 +9.0 +epsilon +5 +6 +9 +9.8 +epsilon +1 +4 +10.2 +epsilon +2 +3 +10.6 +Figure 4: LMLs disconnectivity graphs for the Matérn +(ν = 2.5) kernels. Labels underneath each graph denote +the α parameter for the corresponding GP LML. +The minimum on the LML with the lowest MAE is al- +ways shown on the graphs and, despite its connectivity to +other LML minima changing, is easy to follow. GPs con- +verge towards a “good” model when the Morse parameter +is large enough and all latent function for GPs with α > 1 +perform similarly, as shown by the plateau in figure 3. +The latent functions, given in figure 5, are also similar. +The PES models are shown for the GP with a Morse +parameter of α = 2.0 (as it has been used extensively in +the previous chapter) and for comparison purposes for +the GP with a Morse parameter of α = 5.0 where the +MAE of the best model is reaching a “plateau”. +MAE: 1.58 mHa +0.25 +0.50 +0.75 +exp(−rO-H1/2.0) +0.25 +0.50 +0.75 +exp(−rO-H2/2.0) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +MAE: 1.47 mHa +0.75 +exp(−rO-H1/5.0) +0.75 +exp(−rO-H2/5.0) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +Figure 5: Resulting PES, projected on the Morse trans- +formed O-H nuclear distances, for Matérn kernels trained +on Morse transformed spaces with parameters α =2.0 +(higher graph) and α = 5.0 (lower graph). The magenta +lines are isovalue contours of the kernel function where +the covariance to the highlighted point is equal nσ2/4 +for n = 3, 2, 1 where σ is the amplitude hyperparameter +of equation 1. +The RBF kernel seem to have a much less stable LML +landscapes (see figure 6). However, the lowest MAE(θ) +models is always found to be the lowest minimum on the +LML(θ). As opposed to the Matérn kernel, the optimal +Morse parameter value to minimise the MAE is more +distinct (see figure 8). +The Gaussian process for the optimal value seem to cor- +respond to a value where the training data is not too +compressed and does not allow the RBF kernel to over fit. +Despite the MAE being similar to the Matérn kernel best +performing models, the length scales are shorter and the +latent function, displayed on figure 7, shows that the RBF +model predictions are only reliable close to the training +data and do not “carry” any of the information to longer +bond lengths, like the Matérn kernel does. +5 + +Optimised Morse transform of a Gaussian process feature space +epsilon +15 +16 +17 +0.4 +epsilon +9 +11 +0.6 +epsilon +5 +12 +0.8 +epsilon +4 +11 +1.2 +epsilon +6 +13 +17 +1.4 +epsilon +17 +19 +1.6 +epsilon +5 +12 +16 +1.8 +epsilon +19 +24 +25 +26 +118 +2.2 +epsilon +22 +23 +2.4 +epsilon +1 +4 +2.6 +epsilon +28 +29 +41 +48 +2.8 +epsilon +13 +16 +3.0 +epsilon +13 +18 +3.2 +epsilon +19 +20 +22 +3.6 +epsilon +5 +17 +3.8 +epsilon +13 +16 +4.0 +epsilon +13 +18 +4.4 +epsilon +13 +16 +4.6 +epsilon +5 +13 +5.0 +epsilon +7 +11 +12 +6.2 +epsilon +1 +3 +6 +7 +8 +9 +7.0 +epsilon +1 +4 +8 +10 +11 +12 +15 +16 +17 +18 +7.4 +epsilon +4 +6 +7.8 +epsilon +5 +12 +13 +8.2 +epsilon +1 +4 +7 +9 +10 +11 +8.6 +epsilon +7 +9 +9.4 +epsilon +1 +8 +10 +11 +12 +9.8 +epsilon +7 +9 +10 +10.2 +epsilon +8 +9 +10.6 +Figure 6: LMLs disconnectivity graphs for the RBF ker- +nels. Again, the labels underneath each graph denote the +α parameter for the corresponding GP LML. +Compared to the graphs of the Matérn kernel in figure +4, the graphs for the RBF kernel in figure 6 show more +minima and show stronger changes. This is due to the +tendency for the RBF kernel to over fit data giving a more +complex LML landscape in the region with short length +scales. The TSs are also harder to optimise, in these short +length scale regions, making the graphs rapidly changing. +The most performant GP is found for α = 0.8 and one +can see, in figure 8, that its MAE is similar to the best +Matérn GPs. The resulting latent function is shown in +figure 7 alongside the latent function for the GP trained +with the Morse parameter set to α = 2.0 to compare with +the model of the Matérn kernel in figure 5. One can see +that, for α = 2.0, despite similar MAEs, the RBF kernel +is more “local” and does not predict a meaningful PES at +longer bond lengths. +MAE: 1.60 mHa +0.25 +0.50 +exp(−rO-H1/0.8) +0.25 +0.50 +exp(−rO-H2/0.8) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +MAE: 3.52 mHa +0.25 +0.50 +0.75 +exp(−rO-H1/2.0) +0.25 +0.50 +0.75 +exp(−rO-H2/2.0) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +Figure 7: Resulting PES, projected on the Morse trans- +formed O-H nuclear distances, for RBF kernels trained +on Morse transformed spaces with parameters α = 0.8 +(higher graph) and α = 2.0 (lower graph) respectively. +These correspond to the minimum along the MAE plot +in figure 3 for the RBF and the Matérn kernels. The ma- +genta lines are isovalue contours of the kernel function. +In figure 7, as before, the contours represent an isocon- +tour of the kernel from a given sample↓. One can see +that despite the surfaces covering the same geometry +stretches, for the larger α, the optimised length scales +is much shorter and only allows a sample to span “ in- +fluence” over a small part of the considered space. This +leads to partial over fitting of the training data and a more +complicated PES model. +To summarise both the MAE optimisation of the GPs +trained with RBF and Matérn kernels, we plot the MAE +curves against the Morse parameter. This is the curve +that one minimises in the “best-fit” approach and leads +to selecting α = 0.8 for the RBF kernel and a larger +↓ The first contour is where the covariance function evaluates to 0.75σ2, where σ is the amplitude hyperparameter, while the second +one correspond to 0.5σ2. +6 + +Optimised Morse transform of a Gaussian process feature space +α > 2.0 for the Matérn kernel. The latter produces a +monotonically decreasing line which indicates that the +optimal Morse transform is a linear transform (since the +limit of α → ∞ reduces the Morse transform to the latter, +as explained in equation 8). This is an indication that it +is not optimal , for the Matérn kernel, to do the transfor- +mation and that the initial internuclear distances produce +a better feature space to learn on. +1 +5 +10 +α +2.5 +5.0 +7.5 +MAE [mHa] +Figure 8: MAE curves for the RBF (blue) and Matérn +(red) kernels against the Morse parameter. One can see +that the Matérn curve does not show a minimum and thus +indicates the best transform is the linear transform, i.e. +the limit of the Morse transform when α → ∞. +5 +Optimisable Morse Kernels +We now explore the optimisation of GP with the same +training data projected on the internuclear distances that +are Morse transformed in the kernel, for example as given +by equation 6 for the MorseRBF kernel. As usual the +kernels are scaled by an optimisable CK and have an +added optimisable noise given by a WK. The additional +hyperparameter, α, means we approach the Morse pa- +rameter optimisation in a fully Bayesian manner through +the LML minimisation. As mentioned before this does +remove the testing set in the optimisation and only the +training data affects its optimisation. +For the MorseRBF, multiple minima on the LML are +obtained and, when ranked with their respective MAEs, +the best GP models are found to be in the region of +0.5 < α < 1.0. This is in accordance with the MAE +curve, as seen in figure 8, of the GPs trained with standard +RBF kernels and fixed Morse transforms. As expected, +the MorseRBF GP best models latent function are very +similar to the RBF GP latent function with small α param- +eters↓: the lowest minima on the LML for the MorseRBF +is shown in figure 9. +0.5 +1.5 +2.5 +rO-H1 [˚A] +0.5 +1.5 +2.5 +rO-H2 [˚A] +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +0.5 +1.5 +2.5 +rO-H1 [˚A] +0.5 +1.5 +2.5 +rO-H2 [˚A] +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +Figure 9: Latent functions of GPs trained with a standard +RBF kernel (higher graph) and a fixed Morse parameter +close to the one exhibited by the lowest LML minimum +of the other GP, trained with a MorseRBF kernel (lower +graph). The change in length scale (shown by the kernel +isovalue contour extending further out) is simply a conse- +quence of the small difference in α value and the models +are essentially the same. +Obtaining MorseRBF models that resembles the RBF +ones is important as it tells us that the added hyperpa- +rameter dimension creates a convex LML hypersurface↓ +that can be optimised. The other hyperparameters of the +kernel are quite close to the ones of the kernel that does +not include the transformation when one fixes the latter +with the parameters found by the Morse kernel. +To summarise, the Morse kernels do optimise to lower +values which agree better with the optimal MAE(α) for +the RBF kernel but not for the Matérn kernel. Figure 10 +shows the disparity between the two Morse kernel ability +to replicate the “best-fit” approach.. +1 +5 +10 +α +2.5 +5.0 +7.5 +MAE [mHa] +Figure 10: MAE curves for the RBF (blue) and Matérn +(red) kernels against the Morse parameter. The dots rep- +resent the optimised Morse hyperparameter of the Morse +kernels (blue for MorseRBF and red for MorseMatérn) +with grey line to aid clarity. +6 +Changing the Training Data +Since the feature space is optimised differently with re- +spect to the selected training data, we will consider the +effect of adding data to the previously discussed models. +We still use the MAE of the final GP model but we use +two different testing sets. The new set is also taken from a +↓ The MorseRBF kernel is equal to a RBF kernel and a fixed Morse projection with the optimal Morse hyperparameter. +↓ It should be made clear again that we are technically talking about the −LML surface which we are optimising. On the true LML +surface this would be concave. +7 + +Optimised Morse transform of a Gaussian process feature space +Boltzmann distibution but at a higher temperature which +allows data to be sampled 0.4 Ha above the equilibrium +energy. +The training data is changed by adding data sampled +from NM clusters↓. Two things are interesting to fol- +low: the effect of increasing the size of the dataset on +the MAE(α) curves as well as the progression of LML- +optimised Morse hyperparameters. +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(a) N = 29 +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(b) N = 39 +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(c) N = 49 +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(d) N = 59 +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(e) N = 69 +2 +5 +10 +α +0 +2 +4 +MAElow [mHa] +8 +16 +MAEhigh [mHa] +(f) N = 79 +Figure 11: +Different MAE(α) curves for different +datasets. The two colours represent the MAE on different +testing sets↓ to see the dependency of the minimum of +the MAE(α) with respect to the chosen testing set. There +is no clear choice for an α, although there seem to be +a preference for small α values in the RBF kernel, un- +til training data becomes rather large and most Morse +parameters perform equally. +A first observation that can be made from the curves, +in figure 11, is that a different testing set can lead to +a different optimal Morse parameter. A second impor- +tant aspect is that small changes to the training set (in +this case adding training data) can importantly alter the +curves. For the latter, it is a surprising result since the +new training data does not differ from the original train- +ing data in terms of what it describes. The new sampled +training data does not allow the GP to understand new +patterns in the target function, which were not seen in +the original set. One could expect that consequently the +changes in the training data would not affect the relative +performance of GPs with different Morse parameters. +As data is added to the original training set, the MAE +curves tend to flatten and stop exhibiting a clear mini- +mum. The optimal Morse parameter is not well defined +and GPs, despite having different projections on their +feature spaces, perform similarly in terms of MAE. The +sparsity of training data is reduced, which makes the fea- +ture space less relevant: dense training data is likely to +perform well in any way it is projected. +Consequently, instead of additional data making the min- +imum of the MAE(α) more and more distinct, one sees +the minimum disappearing. +0.2 +0.5 +ρ0 = ρ1 +0.2 +0.5 +ρ2 +5 +15 +25 +35 +45 +RBF +0.5 +1.5 +ρ0 = ρ1 +0.5 +1.5 +ρ2 +5 +15 +25 +35 +45 +Matérn (ν = 2.5) +Figure 12: Trajectories of optimal models in hyperpa- +rameter space for different datasets (each dataset is rep- +resented by a different colour and the dots follow their +respective best minima on the LML landscapes). The +end of the trajectory (where the label is given) is going +towards larger Morse parameters and each trajectory has +points, since the progression is smooth, ordered for in- +creasing α along itself. The colourbar is given as greys +since it has shared by all training sets. +In figure 12, each progression in hyperparameter space +seem to have an initial curved trajectory followed by a +linear trajectory. This linear regime was already observed +in figure 3 as a consequence of the Morse transform limit +(see equation 8). If one considers the end of the trajectory +of the Matérn GPs, one sees that the optimised length +scales of GPs with the same Morse projection increases +with the training set size. The trend is less clear for the +RBF GPs where trajectories are not as well-behaved. +Some latent functions of the Matérn GPs are plotted in +figure 13. Despite the length scale growing larger, the +PESs seem to strongly “oscillate”, as if they had a short +length scale. This is particularly seen away from the data +as seen in panel (e) and (f). +MAE: 3.06 mHa +MAE: 0.73 mHa +MAE: 0.39 mHa +0.25 +0.75 +exp(−rO−H1/2) +0.25 +0.75 +exp(−rO−H2/2) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +(a) N = 29 +0.25 +0.75 +exp(−rO−H1/2) +0.25 +0.75 +exp(−rO−H2/2) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +(b) N = 49 +0.25 +0.75 +exp(−rO−H1/2) +0.25 +0.75 +exp(−rO−H2/2) +-76.2 +-76.0 +-75.8 +-75.6 +-75.4 +-75.2 +(c) N = 69 +Figure 13: Resulting PES, projected on the Morse trans- +formed O-H internuclear distances, for Matérn kernels +trained on Morse transformed spaces with parameters +α =2.0 for different training dataset sizes. +If one thinks of the optimisation in a Bayesian sense, one +would expect the opposite where minima on the LML +become more well-defined as new data, if it still agrees +↓ There is no overlap of the two training set. The additional data is added incrementally to the original training data with batches of +5 random samples drawn from the NM clusters. +8 + +9.0 +6.0 +Q +3.0Optimised Morse transform of a Gaussian process feature space +with the hyperparameters of the model of that particular +minimum, is added. +6.1 +Optimisable Morse kernels for changing data +Gaussian processes trained with a MorseMatérn (ν = +2.5) kernel and a MorseRBF kernel are trained on the +same datasets and compared to the MAE(α) trends of +figure 11. These results do not use the full GMIN imple- +mentation and use a basic sklearn24 L-BFGS approach. +All reported minima have projected gradients converged +to 10−2, which is not as tight a convergence criterion as +LML minima that were found using the GMIN imple- +mentation. Table 1 summaries the results for both the +Morse-transformed kernels with the training data ranging +from the initial set to the fully “merged” one in incre- +ments of 5 data points. +N +MorseRBF: α +MorseRBF: ρ0 = ρ1 +MorseMatérn: α +MorseMatérn: ρ0 = ρ1 +29 +0.534 +0.41 +0.580 +2.08 +34 +0.774 +0.62 +0.960 +0.17 +39 +0.804 +0.39 +0.804 +0.86 +44 +0.937 +0.22 +0.514 +3.05 +49 +0.802 +0.18 +0.539 +0.88 +54 +0.806 +0.12 +0.569 +0.55 +59 +0.440 +0.42 +0.993 +2.40 +64 +0.391 +0.32 +0.622 +2.27 +69 +0.949 +0.23 +0.448 +0.15 +74 +0.750 +0.50 +0.525 +0.72 +79 +0.251 +0.12 +0.721 +0.30 +Table 1: Summary of Morse parameters and length scales for optimised Morse kernels with an increasing size of training +set (number of samples given by N). When the Morse kernel performs better than its “standard” kernel counterpart, the +values are written in green. One can see that the improvement is very rarely seen for the RBF kernel side and only seen +about half the time for the kernel Matérn. +There does not seem to be a smooth transition as data is +added in terms of an optimal α and there does not seem +to a consistently better method for choosing the Morse +parameters in the Bayesian GP framework. +Since N, the size of the training set, is not a continu- +ous variable that changes the LML smoothly, it is not +necessary that the change in α is smooth. One expects +the N parameter to produce smooth changes of the LML +minima. It is clear that MorseMatérn kernels, like the +MorseRBF kernels, tend to favour small α values when +optimised through the LML but it is hard to understand +the reason for the progression seen in the tables above. +These optimal values are not similar to the usual values +used for Morse projections in the literature but it does +not in terms of MAE discredit those choices. +7 +Conclusion +Optimising, with a Bayesian approach, the feature space +of the GP to produce more performant latent functions, +in terms of MAE, is not straight forward. The LML is +made more complex by the additional DOF and there is +a strong correlation between the hyperparameters. Differ- +ent transforms might not necessarily suffer from this but, +for the Morse transform, the relation between the Morse +parameter and the length scales is evident. +For the Morse transform, since the limit of its parameters +tending to a certain value (α → ∞ here) give a linear +transform, optimising the transform parameters can also +inform us on the “usefulness” of the transform. The +curves of figure 8, for example, show that the transform +9 + +Optimised Morse transform of a Gaussian process feature space +is actually producing less performant GP models for this +system. +The “best-fit” approach also produced some interesting +results regarding the choice of testing set to produce the +MAE curves one minimises. A target function is as- +sumed to have an optimal Morse parameter to project +it to a “simpler” surface↓. However, a different testing +set can significantly affect the result of the minimisation +(this cannot be interpreted in the Bayesian approach since +there is no testing set in the LML minimisation). This +should not be the case if the testing set is “complete”, +in the sense that new samples are drawn from the same +distribution. It will be the case if the distribution change↓ +which suggests that one should use testing data that is +suitable for the intended use of the GP model. +Acknowledgments +I would like to thank the Royal Society for funding as +well as the Wales group of the University of Cambridge +for providing access to the GMIN suite17. Moreover, I +would like to thank Angelos Michaelides and Albert Par- +tay Bartòk for fruitful discussion during my PhD viva +that improved this work. +References +(1) +J. Behler, The Journal of Chemical Physics, 2016, +145, 170901. +(2) +F. Noé, A. Tkatchenko, K.-R. 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Brucher, M. Perrot +and E. Duchesnay, Journal of Machine Learning +Research, 2011, 12, 2825–2830. +11 + diff --git a/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/load_file.txt b/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..11a164a186debebe8accc95cddd265418dd67e9e --- /dev/null +++ b/-dA0T4oBgHgl3EQfPP9i/content/tmp_files/load_file.txt @@ -0,0 +1,919 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf,len=918 +page_content='OPTIMISED MORSE TRANSFORM OF A GAUSSIAN PROCESS FEATURE SPACE Fabio E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Albertani∗† , Alex J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Thom∗ January 5, 2023 ABSTRACT Morse projections are well-known in chemistry and allow one, within a Morse potential approximation, to redefine the potential in a simple quadratic form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The latter, being a non-linear transform, is also very helpful for machine learning methods as they improve the performance of models by projecting the feature space onto more well-suited coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Usually, the Morse projection parameters are taken from numerical benchmarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We investigate the effect of changing these parameters latter on the model learning, as well as using the machine learning method itself to make the parameters decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We find that learning is not necessarily improved by the latter and that general Morse projections are extremely susceptible to changes in the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 1 Introduction Machine learning, as in many fields of science, has rev- olutionised the way theoretical chemists approach the interpolation of molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The many methods encompassed by the machine learning framework pro- vide tools to construct models of the former with great accuracy1–3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A particular method that has seen success is the Gaussian process (GP) framework which has seen ex- tensive publications in machine learning potential energy surface applications4–12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The representation of the molecular geometry is an essen- tial part of the ML building process and has seen many “solutions” spring up through the years13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' When using global or local descriptors of the atomic configuration of the system to build a “feature space”, one often uses the internuclear distances as an underlying coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The latter are often transformed to improve the accu- racy of models since ML models do not perform equally when the training data is projected onto different feature spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One known projection of the feature space is the Morse transform of the internuclear distances which often improves one’s ability to learn the surface14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Given the ability of a GPs to learn the underlying pattern of the target function15,16, it is interesting to consider a GP which can change the underlying function in its opti- misation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is done by making the distance fed to the kernel (see next section) transform with GP hyperparam- eters within the kernel itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Many more feature space transformations could be con- sidered (these are also not restricted to transformations based on internuclear distances) but we will here discuss the effect of the added “transformation hyperparameters” on the GP optimisation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 2 Gaussian Processes A Gaussian process is a machine learning regression method and is defined as a collection of random vari- ables, any finite number of which have a joint Gaussian distribution16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' An essential part of a GP model is its kernel function which defines, over a feature space (the input space of the GP), a measure of similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' There are many possible kernel functions one can defined, as they only need to adhere to a few simple rules16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We use here the Matérn class kernel multiplied by a constant kernel (CK) and summed with a White Kernel (WK) to model noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The covariance between two vectors over the feature space, X and X′ here, is given by K(X, X′) = σ2 21−ν Γ(ν) � √ 2ν d ρ �ν Kν � √ 2ν d ρ � + λ2 (1) ∗ Yusuf Hamied Department of Chemistry, University of Cambridge, Cambridge, Lensfield Road, CB2 1EW † fa381@cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='02172v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='chem-ph] 5 Jan 2023 Optimised Morse transform of a Gaussian process feature space where Γ is the gamma function, Kν is the modified Bessel function of the second kind of degree ν, ρ are length scales and d is the Euclidean distance in feature space |X − X′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The ν parameter is not optimised and defines the smoothness of the kernel: a GP with a Matérn kernel of parameter ν = n + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 is n-times differen- tiable↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We also explore an infinitely smooth version of the Matérn kernel with ν → ∞, commonly known as the radial basis function (RBF) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' At a set of query points, forming a matrix Xp of size Np × Nfeatures, a GP model predicts a Gaussian distri- bution with a mean (sometimes called the latent func- tion), here denoted y(Xp), and a variance, here denoted ∆(Xp), which is associated to the model confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' For a set of prediction points, Xp, the predicted distribution are given by16: y(Xp) = Kpt K−1 tt y ∆(Xp) = Kpp − Kpt K−1 tt Ktp (2) where the kernel matrices are subscripted with the ma- trices they evaluate (p for query points and t for train- ing) and the ijth element of the matrix Knm is given by K(Xn,i, Xm,j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A common metric, used by the ML community, to define the confidence in predictions is the ∆95% confidence interval which is given as y ± 2∆ for GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' GPs are optimised by finding the most suited hyperpa- rameters for its kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Using a Bayesian approach, one finds the latter by maximising the log-marginal likelihood (LML) defined as16 LML = −1 2yTK−1 tt y − 1 2log|Ktt| − n 2 log(2π) (3) where Ktt, as before, is the covariance matrix of the training set to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The terms on the LHS of equa- tion 3 can be understood as a fit, a regularisation and a normalisation term respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Practically, the maximisation is done by minimising −LML but we will use the term LML as the surface we minimise and the term “minimum” as a set of hy- perparameters corresponding to a model selected by the GP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The LML exploration is done with the GMIN suite17–19 which allows to give a full description of the minima, both global and local, of the surface as well as their connectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In order to visualise the surfaces, we use disconnectivity graphs20–22 which represent, on a −LML vertical scale, minima as vertical lines connected by tran- sition states shown by connecting those lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 3 Methodology If one takes the coordinates to be specified as a vector, X, of N(N − 1)/2 internuclear distances, then the Morse transformed coordinates form a vector defined as T (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' M = {α, X0}) : RN(N−1)/2 → RN(N−1)/2 Xi �→ exp � − (Xi − X0)/α � (4) where α is the Morse parameter and X0 is the Morse shift parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In order to simplify the notation, we will write the Morse transformed vector XJ as T XJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The reasoning behind this transform is that an analytical Morse potential becomes quadratic when projected onto the coordinates T X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' If one considers non-analytical po- tentials, one expects the potentials to closer to quadratic in T X than X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The simpler PES is better described by a GP since the length scale of the problem becomes more “unique”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Despite some specific bonds having chemically derived optimal Morse parameters, there is not always a straight-forward way to select those parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' There are two ways of optimising those parameters: a numeri- cal optimisation to reduce the error on a testing set (the traditional “best-fit” approach) and a Bayesian approach with a Morse hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As one does not want the number of hyperparameters to be too large and optimise in a very large space, we will set Morse hyparparameters to be equal for all feature dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Taking a basic RBF kernel16, on a Morse transformed feature space, the kernel is evaluated as K( ˜XA = T XA, ˜XB = T XB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' ρ) = exp � − 1 2( ˜XA − ˜XB)TP( ˜XA − ˜XB) � where P = In � ����� ρ−2 1 ρ−2 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' ρ−2 n � ����� (5) where we now used the matrix notation of the RBF ker- nel and where ρi are the length scales along each feature dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We use here the ˜XJ notation to differentiate the fixed Morse projection of the kernel input (the Morse parameters do not appear in the evaluation of the ker- nel in the LHS of equation 5) from the projection taken within the kernel itself, like in equation 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Instead of equation 5, one can use the internuclear dis- tances, X as an input and optimise the Morse parameters ↓ For example, with ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5, one ensures that the GP latent function is physical since both the atomic forces (first derivative) and atomic Hessians (second derivative) are smooth w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' geometrical changes 2 Optimised Morse transform of a Gaussian process feature space inside a “MorseRBF” kernel which is evaluated as K(XA, XB;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' M, ρ) = exp � − 1 2(T XA − T XB)T P (T XA − T XB) � ̸≡ exp � − 1 2( ˜XA − ˜XB)T P ( ˜XA − ˜XB) � (6) where P is the same matrix as the one in equation 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In the MorseRBF approach, the Morse parameters are hy- perparameters of the kernel alongside the length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The interesting approach of the kernel is that it does not, as is common practice, optimise to the Morse parameters that minimises the error on the testing set, the “best-fit” approach, but instead uses a Bayesian approach and a “statistically relevant” set of Morse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Regarding the X0 parameter, one can see in figure 1-2 that the covariance function drops very quickly for data points in the region X < X0 which could potentially lead to a loss of information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' With very compressed kernel length scales around X < X0, data will not affect the latent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Learning was also performed for those surfaces but were not considered further in this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The derivatives of the kernel with respect to each hyper- parameter can be obtained analytically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The derivatives with respect to the length scales are not affected by the Morse transform and are equivalent to simply changing the feature space in the derivatives of the standard RBF kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The derivative with respect to the Morse parame- ter can also be analytical obtained and is given by ∂αK(XA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' XB) = � − 1 2∂α ��T XA − T XB �T� P �T XA − T XB �� K(XA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' XB)+ � − 1 2 �T XA − T XB �T P ∂α ��T XA − T XB ��� K(XA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' XB) = �� XA 2α2 T XA − XB 2α2 T XB �T P �T XA − T XB �� K(XA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' XB)+ ��T XA − T XB �T P � XA 2α2 T XA − XB 2α2 T XB �� K(XA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' XB) (7) where T XI are Morse transformed vectors of the original XI data points (to simplify the notation we did not write the dependency on α and ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In a very similar manner to the MorseRBF kernel, one can define a MorseMatérn kernel starting from equation 1 and, despite analytical definitions of the gradient of the kernel with respect to the Morse hyperparameter being quite complicated, one can use numerical gradients and optimise the Morse transformed kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' To understand the effect of the Morse kernels, we compare the shape of the kernel functions in the non- transformed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Figure 1-2 shows a Matérn and a MorseMatérn, projected back to the non-transformed di- mension, to give a better insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 0 −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 1 −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 2 Figure 1: Covariance of the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel (black lines) compared to the MorseMatérn kernel (red lines), projected back onto X, for different X0 and with α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The covariance is quite unsymmetrical an the forward influence is greater than the backward influence since the transform expands the dataset at large X values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The X0 parameter dampens the strong “elongation” of the covariance at small X values and also strongly contracts the covariance extent at X< X0 where the exponent in equation 4 becomes positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 0 −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 1 −1 0 1 2 3 X 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 K(XA, XB) X0 = 2 Figure 2: Covariance of the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel (black lines) compared to the MorseMatérn kernel (red lines), projected back onto X, for a larger α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As opposed to figure 1, the effect of the X0 does not affect the covariance as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The widening seen from the previous figure is just a consequence of the length scale hyperparameter being equal since one cannot associate the Morse transformed length scale to the linear space one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Morse kernels allow the covariance to be unsymmetrical in the internuclear space, which affects the correlation of data in a very particular way over the feature space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As shown on figure 1, the forward and backward correlation differ and the extent of that effect depends greatly on the α parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This increased flexibility of the kernel in the model optimisation, allows to control the long range effect of the training data for PES modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 3 Optimised Morse transform of a Gaussian process feature space 4 Results We use a training set of 48 water geometries calculated UHF/aug-cc-pVDZ energies, sampled from a Boltzmann distribution using the Metropolis–Hastings algorithm with data up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='3 Ha above the equilibrium energy, using the Q-Chem software23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Firstly, the training data is projected on the 3 internuclear distances (ID) and Morse transformed according to equation 4 to create the feature space of the GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Secondly, the optimisable transforma- tion using GPs with both the MorseRBF kernel and the MorseMatérn class of kernel, we use the twice differen- tiable kernel with ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Finally, in order to assess the performance of each latent function, we define the MAE of predictions on a testing set also sampled from a Boltzmann distribution with data up to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 Ha above the equilibrium energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The Bayesian approach optimises the LML(θ) which only includes the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is very different from optimising the MAE(θ) which only includes the testing data (we do not explore this surface here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Since we use GMIN and explore the whole LML landscape, one can combine those approaches and rank local minima of the LML surface with their respective MAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One then selects the minimum which has the lowest error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This gives an hybrid approach which optimises the MAE(θ | ∂ LML(θ) = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The “best-fit” approach, given we use a single Morse parameter, is a 1D minimisation of the MAE(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One also selects, for each GP trained with a different Morse parameter, the LML minimum with the lowest MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We first look at the results of the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As mentioned before, the better performing minimum of the LML is not always the global minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' For the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel, this is seen in figure 3: from α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 it is a worse performing model that is lower on the LML surface while the best performing minimum can be followed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Even though there is no guarantee that fol- lowing the better performing minimum on the LML as α changes ensures selection of the best model, it also seem unlikely that an eventually better performing minimum at a different and larger α could not be followed back to smaller Morse parameters where it disappears (with the exception of α → 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 2 5 10 α 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 MAE [mHa] 5 10 α 0 1 log(ρ0) = log(ρ1) 0 1 log(ρ2) 1 5 10 α Figure 3: Optimised hyperparameters of the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel along the lengths scales representing the Morse transformed O-H distances (ρ0 = ρ1) and the Morse transformed H-H distance (ρ2) for different minima as well as the MAE (of each respective minima) on a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The blue-green dots represent the lower of the minima on the LML while the red-yellow dots are the second lowest minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The trajectory highlighted in black represents the models with the lowest MAE in both panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Around α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0, the two modes of selec- tion yield different models (the grey area is plotted to aid clarity of the switch between the two regimes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The overall behaviour of the trajectories in hyperparam- eters space of both LML minima represented in panel (b) of figure 3 is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As α increases, the length scales shorten.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A larger Morse parameter compresses the training data, shortening the distance between data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As a consequence, a constant length scale would flatten the GP latent function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is prevented by the data term of the LML, which causes the minima to move towards shorter length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Moreover, the minima trajectory can be observed to be almost linear towards larger Morse parameters as the transform of equation 4 becomes itself more linear since, in the limit of infinite α, the transform is linear: Xi �→ lim α→∞ exp(−Xi/α) = lim α→∞ � 1 − Xi α + X2 i 2α2 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' � ≃ 1 − Xi α (8) This opens the question of redefining length scales to re- flect the change in α (for example as ρi → ρi/α) hyper- parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This does not seem to affect the optimisation process and will not be considered further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Despite figure 3 showing only two minima, the GP has multiple minima on the LML surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' However, only two minima provided PES models with low MAEs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Figure 4 shows the disconnectivity graphs of the LML to show the complexity of the surface for the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' It is surprising that the variations are so large despite the training data being unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 4 Optimised Morse transform of a Gaussian process feature space Disconnectivity graphs show some surprising variations that are sometimes a simple consequence of TSs being very flat and hard to capture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This leads to the latter disap- pearing after small changes to the LML space which has strong consequences on the network of minima that can be shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is the case of the graphs for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 and α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 in figure 4 where the latter finds TSs between LML minima more easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' epsilon 11 12 19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 2 3 4 6 7 8 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 4 5 6 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 5 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 4 6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 5 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 10 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 6 7 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 7 8 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 6 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 1 8 9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 1 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 1 6 7 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 2 10 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 7 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 9 10 11 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 3 4 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 2 3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 10 11 12 14 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 1 7 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 5 6 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 1 4 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 2 3 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 Figure 4: LMLs disconnectivity graphs for the Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Labels underneath each graph denote the α parameter for the corresponding GP LML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The minimum on the LML with the lowest MAE is al- ways shown on the graphs and, despite its connectivity to other LML minima changing, is easy to follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' GPs con- verge towards a “good” model when the Morse parameter is large enough and all latent function for GPs with α > 1 perform similarly, as shown by the plateau in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The latent functions, given in figure 5, are also similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The PES models are shown for the GP with a Morse parameter of α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 (as it has been used extensively in the previous chapter) and for comparison purposes for the GP with a Morse parameter of α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 where the MAE of the best model is reaching a “plateau”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' MAE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='58 mHa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 MAE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='47 mHa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H1/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 Figure 5: Resulting PES, projected on the Morse trans- formed O-H nuclear distances, for Matérn kernels trained on Morse transformed spaces with parameters α =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 (higher graph) and α = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 (lower graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The magenta lines are isovalue contours of the kernel function where the covariance to the highlighted point is equal nσ2/4 for n = 3, 2, 1 where σ is the amplitude hyperparameter of equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The RBF kernel seem to have a much less stable LML landscapes (see figure 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' However, the lowest MAE(θ) models is always found to be the lowest minimum on the LML(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As opposed to the Matérn kernel, the optimal Morse parameter value to minimise the MAE is more distinct (see figure 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The Gaussian process for the optimal value seem to cor- respond to a value where the training data is not too compressed and does not allow the RBF kernel to over fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Despite the MAE being similar to the Matérn kernel best performing models, the length scales are shorter and the latent function, displayed on figure 7, shows that the RBF model predictions are only reliable close to the training data and do not “carry” any of the information to longer bond lengths, like the Matérn kernel does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 5 Optimised Morse transform of a Gaussian process feature space epsilon 15 16 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 9 11 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 5 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 4 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 6 13 17 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 17 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 5 12 16 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 19 24 25 26 118 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 22 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 1 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 28 29 41 48 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 13 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 13 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 19 20 22 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 5 17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 13 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 13 18 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 13 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 5 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 7 11 12 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 1 3 6 7 8 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 epsilon 1 4 8 10 11 12 15 16 17 18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 4 6 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 5 12 13 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 1 4 7 9 10 11 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 epsilon 7 9 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 epsilon 1 8 10 11 12 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 epsilon 7 9 10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 epsilon 8 9 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 Figure 6: LMLs disconnectivity graphs for the RBF ker- nels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Again, the labels underneath each graph denote the α parameter for the corresponding GP LML.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Compared to the graphs of the Matérn kernel in figure 4, the graphs for the RBF kernel in figure 6 show more minima and show stronger changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is due to the tendency for the RBF kernel to over fit data giving a more complex LML landscape in the region with short length scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The TSs are also harder to optimise, in these short length scale regions, making the graphs rapidly changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The most performant GP is found for α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 and one can see, in figure 8, that its MAE is similar to the best Matérn GPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The resulting latent function is shown in figure 7 alongside the latent function for the GP trained with the Morse parameter set to α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 to compare with the model of the Matérn kernel in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One can see that, for α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0, despite similar MAEs, the RBF kernel is more “local” and does not predict a meaningful PES at longer bond lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' MAE: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='60 mHa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 exp(−rO-H1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 exp(−rO-H2/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 MAE: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='52 mHa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO-H2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 Figure 7: Resulting PES, projected on the Morse trans- formed O-H nuclear distances, for RBF kernels trained on Morse transformed spaces with parameters α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 (higher graph) and α = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 (lower graph) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' These correspond to the minimum along the MAE plot in figure 3 for the RBF and the Matérn kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The ma- genta lines are isovalue contours of the kernel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In figure 7, as before, the contours represent an isocon- tour of the kernel from a given sample↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One can see that despite the surfaces covering the same geometry stretches, for the larger α, the optimised length scales is much shorter and only allows a sample to span “ in- fluence” over a small part of the considered space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This leads to partial over fitting of the training data and a more complicated PES model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' To summarise both the MAE optimisation of the GPs trained with RBF and Matérn kernels, we plot the MAE curves against the Morse parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is the curve that one minimises in the “best-fit” approach and leads to selecting α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 for the RBF kernel and a larger ↓ The first contour is where the covariance function evaluates to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75σ2, where σ is the amplitude hyperparameter, while the second one correspond to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 6 Optimised Morse transform of a Gaussian process feature space α > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 for the Matérn kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The latter produces a monotonically decreasing line which indicates that the optimal Morse transform is a linear transform (since the limit of α → ∞ reduces the Morse transform to the latter, as explained in equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is an indication that it is not optimal , for the Matérn kernel, to do the transfor- mation and that the initial internuclear distances produce a better feature space to learn on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 1 5 10 α 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 MAE [mHa] Figure 8: MAE curves for the RBF (blue) and Matérn (red) kernels against the Morse parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One can see that the Matérn curve does not show a minimum and thus indicates the best transform is the linear transform, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' the limit of the Morse transform when α → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 5 Optimisable Morse Kernels We now explore the optimisation of GP with the same training data projected on the internuclear distances that are Morse transformed in the kernel, for example as given by equation 6 for the MorseRBF kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As usual the kernels are scaled by an optimisable CK and have an added optimisable noise given by a WK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The additional hyperparameter, α, means we approach the Morse pa- rameter optimisation in a fully Bayesian manner through the LML minimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As mentioned before this does remove the testing set in the optimisation and only the training data affects its optimisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' For the MorseRBF, multiple minima on the LML are obtained and, when ranked with their respective MAEs, the best GP models are found to be in the region of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 < α < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is in accordance with the MAE curve, as seen in figure 8, of the GPs trained with standard RBF kernels and fixed Morse transforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As expected, the MorseRBF GP best models latent function are very similar to the RBF GP latent function with small α param- eters↓: the lowest minima on the LML for the MorseRBF is shown in figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 rO-H1 [˚A] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 rO-H2 [˚A] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 rO-H1 [˚A] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 rO-H2 [˚A] 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 Figure 9: Latent functions of GPs trained with a standard RBF kernel (higher graph) and a fixed Morse parameter close to the one exhibited by the lowest LML minimum of the other GP, trained with a MorseRBF kernel (lower graph).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The change in length scale (shown by the kernel isovalue contour extending further out) is simply a conse- quence of the small difference in α value and the models are essentially the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Obtaining MorseRBF models that resembles the RBF ones is important as it tells us that the added hyperpa- rameter dimension creates a convex LML hypersurface↓ that can be optimised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The other hyperparameters of the kernel are quite close to the ones of the kernel that does not include the transformation when one fixes the latter with the parameters found by the Morse kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' To summarise, the Morse kernels do optimise to lower values which agree better with the optimal MAE(α) for the RBF kernel but not for the Matérn kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Figure 10 shows the disparity between the two Morse kernel ability to replicate the “best-fit” approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='. 1 5 10 α 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 MAE [mHa] Figure 10: MAE curves for the RBF (blue) and Matérn (red) kernels against the Morse parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The dots rep- resent the optimised Morse hyperparameter of the Morse kernels (blue for MorseRBF and red for MorseMatérn) with grey line to aid clarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 6 Changing the Training Data Since the feature space is optimised differently with re- spect to the selected training data, we will consider the effect of adding data to the previously discussed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' We still use the MAE of the final GP model but we use two different testing sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The new set is also taken from a ↓ The MorseRBF kernel is equal to a RBF kernel and a fixed Morse projection with the optimal Morse hyperparameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' ↓ It should be made clear again that we are technically talking about the −LML surface which we are optimising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' On the true LML surface this would be concave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 7 Optimised Morse transform of a Gaussian process feature space Boltzmann distibution but at a higher temperature which allows data to be sampled 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 Ha above the equilibrium energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The training data is changed by adding data sampled from NM clusters↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Two things are interesting to fol- low: the effect of increasing the size of the dataset on the MAE(α) curves as well as the progression of LML- optimised Morse hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (a) N = 29 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (b) N = 39 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (c) N = 49 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (d) N = 59 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (e) N = 69 2 5 10 α 0 2 4 MAElow [mHa] 8 16 MAEhigh [mHa] (f) N = 79 Figure 11: Different MAE(α) curves for different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The two colours represent the MAE on different testing sets↓ to see the dependency of the minimum of the MAE(α) with respect to the chosen testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' There is no clear choice for an α, although there seem to be a preference for small α values in the RBF kernel, un- til training data becomes rather large and most Morse parameters perform equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A first observation that can be made from the curves, in figure 11, is that a different testing set can lead to a different optimal Morse parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A second impor- tant aspect is that small changes to the training set (in this case adding training data) can importantly alter the curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' For the latter, it is a surprising result since the new training data does not differ from the original train- ing data in terms of what it describes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The new sampled training data does not allow the GP to understand new patterns in the target function, which were not seen in the original set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One could expect that consequently the changes in the training data would not affect the relative performance of GPs with different Morse parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' As data is added to the original training set, the MAE curves tend to flatten and stop exhibiting a clear mini- mum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The optimal Morse parameter is not well defined and GPs, despite having different projections on their feature spaces, perform similarly in terms of MAE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The sparsity of training data is reduced, which makes the fea- ture space less relevant: dense training data is likely to perform well in any way it is projected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Consequently, instead of additional data making the min- imum of the MAE(α) more and more distinct, one sees the minimum disappearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 ρ0 = ρ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 ρ2 5 15 25 35 45 RBF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 ρ0 = ρ1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5 ρ2 5 15 25 35 45 Matérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) Figure 12: Trajectories of optimal models in hyperpa- rameter space for different datasets (each dataset is rep- resented by a different colour and the dots follow their respective best minima on the LML landscapes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The end of the trajectory (where the label is given) is going towards larger Morse parameters and each trajectory has points, since the progression is smooth, ordered for in- creasing α along itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The colourbar is given as greys since it has shared by all training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' In figure 12, each progression in hyperparameter space seem to have an initial curved trajectory followed by a linear trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This linear regime was already observed in figure 3 as a consequence of the Morse transform limit (see equation 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' If one considers the end of the trajectory of the Matérn GPs, one sees that the optimised length scales of GPs with the same Morse projection increases with the training set size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The trend is less clear for the RBF GPs where trajectories are not as well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Some latent functions of the Matérn GPs are plotted in figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Despite the length scale growing larger, the PESs seem to strongly “oscillate”, as if they had a short length scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This is particularly seen away from the data as seen in panel (e) and (f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' MAE: 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='06 mHa MAE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='73 mHa MAE: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='39 mHa 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H1/2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H2/2) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 (a) N = 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H1/2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H2/2) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 (b) N = 49 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H1/2) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='75 exp(−rO−H2/2) 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='8 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='6 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='4 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='2 (c) N = 69 Figure 13: Resulting PES, projected on the Morse trans- formed O-H internuclear distances, for Matérn kernels trained on Morse transformed spaces with parameters α =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 for different training dataset sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' If one thinks of the optimisation in a Bayesian sense, one would expect the opposite where minima on the LML become more well-defined as new data, if it still agrees ↓ There is no overlap of the two training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The additional data is added incrementally to the original training data with batches of 5 random samples drawn from the NM clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 8 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0 Q 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='0Optimised Morse transform of a Gaussian process feature space with the hyperparameters of the model of that particular minimum, is added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='1 Optimisable Morse kernels for changing data Gaussian processes trained with a MorseMatérn (ν = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='5) kernel and a MorseRBF kernel are trained on the same datasets and compared to the MAE(α) trends of figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' These results do not use the full GMIN imple- mentation and use a basic sklearn24 L-BFGS approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' All reported minima have projected gradients converged to 10−2, which is not as tight a convergence criterion as LML minima that were found using the GMIN imple- mentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Table 1 summaries the results for both the Morse-transformed kernels with the training data ranging from the initial set to the fully “merged” one in incre- ments of 5 data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' N MorseRBF: α MorseRBF: ρ0 = ρ1 MorseMatérn: α MorseMatérn: ρ0 = ρ1 29 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='534 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='41 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='580 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='08 34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='774 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='448 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='15 74 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='525 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='72 79 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='251 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='721 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='30 Table 1: Summary of Morse parameters and length scales for optimised Morse kernels with an increasing size of training set (number of samples given by N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' When the Morse kernel performs better than its “standard” kernel counterpart, the values are written in green.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One can see that the improvement is very rarely seen for the RBF kernel side and only seen about half the time for the kernel Matérn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' There does not seem to be a smooth transition as data is added in terms of an optimal α and there does not seem to a consistently better method for choosing the Morse parameters in the Bayesian GP framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Since N, the size of the training set, is not a continu- ous variable that changes the LML smoothly, it is not necessary that the change in α is smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' One expects the N parameter to produce smooth changes of the LML minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' It is clear that MorseMatérn kernels, like the MorseRBF kernels, tend to favour small α values when optimised through the LML but it is hard to understand the reason for the progression seen in the tables above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' These optimal values are not similar to the usual values used for Morse projections in the literature but it does not in terms of MAE discredit those choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 7 Conclusion Optimising, with a Bayesian approach, the feature space of the GP to produce more performant latent functions, in terms of MAE, is not straight forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The LML is made more complex by the additional DOF and there is a strong correlation between the hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Differ- ent transforms might not necessarily suffer from this but, for the Morse transform, the relation between the Morse parameter and the length scales is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' For the Morse transform, since the limit of its parameters tending to a certain value (α → ∞ here) give a linear transform, optimising the transform parameters can also inform us on the “usefulness” of the transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The curves of figure 8, for example, show that the transform 9 Optimised Morse transform of a Gaussian process feature space is actually producing less performant GP models for this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' The “best-fit” approach also produced some interesting results regarding the choice of testing set to produce the MAE curves one minimises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A target function is as- sumed to have an optimal Morse parameter to project it to a “simpler” surface↓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' However, a different testing set can significantly affect the result of the minimisation (this cannot be interpreted in the Bayesian approach since there is no testing set in the LML minimisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' This should not be the case if the testing set is “complete”, in the sense that new samples are drawn from the same distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' It will be the case if the distribution change↓ which suggests that one should use testing data that is suitable for the intended use of the GP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Acknowledgments I would like to thank the Royal Society for funding as well as the Wales group of the University of Cambridge for providing access to the GMIN suite17.' 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M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Diedenhofen, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Jr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=', H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Do, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' D.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Laurent, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Lawler, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Levchenko, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Lin, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Liu, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Livshits, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Lochan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Luenser, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Manohar, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Manzer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='-P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Mao, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Mardirossian, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Marenich, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' ↓ This is not guaranteed to a GP easier to train on that surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' ↓ In the results presented in figure 11, it was the temperature of the Boltzmann distribution that changed between the distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 10 Optimised Morse transform of a Gaussian process feature space Maurer, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content='Neill, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Parkhill, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Perrine, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Peverati, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Prociuk, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Rehn, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Rosta, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Russ, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Sharada, S.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Zhang, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Zhang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Zhao, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Brooks, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Chan, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Chipman, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Cramer, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' III, M.' metadata={'source': 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Brucher, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Perrot and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' Duchesnay, Journal of Machine Learning Research, 2011, 12, 2825–2830.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} +page_content=' 11' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-dA0T4oBgHgl3EQfPP9i/content/2301.02172v1.pdf'} diff --git a/-tA0T4oBgHgl3EQfPP-n/content/tmp_files/2301.02173v1.pdf.txt b/-tA0T4oBgHgl3EQfPP-n/content/tmp_files/2301.02173v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..de621f6f66e6bc68412a97cc031654f4bdef9976 --- /dev/null +++ b/-tA0T4oBgHgl3EQfPP-n/content/tmp_files/2301.02173v1.pdf.txt @@ -0,0 +1,1701 @@ +arXiv:2301.02173v1 [nlin.AO] 22 Dec 2022 +Reconstruction of Phase Dynamics from Macroscopic Observations Based on Linear +and Nonlinear Response Theories +Yoshiyuki Y. Yamaguchi1∗ and Yu Terada2,3,4† +1Department of Applied Mathematics and Physics, +Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan +2Neurobiology Section, Division of Biological Sciences, +University of California San Diego, La Jolla, CA 92093, United States of America +3Institute for Physics of Intelligence, Department of Physics Graduate School of Science, +The University of Tokyo 7-3-1 Hongo, Bunkyo-ku, Tokyo 113-0033, Japan +4Laboratory for Neural Computation and Adaptation, +RIKEN Center for Brain Science, 2-1 Hirosawa, Wako, Saitama 351-0198, Japan +We propose a novel method to reconstruct phase dynamics equations from responses in macro- +scopic variables to weak inputs. Developing linear and nonlinear response theories in coupled phase- +oscillators, we derive formulae which connect the responses with the system parameters including +the time delay in interactions. We examine our method by applying it to two phase models, one +of which describes a mean-field network of the Hodgkin–Huxley type neurons with a nonzero time +delay. +The method does not require much invasiveness nor microscopic observations, and these +advantages highlight its broad applicability in various fields. +Rhythmical phenomena have been ubiquitously ob- +served in nature as well as in engineering systems and +attracted a wide spectrum of interests [1–3]. +Specific +rhythmical dynamics are believed to play crucial func- +tional roles in information processing of the brain [4, 5]. +Theoretical analysis have contributed to understanding +the nature of interacting rhythmical systems. One signif- +icant success in theoretical researches is the phase reduc- +tion, which reduces a high-dimensional rhythmic dynam- +ical system to a one-dimensional phase-oscillator system +by eliminating the other nonessential degrees of freedom +[6–8]. +In this framework, a collective system of inter- +acting units is described by a coupled phase-oscillator +system, which consists of the natural frequency distribu- +tion, coupling function, and time delay in interactions. +A dynamical system behind an observed rhythmic phe- +nomenon in the real world is mostly, however, unknown, +while the knowledge helps to profoundly understand, pre- +dict, and control it. This means high demand to specify +the underlying coupled phase-oscillator system. +As the reconstruction is a central issue in coupled +phase-oscillator systems, many works have proposed re- +construction methods [9–18]. However, there are mainly +two rooms that should be addressed. The first is the as- +sumption of accessibility to individual elements. The pre- +vious works assume that time series of almost all elements +are available, which implausible in some situations. For +example, with electroencephalogram or functional mag- +netic response imaging signals, we can obtain only meso- +scopic or macroscopic activity of the nervous systems. +The second is the inference of the time delay. The exis- +tence of the time delay is in principle inevitable in real +systems, and can drastically change dynamics [19, 20]. It +∗ yyama@amp.i.kyoto-u.ac.jp +† yuterada@ucsd.edu +is therefore a next step to develop a method that can be +implemented with unknown interaction delay. +Here, we utilize the linear response theory for coupled +phase-oscillator systems [21–23] with the aid of a nonlin- +ear response theory. We apply weak external forces into +a system, and observe asymptotic responses of order pa- +rameters, which are macroscopic variables. We note that +it does not require time series of individual elements and +that the time delay is tractable. Further, applied external +forces are assumed substantially weak, since we focus on +a regime where the linear response theory is valid. This +assumption brings another advantage that our approach +possesses, because strong inputs into a system may cause +an undesirable change in states of a system. The essen- +tial assumptions on models are that the system has the +mean-field, all-to-all homogeneous interactions and that +the system lies in the nonsynchronized state. +For the +first assumption, it is worth remarking that the all-to- +all interaction may not be extremely special, because the +criticality in the small-world network [24] belongs to the +universality class of the all-to-all interaction [25, 26]. The +mean-field analysis employed here could be extended by +assuming statistics in couplings [27, 28]. The second as- +sumption comes from the effectiveness of linear response +theory developed in [23] and here. +Based on the phase reduction [29] and following the +first assumption, we describe the underlying coupled +phase-oscillator system by +dθj +dt = ωj + 1 +N +N +� +k=1 +Γ (θj(t) − θk(t − τ)) + H(θj(t), t; ωex). +(1) +The variable θj(t) represents the phase of the jth oscilla- +tor at time t, the constant ωj is the natural frequency +following the natural frequency distribution g(ω), the +function Γ represents the coupling function, the constant +τ is the time delay for the coupling. +The function H + +2 +represents the external force and the constant ωex is its +frequency. The system parameters g(ω), Γ, and τ are +intrinsically determined but unknown, and we will infer +them from observation of responses to the external force +H by varying the controllable frequency ωex. The cou- +pling function Γ(θ) is 2π-periodic and is expanded into +the Fourier series as +Γ (θ) = − +∞ +� +m=1 +Km sin (mθ + αm) , +(2) +where Km is the coupling strength and αm is the phase- +lag parameter for the mth Fourier component of Γ(θ). +We here apply the external force as +H (θ, t; ωex) = −Θ(t) +∞ +� +m=1 +hm sin [m (θ − ωext)] , +(3) +where hm is the amplitude of the mth mode. The func- +tion Θ(t) is the unit step function: The external force is +off for t < 0 and kicks in at t = 0. +The dynamics (A1) are described in the limit N → ∞ +by the equation of continuity [30] governing F(θ, ω, t), +which is the probability density function at the time t +and normalized as +� ∞ +−∞ dω +� 2π +0 +dθ F(θ, ω, t) = 1. +The +nonsynchronized state specified as F0(ω) = g(ω)/(2π), +which corresponds to the uniform distribution over θ, is +a stationary solution to the equation of continuity. The +order parameters, whose responses we observe, are de- +fined by [31] +zn(t) = +� ∞ +−∞ +dω +� 2π +0 +dθ einθF(θ, ω, t). +(4) +Assuming that the external force h = (h1, h2, · · · ) is +sufficiently small, we perturbatively analyze the equation +of continuity by using the Fourier transform in θ and +the Laplace transform in t. Supposing that F0 is stable, +we obtain the asymptotic evolution of zn(t) in the lin- +ear regime as e−inωextzn(t) +t→∞ +−−−→ χn(ωex)hn + O(∥h∥2), +where we suppose n > 0 hereafter [23]. Smallness of h +ensures that observation of e−inωextzn provides a good +approximation of χn(ωex)hn. +Moreover, if we apply +hm (m > 0) and observe e−inωextzn (n ̸= m), then +we have a nonlinear response of order O(∥h∥2). +Our +goal is to obtain formulae that allow to reconstruct τ, +Km’s, αm’s, and g(ω) from observation date of {χn(ωex)} +and nonlinear responses for a set of external frequency, +ωex ∈ {ω1 +ex, · · · , ωS +ex}, where ω1 +ex < · · · < ωS +ex. We call +a sampling reliable, if the range ωS +ex − ω1 +ex is sufficiently +large and the gaps ωi+1 +ex +− ωi +ex are sufficiently small. +The susceptibility χn(ωex) of the linear response reads +[32] +χn(ωex) = +G(ωex) +2 − Ln(ωex)G(ωex) +(n > 0), +(5) +where Ln(ωex) += +Kne−i(αn+nωexτ) +and G(ωex) += +πg(ωex) + i PV +� ∞ +−∞ dω g(ω)/(ω − ωex). The symbol PV +indicates the Cauchy principal value. We remark that +G(ωex) does not depend on the mode number n. Thanks +to this independence, once we obtain one of Lm’s, say Ln, +the other coefficients are obtained thought the relation +Lm(ωex) − Ln(ωex) = +1 +χn(ωex) − +1 +χm(ωex). +(6) +This is the key relation in our method. +An obtained +Lm infers the natural frequency distribution g(ω) from +observation of the susceptibility χm(ωex) as +g(ω) = 1 +π Re G(ω) = 1 +π Re +� +2χm(ω) +1 + Lm(ω)χm(ω) +� +. +(7) +Our method is twofold: inference of τ (Procedure-1) +and the others (Procedure-2). +The latter is further +decomposed into the two cases of τ > 0 (Procedure- +2A) and τ = 0 (Procedure-2B). +Procedure-1 performs a finite Fourier transform +Lmn(t) = +1 +ωSex − ω1ex +� ωS +ex +ω1 +ex +[Lm(ωex) − Ln(ωex)]eiωextdωex. +(8) +If the sampling of ωex is perfectly reliable so as to repro- +duce the integral of (8) in the limit ωS +ex − ω1 +ex → ∞, we +have Lmn(t) +ωS +ex−ω1 +ex→∞ +−−−−−−−−→ Kme−iαmδt,mτ −Kne−iαnδt,nτ, +where δt,t′ is the Kronecker delta. The absolute value +|Lmn(t)| has one (τ = 0) or two (τ ̸= 0) peaks at t = mτ +and t = nτ, and the peak positions infer the time delay τ. +An actual sampling induces two types of errors from the +above limit: One comes from boundedness of ωS +ex − ω1 +ex, +and the other from finiteness of the sample number. The +latter type concerns errors of the numerical integration. +Nevertheless, large peaks appear at t = mτ and t = nτ if +the sampling is sufficiently reliable, and Km and Kn are +sufficiently large comparing with the errors. +Procedure-2A +uses +the +relation +Lmn(mτ) += +Kme−iαm under a reliable sampling of ωex to infer Km +and αnm. They with τ give the factor Lm(ω), and the +natural frequency distribution g(ω) is inferred by (7). We +remark that we solely used linear responses up to this +procedure. +Procedure-2B is for τ = 0, since the peak at t = 0 +mixes the modes m and n, Lmn(0) = Kme−iαm − +Kne−iαn. +The linear equations for Kme−iαm (m = +1, 2, 3) obtained from L12(0), L13(0), and L23(0), for in- +stance, are degenerate. We thus use a nonlinear response +to infer, for example, L1: +z2 in O(∥h∥2) can be ob- +served by applying the external force in the first mode +h = (h1, 0, 0, · · · ) as e−i2ωextz2(t) +t→∞ +−−−→ χ11 +2 (ωex)h2 +1. The +nonlinear response coefficient is theoretically obtained as +[32] +χ11 +2 (ωex) = +2iG′(ωex) +[2 − L2G(ωex)][2 − L1G(ωex)]2 , +(9) +where G′(ωex) is the derivative of G(ωex) with respect +to ωex. Solving (9) we have one expression of G′(ωex). + +3 +TABLE I. True and inferred parameter values of Model-1 +and Model-2. The inferred values are given for each sample +set. NI means noninferred values, because there is no clear +peak around t = 3τ in neither |L34| nor |L35|. Procedure-1 +implies that K4 should be sufficiently small from absence of +clear peak of |L45(t)| [see Fig. 1(d)]. +Model-1 τ +K1 +α1 +K2 +α2 +K3 +α3 +Truth +2 +1.379 0.7884 0.568 -3.0316 0.154 -0.7546 +Ω50 +1 +1.987 +1.383 0.820 +0.596 -3.016 +0.153 -0.864 +Ω25 +1 +1.995 +1.381 0.793 +0.582 -3.111 +NI +NI +Model-2 τ +K1 +α1 +K2 +α2 +Truth +0 +1 +1 +0 +0 +Ω81 +2 +0.001 +0.958 1.001 +0.044 -2.119 +Ω41 +2 +-0.001 1.063 0.497 +0.521 -0.706 +We independently have another expression of G′(ωex) +through solving (A31) by G and derivating it. The com- +bination of the above two expressions of G′(ωex) gives +L1 = K1e−iα1 = +2χ11 +2 (ωex) +iχ2(ωex)χ′ +1(ωex) − +1 +χ1(ωex) +(10) +for τ = 0 [32]. We take the average over S estimated +values of L1 from ω1 +ex, · · · , ωS +ex. +The other coefficients +Lm (m > 1) are estimated from (6) by taking the av- +erage. We remark that Procedure-2B is also applica- +ble for τ > 0, where L1 is obtained as a solution to a +quadratic equation. However, Procedure-2A provides +higher performance in inference for a nonzero time-delay +case as compared in an application [32]. +By employing the theory developed above, we tackle +a reconstruction problem in two models: Model-1 has +a delay, that is, τ > 0 and Procedure-2A is applied, +while Model-2 does not and Procedure-2B is in use. +Their system parameters are arranged in Table I. Nu- +merical simulations of (A1) are performed in the use of +the second-order Runge-Kutta algorithm with the time +step ∆t = 0.01. Responses of order parameters are ob- +tained as the average in the time interval (50, 150]. The +number of oscillators is N = 105. All the numerical sim- +ulations are performed by activating only one mode in +h with strength 0.1: hm = 0.1 and hn = 0 (n ̸= m) +for the mth mode. This strength is sufficiently small for +the linear response but sufficiently large for overcoming +finite-size fluctuation of order O(1/ +√ +N) by the second- +order response of order O(∥h∥2). +Model-1 is motivated by neurobiological systems and +is connected directly to a network of the Hodgkin–Huxley +neurons. As in [33, 34], the Fourier components of the +modes m (m ≥ 4) are zero. +The time delay is set as +τ = 2, which is compatible with experimental observa- +tions [35]. Taking another experimental observation [36] +into account, we assume the log-normal natural frequency +distribution +g1(ω) = +1 +ω +� +2πσ2 +1 +exp +� +−(ln ω − µ1)2 +2σ2 +1 +� +(11) +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +1.4 +0 +2 +4 +6 +8 +10 +(a) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0 +2 +4 +6 +8 +10 +(b) +0 +0.05 +0.1 +0.15 +0.2 +0 +2 +4 +6 +8 +10 +(c) +0 +0.05 +0.1 +0.15 +0.2 +0 +2 +4 +6 +8 +10 +(d) +|L1n(t)| +t +|L2n(t)| +t +|L3n(t)| +t +|L4n(t)| +t +FIG. 1. Procedure-1 in Model-1. |Lmn(t)| (8) computed +from the sample set Ω50 +1 . (a) m = 1 and n ∈ {2, 3, 4, 5}. (b) +m = 2 and n ∈ {3, 4, 5}. (c) m = 3 and n ∈ {4, 5}. (c) m = 4 +and n ∈ {5}. The lines are n = 2 (purple chain), n = 3 (green +broken), n = 4 (blue dotted), and n = 5 (orange solid). The +vertical dashed black lines mark the inferred time-delay mτ, +and the horizontal solid black lines the inferred Km. +−1 +−0.5 +0 +0.5 +1 +1.5 +2 +−3 +−2 +−1 +0 +1 +2 +3 +(a) +0 +0.02 +0.04 +0.06 +0.08 +0.1 +0.12 +0.14 +0.16 +0 +2 +4 +6 +8 +10 +(b) +Γ1(θ) +θ +Truth +Ω50 +1 +Ω25 +1 +g1(ω) +ω +Truth +From L1 +From L2 +From L3 +FIG. 2. +Comparison between the truth (purple solid line) +and the inference in Model-1 having τ > 0. (a) The coupling +function Γ1(θ). The sample sets are Ω50 +1 +(green broken line) +and Ω25 +1 +(blue chain line). (b) The natural frequency distri- +bution g1(ω) (11) obtained from the inferred L1 (green filled +circles), L2 (blue open circles), and L3 (orange triangles) by +(7). The sample set is Ω50 +1 . +with µ1 = ln 5 and σ1 = 1. The external frequency is +sampled from the interval [0.2, 10] with the step ∆ωex = +0.2 for the sample set Ω50 +1 +(S = 50), and ∆ωex = 0.4 +for the set Ω25 +1 +(S = 25). We start from Procedure- +1. We approximately compute Lmn(t) (8) by using the +midpoint algorithm, where a sampling point ωi +ex is the +midpoint. Absolute values |Lmn(t)| for the set Ω50 +1 +are +reported in Fig. 1. We obtain the estimate τ = 1.987 +by taking the average over the largest peak positions for +the pairs (m, n) = (3, 4) and (m′, n′) (m′ = 1, 2; n′ = +m′ + 1, · · · , 5). A graph should have two large peaks at +t = mτ and t = nτ, but some peaks are not visible in +Fig. 1. No clear peak at t = nτ implies that Kn is smaller +than the error level. Indeed, no clear peak of |L45(t)| in +Fig. 1(d) is consistent with K4 = K5 = 0. Procedure- +2A infers the coefficients Lm’s from the value of Lmn(t) +at the peak position, where the above mentioned pairs + +4 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +−10 −8 −6 −4 −2 +0 +2 +4 +6 +8 +10 +(a) +−8 +−6 +−4 +−2 +0 +2 +4 +6 +8 +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +(b) +|L12(t)| +t +ReL1, ImL1 +ωex +ReL1 +ImL1 +0.5165 +-0.8068 +FIG. 3. +Model-2. +(a) Procedure-1. +The peak position +is τ = 0.001 and the peak height is 1.014. (b) Procedure- +2B to infer L1 by (10) for each external frequency ωex. The +real part ReLm (purple filled circles) and the imaginary part +ImLm (green open circles). The purple and green horizontal +solid lines mark the averaged values. The sample set is Ω81 +2 . +are in use to take the average. Performing the same pro- +cedure but using the set Ω25 +1 , we obtain another set of +inferences. The inferences are compared with the true +values in Table I. The coupling function Γ1(θ) is directly +obtained from Lm’s, and the natural frequency distribu- +tion g1(ω) is inferred through the relation (7). They are +in good agreement with the true ones for the set Ω50 +1 +as +exhibited in Fig. 2. Increasing the number of samples im- +proves the inference, because the sampling set becomes +more reliable. +Model-2 is the Sakaguchi–Kuramoto model [37] which +is specified by the parameter set (K1, α1) = (1, 1) and the +other Fourier modes are zero. To demonstrate the ability +of the proposed method for general natural frequency dis- +tributions, a nonunimodal and asymmetric natural fre- +quency distribution is assumed as +g2(ω) = ae−(x−µ2)2/(2σ2 +2) + (1 − a)e−(x+µ2)2/(2σ2 +2) +√ +2π +, +(12) +where a = 0.8, µ2 = 2, and σ2 = 1. The external fre- +quency is sampled from [−4, 4] with the step ∆ωex = 0.1 +for the sample set Ω81 +2 +(S = 81) and ∆ωex = 0.2 for the +set Ω41 +2 (S = 41). To compute the derivative χ′ +1(ωex), we +use the central difference except for the head and the end +points, namely ω1 +ex and ωS +ex, for which the forward and +backward differences are in use, respectively. +From now on, we concentrate on inferences of L1 and +L2. Procedure-1 confirms that |L12(t)| has a large peak +at t = 0.001 [see Fig. 3(a)], and hence we conclude no +time-delay, τ = 0. The peak height 1.014 corresponds to +|K1e−iα1 − K2e−iα2|, and the fact K2 = 0 implies that +the peak height approximately infers the value of K1 = 1. +However, we do not know the value of K2 a priori, and we +cannot determine K1 yet. We thus use Procedure-2B, +(10), for inferring L1, and (6) for L2. They are obtained +as functions of ωex, and L1(ωex) is reported in Fig. 3(b). +We determine the inferred values of the constants L1 and +L2 by taking the average over ωex, and the constants +Km and αm (m = 1, 2) from the averaged Lm. +The +inferred values are arranged in Table I. The set Ω81 +2 infers +good values, while the set Ω41 +2 +does not provide good +−1.5 +−1 +−0.5 +0 +0.5 +1 +−3 +−2 +−1 +0 +1 +2 +3 +(a) +0 +0.05 +0.1 +0.15 +0.2 +0.25 +0.3 +0.35 +−4 +−3 +−2 +−1 +0 +1 +2 +3 +4 +(b) +Γ2(θ) +θ +Truth +Ω81 +2 +Ω41 +2 +g2(ω) +ω +Truth +From L1 +From L2 +FIG. 4. +Comparison between the truth (purple solid line) +and the inference in Model-2 having τ = 0. (a) The cou- +pling function Γ2(θ). The sample sets are Ω81 +2 +(green broken +line) and Ω41 +2 +(blue chain line). (b) The natural frequency +distribution g2(ω) (12) obtained from the inferred L1 (green +filled circles) and L2 (blue open circles) through (7). +The +sample set is Ω81 +2 . +inferences, due to the lack of precision in computation of +the derivative χ′ +1(ωex). The inferred coupling function Γ2 +and the natural frequency distribution g2(ω) agree with +the true ones as reported in Fig. 4. +In summary, we proposed a method to reconstruct the +underlying coupled phase-oscillator model of a collective +rhythmic system by observing responses in order param- +eters to a weak external force with varying its frequency. +Non-invasivity is respected due to weakness of the exter- +nal force, and we do not need to know activity of indi- +vidual elements of the system. The proposed method is +examined through numerical simulations in two models. +The unknown system parameters including the time de- +lay in interactions have been successfully inferred, when +the sampling of the external frequency lies on a suffi- +ciently large range with sufficiently small gaps. Finally, +we remark on potential directions of development: ex- +tensions to synchronized states, to noisy systems, and to +network systems. +Y.Y.Y. acknowledges the support of JSPS KAKENHI +Grants No. 16K05472 and No. 21K03402. Y.T. is sup- +ported by the Special Postdoctoral Research Program at +RIKEN and JSPS KAKENHI Grant No. 19K20365. + +5 +Appendix A: Linear and nonlinear response theories +1. +Equations to analyze +We consider the equation of motion +dθj +dt = ωj + 1 +N +N +� +k=1 +Γ (θj(t) − θk(t − τ)) + H(θj, t; ωex), +(j = 1, · · · , N). +(A1) +The variable θj is the phase of the jth phase-oscillator. The natural frequency ωj follows the natural frequency +distribution g(ω). The function Γ is the coupling function and the constant τ is the time delay. We assume that the +external force H is sufficiently small, i.e. ∥H∥ ≪ 1, where ∥H∥ is a certain norm of the function H. Dynamics of +(A1) are described in the limit N → ∞ by the equation of continuity +∂F +∂t + ∂ +∂θ {[ω + v[F] + H(θ, t; ωex)] F} = 0, +(A2) +where +v[F](θ, t; τ) = +� ∞ +−∞ +dω +� 2π +0 +dθ Γ(θ − θ′)F(θ′, ω, t − τ). +(A3) +Suppose that the nonsynchronized state F0(ω) = g(ω)/(2π) is stable stationary under H ≡ 0. We expand F around +F0 as +F(θ, ω, t) = F0(ω) + f (1)(θ, ω, t) + f (2)(θ, ω, t) + · · · , +(A4) +where f (k) = O(∥H∥k). Substituting the expansion (A4) into the equation of continuity (A2), we have +∂f (1) +∂t ++ ∂ +∂θ +� +ωf (1) + +� +v[f (1)] + H +� +F0 +� += 0 +(A5) +in the order of O(∥H∥), and +∂f (2) +∂t ++ ∂ +∂θ +� +ωf (2) + v[f (2)]F0 + +� +v[f (1)] + H +� +f (1)� += 0 +(A6) +in the order of O(∥H∥2). We analyze (A5) and (A6) through the Fourier series expansion in θ and the Laplace +transform in t. +2. +Fourier series expansion +The coupling function Γ, the external force H, and the perturbations f (k) are 2π-periodic functions with respect +to θ, and they are expanded into the Fourier series as +Γ (θ) = − +∞ +� +m=1 +Km sin (mθ + αm) = − +� +n̸=0 +Γneinθ, +(A7) +H (θ, t; ωex) = −Θ(t) +∞ +� +m=1 +hm sin [m (θ − ωext)] = − +� +n̸=0 +einθHn(t; ωex), +(A8) +and +f (k)(θ, ω, t) = +� +n̸=0 +einθf (k) +n (ω, t). +(A9) + +6 +Here, we have the relations +Γn = iKn +2 eiαn, +Γ−n = Γ∗ +n +(n > 0) +(A10) +and +Hn(t; ωex) = ihn +2 Θ(t)e−inωext, +H−n = H∗ +n +(n > 0) +(A11) +where the superscript ∗ represents the complex conjugate. We assume that Γ0 = 0, since it is renormalized into ω, in +other words, into a shift of the natural frequency distribution g(ω). Note that there is no external force of the zeroth +mode: H0 ≡ 0. The order parameter functionals zn[f]’s are defined by +zn[f](t) = +� ∞ +−∞ +dω +� 2π +0 +dθ einθf(θ, ω, t) = 2π +� ∞ +−∞ +f−n(ω, t). +(A12) +The Fourier series expansions give +∂f (1) +n +∂t ++ in +� +ωf (1) +n ++ +� +Γnz(1) +−n(t − τ) + Hn +� +F0 +� += 0 +(A13) +in O(∥H∥) and +∂f (2) +n +∂t ++ in +� +ωf (2) +n ++ Γnz(2) +−n(t − τ)F0 + N (2) +n +� += 0 +(A14) +in O(∥H∥2). The symbol z(k) +−n(t) = z−n[f (k)](t) was introduced to simplify the notation. The second-order nonlinear +term N (2) +n +is defined by +N (2) +n (ω, t) = +� +m +� +Γmz(1) +−m(t − τ) + Hm(t) +� +f (1) +n−m(ω, t). +(A15) +3. +Laplace transform +From now on, the Laplace transform of a function is indicated by the upper hat symbol. For an arbitrary analytic +function ϕ(t), the Laplace transform is defined by +�ϕ(s) = +� ∞ +0 +e−stϕ(t)dt, +Re(s) > 0, +(A16) +where the domain Re(s) > 0 is introduced to ensure the convergence of integral. The perturbation f is zero at +t = 0, since F0 is stable stationary and no external force is applied in t < 0. We hence have the Laplace transformed +equations as +(s + inω) �f (1) +n ++ in +� +Γne−sτ �z(1) +−n + �Hn +� +F0 = 0 +(A17) +in O(∥H∥) and +(s + inω) �f (2) +n ++ in +� +Γne−sτ �z(2) +−nF0 + � +N (2) +n +� += 0 +(A18) +in O(∥H∥2). +4. +Linear response : O(∥H∥) +The equation (A17) is solved algebraically. Dividing s + inω, multiplying by 2π, and integrating over ω, we have +�z(1) +−n(s) = − +�Hn(s) +Λn(s) In(s), +Re(s) > 0. +(A19) + +7 +where the spectrum function Λn(s) (n ̸= 0) is +Λn(s) = 1 + Γne−sτIn(s), +Re(s) > 0. +(A20) +and the integral In(s) is +In(s) = +� ∞ +−∞ +g(ω) +ω − is/n, +Re(s) > 0. +(A21) +The domain Re(s) > 0 comes from the domain of the Laplace transform (A16). +In(s), and Λn(s) and z(1) +−n(s) accordingly, are analytically continued to the whole complex s plane as follows. The +integrand of In(s) has the singularity at ω = is/n, which is located on the upper (lower) half of the complex ω plane +for Re(s) > 0 and n > 0 (n < 0). Moving the singularity to the other half, we smoothly modify the integral contour, +the real axis, so as to avoid the singularity. As a result, the residue is added, because the modified contour, denoted +by L, encloses the singularity entirely for Re(s) < 0 and half for Re(s) = 0. The continued integral In(s) is therefore +In(s) = +� +L +g(ω) +ω − is/ndω = + + + + + + + + + + + + + + + +� ∞ +−∞ +g(ω) +ω − is/ndω +(Re(s) > 0) +PV +� ∞ +−∞ +g(ω) +ω − is/ndω + sgn(n)iπg(is/n) (Re(s) = 0) +� ∞ +−∞ +g(ω) +ω − is/ndω + sgn(n)i2πg(is/n) +(Re(s) < 0) +(A22) +where PV represents the Cauchy principal value, and sgn(n) is the sign of n representing the direction of the integral +counter enclosing the singularity. +Temporal evolution of z(1) +−n(t) is obtained by performing the inverse Laplace transform as +z(1) +−n(t) = +1 +2πi +� σ+i∞ +σ−i∞ +est�z(1) +−n(s)ds, +(A23) +where σ ∈ R is larger than the real parts of any singularities of �z(1) +−n(s). The continuation of �z(1) +−n(s) permits us to use +the residue theorem by adding the half-circle lying in left-half of the complex s plane; The inverse Laplace transform +picks up the singularity of �z(1) +−n(s). The asymptotic behavior is determined by the pole of �z−n(s) which has the largest +real part. Since we assumed that the reference state F0 is stable, all the roots of Λn(s) are in the region Re(s) < 0, +which induce the Landau damping. The asymptotic behavior is hence determined by the poles of �Hn(s) and �H−n(s), +which are +�Hn(s) = ihn +2 +1 +s + inωex +, +�H−n(s) = −ihn +2 +1 +s − inωex +, +(n > 0). +(A24) +The continued integrals In(s) at the poles are +In(−inωex) = iG∗(ωex), +I−n(inωex) = −iG(ωex), +(n > 0) +(A25) +where +G(ωex) = πg(ωex) + iPV +� ∞ +−∞ +g(ω) +ω − ωex +dω. +(A26) +The spectrum functions at the poles are +Λn(−inωex) = 1 +2 [2 − L∗ +nG∗(ωex)] , +Λ−n(inωex) = 1 +2 [2 − LnG(ωex)] , +(n > 0) +(A27) +where Ln = Kne−i(αn+nωexτ). +Putting all together, the asymptotic temporal evolution is for n > 0 is +z(1) +−n(t) +t→∞ +−−−→ e−inωext +G∗(ωex) +2 − L∗nG∗(ωex)hn, +z(1) +n (t) +t→∞ +−−−→ einωext +G(ωex) +2 − LnG(ωex)hn. +(A28) + +8 +The susceptibility χm +n (ωex) defined by +e−inωextz(1) +n (t) +t→∞ +−−−→ +� +m +χm +n (ωex)hm + O(∥H∥2), +einωextz(1) +−n(t) +t→∞ +−−−→ +� +m +χ−m +−n (ωex)h−m + O(∥H∥2), +(n > 0) +(A29) +is hence +χm +n (ωex) = χn(ωex)δnm, +χ−m +−n (ωex) = χ−n(ωex)δnm, +(n > 0), +(A30) +where +χn(ωex) = +G(ωex) +2 − LnG(ωex), +χ−n(ωex) = +G∗(ωex) +2 − L∗nG∗(ωex), +(n > 0). +(A31) +5. +Nonlinear response : O(∥H∥2) +The same way as O(∥H∥) gives the Laplace transform �z(2) +−n(s) as +�z(2) +−n(s) = −2π +Λn(s) +� ∞ +−∞ +� +N (2) +n (ω, s) +ω − is/n dω. +(A32) +We need the Laplace transform of products, which appear in � +N (2) +n . +a. +Laplace transform of a product function +For analytic functions f(t) and g(t), we have the relation +� +fg(s) = +1 +2πi +� σg+i∞ +σg−i∞ +�f(s − s′)�g(s′)ds′, +(A33) +where σg ∈ R is larger than the real parts of any singularities of �g(s). A proof of (A33) is straightforward. We denote +the inverse Laplace transforms of �f(s) and �g(s) as +f(t) = +1 +2πi +� σf +i∞ +σf −i∞ +es1t �f(s1)ds1, +(A34) +where σf ∈ R is larger than the real parts of any singularities of �f(s), and +g(t) = +1 +2πi +� σg+i∞ +σg−i∞ +es2t�g(s2)ds2. +(A35) +Changing the variables as (s, s′) = (s1 + s2, s2), the product function (fg)(t) is expressed as +(fg)(t) = +1 +2πi +� σf +σg+i∞ +σf +σg−i∞ +ds est +� +1 +2πi +� σg+i∞ +σg−i∞ +ds′ �f(s − s′)�g(s′) +� +. +(A36) +The integral over s is the inverse Laplace transform of the inside of the square brackets, and hence we have the relation +(A33). +We note that we pick up the singularities of �g only in the integral with respect to s′. Let a be a pole of �f(s), and +b of �g(s). By the definitions, we have Re(a) < σ1 and Re(b) < σ2. The convolution yields a pole of �f which lies on +the right-side of the line Re(s′) = σg, since s′ = s − a = σf + σg − a > σg. Therefore, this singularity is not enclosed +by the integral counter, which consists of the line Re(s′) = σg and the left half-circle passing through the point at +infinity on the left-half complex s′ plane. + +9 +b. +Convolution in � +N (2) +n +Let us denote +Vm(t) = Γmz(1) +−m(t − τ) + Hm(t), +(A37) +which rewrite the nonlinear term N (2) +n +into +N (2) +n (ω, t) = +� +m +Vm(t)f (1) +n−m(ω, t). +(A38) +The Laplace transform �z(2) +−n(s) is expressed as +�z(2) +−n(s) = −2π +Λn(s) +� +m +� ∞ +−∞ +L[Vmf (1) +n−m](s) +ω − is/n +dω, +(A39) +where L represents the Laplace transform operator. +The Laplace transform of Vm is +�Vm(s) = Γme−sτ �z(1) +−m(s) + �Hm(s) = +�Hm(s) +Λm(s) , +(A40) +where we used (A19) and (A20). The Laplace transform �f (1) +m (ω, s) is then from (A17) +�f (1) +m (ω, s) = − +F0(ω) +ω − is/m +�Hm(s) +Λm(s) . +(A41) +The Laplace transform of Vmf (1) +n−m is +L[Vmf (1) +n−m](s) = +1 +2πi +� σ2+i∞ +σ2−i∞ +�Hm(s′) +Λm(s′) +F0(ω) +ω − i s−s′ +n−m +�Hn−m(s − s′) +Λn−m(s − s′) ds′. +(A42) +Remembering the note at the end of Sec. A 5 a and keeping in mind that we are interested in the asymptotic temporal +evolution, we pick up the pole of �Hm(s′) which is at s′ = −imωex. The principal part of the Laplace transform is +then +PPL[Vmf (1) +n−m](s) = +Res( �Hm) +Λm(−imωex) +�Hn−m(s + imωex) +Λn−m(s + imωex) +F0(ω) +ω − i s+imωex +n−m +, +(A43) +where PP represents the principal part surviving in the limit t → ∞, and Res( �Hm) = sgn(m)ihm/2 is the residue of +�Hm. Substituting the above expression into (A44), we have +PP�z(2) +−n(s) = +−1 +Λn(s) +� +m +Res( �Hm) +Λm(−imωex) +�Hn−m(s + imωex) +Λn−m(s + imωex) Tn,m(s), +(A44) +where +Tn,m(s) = +� +L +g(ω) +� +ω − i s+imωex +n−m +� � +ω − i s +n +�dω. +(A45) +We pick up the pole of �Hn−m(s + imωex), which is at s = −inωex, for the asymptotic temporal evolution. Then, +einωextz(2) +−n(t) +t→∞ +−−−→ +−1 +Λn(−inωex) +� +m +Res( �Hm)Res( �Hn−m)Tn,m(−inωex) +Λm(−imωex)Λn−m(−i(n − m)ωex). +(A46) +We have to be careful for the value Tn,m(−inωex), because the integrand of Tn,m(−inωex) has the pole of order two +at ω = ωex. + +10 +c. +Nonlinear response coefficient +From now on, we focus on the linear response of the mode 2 induced by the external force of the mode 1, i.e. h1 > 0 +and hl = 0 (l > 1). Setting n = 2 and m = 1 in (A46), we have +e2iωextz(2) +−2(t) +t→∞ +−−−→ +T2,1(−2iωex) +4Λ2(−2iωex)[Λ1(−iωex)]2 h2 +1. +(A47) +To obtain the value T2,1(−2iωex), we first perform the partial fraction decomposition as +T2,1(s) = +2 +i(s + 2iωex) [I1(s + iωex) − I2(s)] . +(A48) +In the limit s → −2iω′ +ex (ω′ +ex ̸= ωex) from the upper-half s plane, we have +T2,1(−2iω′ +ex) = +i +ω′ex − ωex +[G∗(2ω′ +ex − ωex) − G∗(ω′ +ex)] . +(A49) +Further taking the limit ω′ +ex → ωex, we have +T2,1(−2iωex) = i (G∗)′ (ωex). +(A50) +The asymptotic temporal evolution of z(2) +2 (t) is hence +e−2iωextz(2) +2 (t) +t→∞ +−−−→ χ11 +2 (ωex)h2 +1 + O(∥H∥3), +(A51) +where +χ11 +2 (ωex) = +iG′(ωex) +4Λ∗ +2(−2iωex)[Λ∗ +1(−iωex)]2 . +(A52) +Substituting (A27) into the above expression, we have +χ11 +2 (ωex) = +2iG′(ωex) +[2 − L2(ωex)G(ωex)][2 − L1(ωex)G(ωex)]2 = 2iG′(ωex) +[G(ωex)]3 χ2(ωex)[χ1(ωex)]2, +(A53) +where we used (A31). +Appendix B: Inference of L1 +The nonlinear response coefficient (A53) gives +G′(ωex) = +χ11 +2 (ωex)[G(ωex)]3 +2iχ2(ωex)[χ1(ωex)]2 . +(B1) +Another expression of G′(ωex) is obtained by solving (A31) by G(ωex) as +G(ωex) = +2χn(ωex) +1 + Ln(ωex)χn(ωex) +(B2) +and derivating it with respect to ωex as +G′(ωex) = 2χ′ +n[1 + Lnχn] − χn[Lnχn]′ +[1 + Lnχn]2 += χ′ +n(ωex) + inτLn[χn(ωex)]2 +2[χn(ωex)]2 +[G(ωex)]2. +(B3) +where we used the definition Ln = Kne−i(αn+nωexτ). The combination between (B1) and (B3) provides for n = 1 +G(ωex) = iχ2(ωex)[χ′ +1(ωex) + iτL1[χ1(ωex)]2] +χ11 +2 (ωex) +. +(B4) +This expression and (B2) for n = 1 give the equality +1 + L1(ωex)χ1(ωex) +2χ1(ωex) += +χ11 +2 (ωex) +iχ2(ωex){χ′ +1(ωex) + iτL1[χ1(ωex)]2}. +(B5) +This is the equation for determining L1. + +11 +1. +For τ = 0 +In particular, L1 is uniquely determined for τ = 0 as +L1 = K1e−iα1 = +2χ11 +2 (ωex) +iχ2(ωex)χ′ +1(ωex) − +1 +χ1(ωex). +(B6) +2. +For τ > 0 +We can infer L1 from the quadratic equation (B5) for τ > 0 as well as for τ = 0. The quadratic equation is rewritten +into +AL2 +1 + BL1 + C = 0, +(B7) +where +A(ωex) = iτ [χ1(ωex)]2 +χ′ +1(ωex) , +B(ωex) = 1 + iτ χ1(ωex) +χ′ +1(ωex), +C(ωex) = +1 +χ1(ωex) − +2χ11 +2 (ωex) +iχ2(ωex)χ′ +1(ωex). +(B8) +We have the two solutions to (B7), and we select the solution +L1(ωex) = − B(ωex) +2A(ωex) +� +1 − +� +1 − 4A(ωex)C(ωex) +[B(ωex)]2 +� +(B9) +to have (B6) in the limit τ → 0, namely A → 0. The inferred L1 induces the other inferences of Lm’s through the +relation +Lm(ωex) − L1(ωex) = +1 +χ1(ωex) − +1 +χm(ωex) +(m ≥ 2). +(B10) +The inferred parameter values are summarized in Table II for Model-1. The inferred coupling function Γ1(θ) and +the natural frequency distribution g1(ω) are compared with the true ones in Fig. 5. We observe rather large errors in +higher order modes in Γ1(θ), and precision is improved by truncating the Fourier series up to the mode-3. Moreover, +the errors tend to decrease as the number of samples increases, and g1(ω) is well inferred irrespective of used modes. +TABLE II. True and inferred parameter values of Model-1 from (B9) and (B10), by taking the average over ωex. 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Department of Physics Graduate School of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The University of Tokyo 7-3-1 Hongo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Bunkyo-ku,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Tokyo 113-0033,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Japan 4Laboratory for Neural Computation and Adaptation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' RIKEN Center for Brain Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 2-1 Hirosawa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Wako,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Saitama 351-0198,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Japan We propose a novel method to reconstruct phase dynamics equations from responses in macro- scopic variables to weak inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Developing linear and nonlinear response theories in coupled phase- oscillators, we derive formulae which connect the responses with the system parameters including the time delay in interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We examine our method by applying it to two phase models, one of which describes a mean-field network of the Hodgkin–Huxley type neurons with a nonzero time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The method does not require much invasiveness nor microscopic observations, and these advantages highlight its broad applicability in various fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Rhythmical phenomena have been ubiquitously ob- served in nature as well as in engineering systems and attracted a wide spectrum of interests [1–3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Specific rhythmical dynamics are believed to play crucial func- tional roles in information processing of the brain [4, 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Theoretical analysis have contributed to understanding the nature of interacting rhythmical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' One signif- icant success in theoretical researches is the phase reduc- tion, which reduces a high-dimensional rhythmic dynam- ical system to a one-dimensional phase-oscillator system by eliminating the other nonessential degrees of freedom [6–8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' In this framework, a collective system of inter- acting units is described by a coupled phase-oscillator system, which consists of the natural frequency distribu- tion, coupling function, and time delay in interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' A dynamical system behind an observed rhythmic phe- nomenon in the real world is mostly, however, unknown, while the knowledge helps to profoundly understand, pre- dict, and control it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' This means high demand to specify the underlying coupled phase-oscillator system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' As the reconstruction is a central issue in coupled phase-oscillator systems, many works have proposed re- construction methods [9–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' However, there are mainly two rooms that should be addressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The first is the as- sumption of accessibility to individual elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The pre- vious works assume that time series of almost all elements are available, which implausible in some situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' For example, with electroencephalogram or functional mag- netic response imaging signals, we can obtain only meso- scopic or macroscopic activity of the nervous systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The second is the inference of the time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The exis- tence of the time delay is in principle inevitable in real systems, and can drastically change dynamics [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' It ∗ yyama@amp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='kyoto-u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='jp † yuterada@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='edu is therefore a next step to develop a method that can be implemented with unknown interaction delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Here, we utilize the linear response theory for coupled phase-oscillator systems [21–23] with the aid of a nonlin- ear response theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We apply weak external forces into a system, and observe asymptotic responses of order pa- rameters, which are macroscopic variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We note that it does not require time series of individual elements and that the time delay is tractable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Further, applied external forces are assumed substantially weak, since we focus on a regime where the linear response theory is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' This assumption brings another advantage that our approach possesses, because strong inputs into a system may cause an undesirable change in states of a system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The essen- tial assumptions on models are that the system has the mean-field, all-to-all homogeneous interactions and that the system lies in the nonsynchronized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' For the first assumption, it is worth remarking that the all-to- all interaction may not be extremely special, because the criticality in the small-world network [24] belongs to the universality class of the all-to-all interaction [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The mean-field analysis employed here could be extended by assuming statistics in couplings [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The second as- sumption comes from the effectiveness of linear response theory developed in [23] and here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Based on the phase reduction [29] and following the first assumption, we describe the underlying coupled phase-oscillator system by dθj dt = ωj + 1 N N � k=1 Γ (θj(t) − θk(t − τ)) + H(θj(t), t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (1) The variable θj(t) represents the phase of the jth oscilla- tor at time t, the constant ωj is the natural frequency following the natural frequency distribution g(ω), the function Γ represents the coupling function, the constant τ is the time delay for the coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The function H 2 represents the external force and the constant ωex is its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The system parameters g(ω), Γ, and τ are intrinsically determined but unknown, and we will infer them from observation of responses to the external force H by varying the controllable frequency ωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The cou- pling function Γ(θ) is 2π-periodic and is expanded into the Fourier series as Γ (θ) = − ∞ � m=1 Km sin (mθ + αm) , (2) where Km is the coupling strength and αm is the phase- lag parameter for the mth Fourier component of Γ(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We here apply the external force as H (θ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex) = −Θ(t) ∞ � m=1 hm sin [m (θ − ωext)] , (3) where hm is the amplitude of the mth mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The func- tion Θ(t) is the unit step function: The external force is off for t < 0 and kicks in at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The dynamics (A1) are described in the limit N → ∞ by the equation of continuity [30] governing F(θ, ω, t), which is the probability density function at the time t and normalized as � ∞ −∞ dω � 2π 0 dθ F(θ, ω, t) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The nonsynchronized state specified as F0(ω) = g(ω)/(2π), which corresponds to the uniform distribution over θ, is a stationary solution to the equation of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The order parameters, whose responses we observe, are de- fined by [31] zn(t) = � ∞ −∞ dω � 2π 0 dθ einθF(θ, ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (4) Assuming that the external force h = (h1, h2, · · · ) is sufficiently small, we perturbatively analyze the equation of continuity by using the Fourier transform in θ and the Laplace transform in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Supposing that F0 is stable, we obtain the asymptotic evolution of zn(t) in the lin- ear regime as e−inωextzn(t) t→∞ −−−→ χn(ωex)hn + O(∥h∥2), where we suppose n > 0 hereafter [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Smallness of h ensures that observation of e−inωextzn provides a good approximation of χn(ωex)hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Moreover, if we apply hm (m > 0) and observe e−inωextzn (n ̸= m), then we have a nonlinear response of order O(∥h∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Our goal is to obtain formulae that allow to reconstruct τ, Km’s, αm’s, and g(ω) from observation date of {χn(ωex)} and nonlinear responses for a set of external frequency, ωex ∈ {ω1 ex, · · · , ωS ex}, where ω1 ex < · · · < ωS ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We call a sampling reliable, if the range ωS ex − ω1 ex is sufficiently large and the gaps ωi+1 ex − ωi ex are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The susceptibility χn(ωex) of the linear response reads [32] χn(ωex) = G(ωex) 2 − Ln(ωex)G(ωex) (n > 0), (5) where Ln(ωex) = Kne−i(αn+nωexτ) and G(ωex) = πg(ωex) + i PV � ∞ −∞ dω g(ω)/(ω − ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The symbol PV indicates the Cauchy principal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We remark that G(ωex) does not depend on the mode number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Thanks to this independence, once we obtain one of Lm’s, say Ln, the other coefficients are obtained thought the relation Lm(ωex) − Ln(ωex) = 1 χn(ωex) − 1 χm(ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (6) This is the key relation in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' An obtained Lm infers the natural frequency distribution g(ω) from observation of the susceptibility χm(ωex) as g(ω) = 1 π Re G(ω) = 1 π Re � 2χm(ω) 1 + Lm(ω)χm(ω) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (7) Our method is twofold: inference of τ (Procedure-1) and the others (Procedure-2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The latter is further decomposed into the two cases of τ > 0 (Procedure- 2A) and τ = 0 (Procedure-2B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-1 performs a finite Fourier transform Lmn(t) = 1 ωSex − ω1ex � ωS ex ω1 ex [Lm(ωex) − Ln(ωex)]eiωextdωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (8) If the sampling of ωex is perfectly reliable so as to repro- duce the integral of (8) in the limit ωS ex − ω1 ex → ∞, we have Lmn(t) ωS ex−ω1 ex→∞ −−−−−−−−→ Kme−iαmδt,mτ −Kne−iαnδt,nτ, where δt,t′ is the Kronecker delta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The absolute value |Lmn(t)| has one (τ = 0) or two (τ ̸= 0) peaks at t = mτ and t = nτ, and the peak positions infer the time delay τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' An actual sampling induces two types of errors from the above limit: One comes from boundedness of ωS ex − ω1 ex, and the other from finiteness of the sample number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The latter type concerns errors of the numerical integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nevertheless, large peaks appear at t = mτ and t = nτ if the sampling is sufficiently reliable, and Km and Kn are sufficiently large comparing with the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-2A uses the relation Lmn(mτ) = Kme−iαm under a reliable sampling of ωex to infer Km and αnm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' They with τ give the factor Lm(ω), and the natural frequency distribution g(ω) is inferred by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We remark that we solely used linear responses up to this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-2B is for τ = 0, since the peak at t = 0 mixes the modes m and n, Lmn(0) = Kme−iαm − Kne−iαn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The linear equations for Kme−iαm (m = 1, 2, 3) obtained from L12(0), L13(0), and L23(0), for in- stance, are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We thus use a nonlinear response to infer, for example, L1: z2 in O(∥h∥2) can be ob- served by applying the external force in the first mode h = (h1, 0, 0, · · · ) as e−i2ωextz2(t) t→∞ −−−→ χ11 2 (ωex)h2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The nonlinear response coefficient is theoretically obtained as [32] χ11 2 (ωex) = 2iG′(ωex) [2 − L2G(ωex)][2 − L1G(ωex)]2 , (9) where G′(ωex) is the derivative of G(ωex) with respect to ωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Solving (9) we have one expression of G′(ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 3 TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' True and inferred parameter values of Model-1 and Model-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferred values are given for each sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' NI means noninferred values, because there is no clear peak around t = 3τ in neither |L34| nor |L35|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-1 implies that K4 should be sufficiently small from absence of clear peak of |L45(t)| [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 1(d)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Model-1 τ K1 α1 K2 α2 K3 α3 Truth 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='7884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='568 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='0316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='154 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='7546 Ω50 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='987 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='383 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='820 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='596 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='153 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='864 Ω25 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='995 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='381 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='793 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='582 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='111 NI NI Model-2 τ K1 α1 K2 α2 Truth 0 1 1 0 0 Ω81 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='958 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='001 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='044 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='119 Ω41 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='001 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='063 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='497 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='521 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='706 We independently have another expression of G′(ωex) through solving (A31) by G and derivating it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The com- bination of the above two expressions of G′(ωex) gives L1 = K1e−iα1 = 2χ11 2 (ωex) iχ2(ωex)χ′ 1(ωex) − 1 χ1(ωex) (10) for τ = 0 [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We take the average over S estimated values of L1 from ω1 ex, · · · , ωS ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The other coefficients Lm (m > 1) are estimated from (6) by taking the av- erage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We remark that Procedure-2B is also applica- ble for τ > 0, where L1 is obtained as a solution to a quadratic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' However, Procedure-2A provides higher performance in inference for a nonzero time-delay case as compared in an application [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' By employing the theory developed above, we tackle a reconstruction problem in two models: Model-1 has a delay, that is, τ > 0 and Procedure-2A is applied, while Model-2 does not and Procedure-2B is in use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Their system parameters are arranged in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nu- merical simulations of (A1) are performed in the use of the second-order Runge-Kutta algorithm with the time step ∆t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Responses of order parameters are ob- tained as the average in the time interval (50, 150].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The number of oscillators is N = 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' All the numerical sim- ulations are performed by activating only one mode in h with strength 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1: hm = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 and hn = 0 (n ̸= m) for the mth mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' This strength is sufficiently small for the linear response but sufficiently large for overcoming finite-size fluctuation of order O(1/ √ N) by the second- order response of order O(∥h∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Model-1 is motivated by neurobiological systems and is connected directly to a network of the Hodgkin–Huxley neurons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' As in [33, 34], the Fourier components of the modes m (m ≥ 4) are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The time delay is set as τ = 2, which is compatible with experimental observa- tions [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Taking another experimental observation [36] into account, we assume the log-normal natural frequency distribution g1(ω) = 1 ω � 2πσ2 1 exp � −(ln ω − µ1)2 2σ2 1 � (11) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='4 0 2 4 6 8 10 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='8 0 2 4 6 8 10 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0 2 4 6 8 10 (c) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0 2 4 6 8 10 (d) |L1n(t)| t |L2n(t)| t |L3n(t)| t |L4n(t)| t FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-1 in Model-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' |Lmn(t)| (8) computed from the sample set Ω50 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (a) m = 1 and n ∈ {2, 3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (b) m = 2 and n ∈ {3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (c) m = 3 and n ∈ {4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (c) m = 4 and n ∈ {5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The lines are n = 2 (purple chain), n = 3 (green broken), n = 4 (blue dotted), and n = 5 (orange solid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The vertical dashed black lines mark the inferred time-delay mτ, and the horizontal solid black lines the inferred Km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 2 −3 −2 −1 0 1 2 3 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='16 0 2 4 6 8 10 (b) Γ1(θ) θ Truth Ω50 1 Ω25 1 g1(ω) ω Truth From L1 From L2 From L3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Comparison between the truth (purple solid line) and the inference in Model-1 having τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (a) The coupling function Γ1(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The sample sets are Ω50 1 (green broken line) and Ω25 1 (blue chain line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (b) The natural frequency distri- bution g1(ω) (11) obtained from the inferred L1 (green filled circles), L2 (blue open circles), and L3 (orange triangles) by (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The sample set is Ω50 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' with µ1 = ln 5 and σ1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The external frequency is sampled from the interval [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2, 10] with the step ∆ωex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 for the sample set Ω50 1 (S = 50), and ∆ωex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='4 for the set Ω25 1 (S = 25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We start from Procedure- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We approximately compute Lmn(t) (8) by using the midpoint algorithm, where a sampling point ωi ex is the midpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Absolute values |Lmn(t)| for the set Ω50 1 are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We obtain the estimate τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='987 by taking the average over the largest peak positions for the pairs (m, n) = (3, 4) and (m′, n′) (m′ = 1, 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' n′ = m′ + 1, · · · , 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' A graph should have two large peaks at t = mτ and t = nτ, but some peaks are not visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' No clear peak at t = nτ implies that Kn is smaller than the error level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Indeed, no clear peak of |L45(t)| in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 1(d) is consistent with K4 = K5 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure- 2A infers the coefficients Lm’s from the value of Lmn(t) at the peak position, where the above mentioned pairs 4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 −10 −8 −6 −4 −2 0 2 4 6 8 10 (a) −8 −6 −4 −2 0 2 4 6 8 −4 −3 −2 −1 0 1 2 3 4 (b) |L12(t)| t ReL1, ImL1 ωex ReL1 ImL1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5165 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='8068 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Model-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (a) Procedure-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The peak position is τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='001 and the peak height is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (b) Procedure- 2B to infer L1 by (10) for each external frequency ωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The real part ReLm (purple filled circles) and the imaginary part ImLm (green open circles).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The purple and green horizontal solid lines mark the averaged values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The sample set is Ω81 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' are in use to take the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Performing the same pro- cedure but using the set Ω25 1 , we obtain another set of inferences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferences are compared with the true values in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The coupling function Γ1(θ) is directly obtained from Lm’s, and the natural frequency distribu- tion g1(ω) is inferred through the relation (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' They are in good agreement with the true ones for the set Ω50 1 as exhibited in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Increasing the number of samples im- proves the inference, because the sampling set becomes more reliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Model-2 is the Sakaguchi–Kuramoto model [37] which is specified by the parameter set (K1, α1) = (1, 1) and the other Fourier modes are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' To demonstrate the ability of the proposed method for general natural frequency dis- tributions, a nonunimodal and asymmetric natural fre- quency distribution is assumed as g2(ω) = ae−(x−µ2)2/(2σ2 2) + (1 − a)e−(x+µ2)2/(2σ2 2) √ 2π , (12) where a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='8, µ2 = 2, and σ2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The external fre- quency is sampled from [−4, 4] with the step ∆ωex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 for the sample set Ω81 2 (S = 81) and ∆ωex = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 for the set Ω41 2 (S = 41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' To compute the derivative χ′ 1(ωex), we use the central difference except for the head and the end points, namely ω1 ex and ωS ex, for which the forward and backward differences are in use, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' From now on, we concentrate on inferences of L1 and L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Procedure-1 confirms that |L12(t)| has a large peak at t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='001 [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 3(a)], and hence we conclude no time-delay, τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The peak height 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='014 corresponds to |K1e−iα1 − K2e−iα2|, and the fact K2 = 0 implies that the peak height approximately infers the value of K1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' However, we do not know the value of K2 a priori, and we cannot determine K1 yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We thus use Procedure-2B, (10), for inferring L1, and (6) for L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' They are obtained as functions of ωex, and L1(ωex) is reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We determine the inferred values of the constants L1 and L2 by taking the average over ωex, and the constants Km and αm (m = 1, 2) from the averaged Lm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferred values are arranged in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The set Ω81 2 infers good values, while the set Ω41 2 does not provide good −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 1 −3 −2 −1 0 1 2 3 (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='35 −4 −3 −2 −1 0 1 2 3 4 (b) Γ2(θ) θ Truth Ω81 2 Ω41 2 g2(ω) ω Truth From L1 From L2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Comparison between the truth (purple solid line) and the inference in Model-2 having τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (a) The cou- pling function Γ2(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The sample sets are Ω81 2 (green broken line) and Ω41 2 (blue chain line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (b) The natural frequency distribution g2(ω) (12) obtained from the inferred L1 (green filled circles) and L2 (blue open circles) through (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The sample set is Ω81 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' inferences, due to the lack of precision in computation of the derivative χ′ 1(ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferred coupling function Γ2 and the natural frequency distribution g2(ω) agree with the true ones as reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' In summary, we proposed a method to reconstruct the underlying coupled phase-oscillator model of a collective rhythmic system by observing responses in order param- eters to a weak external force with varying its frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Non-invasivity is respected due to weakness of the exter- nal force, and we do not need to know activity of indi- vidual elements of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The proposed method is examined through numerical simulations in two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The unknown system parameters including the time de- lay in interactions have been successfully inferred, when the sampling of the external frequency lies on a suffi- ciently large range with sufficiently small gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Finally, we remark on potential directions of development: ex- tensions to synchronized states, to noisy systems, and to network systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' acknowledges the support of JSPS KAKENHI Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 16K05472 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 21K03402.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' is sup- ported by the Special Postdoctoral Research Program at RIKEN and JSPS KAKENHI Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 19K20365.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 5 Appendix A: Linear and nonlinear response theories 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Equations to analyze We consider the equation of motion dθj dt = ωj + 1 N N � k=1 Γ (θj(t) − θk(t − τ)) + H(θj, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex), (j = 1, · · · , N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A1) The variable θj is the phase of the jth phase-oscillator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The natural frequency ωj follows the natural frequency distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The function Γ is the coupling function and the constant τ is the time delay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We assume that the external force H is sufficiently small, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ∥H∥ ≪ 1, where ∥H∥ is a certain norm of the function H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Dynamics of (A1) are described in the limit N → ∞ by the equation of continuity ∂F ∂t + ∂ ∂θ {[ω + v[F] + H(θ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex)] F} = 0, (A2) where v[F](θ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' τ) = � ∞ −∞ dω � 2π 0 dθ Γ(θ − θ′)F(θ′, ω, t − τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A3) Suppose that the nonsynchronized state F0(ω) = g(ω)/(2π) is stable stationary under H ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We expand F around F0 as F(θ, ω, t) = F0(ω) + f (1)(θ, ω, t) + f (2)(θ, ω, t) + · · · , (A4) where f (k) = O(∥H∥k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Substituting the expansion (A4) into the equation of continuity (A2), we have ∂f (1) ∂t + ∂ ∂θ � ωf (1) + � v[f (1)] + H � F0 � = 0 (A5) in the order of O(∥H∥), and ∂f (2) ∂t + ∂ ∂θ � ωf (2) + v[f (2)]F0 + � v[f (1)] + H � f (1)� = 0 (A6) in the order of O(∥H∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We analyze (A5) and (A6) through the Fourier series expansion in θ and the Laplace transform in t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Fourier series expansion The coupling function Γ, the external force H, and the perturbations f (k) are 2π-periodic functions with respect to θ, and they are expanded into the Fourier series as Γ (θ) = − ∞ � m=1 Km sin (mθ + αm) = − � n̸=0 Γneinθ, (A7) H (θ, t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex) = −Θ(t) ∞ � m=1 hm sin [m (θ − ωext)] = − � n̸=0 einθHn(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex), (A8) and f (k)(θ, ω, t) = � n̸=0 einθf (k) n (ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A9) 6 Here, we have the relations Γn = iKn 2 eiαn, Γ−n = Γ∗ n (n > 0) (A10) and Hn(t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' ωex) = ihn 2 Θ(t)e−inωext, H−n = H∗ n (n > 0) (A11) where the superscript ∗ represents the complex conjugate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We assume that Γ0 = 0, since it is renormalized into ω, in other words, into a shift of the natural frequency distribution g(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Note that there is no external force of the zeroth mode: H0 ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The order parameter functionals zn[f]’s are defined by zn[f](t) = � ∞ −∞ dω � 2π 0 dθ einθf(θ, ω, t) = 2π � ∞ −∞ f−n(ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A12) The Fourier series expansions give ∂f (1) n ∂t + in � ωf (1) n + � Γnz(1) −n(t − τ) + Hn � F0 � = 0 (A13) in O(∥H∥) and ∂f (2) n ∂t + in � ωf (2) n + Γnz(2) −n(t − τ)F0 + N (2) n � = 0 (A14) in O(∥H∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The symbol z(k) −n(t) = z−n[f (k)](t) was introduced to simplify the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The second-order nonlinear term N (2) n is defined by N (2) n (ω, t) = � m � Γmz(1) −m(t − τ) + Hm(t) � f (1) n−m(ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A15) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Laplace transform From now on, the Laplace transform of a function is indicated by the upper hat symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' For an arbitrary analytic function ϕ(t), the Laplace transform is defined by �ϕ(s) = � ∞ 0 e−stϕ(t)dt, Re(s) > 0, (A16) where the domain Re(s) > 0 is introduced to ensure the convergence of integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The perturbation f is zero at t = 0, since F0 is stable stationary and no external force is applied in t < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We hence have the Laplace transformed equations as (s + inω) �f (1) n + in � Γne−sτ �z(1) −n + �Hn � F0 = 0 (A17) in O(∥H∥) and (s + inω) �f (2) n + in � Γne−sτ �z(2) −nF0 + � N (2) n � = 0 (A18) in O(∥H∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Linear response : O(∥H∥) The equation (A17) is solved algebraically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Dividing s + inω, multiplying by 2π, and integrating over ω, we have �z(1) −n(s) = − �Hn(s) Λn(s) In(s), Re(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A19) 7 where the spectrum function Λn(s) (n ̸= 0) is Λn(s) = 1 + Γne−sτIn(s), Re(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A20) and the integral In(s) is In(s) = � ∞ −∞ g(ω) ω − is/n, Re(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A21) The domain Re(s) > 0 comes from the domain of the Laplace transform (A16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' In(s), and Λn(s) and z(1) −n(s) accordingly, are analytically continued to the whole complex s plane as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The integrand of In(s) has the singularity at ω = is/n, which is located on the upper (lower) half of the complex ω plane for Re(s) > 0 and n > 0 (n < 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Moving the singularity to the other half, we smoothly modify the integral contour, the real axis, so as to avoid the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' As a result, the residue is added, because the modified contour, denoted by L, encloses the singularity entirely for Re(s) < 0 and half for Re(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The continued integral In(s) is therefore In(s) = � L g(ω) ω − is/ndω = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f4 \uf8f3 � ∞ −∞ g(ω) ω − is/ndω (Re(s) > 0) PV � ∞ −∞ g(ω) ω − is/ndω + sgn(n)iπg(is/n) (Re(s) = 0) � ∞ −∞ g(ω) ω − is/ndω + sgn(n)i2πg(is/n) (Re(s) < 0) (A22) where PV represents the Cauchy principal value, and sgn(n) is the sign of n representing the direction of the integral counter enclosing the singularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Temporal evolution of z(1) −n(t) is obtained by performing the inverse Laplace transform as z(1) −n(t) = 1 2πi � σ+i∞ σ−i∞ est�z(1) −n(s)ds, (A23) where σ ∈ R is larger than the real parts of any singularities of �z(1) −n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The continuation of �z(1) −n(s) permits us to use the residue theorem by adding the half-circle lying in left-half of the complex s plane;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inverse Laplace transform picks up the singularity of �z(1) −n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The asymptotic behavior is determined by the pole of �z−n(s) which has the largest real part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Since we assumed that the reference state F0 is stable, all the roots of Λn(s) are in the region Re(s) < 0, which induce the Landau damping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The asymptotic behavior is hence determined by the poles of �Hn(s) and �H−n(s), which are �Hn(s) = ihn 2 1 s + inωex , �H−n(s) = −ihn 2 1 s − inωex , (n > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A24) The continued integrals In(s) at the poles are In(−inωex) = iG∗(ωex), I−n(inωex) = −iG(ωex), (n > 0) (A25) where G(ωex) = πg(ωex) + iPV � ∞ −∞ g(ω) ω − ωex dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A26) The spectrum functions at the poles are Λn(−inωex) = 1 2 [2 − L∗ nG∗(ωex)] , Λ−n(inωex) = 1 2 [2 − LnG(ωex)] , (n > 0) (A27) where Ln = Kne−i(αn+nωexτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Putting all together, the asymptotic temporal evolution is for n > 0 is z(1) −n(t) t→∞ −−−→ e−inωext G∗(ωex) 2 − L∗nG∗(ωex)hn, z(1) n (t) t→∞ −−−→ einωext G(ωex) 2 − LnG(ωex)hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A28) 8 The susceptibility χm n (ωex) defined by e−inωextz(1) n (t) t→∞ −−−→ � m χm n (ωex)hm + O(∥H∥2), einωextz(1) −n(t) t→∞ −−−→ � m χ−m −n (ωex)h−m + O(∥H∥2), (n > 0) (A29) is hence χm n (ωex) = χn(ωex)δnm, χ−m −n (ωex) = χ−n(ωex)δnm, (n > 0), (A30) where χn(ωex) = G(ωex) 2 − LnG(ωex), χ−n(ωex) = G∗(ωex) 2 − L∗nG∗(ωex), (n > 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A31) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nonlinear response : O(∥H∥2) The same way as O(∥H∥) gives the Laplace transform �z(2) −n(s) as �z(2) −n(s) = −2π Λn(s) � ∞ −∞ � N (2) n (ω, s) ω − is/n dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A32) We need the Laplace transform of products, which appear in � N (2) n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Laplace transform of a product function For analytic functions f(t) and g(t), we have the relation � fg(s) = 1 2πi � σg+i∞ σg−i∞ �f(s − s′)�g(s′)ds′, (A33) where σg ∈ R is larger than the real parts of any singularities of �g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' A proof of (A33) is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We denote the inverse Laplace transforms of �f(s) and �g(s) as f(t) = 1 2πi � σf +i∞ σf −i∞ es1t �f(s1)ds1, (A34) where σf ∈ R is larger than the real parts of any singularities of �f(s), and g(t) = 1 2πi � σg+i∞ σg−i∞ es2t�g(s2)ds2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A35) Changing the variables as (s, s′) = (s1 + s2, s2), the product function (fg)(t) is expressed as (fg)(t) = 1 2πi � σf +σg+i∞ σf +σg−i∞ ds est � 1 2πi � σg+i∞ σg−i∞ ds′ �f(s − s′)�g(s′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A36) The integral over s is the inverse Laplace transform of the inside of the square brackets, and hence we have the relation (A33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We note that we pick up the singularities of �g only in the integral with respect to s′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Let a be a pole of �f(s), and b of �g(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' By the definitions, we have Re(a) < σ1 and Re(b) < σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The convolution yields a pole of �f which lies on the right-side of the line Re(s′) = σg, since s′ = s − a = σf + σg − a > σg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Therefore, this singularity is not enclosed by the integral counter, which consists of the line Re(s′) = σg and the left half-circle passing through the point at infinity on the left-half complex s′ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 9 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Convolution in � N (2) n Let us denote Vm(t) = Γmz(1) −m(t − τ) + Hm(t), (A37) which rewrite the nonlinear term N (2) n into N (2) n (ω, t) = � m Vm(t)f (1) n−m(ω, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A38) The Laplace transform �z(2) −n(s) is expressed as �z(2) −n(s) = −2π Λn(s) � m � ∞ −∞ L[Vmf (1) n−m](s) ω − is/n dω, (A39) where L represents the Laplace transform operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The Laplace transform of Vm is �Vm(s) = Γme−sτ �z(1) −m(s) + �Hm(s) = �Hm(s) Λm(s) , (A40) where we used (A19) and (A20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The Laplace transform �f (1) m (ω, s) is then from (A17) �f (1) m (ω, s) = − F0(ω) ω − is/m �Hm(s) Λm(s) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A41) The Laplace transform of Vmf (1) n−m is L[Vmf (1) n−m](s) = 1 2πi � σ2+i∞ σ2−i∞ �Hm(s′) Λm(s′) F0(ω) ω − i s−s′ n−m �Hn−m(s − s′) Λn−m(s − s′) ds′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A42) Remembering the note at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' A 5 a and keeping in mind that we are interested in the asymptotic temporal evolution, we pick up the pole of �Hm(s′) which is at s′ = −imωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The principal part of the Laplace transform is then PPL[Vmf (1) n−m](s) = Res( �Hm) Λm(−imωex) �Hn−m(s + imωex) Λn−m(s + imωex) F0(ω) ω − i s+imωex n−m , (A43) where PP represents the principal part surviving in the limit t → ∞, and Res( �Hm) = sgn(m)ihm/2 is the residue of �Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Substituting the above expression into (A44), we have PP�z(2) −n(s) = −1 Λn(s) � m Res( �Hm) Λm(−imωex) �Hn−m(s + imωex) Λn−m(s + imωex) Tn,m(s), (A44) where Tn,m(s) = � L g(ω) � ω − i s+imωex n−m � � ω − i s n �dω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A45) We pick up the pole of �Hn−m(s + imωex), which is at s = −inωex, for the asymptotic temporal evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Then, einωextz(2) −n(t) t→∞ −−−→ −1 Λn(−inωex) � m Res( �Hm)Res( �Hn−m)Tn,m(−inωex) Λm(−imωex)Λn−m(−i(n − m)ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A46) We have to be careful for the value Tn,m(−inωex), because the integrand of Tn,m(−inωex) has the pole of order two at ω = ωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 10 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nonlinear response coefficient From now on, we focus on the linear response of the mode 2 induced by the external force of the mode 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' h1 > 0 and hl = 0 (l > 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Setting n = 2 and m = 1 in (A46), we have e2iωextz(2) −2(t) t→∞ −−−→ T2,1(−2iωex) 4Λ2(−2iωex)[Λ1(−iωex)]2 h2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A47) To obtain the value T2,1(−2iωex), we first perform the partial fraction decomposition as T2,1(s) = 2 i(s + 2iωex) [I1(s + iωex) − I2(s)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A48) In the limit s → −2iω′ ex (ω′ ex ̸= ωex) from the upper-half s plane, we have T2,1(−2iω′ ex) = i ω′ex − ωex [G∗(2ω′ ex − ωex) − G∗(ω′ ex)] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A49) Further taking the limit ω′ ex → ωex, we have T2,1(−2iωex) = i (G∗)′ (ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A50) The asymptotic temporal evolution of z(2) 2 (t) is hence e−2iωextz(2) 2 (t) t→∞ −−−→ χ11 2 (ωex)h2 1 + O(∥H∥3), (A51) where χ11 2 (ωex) = iG′(ωex) 4Λ∗ 2(−2iωex)[Λ∗ 1(−iωex)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (A52) Substituting (A27) into the above expression, we have χ11 2 (ωex) = 2iG′(ωex) [2 − L2(ωex)G(ωex)][2 − L1(ωex)G(ωex)]2 = 2iG′(ωex) [G(ωex)]3 χ2(ωex)[χ1(ωex)]2, (A53) where we used (A31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Appendix B: Inference of L1 The nonlinear response coefficient (A53) gives G′(ωex) = χ11 2 (ωex)[G(ωex)]3 2iχ2(ωex)[χ1(ωex)]2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B1) Another expression of G′(ωex) is obtained by solving (A31) by G(ωex) as G(ωex) = 2χn(ωex) 1 + Ln(ωex)χn(ωex) (B2) and derivating it with respect to ωex as G′(ωex) = 2χ′ n[1 + Lnχn] − χn[Lnχn]′ [1 + Lnχn]2 = χ′ n(ωex) + inτLn[χn(ωex)]2 2[χn(ωex)]2 [G(ωex)]2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B3) where we used the definition Ln = Kne−i(αn+nωexτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The combination between (B1) and (B3) provides for n = 1 G(ωex) = iχ2(ωex)[χ′ 1(ωex) + iτL1[χ1(ωex)]2] χ11 2 (ωex) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B4) This expression and (B2) for n = 1 give the equality 1 + L1(ωex)χ1(ωex) 2χ1(ωex) = χ11 2 (ωex) iχ2(ωex){χ′ 1(ωex) + iτL1[χ1(ωex)]2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B5) This is the equation for determining L1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' For τ = 0 In particular, L1 is uniquely determined for τ = 0 as L1 = K1e−iα1 = 2χ11 2 (ωex) iχ2(ωex)χ′ 1(ωex) − 1 χ1(ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B6) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' For τ > 0 We can infer L1 from the quadratic equation (B5) for τ > 0 as well as for τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The quadratic equation is rewritten into AL2 1 + BL1 + C = 0, (B7) where A(ωex) = iτ [χ1(ωex)]2 χ′ 1(ωex) , B(ωex) = 1 + iτ χ1(ωex) χ′ 1(ωex), C(ωex) = 1 χ1(ωex) − 2χ11 2 (ωex) iχ2(ωex)χ′ 1(ωex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B8) We have the two solutions to (B7), and we select the solution L1(ωex) = − B(ωex) 2A(ωex) � 1 − � 1 − 4A(ωex)C(ωex) [B(ωex)]2 � (B9) to have (B6) in the limit τ → 0, namely A → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferred L1 induces the other inferences of Lm’s through the relation Lm(ωex) − L1(ωex) = 1 χ1(ωex) − 1 χm(ωex) (m ≥ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (B10) The inferred parameter values are summarized in Table II for Model-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The inferred coupling function Γ1(θ) and the natural frequency distribution g1(ω) are compared with the true ones in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' We observe rather large errors in higher order modes in Γ1(θ), and precision is improved by truncating the Fourier series up to the mode-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Moreover, the errors tend to decrease as the number of samples increases, and g1(ω) is well inferred irrespective of used modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' TABLE II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' True and inferred parameter values of Model-1 from (B9) and (B10), by taking the average over ωex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' The time delay τ is inferred by Procedure-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Model-1 τ K1 α1 K2 α2 K3 α3 K4 α4 K5 α5 Truth 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='7884 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='568 -3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='0316 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='154 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='7546 0 – 0 – Ω50 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='987 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='215 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='925 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='683 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='663 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='257 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='694 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='119 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='108 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='991 Ω25 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='995 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='857 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='806 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='956 -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='584 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='414 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='253 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='190 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='389 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='407 [1] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Winfree, The Geometry of 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Rosenblum, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Kurths, Synchroniza- tion: a universal concept in nonlinear sciences (Cam- bridge University Press, Cambridge, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Palmigiano, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Geisel, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Wolf, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Battaglia, Flexible information routing by transient synchrony, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 20, 1014-1022 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' [5] G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Buzs´aki and E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Moser, Memory, navigation and theta rhythm in the hippocampal-entorhinal system, Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Neurosci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 16, 130-138 (2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' [6] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Kuramoto, Chemical oscillations, waves, and turbu- lence (Dover, New York, 2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' [7] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nakao, Phase reduction approach to synchronisation of nonlinear oscillators, Contemp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 57, 188 (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' [8] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Kuramoto and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Nakao, On the concept of dynamical reduction: the case of coupled oscillators, Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 12 −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 2 −3 −2 −1 0 1 2 3 (a) −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 −1 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='5 2 −3 −2 −1 0 1 2 3 (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content='16 0 2 4 6 8 10 (c) Γ1(θ) θ Truth Sample set Ω50 1 Sample set Ω25 1 Γ1(θ) θ Truth Sample set Ω50 1 (up to mode-3) Sample set Ω25 1 (up to mode-3) g1(ω) ω Truth From L1 From L2 From L3 From L4 From L5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' Comparison between the truth and the inference in Model-1 having τ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (a) The coupling function Γ1(θ) produced from the sample set Ω50 1 (green broken line), Ω25 1 (blue chain line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (b) Same as (a) but the inferred Γ1(θ) are truncated up to the Fourier mode-3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/-tA0T4oBgHgl3EQfPP-n/content/2301.02173v1.pdf'} +page_content=' (c) The natural frequency distribution g1(ω) obtained from the inferred L1 (green filled circles), L2 (blue open circles), L3 (orange triangles), L4 (yellow inverse triangles), and L5 (dark-blue diamonds).' 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Shen,1, ∗ J. Sears,1 G. Fabbris,2 J. Li,3 J. Pelliciari,3 M. Mitrano,4 W. He,1 Junjie +Zhang,5, 6 J. F. Mitchell,5 V. Bisogni,3 M. R. Norman,5 S. Johnston,7, 8 and M. P. M. Dean1, † +1Condensed Matter Physics and Materials Science Department, +Brookhaven National Laboratory, Upton, New York 11973, USA +2Advanced Photon Source, Argonne National Laboratory, Lemont, Illinois 60439, USA +3National Synchrotron Light Source II, Brookhaven National Laboratory, Upton, New York 11973, USA +4Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA +5Materials Science Division, Argonne National Laboratory, Lemont, Illinois 60439, USA +6Institute of Crystal Materials, Shandong University, Jinan, Shandong 250100, China +7Department of Physics and Astronomy, The University of Tennessee, Knoxville, Tennessee 37966, USA +8Institute of Advanced Materials and Manufacturing, The University of Tennessee, Knoxville, Tennessee 37996, USA +(Dated: January 12, 2023) +Charge order is a central feature of the physics of cuprate superconductors and is known to arise +from a modulation of holes with primarily oxygen character. Low-valence nickelate superconductors +also host charge order, but the electronic character of this symmetry breaking is unsettled. Here, +using resonant inelastic x-ray scattering at the Ni L2-edge, we identify intertwined involvements of +Ni 3dx2−y2, 3d3z2−r2, and O 2pσ orbitals in the formation of diagonal charge order in an overdoped +low-valence nickelate La4Ni3O8. The Ni 3dx2−y2 orbitals, strongly hybridized with planar O 2pσ, +largely shape the spatial charge distribution and lead to Ni site-centered charge order. The 3d3z2−r2 +orbitals play a small, but non-negligible role in the charge order as they hybridize with the rare- +earth 5d orbitals. Our results reveal that the low-energy physics and ground-state character of these +nickelates are more complex than those in cuprates. +I. +INTRODUCTION +One of the common threads linking different classes +of unconventional superconductors is their propensity to +host proximate competing orders such as charge and spin +stripes [1, 2]. +For example, the cuprate superconduc- +tors exhibit diagonal (with respect to the Cu-O bonds) +spin stripes when underdoped [3–5], while Cu-O bond +oriented (parallel) charge order dominates the rest of the +phase diagram [6, 7]. The detection of superconductivity +and charge order in the square-planar low-valence fam- +ily of nickelates therefore presents a fascinating oppor- +tunity to study the degree of similarity between differ- +ent unconventional superconducting families [8–17]. In- +triguingly, different nickelates within the structural se- +ries of Rn+1NinO2n+2 (R stands for a rare earth and n is +the number of neighboring NiO2 layers) also host differ- +ent charge ordered phases. Underdoped materials with +n = ∞ and R = La, Nd exhibit parallel charge order +[15–17], whereas n = 3 material La4Ni3O8, which is ef- +fectively 1/3 overdoped, manifests diagonal charge order +[14]. Many researchers have emphasized that charge or- +der plays an important role in the physics of cuprates +[18–21]. +In particular, there is good evidence showing +that charge/spin order is a fundamental feature of min- +imal Hubbard model descriptions of the cuprates [22– +24]. +Some researchers have suggested that charge and +spin order can intertwine with superconductivity to form +pair density waves [25, 26], or that dynamic charge/spin +∗ yshen@bnl.gov +† mdean@bnl.gov +fluctuations might promote superconductivity [27–29]. +Others have associated charge order fluctuations with +the anomalous “strange metal” electronic transport in +cuprates [30]. +Understanding the electronic states in- +volved in charge order formation is a prerequisite to test- +ing all these scenarios in low-valence nickelates and is also +important more generally for understanding charge order +as a prevalent feature of correlated quantum materials. +Here, we use Ni L2-edge RIXS to determine the elec- +tronic character of the charge order in La4Ni3O8. We +find that both the Ni 3dx2−y2 and 3d3z2−r2 orbitals are +involved in charge order formation. +The former con- +tributes most of the charge modulation while the latter +dominates the RIXS spectra in the post-edge regime and +so plays a less important role. +As the charge-transfer +energy of these nickelates is larger than that of cuprates +but comparable to the on-site Coulomb interaction, the +holes involved in the charge modulation reside predomi- +nately on Ni sites, despite an appreciable amount of holes +occupying the O orbitals. Our results indicate that the +low-energy electronic structure and charge order of low- +valence nickelates is largely shaped by hybridized 3dx2−y2 +and planar O 2pσ orbitals, similar to cuprates, while +some differences exist due to the multi-band physics in- +troduced by Ni 3d3z2−r2 orbitals hybridized with rare- +earth 5d states. +II. +RESULTS +The La4Ni3O8 nickelate samples studied here were pre- +pared by reducing single crystals synthesized via the +floating zone method (see the Appendix A for details), +arXiv:2301.04184v1 [cond-mat.str-el] 10 Jan 2023 + +2 +tpd +tpp +Ni1 +Ni2 +Ni3 +a +b +c +σ +Ni L2-edge +� +θ +(a) +Nickel +Oxygen +NiO2 plane +La4Ni3O8 sample +(b) +(c) +(d) +(e) +(f) +-0.2 +-0.1 +0 +0.1 +0.2 +Energy loss (eV) +0.32 +0.36 +0.34 +0.32 +0.34 +0.32 +0.34 +0.32 +0.34 +0.32 +0.34 +(H, H) (r.l.u.) +(H, H) (r.l.u.) +(H, H) (r.l.u.) +(H, H) (r.l.u.) +(H, H) (r.l.u.) +T = 40 K +T = 50 K +T = 70 K +T = 90 K +T = 110 K +×10 +0 +25 +50 +75 +100 +Intensity (arb. units) +(g) +(h) +40 K +50 K +70 K +90 K +110 K +0.32 +0.33 +0.34 +0.35 +Q║=(H, H) (r.l.u.) +40 +60 +80 +100 120 +Temperature (K) +0 +2 +4 +6 +Intensity (arb. units) +FIG. 1. Charge order transition in La4Ni3O8. (a) Schematic of the Ni L2-edge RIXS experimental setup. A single NiO2 layer +is presented with stripes running vertically. A Ni3O10 cluster composed of Ni 3dx2−y2 and planar O 2pσ orbitals is embedded +in it, tracing the charge order motif, in which hole poor Ni1 and Ni3 sites, shown in red, flank the hole rich Ni2 site depicted +in purple. (b)–(f) RIXS intensity maps with σ polarized incident photons at the indicated temperatures obtained by changing +the in-plane sample angle θ. (g) Quasi-elastic-line amplitudes extracted from the data presented in (b)–(f) as a function of +in-plane momentum transfer in reciprocal lattice units (r.l.u.). The solid lines are fitting curves with pseudo-Voigt profiles. (h) +Temperature dependence of the fitted peak amplitudes. The bold gray line is a guide to the eye. +and will be indexed in terms of scattering vector Q = +(2π/a, 2π/a, 2π/c) with a = b = 3.97 ˚A, c = 26.092 ˚A. +As the n = 3 member of the low-valence nickelate family, +it possesses a trilayer structure with a nominal 3d8+2/3 +valence. +This leads to a 1/3-hole self-doping with re- +spect to the undoped 3d9 state, putting it in the over- +doped regime of the phase diagram [13, 31]. It shares +the same structural motif as infinite-layer nickelates with +square-planar NiO2 layers stacked without apical oxy- +gens, leading to dominant Ni 3dx2−y2 character near +the Fermi energy. Although La4Ni3O8 has two inequiv- +alent NiO2 layers, they are expected to show similar +electronic structure as indicated by theoretical calcula- +tions [32, 33], which is further supported by the obser- +vation that the same charge order pattern is formed in +both layers [14]. We study their properties using Ni L2- +edge RIXS in order to avoid interference from the La +M4-edge, which overlaps the Ni L3-edge (see the Ap- +pendix B for details). As shown in Fig. 1(a), charge or- +der in La4Ni3O8 is quasi-two-dimensional in nature and +occurs at Q∥ = QCO = (1/3, 1/3), where a strong peak +is observed in the quasi-elastic region of the RIXS inten- +sity map at 40 K [see Fig. 1(b)]. The in-plane correlation +length is larger than 100 nm, which might be limited by +the sample mosaic, suggesting the long range nature of +the charge order [14]. This charge order peak persists up +to 90 K and disappears above 110 K, indicating a tran- +sition temperature of around 100 K [see Figs. 1(c)–(h)], +consistent with the reported charge order from hard x-ray +diffraction measurements [14]. No indication of charge +order is apparent in equivalent measurements of metallic +Pr4Ni3O8 samples prepared in the same way (Supple- +mental Material Sec. I [34]). +We begin by identifying the active electronic states +in La4Ni3O8 using x-ray spectroscopy. Figure 2(a) and +2(b) show the L2-edge RIXS energy maps taken with +σ x-ray polarization in the ab-plane and π x-ray po- +larization approximately parallel to the c-axis, respec- +tively. The RIXS maps mainly comprise dd and charge- +transfer excitations that are predominantly localized and +resonate at the Ni L2-edge, and diagonal fluorescence +features (Supplemental Material Sec. II [34]). To distin- +guish among these contributions, we integrated the RIXS +spectra along the incident energy axis and show the re- +sult in Fig. 2(c). With σ polarization, the spectra above +4 eV energy loss are dominated by mostly featureless flu- +orescence originating from particle-hole excitations that +can be understood from an itinerant framework involv- +ing transitions from extended electronic bands spanning +many unit cells [35]. Charge transfer excitations are also + +3 +CT +dd +FL +dd +FL +(a) +(b) +0 +1 +2 +3 +4 +5 +6 +Energy loss (eV) +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0 +1 +2 +3 +4 +5 +6 +Intensity (arb. units) +σ-pol. +�-pol. +0 +2 +4 +6 +8 +Energy loss (eV) +0.0 +0.2 +0.4 +0.6 +Intensity (arb. units) +0.8 +dd +CT ++ +FL +(c) +σ-pol. +�-pol. +Integral over +incident energy +0.00 +0.01 +0.02 +0.03 +Intensity (arb. units) +865 +870 +875 +880 +Incident energy (eV) +CT +FL +(d) +5.5 ≤ Eloss ≤ 6 +σ-pol. +�-pol. +x0.1, CO +CO +866 +868 +870 +872 +874 +Incident energy (eV) +866 +868 +870 +872 +874 +Incident energy (eV) +-0.10 +-0.05 +0.00 +0.05 +0.10 +Energy loss (eV) +0 +4 +8 +12 +16 +-0.10 +-0.05 +0.00 +0.05 +0.10 +0 +4 +8 +12 +16 +Intensity (arb. units) +(e) +(f) +FIG. 2. +RIXS energy maps and the resonant behaviors of the charge order (CO) peak. +(a, b) RIXS intensity maps as a +function of incident photon energy with (a) σ x-ray polarization in the ab plane of the sample and (b) π x-ray polarization +approximately parallel to the c-axis. Several components can be identified: charge transfer excitations (CT), dd excitations (dd) +and constant-emission-energy fluorescence (FL). (c) Integral of the RIXS spectra along the incident energy axis. The dashed +lines are guides to the eye. (d) Incident energy dependence of the integrated RIXS spectra between 5.5 and 6 eV energy loss. +(e, f) RIXS intensity maps around the quasi-elastic regime with Q fixed at QCO. Note that the intensity in (e) is multiplied +by 0.1 for clarity in visualizing the signal. +visible above 4 eV but only at resonance. Below 4 eV, +prominent dd excitations emerge that dominate over the +featureless fluorescence (dashed lines). With π polariza- +tion, the fluorescence contributes most of the spectral +weight and the dd excitations are much weaker. +The +strong dichroism of dd excitations reflects the dominant +Ni 3dx2−y2 orbital character near the Fermi energy in +low-valence nickelates. +To further distinguish between charge-transfer excita- +tions and fluorescence, we inspect the RIXS spectra be- +tween 5.5 and 6 eV energy loss, well above the dd excita- +tion threshold. As shown in Fig. 2(d), the charge-transfer +excitations and fluorescence are separated in the incident +energy axis, with the former stronger in the σ polariza- +tion channel, indicating appreciable dx2−y2-pσ hybridiza- +tion where pσ indicates O orbitals that are parallel to the +Ni-O bonds. In contrast, the fluorescence is stronger in +the π polarization channel, suggesting that states involv- +ing Ni 3d3z2−r2 orbitals dominate the fluorescence for a +broad range of energy losses above ∼3 eV. The broadness +of these states is in contrast with cuprates, and suggests +that although the Ni 3d3z2−r2 orbitals are mostly oc- +cupied and localized, their unoccupied components are +hybridized with the rare earth 5d orbitals and thus con- +tribute to dispersive states. This conclusion is consistent +with density functional theory (DFT)+dynamical mean +field theory (DMFT) calculations [32], as well as RIXS +simulations for RNiO2 that studied the effect of switching +on and off the rare-earth hybridization [36]. Meanwhile, +the Ni 3dx2−y2 orbitals exhibit less hybridization with +the rare earth 5d orbitals and are more localized. Here, +since we are measuring at the Ni L edge and the Ni t2g +orbitals are expected to lie well below the Fermi energy, +we only consider Ni eg orbitals [34]. +Based on the resonant behavior of the different states +identified, we now examine how the 3dx2−y2 and 3d3z2−r2 +orbitals participate in the charge order. Figure 2(e) and +2(f) show the RIXS energy maps around the quasi-elastic +regime at QCO, i.e. the resonant elastic x-ray scatter- +ing (REXS) signals. +The peak intensity strongly res- +onates at the Ni L2-edge in the σ polarization channel +[see Fig. 2(e)], confirming that the (1/3, 1/3) Bragg peak +in La4Ni3O8 involves a charge modulation and is not +purely structural. Surprisingly, the charge order peak in +the π polarization channel, although much weaker, res- +onates at the pre- and post-edge regimes but not at the +main edge [see Fig. 2(f)], distinct from that in cuprates +[37–39]. First, this observation indicates that both the +3dx2−y2 and 3d3z2−r2 orbitals are involved in charge order +formation with the latter much less prominent. Second, + +4 +(a) +868 +870 +872 +Incident energy (eV) +σ-pol. +(b) +868 +870 +872 +Incident energy (eV) +�-pol. +0 +1 +2 +3 +Energy loss (eV) +Intensity (arb. units) +0 +0.25 +FIG. 3. Low-energy electronic states in La4Ni3O8. Calcula- +tions of the RIXS energy maps at the Ni L2-edge for (a) σ and +(b) π incident x-ray polarization. The calculations reproduce +the experimental energy-scale and polarization of the dd exci- +tations evincing an appropriate minimal model for La4Ni3O8. +the charge order peak in the post-edge regime with π po- +larization suggests that the states far above the Fermi +energy also show charge modulation, which is mostly +contributed by 3d3z2−r2 orbitals. Considering that the +3d3z2−r2 density of states in the post-edge regime is likely +caused by hybridization with the rare-earth 5d orbitals, +this indicates potential involvement of rare-earth orbitals +in the charge order formation. Similarly, the weak pre- +edge charge order peak with π polarization indicates that +the 3d3z2−r2 density of states near the Fermi energy is +nonzero but small. +Having established the involvement of Ni orbitals in +the charge order formation, now we look at the role of +oxygen states. To do this, we use exact diagonalization +(ED) methods which allow us to solve the resonant cross- +section and break down the contributions from different +states. Since the charge order is commensurate with a +period of three Ni sites and there is a strong hybridiza- +tion between the Ni and O orbitals, the smallest cluster +one can use to describe the charge-ordered state involves +three Ni-O plaquettes, which we label 1, 2, & 3. +We +choose a bond-oriented cluster, as illustrated in Fig. 1(a), +given that the Ni-O hopping dominates the kinetic en- +ergy. +In order to compute REXS we use the atomic +scattering factors from the cluster and add these am- +plitudes to simulate an effective two-dimensional NiO2 +plane as shown in Fig. 1(a). The appropriate parame- +ters for this cluster, and in particular the charge-transfer +energy ∆ = 5.6 eV and the on-site Coulomb repulsion +Udd = 6.5 eV, have been empirically determined by prior +x-ray measurements of this material at the O K-edge +[40]. We use open boundary conditions and construct the +Hamiltonian in the hole language (see the Appendix C +for details). +Four holes are introduced to the cluster, +which is appropriate for the d9−1/3 electronic configu- +ration of La4Ni3O8. Without any additional constraints, +the holes will be evenly distributed among different NiO4 +plaquettes with minimal charge disproportionation and +no symmetry breaking is expected. To realize the charge +order observed in La4Ni3O8, we manually introduce a +potential difference [41], ∆ϵd, for different Ni sites by +lowering the orbital energies of Ni2 by 2∆ϵd/3 and rais- +ing those of Ni1 and Ni3 by ∆ϵd/3. Based on the sim- +ilar magnetic exchange of charge ordered La4Ni3O8 and +metallic Pr4Ni3O8 [42], ∆ϵd must be significantly smaller +than the charge-transfer energy. Thus, we choose it to +be ∆ϵd = 0.8 eV while noting that apart from modu- +lating the intensity of the charge order peak, the results +are similar provided ∆ϵd is not made unfeasibly large +(Supplemental Material Fig. S5 [34]). This choice leads +to a charge disproportionation of ∆n = 0.32, which is +of a similar order of magnitude as that in cuprates [37]. +This value is much smaller than the fully disproportion- +ate limit ∆n = 1, consistent with DFT calculations that +indicate a small charge modulation upon charge ordering +in this system [31]. When examining the electronic con- +figuration of the cluster, we find that the ground state is +a singlet, and the first excited state is a triplet, which is +around 70 meV above the ground state, consistent with +the magnetic excitations found in La4Ni3O8 [42]. +Figure 3 shows the calculated Ni L2-edge RIXS en- +ergy maps with all the Ni 3d and O 2p orbitals included, +which qualitatively reproduce the localized dd excitations +observed experimentally. Note that the small cluster size +means that we can only capture a limited number of dis- +crete states. For this reason, fluorescence features are not +fully captured, which would require a continuous distri- +bution of states. This can be seen more clearly in the +π polarization channel where the fluorescence dominates +the spectra in experimental data [see Fig. 2(b)] but only +the weak dd excitations are present in our cluster calcu- +lations [see Fig. 3(b)]. +Having verified the relevant parameters via the RIXS +maps, we computed the x-ray absorption spectrum +(XAS) and REXS response of La4Ni3O8 using a simi- +lar ED approach and identical parameters and plot the +results in Fig. 4 (Supplemental Material Sec. V [34]). +The charge disproportionation in the cluster implies a +REXS response at QCO. The predicted REXS resonance +shown in Fig. 4(c) nicely captures the main two peak +structure of the experimental REXS resonance shown in +Fig. 4(b). The same applies for the XAS as shown in +Fig. 4(a). In fact, the lineshape of the resonant profile of +the charge order peak is sensitive to the charge-transfer +energy, and neither the pure charge-transfer nor Mott- +Hubbard scenarios can describe the observed resonant +behaviors (Supplemental Material Fig. S6 [34]), demon- + +5 +strating the mixed charge-transfer/Mott-Hubbard char- +acters of charge order in this material. To understand +the nature of the two resonant features, we projected +the wavefunctions of the RIXS intermediate states onto +the Fock basis which specifies the location of the holes. +Two main manifolds are seen for each Ni site. The first +manifold is primarily attributed to transitions resonant +with d10L0 states, where L stands for ligand holes on +the four oxygen σ orbitals surrounding the Ni site. The +second manifold is mainly resonant with d9L0 and d10L1 +states caused by the doped holes, similar to the cuprates +[43, 44]. With nonzero ∆ϵd, the manifolds of different +Ni sites split along the incident energy axis, as shown in +Fig. 4(c). The successful description of the charge order +in La4Ni3O8 using our cluster model indicates that about +70% of the holes participating in the charge modulation +are on Ni, with the remaining 30% on oxygen, as depicted +in the inset to Fig. 4(b). +III. +DISCUSSION +Our Ni-dominant charge order distribution is quite dif- +ferent from cuprates, in which the charge order has dom- +inant oxygen character [37, 45]. This difference mainly +arises from the larger charge transfer energy in nicke- +lates compared to cuprates. Another difference is that in +cuprates, the 3d3z2−r2 orbitals are strongly localized at +energies more than 1.5 eV away from the 3dx2−y2 orbitals +[46], and thus not involved in the low-energy physics. For +square-planar nickelates, our analysis of La4Ni3O8 indi- +cates that the 3d3z2−r2 density of states, though small, is +spread out over an extended energy range, likely due to +hybridization with the rare earth 5d orbitals. It should be +noted that although the 3d3z2−r2 orbital involvement in +the charge order formation is nonzero, its contribution is +much less than the hybridized 3dx2−y2 and 2pσ orbitals, +as indicated by the stronger charge order peak in the σ +polarization channel. These factors mean that minimal +theoretical models of charge order in nickelates must ex- +plicitly include both Ni and O states alongside strong +correlations. +Another result of our model is that the +doped sites in charge ordered nickelates are much closer +to a low-spin S = 0 state than to a high-spin S = 1 state, +unlike La2−xSrxNiO4, whose high-spin physics drives in- +sulating behavior across the vast majority of its phase +diagram [47]. +Recently, RIXS measurements in infinite-layer nicke- +late films have discovered and studied charge order at +Q∥ = (1/3, 0) in undoped and underdoped samples [15– +17], resembling the charge order in cuprates, but differing +from the diagonal charge order in La4Ni3O8. In terms +of these differing wavevectors, theoretical model studies +in the cuprates have shown that charge order at (Q, 0) +and (Q, Q) are close in energy, the eventual choice of the +charge order wavevector being sensitive to details of the +electronic structure and correlations [48, 49]. This idea is +supported by the experimental observation that the dop- +868 +869 +870 +871 +872 +873 +Incident energy (eV) +2 +1 +0 +Intensity (arb. units) +Ni1: d 10L0 +Ni1: d 10L1 +Ni1: d 9L0 +Ni2: d 10L0 +Ni2: d 10L1 +Ni2: d 9L0 +REXS calculation +(c) +6 +4 +2 +0 +Intensity (arb. units) +Ni1 +Ni3 +Ni2 +O +σ-pol. +�-pol. +REXS data +(b) +(a) +1.2 +0.8 +0.4 +0 +Intensity (arb. units) +XAS, σ-pol. +Data +Ni1+Ni3 +Ni2 +Calculation +FIG. 4. Electronic character of charge order. (a) x-ray ab- +sorption spectrum (XAS) data at the Ni L2 edge in the σ +polarization channel along with the calculation results with +∆ϵd = 0.8 eV. Note that Ni1 and Ni3 are symmetry-related. +(b) Fitted peak amplitudes of the quasi-elastic intensities pre- +sented in Fig. 2(e)&(f), representing the resonant behaviors of +the charge order peak. Inset is a schematic of the electronic +character of the charge order showing a dominant modula- +tion of Ni orbitals along with an appreciable modulation of +the oxygen orbitals. +(c) Simulation of the incident energy +dependence of the charge order peak intensity with σ inci- +dent polarization and ∆ϵd = 0.8 eV. The vertical bars are +weights of different configurations of the RIXS intermediate +states, the total height of which is normalized according to +the simulated charge order peak intensity of each state. The +accurate simulation of the Ni 3d and O 2p components of the +resonance verifies our model, which is used to extract the elec- +tronic character of the charge order illustrated in the inset to +panel (b). +ing dependent charge order wavevector varies in different +cuprate families [20], similar to what has been seen more +recently in the infinite-layer nickelates [15]. In view of +this, the difference in wavevector probably does not re- +flect a difference in the mechanisms at play in charge +order formation. It should, however, be noted that the +parallel charge order seen in infinite-layer materials oc- +curs at a lower hole concentration. +More information can be obtained by comparing the + +6 +states involved in charge order formation for different +low-valence nickelates [15–17]. +All these recent works +support an appreciable role for Ni in charge order forma- +tion. However, controversy exists regarding whether the +rare-earth-Ni hybridization is crucial for charge order for- +mation [16], or whether the charge modulation on rare- +earth states only plays a secondary parasitic role [15]. +Our results support the latter scenario in La4Ni3O8. Re- +garding the involvement of oxygen states, we provide the +first spectroscopic modeling that allows this question to +be addressed quantitatively. We deduce a mixed charge- +transfer/Mott-Hubbard picture for the charge order and +70%/30% split of Ni vs. O contributions to the charge +modulation. This contradicts some of the previous sug- +gestions for infinite-layer nickelates, which propose a neg- +ligible role for oxygen in charge order formation and that +in-plane and out-of-plane Ni states contribute roughly +equally [16]. +These differences are puzzling consider- +ing that different members of the Rn+1NinO2n+2 family +share similar Ni-O bonding, magnetic exchange [42, 50], +superconducting transition temperatures [12, 13, 51, 52], +and calculated electronic structures [53]. Part of the chal- +lenge of making this comparison is that RIXS maps of +infinite-layer films, as well as their charge order proper- +ties, vary substantially between different samples of nom- +inally the same composition [15–17]. In this regard, our +quantitative spectroscopic analysis on single crystals is +valuable considering that these samples show more con- +sistent spectral properties than films of infinite layer ma- +terials [15–17]. +IV. +CONCLUSION +In summary, we have used RIXS measurements at +the Ni L2-edge to study the character of the electronic +structure and charge order in the low-valence nickelate +La4Ni3O8. Our work is unique in providing a realistic +quantitative empirical model for charge order and vali- +dating it using Q-resolved spectroscopy at the charge or- +der wavevector. Different from cuprates where the spatial +charge modulation dominantly resides on ligand orbitals, +the charge order in La4Ni3O8 is mostly contributed by +the Ni sites due to the larger charge transfer energy in +low-valence nickelates. In addition to the dominant role +of in-plane Ni 3dx2−y2 and O 2pσ orbitals, the out-of- +plane Ni 3d3z2−r2 orbitals also participate in the charge +order, this being enabled by their hybridization with +the rare-earth 5d orbitals. Thus, our results reveal that +the overall low-energy physical properties of low-valence +nickelates are shaped by Ni 3dx2−y2 and O 2pσ orbitals, +while the detailed electronic structure is fine tuned by +Ni 3d3z2−r2 and rare-earth 5d orbitals. This reveals that +multi-orbital physics is crucial to low-valence nickelates, +indicating that several different ground states are close +in energy. This observation points to a more complex, +and perhaps an even richer, phenomenology than their +cuprate cousins, while charge order remains an intrinsic +character of these strongly correlated materials. +The RIXS data generated in this study have been de- +posited in the Zenodo database under accession code [to +be assigned]. +ACKNOWLEDGMENTS +Work at Brookhaven and the University of Tennessee +(RIXS measurements and the interpretation and model +Hamiltonian calculations) was supported by the U.S. De- +partment of Energy, Office of Science, Office of Basic +Energy Sciences, under Award Number DE-SC0022311. +Work at Argonne was supported by the U.S. DOE, Of- +fice of Science, Basic Energy Sciences, Materials Science +and Engineering Division (nickelate sample synthesis and +first principles calculations). +Work performed at Har- +vard University (data interpretation and paper writing) +was supported by the US Department of Energy, Division +of Materials Science, under Contract No. DESC0012704. +This research used resources at the SIX beamline of the +National Synchrotron Light Source II, a U.S. DOE Office +of Science User Facility operated for the DOE Office of +Science by Brookhaven National Laboratory under Con- +tract No. DE-SC0012704. +Appendix A: Sample synthesis +Parent Ruddlesden-Popper La4Ni3O10 and Pr4Ni3O10 +were prepared using the high-pressure optical floating +zone method. Sample reduction was performed by cleav- +ing small crystals from the boules and heating them +in a flowing H2/Ar gas mixture as described previously +[31]. We adopt the tetragonal notation with space group +I4/mmm and lattice constants of a = b = 3.97 ˚A, +c = 26.092 ˚A to describe reciprocal space. +Using this +notation, the samples had a c-axis surface normal. The +high quality of these samples is confirmed by prior stud- +ies [40, 42]. Single crystals of La4Ni3O8 are particularly +suitable for this study as they exhibit more consistent +XAS spectra and charge order properties than thin films +of infinite-layer nickelates [15–17]]. +Appendix B: RIXS measurements +High-energy-resolution RIXS measurements were per- +formed at the SIX beamline at the NSLS-II. Although +the sample geometry and the energy of the Ni L2-edge +resonance limits reciprocal space access, charge order in +La4Ni3O8 has a c-axis correlation length of less than one +unit cell, which means that the charge order Bragg peaks +are accessible for a wide range of L values [14]. We chose +to measure at the Ni L2-edge instead of the L3 edge to +avoid contamination from the La M-edge which is very +close to the Ni L3-edge and can strongly distort the reso- +nant process [54]. In view of this, we fixed the spectrome- + +7 +ter angle at its maximum value of 2Θ = 153◦ throughout +the measurements of the charge order peak. The sam- +ples were aligned with the crystalline [0, 0, L] and [H, +H, 0] directions lying in the horizontal scattering plane +to access the charge order peak with momentum transfer +QCO = (1/3, 1/3, L) where L ≈ 1.75. In this geometry, +the x-ray intensity is dominated by charge, rather than +spin, scattering (Supplemental Material Sec. IV [34]). +Spectra designed to study the charge order resonance +in the σ polarization channel, such as Fig. 2(e), were +taken with 24 meV energy resolution. +For the charge +order in the π polarization channel, such as Fig. 2(f), +a relaxed energy resolution of 32 meV was used to in- +crease throughput. Whenever the energy was changed, +the sample was rotated in order to remain at the same +in-plane scattering vector. In order to study the high- +energy features, as done in Fig. 2(a) and 2(b), the en- +ergy resolution was further relaxed to 48 meV and the +sample and spectrometer were slightly offset from the +diffraction condition with a sample angle of 14.3◦ and +a spectrometer angle of 2Θ = 147◦ to avoid saturating +the detector. Note that the strong elastic intensity over- +whelms the low-energy inelastic signals such as that from +the magnetic excitations studied previously [42]. Data +collected with different energy-resolution configurations +were normalized by the dd excitations measured with the +same sample geometry. +Upon illumination by very strong elastic scattering +from charge order, a weak periodic error was identified in +the spectrometer grating which created the weak feature +in the energy gain side of Fig. 2(a). This was confirmed +by measuring reference elastic scattering. +Appendix C: Exact diagonalization calculations +The RIXS spectra and REXS responses presented here +were calculated using the Kramers-Heisenberg formula in +the dipole approximation through the EDRIXS software +[55, 56]. The eigenstates for the initial/final and interme- +diate states are obtained from exact diagonalization of a +Ni3O10 cluster with four holes and open boundary con- +ditions. To fully take into account the many-body and +multi-orbital effects, we explicitly include the Coulomb +interactions and nearest-neighbor inter-atomic hoppings +in our model, and construct the Hamiltonian in hole lan- +guage. We use the same parameters as those used in the +O K-edge calculations which are proved to well describe +the RIXS data [40]. +By doing so, the charge-transfer +energy ∆ is set to 5.6 eV and the on-site Coulomb repul- +sion to 6.5 eV, locating the material in the mixed charge- +transfer/Mott-Hubbard regime of the Zaanen-Sawatzky- +Allen (ZSA) scheme. We also include the spin-orbit cou- +pling for the Ni 3d electrons, which is very small and is +expected to play a minimal role. For simplicity, the scat- +tering angle 2Θ is kept at 150◦ and the sample angle is +fixed to θ = 15◦. +The total RIXS scattering amplitude is calculated via +F = +� +i +FieiQ·ri +(C1) +where Fi and ri are the scattering amplitude and position +of each Ni site, respectively. 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Figure S1 shows the RIXS spectra in the quasi- +elastic regime with σ polarized incident photons. No superlattice peaks are found but only background evolving +smoothly with the in-plane sample angle θ, which is primarily caused by the self-absorption effect. Note that despite +the absence of long-range or short-range stripe order indicated here, stripe related spin fluctuations are distinguished +in the inelastic regime [2]. +0.2 +0.1 +0.0 +0.1 +0.2 +Energy loss (eV) +(a) +10 +15 +20 +25 +30 + (deg) +0.0 +0.2 +0.4 +0.6 +Intensity (arb. units) +(b) +0 +2 +4 +6 +8 +10 +Intensity (arb. units) +FIG. S1. Absence of charge order in Pr4Ni3O8 at 40 K. (a) RIXS intensity map around Q∥ = (1/3, 1/3) in the quasi-elastic +regime at the Ni L3-edge. The experimental configuration is the same as that for La4Ni3O8. (b) Quasi-elastic amplitudes +extracted from (a). +∗ yshen@bnl.gov +† mdean@bnl.gov +arXiv:2301.04184v1 [cond-mat.str-el] 10 Jan 2023 + +2 +II. +RIXS PROCESS FOR DIFFERENT EXCITATIONS +Here we discuss the RIXS process for different excitations. Due to the presence of the strong core-hole potential in +the RIXS intermediate states, the electron that is excited from the core level is constrained to a few unit cells near the +Ni site where the x-ray absorption takes place. This effect competes with the kinetic energy of the electron and leads +to intertwined excitations in the RIXS spectra. In a simplified picture, the orbital states during the RIXS process can +be divided into three categories based on how they are affected by the core-hole potential [see Fig. S2(a)]. The first +one involves the Ni 3d orbitals that are strongly localized at the core-hole site. The second one involves the ligand +orbitals that surround the Ni site and strongly hybridize with the Ni 3d orbitals. They are largely localized but could +show a finite bandwidth. The third one involves continuous electronic bands that are mostly unperturbed by the +core-hole potential and behave itinerantly with an appreciable bandwidth. The localized Ni 3d orbitals can hybridize +with the continuous bands in an orbital dependent fashion. At the Ni L-edge, the core electron is predominantly +excited to the unoccupied localized Ni 3d orbitals [see Fig. S2(b)]. During the photon emission process, either an +electron from another 3d orbital deexcites to fill the core hole, leading to dd multiplet excitations [see Fig. S2(d)], or +an electron from the ligand orbitals hops to the Ni site, resulting in charge-transfer excitations [Fig. S2(e)]. Since the +ligand orbitals normally lie at a lower energy, the charge-transfer excitations usually occur with a larger energy loss +than the dd excitations and are much weaker at the Ni L-edge as they are made possible through hybridization. In the +post-edge regime, the core electron is excited to the unoccupied states in the continuous bands through hybridization +in the intermediate state [see Fig. S2(c)], and during the photon emission an electron below the Fermi level deexcites +to fill the core hole, leading to the fluorescence [see Fig. S2(f)]. As the deexciting process is dominated by electrons +near the Fermi energy, fluorescence tends to present a constant emission photon energy. Note that at the Ni L-edge +RIXS process, contributions from rare earth and oxygen states are seen via their hybridization with atomic Ni 3d +orbitals. In real materials, there are no clear boundaries between the localized orbitals and continuous bands. Thus, +different excitations are also intertwined but the weights are quite different, which helps us distinguish them in the +RIXS spectra. +electron +hole +Ni site +Core level +Multiplets +Ligand +Continuous bands +Energy +Distance +Fermi +level +Final states +Intermediate states +fluorescence +Post-edge +Charge-transfer excitations +dd excitations +At edge +(a) +(b) +(d) +(e) +(c) +(f) +FIG. S2. RIXS process for different excitations. (a) Legend for each symbol. (b, c) Photon absorption process and corresponding +intermediate states. (d)–(f) Photon emission process and corresponding final states. For the fluorescence excitation scenario, +the multiplets are not well defined so they are replaced by dashed lines. + +3 +0 +30 +60 +90 +120 +150 +180 + (deg) +0 +1 +2 +3 +4 +Absorption ratio for vs polarization +d3z2 +r2 +dxz/dyz +dx2 +y2/dxy +FIG. S3. Intensity ratio for dipole absorption between π and σ polarization channels as a function of sample angle θ. The +vertical dashed line indicates the experimental configuration we used for charge order measurements while the horizontal dotted +line denotes a unit ratio. +III. +POLARIZATION DEPENDENCE OF FLUORESCENCE +The polarization dependence of dd excitations in cuprates and nickelates has been widely discussed [3, 4]. Here, +we focus on fluorescence to show how the orbital information can be extracted by comparing the RIXS intensity in +different polarization channels. Since we are measuring at the Ni L edge, the RIXS signal can only arise from either +Ni orbitals or Ni states hybridized with other orbitals. For the fluorescence features, the photon emission process +is quite similar, with electrons from the crystalline environment surrounding the Ni site deexciting to fill the core +hole. Hence, the main intensity difference between these two polarization channels comes from the photon absorption +process, the cross-section of which can be simulated in the dipole approximation. Fig. S3 presents the x-ray dipole +absorption intensity ratio between π and σ polarization channels as a function of sample angle θ. For the experimental +configuration we used (vertical dashed line), the biggest contribution to the π over σ polarization intensity ratio is +the Ni 3d3z2−r2 orbitals while 3dx2−y2 and 3dxy contribute equally to σ over π polarization intensity ratio. Since the +3dx2−y2 orbitals dominate near the Fermi level and are expected to show stronger hybridization with oxygen orbitals +[5], 3dxy orbitals are expected to play a less important role. Moreover, the t2g states do not make any significant +contribution to the unoccupied states. Thus, we focus on Ni eg orbitals during the discussion in the main text, which +are the subject of most of the debates over the appropriate theoretical models. +IV. +MINIMAL CONTRIBUTION OF SPIN ORDER TO THE REXS SIGNAL +In La4Ni3O8, spin order takes place concomitantly with the charge order and shares the same Q∥. Hence we need +to invoke cross-section considerations to separate the possible contribution of charge and spin order [6]. +With π incident x-ray polarization, charge order contributes to the measured signal in the π-π′ scattering channel +while the spin order is responsible for the π-σ′ channel. The resonant elastic x-ray scattering (REXS) intensity ratio +between these channels can be estimated by (ki · kf)2/(ϵi × ϵ′ +f · M)2 = cos2 2Θ/ sin2 θ ≈ 11.8, where ki (kf) is the +initial (final) x-ray wavevector, ϵi (ϵ′ +f) is the initial (final) x-ray polarization, and M is the spin direction, which is +parallel to the c-axis in this case. Based on this formula we can see that the REXS signal with π incident polarization +is dominantly of charge order origin. +Regarding the spin order contribution with σ incident x-ray polarization, we can compare the peak intensity with +grazing-in and grazing-out conditions. Since the charge order composes the σ-σ′ channel, its intensity is expected to +be the same in these two geometries. For spin order signal that is only observable in the σ-π′ channel, the intensity + +4 +0.32 +0.33 +0.34 +0.35 +(H, H) (r.l.u.) +0 +2 +4 +6 +8 +Intensity (arb. units) +grazing-in +0.32 +0.33 +0.34 +0.35 +grazing-out +FIG. S4. Comparison of the superlattice peak intensity with grazing-in and grazing-out conditions. The scattering angle 2Θ +was fixed to 153◦ and the data were collected in σ polarization channel at 40 K. The solid lines are guides to the eye. Both +peaks are found to have essentially the same intensity, which confirms that the peak arises from charge, rather than spin, order. +TABLE S1. Full list of parameters used for the ED calculations. The on-site orbital energies, hopping integrals and Coulomb +interactions are kept the same as those used in the O K-edge calculations [7], and Vpdπ = −Vpdσ/2, Vppπ = −Vppσ/4. The +potential difference, ∆ϵd, only applies to the Ni 3d orbitals. Note that the crystal field splitting that is instead used in a Ni +atomic model is a combination of point charge potential and orbital hybridization, which can be estimated through ligand field +theory [5]. The resulting effective crystal field splitting gives 10Dq = 0.971, ∆eg = 1.041, ∆t2g = 0.342 eV, which are of a +similar energy scale as the dd excitations observed in the RIXS measurements. ζi and ζn are spin-orbit coupling parameters of +the Ni 3d electrons for the initial and intermediate states, respectively, and ζc is the spin-orbit coupling strength for the Ni 2p +core electrons. The core-hole lifetime is set to be 0.6 eV. All parameters are in units of eV. +On-site orbital energies +Hopping integrals +ϵdx2−y2 +ϵd3z2−r2 +ϵdxy +ϵdxz/yz +ϵpσ +ϵpπ/pz +Vpdσ +Vppσ +0 +0.2 +0.1 +0.3 +5.6 +6.1 +1.57 +0.6 +Spin-orbit coupling +On-site Coulomb interactions +ζi +ζn +ζc +F 0 +dd +F 2 +dd +F 4 +dd +F 0 +pp +F 2 +pp +0.083 +0.102 +11.507 +5.58 +6.89 +4.31 +3.3 +5 +Inter-site Coulomb interactions +Core-hole potential +F 0 +dp +F 2 +dp +G1 +dp +G3 +dp +F 0 +dp +F 2 +dp +G1 +dp +G3 +dp +1 +0 +0 +0 +7.869 +5.405 +4.051 +2.304 +ratio between grazing-in and grazing-out conditions is (kf, grazing−in ·M)2/(kf, grazing−out ·M)2 ≈ 5.6, indicating that +the spin order signal should be strongly suppressed with grazing-out condition. Figure S4 shows the Q dependence +of the superlattice peak with both conditions, which are comparable with each other, proving that the superlattice +peak observed with σ incident x-ray polarization is also dominantly of charge order origin. +V. +CHARGE ORDER IN ED CALCULATIONS. +We use cluster ED to study the charge order in the low-valence nickelate La4Ni3O8. The full list of the parameters +used is presented in Table S1. The validity of our cluster model and parameters has been verified by calculating +the RIXS energy maps and confirming that they capture the main features of the measurements as shown in the +main text. In the calculations, we include all the Ni 3d and O 2p orbitals, which leads to a large Hilbert space and +correspondingly only a limited number of states can be solved for. Fortunately, the accessible energy range covers +the dd excitations so that we can make a direct comparison with the experimental data. The calculated results are +broadened using a Gaussian profile with a full width at half maximum of 0.3 eV and are shown in Fig. 3 of the main +text. +To fully explore the charge order character in the ED calculations, we need to cover a large incident energy range +but only the ground state is needed to calculate the REXS signals. Thus, we only include the Ni 3dx2−y2 and O 2pσ + +5 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +Hole occupation +(a) +Ni1 +Ni1L +Ni2 +Ni2L +0.0 +0.4 +0.8 +1.2 +1.6 +2.0 +d (eV) +0.00 +0.25 +0.50 +0.75 +Charge disproportionation +(b) +(Ni2+Ni2L)-(Ni1+Ni1L) +10 +12 +14 +16 +18 +Incident energy (eV) +0.0 +0.5 +1.0 +1.5 +2.0 +Intensity (arb. units) +(c) +REXS@QCO, -pol. +d=0.0 +d=0.1 +d=0.2 +d=0.3 +d=0.4 +d=0.5 +d=0.6 +d=0.7 +d=0.8 +d=0.9 +d=1.0 +FIG. S5. The emergence of charge order by introducing the potential difference term ∆ϵd. (a) Hole occupations of different +sites as a function of ∆ϵd. Ni1L stands for the ligand orbitals for Ni1 (the surrounding four oxygens). Correspondingly, one +oxygen is shared by Ni1L and Ni2L. (b) Charge disproportionation defined as the hole occupation difference of Ni1+Ni1L and +Ni2+Ni2L. (c) Calculated REXS signals at QCO with different ∆ϵd. All the calculations are performed with ∆ = 5.6 eV. +orbitals during the calculations of the charge order, which dominate the ground state, so that a tractable basis size +is realized. To trigger charge order in the Ni3O10 cluster, we introduce a potential difference ∆ϵd as described in the +main text. In a microscopic model like we use here, the onsite energy shift and charge occupation are intrinsically +coupled, which is different from a phenomenological model where these two factors can be tuned independently [8, 9]. +As shown in Fig. S5, when ∆ϵd is zero, the hole occupations on different Ni sites are almost the same while the hole +occupations of ligand orbitals are slightly imbalanced since Ni2 shares oxygens with both Ni1 and Ni3, leading to a +small charge disproportionation. With increasing ∆ϵd, the charge imbalances on both the Ni and ligand orbitals are +enhanced with the former much more prominent, indicating that most of the spatial charge modulation resides on +the Ni sites, leading to a Ni site-centered charge order. Correspondingly, a charge-order peak emerge in the REXS +calculations, the intensity of which increases with increasing charge disproportionation while the lineshape only evolves +by a little. +After testing the effect of ∆ϵd, here we compare results with different charge-transfer energy ∆ in addition to the +calculated results presented in the main text. As shown in Fig. S6, in the charge-transfer regime (∆ ≪ Udd), a +sharp resonant peak is obtained, resembling the experimental observations in cuprates. With increasing ∆, the REXS +lineshape evolves correspondingly. In the Mott-Hubbard limit (∆ ≫ Udd), the charge order peak becomes broader and +shows multiple peak features. Compared with the data presented in the main text, we conclude that a charge-transfer +energy with an intermediate strength (∆ ≈ Udd) matches the experimental results the best. +[1] Junjie Zhang, A.S. Botana, J.W. Freeland, D. Phelan, Hong Zheng, V. Pardo, M.R. Norman, and J.F. Mitchell, “Large +orbital polarization in a metallic square-planar nickelate,” Nature Physics 13, 864–869 (2017). +[2] J. Q. Lin, P. Villar Arribi, G. Fabbris, A. S. Botana, D. Meyers, H. Miao, Y. Shen, D. G. Mazzone, J. Feng, S. G. Chiuzb˘aian, +A. Nag, A. C. Walters, M. Garc´ıa-Fern´andez, Ke-Jin Zhou, J. Pelliciari, I. Jarrige, J. W. Freeland, Junjie Zhang, J. F. +Mitchell, V. Bisogni, X. Liu, M. R. Norman, and M. P. M. Dean, “Strong superexchange in a d9−δ nickelate revealed by +resonant inelastic x-ray scattering,” Physical Review Letters 126, 087001 (2021). +[3] M. Rossi, H. Lu, A. Nag, D. Li, M. Osada, K. Lee, B. Y. Wang, S. Agrestini, M. Garcia-Fernandez, J. J. Kas, Y.-D. Chuang, +Z. X. Shen, H. Y. Hwang, B. Moritz, Ke-Jin Zhou, T. P. Devereaux, and W. S. Lee, “Orbital and spin character of doped +carriers in infinite-layer nickelates,” Physical Review B 104, L220505 (2021). +[4] M. Moretti Sala, V. Bisogni, C. Aruta, G. Balestrino, H. Berger, N. B. Brookes, G. M. de Luca, D. Di Castro, M. Grioni, + +6 +10 +12 +14 +16 +18 +Incident energy (eV) +0 +1 +2 +3 +4 +5 +Intensity (arb. units) +REXS@QCO, -pol. +=2.5 +=3.5 +=4.5 +=5.6 +=6.7 +=7.8 +=8.9 +FIG. S6. +Calculated REXS signals at the charge order wavevector QCO with different charge-transfer energy ∆. +All the +calculations are performed with ∆ϵd = 0.8 eV and U = 6.5 eV. +M. Guarise, P. G. Medaglia, F. Miletto Granozio, M. Minola, P. Perna, M. Radovic, M. Salluzzo, T. Schmitt, K. J. Zhou, +L. Braicovich, and G. Ghiringhelli, “Energy and symmetry of dd excitations in undoped layered cuprates measured by Cu +L3 resonant inelastic x-ray scattering,” New Journal of Physics 13, 043026 (2011). +[5] M. W. Haverkort, M. Zwierzycki, +and O. K. Andersen, “Multiplet ligand-field theory using wannier orbitals,” Physical +Review B 85, 165113 (2012). +[6] M. W. Haverkort, “Theory of resonant inelastic x-ray scattering by collective magnetic excitations,” Physical Review Letters +105, 167404 (2010). +[7] Y. Shen, J. Sears, G. Fabbris, J. Li, J. Pelliciari, I. Jarrige, Xi He, I. Boˇzovi´c, M. Mitrano, Junjie Zhang, J. F. Mitchell, +A. S. Botana, V. Bisogni, M. R. Norman, S. Johnston, +and M. P. M. Dean, “Role of oxygen states in the low valence +nickelate La4Ni3O8,” Physical Review X 12, 011055 (2022). +[8] A. J. Achkar, R. Sutarto, X. Mao, F. He, A. Frano, S. Blanco-Canosa, M. Le Tacon, G. Ghiringhelli, L. Braicovich, +M. Minola, M. Moretti Sala, C. Mazzoli, Ruixing Liang, D. A. Bonn, W. N. Hardy, B. Keimer, G. A. Sawatzky, and D. G. +Hawthorn, “Distinct charge orders in the planes and chains of ortho-III-ordered YBa2Cu3O6+δ superconductors identified +by resonant elastic x-ray scattering,” Physical Review Letters 109, 167001 (2012). +[9] A. J. Achkar, F. He, R. Sutarto, J. Geck, H. Zhang, Y.-J. Kim, and D. G. Hawthorn, “Resonant x-ray scattering measure- +ments of a spatial modulation of the Cu 3d and O 2p energies in stripe-ordered cuprate superconductors,” Physical Review +Letters 110, 017001 (2013). + diff --git a/0tE2T4oBgHgl3EQf4wij/content/tmp_files/load_file.txt b/0tE2T4oBgHgl3EQf4wij/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7775bdd112f430e007ab1e960766f32ace76a639 --- /dev/null +++ b/0tE2T4oBgHgl3EQf4wij/content/tmp_files/load_file.txt @@ -0,0 +1,1503 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf,len=1502 +page_content='Electronic character of charge order in square planar low valence nickelates Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Shen,1, ∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Sears,1 G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Fabbris,2 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Li,3 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Pelliciari,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Mitrano,4 W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' He,1 Junjie Zhang,5, 6 J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Mitchell,5 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Bisogni,3 M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Norman,5 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Johnston,7, 8 and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Dean1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' † 1Condensed Matter Physics and Materials Science Department,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' New York 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 2Advanced Photon Source,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Lemont,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Illinois 60439,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 3National Synchrotron Light Source II,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Brookhaven National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Upton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' New York 11973,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 4Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Harvard University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Cambridge,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Massachusetts 02138,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 5Materials Science Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Argonne National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Lemont,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Illinois 60439,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 6Institute of Crystal Materials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Shandong University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Jinan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Shandong 250100,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' China 7Department of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Tennessee 37966,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA 8Institute of Advanced Materials and Manufacturing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The University of Tennessee,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Knoxville,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Tennessee 37996,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' USA (Dated: January 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2023) Charge order is a central feature of the physics of cuprate superconductors and is known to arise from a modulation of holes with primarily oxygen character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Low-valence nickelate superconductors also host charge order, but the electronic character of this symmetry breaking is unsettled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Here, using resonant inelastic x-ray scattering at the Ni L2-edge, we identify intertwined involvements of Ni 3dx2−y2, 3d3z2−r2, and O 2pσ orbitals in the formation of diagonal charge order in an overdoped low-valence nickelate La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The Ni 3dx2−y2 orbitals, strongly hybridized with planar O 2pσ, largely shape the spatial charge distribution and lead to Ni site-centered charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The 3d3z2−r2 orbitals play a small, but non-negligible role in the charge order as they hybridize with the rare- earth 5d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Our results reveal that the low-energy physics and ground-state character of these nickelates are more complex than those in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' INTRODUCTION One of the common threads linking different classes of unconventional superconductors is their propensity to host proximate competing orders such as charge and spin stripes [1, 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For example, the cuprate superconduc- tors exhibit diagonal (with respect to the Cu-O bonds) spin stripes when underdoped [3–5], while Cu-O bond oriented (parallel) charge order dominates the rest of the phase diagram [6, 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The detection of superconductivity and charge order in the square-planar low-valence fam- ily of nickelates therefore presents a fascinating oppor- tunity to study the degree of similarity between differ- ent unconventional superconducting families [8–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In- triguingly, different nickelates within the structural se- ries of Rn+1NinO2n+2 (R stands for a rare earth and n is the number of neighboring NiO2 layers) also host differ- ent charge ordered phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Underdoped materials with n = ∞ and R = La, Nd exhibit parallel charge order [15–17], whereas n = 3 material La4Ni3O8, which is ef- fectively 1/3 overdoped, manifests diagonal charge order [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Many researchers have emphasized that charge or- der plays an important role in the physics of cuprates [18–21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In particular, there is good evidence showing that charge/spin order is a fundamental feature of min- imal Hubbard model descriptions of the cuprates [22– 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Some researchers have suggested that charge and spin order can intertwine with superconductivity to form pair density waves [25, 26], or that dynamic charge/spin ∗ yshen@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='gov † mdean@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='gov fluctuations might promote superconductivity [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Others have associated charge order fluctuations with the anomalous “strange metal” electronic transport in cuprates [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Understanding the electronic states in- volved in charge order formation is a prerequisite to test- ing all these scenarios in low-valence nickelates and is also important more generally for understanding charge order as a prevalent feature of correlated quantum materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Here, we use Ni L2-edge RIXS to determine the elec- tronic character of the charge order in La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We find that both the Ni 3dx2−y2 and 3d3z2−r2 orbitals are involved in charge order formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The former con- tributes most of the charge modulation while the latter dominates the RIXS spectra in the post-edge regime and so plays a less important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As the charge-transfer energy of these nickelates is larger than that of cuprates but comparable to the on-site Coulomb interaction, the holes involved in the charge modulation reside predomi- nately on Ni sites, despite an appreciable amount of holes occupying the O orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Our results indicate that the low-energy electronic structure and charge order of low- valence nickelates is largely shaped by hybridized 3dx2−y2 and planar O 2pσ orbitals, similar to cuprates, while some differences exist due to the multi-band physics in- troduced by Ni 3d3z2−r2 orbitals hybridized with rare- earth 5d states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' RESULTS The La4Ni3O8 nickelate samples studied here were pre- pared by reducing single crystals synthesized via the floating zone method (see the Appendix A for details), arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='04184v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='str-el] 10 Jan 2023 2 tpd tpp Ni1 Ni2 Ni3 a b c σ Ni L2-edge � θ (a) Nickel Oxygen NiO2 plane La4Ni3O8 sample (b) (c) (d) (e) (f) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 Energy loss (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') T = 40 K T = 50 K T = 70 K T = 90 K T = 110 K ×10 0 25 50 75 100 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) (g) (h) 40 K 50 K 70 K 90 K 110 K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='35 Q║=(H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') 40 60 80 100 120 Temperature (K) 0 2 4 6 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Charge order transition in La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a) Schematic of the Ni L2-edge RIXS experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' A single NiO2 layer is presented with stripes running vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' A Ni3O10 cluster composed of Ni 3dx2−y2 and planar O 2pσ orbitals is embedded in it, tracing the charge order motif, in which hole poor Ni1 and Ni3 sites, shown in red, flank the hole rich Ni2 site depicted in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b)–(f) RIXS intensity maps with σ polarized incident photons at the indicated temperatures obtained by changing the in-plane sample angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (g) Quasi-elastic-line amplitudes extracted from the data presented in (b)–(f) as a function of in-plane momentum transfer in reciprocal lattice units (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The solid lines are fitting curves with pseudo-Voigt profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (h) Temperature dependence of the fitted peak amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The bold gray line is a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' and will be indexed in terms of scattering vector Q = (2π/a, 2π/a, 2π/c) with a = b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='97 ˚A, c = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='092 ˚A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As the n = 3 member of the low-valence nickelate family, it possesses a trilayer structure with a nominal 3d8+2/3 valence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This leads to a 1/3-hole self-doping with re- spect to the undoped 3d9 state, putting it in the over- doped regime of the phase diagram [13, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' It shares the same structural motif as infinite-layer nickelates with square-planar NiO2 layers stacked without apical oxy- gens, leading to dominant Ni 3dx2−y2 character near the Fermi energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Although La4Ni3O8 has two inequiv- alent NiO2 layers, they are expected to show similar electronic structure as indicated by theoretical calcula- tions [32, 33], which is further supported by the obser- vation that the same charge order pattern is formed in both layers [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We study their properties using Ni L2- edge RIXS in order to avoid interference from the La M4-edge, which overlaps the Ni L3-edge (see the Ap- pendix B for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(a), charge or- der in La4Ni3O8 is quasi-two-dimensional in nature and occurs at Q∥ = QCO = (1/3, 1/3), where a strong peak is observed in the quasi-elastic region of the RIXS inten- sity map at 40 K [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The in-plane correlation length is larger than 100 nm, which might be limited by the sample mosaic, suggesting the long range nature of the charge order [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This charge order peak persists up to 90 K and disappears above 110 K, indicating a tran- sition temperature of around 100 K [see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(c)–(h)], consistent with the reported charge order from hard x-ray diffraction measurements [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' No indication of charge order is apparent in equivalent measurements of metallic Pr4Ni3O8 samples prepared in the same way (Supple- mental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' I [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We begin by identifying the active electronic states in La4Ni3O8 using x-ray spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Figure 2(a) and 2(b) show the L2-edge RIXS energy maps taken with σ x-ray polarization in the ab-plane and π x-ray po- larization approximately parallel to the c-axis, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The RIXS maps mainly comprise dd and charge- transfer excitations that are predominantly localized and resonate at the Ni L2-edge, and diagonal fluorescence features (Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' II [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To distin- guish among these contributions, we integrated the RIXS spectra along the incident energy axis and show the re- sult in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With σ polarization, the spectra above 4 eV energy loss are dominated by mostly featureless flu- orescence originating from particle-hole excitations that can be understood from an itinerant framework involv- ing transitions from extended electronic bands spanning many unit cells [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Charge transfer excitations are also 3 CT dd FL dd FL (a) (b) 0 1 2 3 4 5 6 Energy loss (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 0 1 2 3 4 5 6 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' �-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 0 2 4 6 8 Energy loss (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 dd CT + FL (c) σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' �-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Integral over incident energy 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='03 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) 865 870 875 880 Incident energy (eV) CT FL (d) 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 ≤ Eloss ≤ 6 σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' �-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1, CO CO 866 868 870 872 874 Incident energy (eV) 866 868 870 872 874 Incident energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 Energy loss (eV) 0 4 8 12 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='10 0 4 8 12 16 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) (e) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' RIXS energy maps and the resonant behaviors of the charge order (CO) peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a, b) RIXS intensity maps as a function of incident photon energy with (a) σ x-ray polarization in the ab plane of the sample and (b) π x-ray polarization approximately parallel to the c-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Several components can be identified: charge transfer excitations (CT), dd excitations (dd) and constant-emission-energy fluorescence (FL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (c) Integral of the RIXS spectra along the incident energy axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The dashed lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (d) Incident energy dependence of the integrated RIXS spectra between 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 and 6 eV energy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (e, f) RIXS intensity maps around the quasi-elastic regime with Q fixed at QCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that the intensity in (e) is multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 for clarity in visualizing the signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' visible above 4 eV but only at resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Below 4 eV, prominent dd excitations emerge that dominate over the featureless fluorescence (dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With π polariza- tion, the fluorescence contributes most of the spectral weight and the dd excitations are much weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The strong dichroism of dd excitations reflects the dominant Ni 3dx2−y2 orbital character near the Fermi energy in low-valence nickelates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To further distinguish between charge-transfer excita- tions and fluorescence, we inspect the RIXS spectra be- tween 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 and 6 eV energy loss, well above the dd excita- tion threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(d), the charge-transfer excitations and fluorescence are separated in the incident energy axis, with the former stronger in the σ polariza- tion channel, indicating appreciable dx2−y2-pσ hybridiza- tion where pσ indicates O orbitals that are parallel to the Ni-O bonds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In contrast, the fluorescence is stronger in the π polarization channel, suggesting that states involv- ing Ni 3d3z2−r2 orbitals dominate the fluorescence for a broad range of energy losses above ∼3 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The broadness of these states is in contrast with cuprates, and suggests that although the Ni 3d3z2−r2 orbitals are mostly oc- cupied and localized, their unoccupied components are hybridized with the rare earth 5d orbitals and thus con- tribute to dispersive states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This conclusion is consistent with density functional theory (DFT)+dynamical mean field theory (DMFT) calculations [32], as well as RIXS simulations for RNiO2 that studied the effect of switching on and off the rare-earth hybridization [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Meanwhile, the Ni 3dx2−y2 orbitals exhibit less hybridization with the rare earth 5d orbitals and are more localized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Here, since we are measuring at the Ni L edge and the Ni t2g orbitals are expected to lie well below the Fermi energy, we only consider Ni eg orbitals [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Based on the resonant behavior of the different states identified, we now examine how the 3dx2−y2 and 3d3z2−r2 orbitals participate in the charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Figure 2(e) and 2(f) show the RIXS energy maps around the quasi-elastic regime at QCO, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' the resonant elastic x-ray scatter- ing (REXS) signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The peak intensity strongly res- onates at the Ni L2-edge in the σ polarization channel [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(e)], confirming that the (1/3, 1/3) Bragg peak in La4Ni3O8 involves a charge modulation and is not purely structural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Surprisingly, the charge order peak in the π polarization channel, although much weaker, res- onates at the pre- and post-edge regimes but not at the main edge [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(f)], distinct from that in cuprates [37–39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' First, this observation indicates that both the 3dx2−y2 and 3d3z2−r2 orbitals are involved in charge order formation with the latter much less prominent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Second, 4 (a) 868 870 872 Incident energy (eV) σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b) 868 870 872 Incident energy (eV) �-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 0 1 2 3 Energy loss (eV) Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Low-energy electronic states in La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Calcula- tions of the RIXS energy maps at the Ni L2-edge for (a) σ and (b) π incident x-ray polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The calculations reproduce the experimental energy-scale and polarization of the dd exci- tations evincing an appropriate minimal model for La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' the charge order peak in the post-edge regime with π po- larization suggests that the states far above the Fermi energy also show charge modulation, which is mostly contributed by 3d3z2−r2 orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Considering that the 3d3z2−r2 density of states in the post-edge regime is likely caused by hybridization with the rare-earth 5d orbitals, this indicates potential involvement of rare-earth orbitals in the charge order formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Similarly, the weak pre- edge charge order peak with π polarization indicates that the 3d3z2−r2 density of states near the Fermi energy is nonzero but small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Having established the involvement of Ni orbitals in the charge order formation, now we look at the role of oxygen states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To do this, we use exact diagonalization (ED) methods which allow us to solve the resonant cross- section and break down the contributions from different states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Since the charge order is commensurate with a period of three Ni sites and there is a strong hybridiza- tion between the Ni and O orbitals, the smallest cluster one can use to describe the charge-ordered state involves three Ni-O plaquettes, which we label 1, 2, & 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We choose a bond-oriented cluster, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(a), given that the Ni-O hopping dominates the kinetic en- ergy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In order to compute REXS we use the atomic scattering factors from the cluster and add these am- plitudes to simulate an effective two-dimensional NiO2 plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The appropriate parame- ters for this cluster, and in particular the charge-transfer energy ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 eV and the on-site Coulomb repulsion Udd = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 eV, have been empirically determined by prior x-ray measurements of this material at the O K-edge [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We use open boundary conditions and construct the Hamiltonian in the hole language (see the Appendix C for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Four holes are introduced to the cluster, which is appropriate for the d9−1/3 electronic configu- ration of La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Without any additional constraints, the holes will be evenly distributed among different NiO4 plaquettes with minimal charge disproportionation and no symmetry breaking is expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To realize the charge order observed in La4Ni3O8, we manually introduce a potential difference [41], ∆ϵd, for different Ni sites by lowering the orbital energies of Ni2 by 2∆ϵd/3 and rais- ing those of Ni1 and Ni3 by ∆ϵd/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Based on the sim- ilar magnetic exchange of charge ordered La4Ni3O8 and metallic Pr4Ni3O8 [42], ∆ϵd must be significantly smaller than the charge-transfer energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Thus, we choose it to be ∆ϵd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 eV while noting that apart from modu- lating the intensity of the charge order peak, the results are similar provided ∆ϵd is not made unfeasibly large (Supplemental Material Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S5 [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This choice leads to a charge disproportionation of ∆n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32, which is of a similar order of magnitude as that in cuprates [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This value is much smaller than the fully disproportion- ate limit ∆n = 1, consistent with DFT calculations that indicate a small charge modulation upon charge ordering in this system [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' When examining the electronic con- figuration of the cluster, we find that the ground state is a singlet, and the first excited state is a triplet, which is around 70 meV above the ground state, consistent with the magnetic excitations found in La4Ni3O8 [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Figure 3 shows the calculated Ni L2-edge RIXS en- ergy maps with all the Ni 3d and O 2p orbitals included, which qualitatively reproduce the localized dd excitations observed experimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that the small cluster size means that we can only capture a limited number of dis- crete states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For this reason, fluorescence features are not fully captured, which would require a continuous distri- bution of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This can be seen more clearly in the π polarization channel where the fluorescence dominates the spectra in experimental data [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(b)] but only the weak dd excitations are present in our cluster calcu- lations [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Having verified the relevant parameters via the RIXS maps, we computed the x-ray absorption spectrum (XAS) and REXS response of La4Ni3O8 using a simi- lar ED approach and identical parameters and plot the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4 (Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' V [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The charge disproportionation in the cluster implies a REXS response at QCO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The predicted REXS resonance shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4(c) nicely captures the main two peak structure of the experimental REXS resonance shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The same applies for the XAS as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In fact, the lineshape of the resonant profile of the charge order peak is sensitive to the charge-transfer energy, and neither the pure charge-transfer nor Mott- Hubbard scenarios can describe the observed resonant behaviors (Supplemental Material Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S6 [34]), demon- 5 strating the mixed charge-transfer/Mott-Hubbard char- acters of charge order in this material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To understand the nature of the two resonant features, we projected the wavefunctions of the RIXS intermediate states onto the Fock basis which specifies the location of the holes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Two main manifolds are seen for each Ni site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The first manifold is primarily attributed to transitions resonant with d10L0 states, where L stands for ligand holes on the four oxygen σ orbitals surrounding the Ni site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The second manifold is mainly resonant with d9L0 and d10L1 states caused by the doped holes, similar to the cuprates [43, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With nonzero ∆ϵd, the manifolds of different Ni sites split along the incident energy axis, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The successful description of the charge order in La4Ni3O8 using our cluster model indicates that about 70% of the holes participating in the charge modulation are on Ni, with the remaining 30% on oxygen, as depicted in the inset to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' DISCUSSION Our Ni-dominant charge order distribution is quite dif- ferent from cuprates, in which the charge order has dom- inant oxygen character [37, 45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This difference mainly arises from the larger charge transfer energy in nicke- lates compared to cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Another difference is that in cuprates, the 3d3z2−r2 orbitals are strongly localized at energies more than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 eV away from the 3dx2−y2 orbitals [46], and thus not involved in the low-energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For square-planar nickelates, our analysis of La4Ni3O8 indi- cates that the 3d3z2−r2 density of states, though small, is spread out over an extended energy range, likely due to hybridization with the rare earth 5d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' It should be noted that although the 3d3z2−r2 orbital involvement in the charge order formation is nonzero, its contribution is much less than the hybridized 3dx2−y2 and 2pσ orbitals, as indicated by the stronger charge order peak in the σ polarization channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' These factors mean that minimal theoretical models of charge order in nickelates must ex- plicitly include both Ni and O states alongside strong correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Another result of our model is that the doped sites in charge ordered nickelates are much closer to a low-spin S = 0 state than to a high-spin S = 1 state, unlike La2−xSrxNiO4, whose high-spin physics drives in- sulating behavior across the vast majority of its phase diagram [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Recently, RIXS measurements in infinite-layer nicke- late films have discovered and studied charge order at Q∥ = (1/3, 0) in undoped and underdoped samples [15– 17], resembling the charge order in cuprates, but differing from the diagonal charge order in La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In terms of these differing wavevectors, theoretical model studies in the cuprates have shown that charge order at (Q, 0) and (Q, Q) are close in energy, the eventual choice of the charge order wavevector being sensitive to details of the electronic structure and correlations [48, 49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This idea is supported by the experimental observation that the dop- 868 869 870 871 872 873 Incident energy (eV) 2 1 0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) Ni1: d 10L0 Ni1: d 10L1 Ni1: d 9L0 Ni2: d 10L0 Ni2: d 10L1 Ni2: d 9L0 REXS calculation (c) 6 4 2 0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) Ni1 Ni3 Ni2 O σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' �-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' REXS data (b) (a) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='4 0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) XAS, σ-pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Data Ni1+Ni3 Ni2 Calculation FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Electronic character of charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a) x-ray ab- sorption spectrum (XAS) data at the Ni L2 edge in the σ polarization channel along with the calculation results with ∆ϵd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that Ni1 and Ni3 are symmetry-related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b) Fitted peak amplitudes of the quasi-elastic intensities pre- sented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(e)&(f), representing the resonant behaviors of the charge order peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Inset is a schematic of the electronic character of the charge order showing a dominant modula- tion of Ni orbitals along with an appreciable modulation of the oxygen orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (c) Simulation of the incident energy dependence of the charge order peak intensity with σ inci- dent polarization and ∆ϵd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The vertical bars are weights of different configurations of the RIXS intermediate states, the total height of which is normalized according to the simulated charge order peak intensity of each state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The accurate simulation of the Ni 3d and O 2p components of the resonance verifies our model, which is used to extract the elec- tronic character of the charge order illustrated in the inset to panel (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' ing dependent charge order wavevector varies in different cuprate families [20], similar to what has been seen more recently in the infinite-layer nickelates [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In view of this, the difference in wavevector probably does not re- flect a difference in the mechanisms at play in charge order formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' It should, however, be noted that the parallel charge order seen in infinite-layer materials oc- curs at a lower hole concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' More information can be obtained by comparing the 6 states involved in charge order formation for different low-valence nickelates [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' All these recent works support an appreciable role for Ni in charge order forma- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' However, controversy exists regarding whether the rare-earth-Ni hybridization is crucial for charge order for- mation [16], or whether the charge modulation on rare- earth states only plays a secondary parasitic role [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Our results support the latter scenario in La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Re- garding the involvement of oxygen states, we provide the first spectroscopic modeling that allows this question to be addressed quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We deduce a mixed charge- transfer/Mott-Hubbard picture for the charge order and 70%/30% split of Ni vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' O contributions to the charge modulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This contradicts some of the previous sug- gestions for infinite-layer nickelates, which propose a neg- ligible role for oxygen in charge order formation and that in-plane and out-of-plane Ni states contribute roughly equally [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' These differences are puzzling consider- ing that different members of the Rn+1NinO2n+2 family share similar Ni-O bonding, magnetic exchange [42, 50], superconducting transition temperatures [12, 13, 51, 52], and calculated electronic structures [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Part of the chal- lenge of making this comparison is that RIXS maps of infinite-layer films, as well as their charge order proper- ties, vary substantially between different samples of nom- inally the same composition [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In this regard, our quantitative spectroscopic analysis on single crystals is valuable considering that these samples show more con- sistent spectral properties than films of infinite layer ma- terials [15–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' CONCLUSION In summary, we have used RIXS measurements at the Ni L2-edge to study the character of the electronic structure and charge order in the low-valence nickelate La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Our work is unique in providing a realistic quantitative empirical model for charge order and vali- dating it using Q-resolved spectroscopy at the charge or- der wavevector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Different from cuprates where the spatial charge modulation dominantly resides on ligand orbitals, the charge order in La4Ni3O8 is mostly contributed by the Ni sites due to the larger charge transfer energy in low-valence nickelates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In addition to the dominant role of in-plane Ni 3dx2−y2 and O 2pσ orbitals, the out-of- plane Ni 3d3z2−r2 orbitals also participate in the charge order, this being enabled by their hybridization with the rare-earth 5d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Thus, our results reveal that the overall low-energy physical properties of low-valence nickelates are shaped by Ni 3dx2−y2 and O 2pσ orbitals, while the detailed electronic structure is fine tuned by Ni 3d3z2−r2 and rare-earth 5d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This reveals that multi-orbital physics is crucial to low-valence nickelates, indicating that several different ground states are close in energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This observation points to a more complex, and perhaps an even richer, phenomenology than their cuprate cousins, while charge order remains an intrinsic character of these strongly correlated materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The RIXS data generated in this study have been de- posited in the Zenodo database under accession code [to be assigned].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' ACKNOWLEDGMENTS Work at Brookhaven and the University of Tennessee (RIXS measurements and the interpretation and model Hamiltonian calculations) was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' De- partment of Energy, Office of Science, Office of Basic Energy Sciences, under Award Number DE-SC0022311.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Work at Argonne was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' DOE, Of- fice of Science, Basic Energy Sciences, Materials Science and Engineering Division (nickelate sample synthesis and first principles calculations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Work performed at Har- vard University (data interpretation and paper writing) was supported by the US Department of Energy, Division of Materials Science, under Contract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' DESC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This research used resources at the SIX beamline of the National Synchrotron Light Source II, a U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' DOE Office of Science User Facility operated for the DOE Office of Science by Brookhaven National Laboratory under Con- tract No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' DE-SC0012704.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Appendix A: Sample synthesis Parent Ruddlesden-Popper La4Ni3O10 and Pr4Ni3O10 were prepared using the high-pressure optical floating zone method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Sample reduction was performed by cleav- ing small crystals from the boules and heating them in a flowing H2/Ar gas mixture as described previously [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We adopt the tetragonal notation with space group I4/mmm and lattice constants of a = b = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='97 ˚A, c = 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='092 ˚A to describe reciprocal space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Using this notation, the samples had a c-axis surface normal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The high quality of these samples is confirmed by prior stud- ies [40, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Single crystals of La4Ni3O8 are particularly suitable for this study as they exhibit more consistent XAS spectra and charge order properties than thin films of infinite-layer nickelates [15–17]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Appendix B: RIXS measurements High-energy-resolution RIXS measurements were per- formed at the SIX beamline at the NSLS-II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Although the sample geometry and the energy of the Ni L2-edge resonance limits reciprocal space access, charge order in La4Ni3O8 has a c-axis correlation length of less than one unit cell, which means that the charge order Bragg peaks are accessible for a wide range of L values [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We chose to measure at the Ni L2-edge instead of the L3 edge to avoid contamination from the La M-edge which is very close to the Ni L3-edge and can strongly distort the reso- nant process [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In view of this, we fixed the spectrome- 7 ter angle at its maximum value of 2Θ = 153◦ throughout the measurements of the charge order peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The sam- ples were aligned with the crystalline [0, 0, L] and [H, H, 0] directions lying in the horizontal scattering plane to access the charge order peak with momentum transfer QCO = (1/3, 1/3, L) where L ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In this geometry, the x-ray intensity is dominated by charge, rather than spin, scattering (Supplemental Material Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' IV [34]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Spectra designed to study the charge order resonance in the σ polarization channel, such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(e), were taken with 24 meV energy resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For the charge order in the π polarization channel, such as Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(f), a relaxed energy resolution of 32 meV was used to in- crease throughput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Whenever the energy was changed, the sample was rotated in order to remain at the same in-plane scattering vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In order to study the high- energy features, as done in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(a) and 2(b), the en- ergy resolution was further relaxed to 48 meV and the sample and spectrometer were slightly offset from the diffraction condition with a sample angle of 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='3◦ and a spectrometer angle of 2Θ = 147◦ to avoid saturating the detector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that the strong elastic intensity over- whelms the low-energy inelastic signals such as that from the magnetic excitations studied previously [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Data collected with different energy-resolution configurations were normalized by the dd excitations measured with the same sample geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Upon illumination by very strong elastic scattering from charge order, a weak periodic error was identified in the spectrometer grating which created the weak feature in the energy gain side of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This was confirmed by measuring reference elastic scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Appendix C: Exact diagonalization calculations The RIXS spectra and REXS responses presented here were calculated using the Kramers-Heisenberg formula in the dipole approximation through the EDRIXS software [55, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The eigenstates for the initial/final and interme- diate states are obtained from exact diagonalization of a Ni3O10 cluster with four holes and open boundary con- ditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To fully take into account the many-body and multi-orbital effects, we explicitly include the Coulomb interactions and nearest-neighbor inter-atomic hoppings in our model, and construct the Hamiltonian in hole lan- guage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We use the same parameters as those used in the O K-edge calculations which are proved to well describe the RIXS data [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' By doing so, the charge-transfer energy ∆ is set to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 eV and the on-site Coulomb repul- sion to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 eV, locating the material in the mixed charge- transfer/Mott-Hubbard regime of the Zaanen-Sawatzky- Allen (ZSA) scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We also include the spin-orbit cou- pling for the Ni 3d electrons, which is very small and is expected to play a minimal role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For simplicity, the scat- tering angle 2Θ is kept at 150◦ and the sample angle is fixed to θ = 15◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The total RIXS scattering amplitude is calculated via F = � i FieiQ·ri (C1) where Fi and ri are the scattering amplitude and position of each Ni site, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The charge order peak was then calculated by combining the atomic scattering am- plitudes with the phases appropriate for tiling the cluster into the NiO2 plane as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Norman, The challenge of unconventional super- conductivity, Science 332, 196 (2011).' metadata={'source': 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+page_content=' Keimer, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Sawatzky, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Hawthorn, Distinct charge orders in the planes and chains of ortho-III-ordered YBa2Cu3O6+δ superconductors identified by resonant elastic x-ray scat- tering, Physical Review Letters 109, 167001 (2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' [60] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' LaBollita and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Botana, Electronic structure and magnetic properties of higher-order layered nickelates: Lan+1NinO2n+2(n = 4 − 6), Physical Review B 104, 035148 (2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Supplemental Material: Electronic character of charge order in square planar low valence nickelates Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Shen,∗ J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Sears, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Fabbris, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Pelliciari, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Mitrano, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' He, Junjie Zhang, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Mitchell, V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Bisogni, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Norman, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Johnston, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Dean† (Dated: January 12, 2023) I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' ABSENCE OF DIAGONAL CHARGE ORDER IN Pr4Ni3O8 To confirm the absence of diagonal charge order in metallic Pr4Ni3O8 [1], we performed resonant inelastic x-ray scattering (RIXS) measurements near Q∥ = (1/3, 1/3) in Pr4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Figure S1 shows the RIXS spectra in the quasi- elastic regime with σ polarized incident photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' No superlattice peaks are found but only background evolving smoothly with the in-plane sample angle θ, which is primarily caused by the self-absorption effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that despite the absence of long-range or short-range stripe order indicated here, stripe related spin fluctuations are distinguished in the inelastic regime [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 Energy loss (eV) (a) 10 15 20 25 30 (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) (b) 0 2 4 6 8 10 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Absence of charge order in Pr4Ni3O8 at 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a) RIXS intensity map around Q∥ = (1/3, 1/3) in the quasi-elastic regime at the Ni L3-edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The experimental configuration is the same as that for La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b) Quasi-elastic amplitudes extracted from (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' ∗ yshen@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='gov † mdean@bnl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='gov arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='04184v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='str-el] 10 Jan 2023 2 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' RIXS PROCESS FOR DIFFERENT EXCITATIONS Here we discuss the RIXS process for different excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Due to the presence of the strong core-hole potential in the RIXS intermediate states, the electron that is excited from the core level is constrained to a few unit cells near the Ni site where the x-ray absorption takes place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' This effect competes with the kinetic energy of the electron and leads to intertwined excitations in the RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In a simplified picture, the orbital states during the RIXS process can be divided into three categories based on how they are affected by the core-hole potential [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The first one involves the Ni 3d orbitals that are strongly localized at the core-hole site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The second one involves the ligand orbitals that surround the Ni site and strongly hybridize with the Ni 3d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' They are largely localized but could show a finite bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The third one involves continuous electronic bands that are mostly unperturbed by the core-hole potential and behave itinerantly with an appreciable bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The localized Ni 3d orbitals can hybridize with the continuous bands in an orbital dependent fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' At the Ni L-edge, the core electron is predominantly excited to the unoccupied localized Ni 3d orbitals [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' During the photon emission process, either an electron from another 3d orbital deexcites to fill the core hole, leading to dd multiplet excitations [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(d)], or an electron from the ligand orbitals hops to the Ni site, resulting in charge-transfer excitations [Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Since the ligand orbitals normally lie at a lower energy, the charge-transfer excitations usually occur with a larger energy loss than the dd excitations and are much weaker at the Ni L-edge as they are made possible through hybridization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In the post-edge regime, the core electron is excited to the unoccupied states in the continuous bands through hybridization in the intermediate state [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(c)], and during the photon emission an electron below the Fermi level deexcites to fill the core hole, leading to the fluorescence [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2(f)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As the deexciting process is dominated by electrons near the Fermi energy, fluorescence tends to present a constant emission photon energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that at the Ni L-edge RIXS process, contributions from rare earth and oxygen states are seen via their hybridization with atomic Ni 3d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In real materials, there are no clear boundaries between the localized orbitals and continuous bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Thus, different excitations are also intertwined but the weights are quite different, which helps us distinguish them in the RIXS spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' electron hole Ni site Core level Multiplets Ligand Continuous bands Energy Distance Fermi level Final states Intermediate states fluorescence Post-edge Charge-transfer excitations dd excitations At edge (a) (b) (d) (e) (c) (f) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' RIXS process for different excitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a) Legend for each symbol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b, c) Photon absorption process and corresponding intermediate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (d)–(f) Photon emission process and corresponding final states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For the fluorescence excitation scenario, the multiplets are not well defined so they are replaced by dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 3 0 30 60 90 120 150 180 (deg) 0 1 2 3 4 Absorption ratio for vs polarization d3z2 r2 dxz/dyz dx2 y2/dxy FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Intensity ratio for dipole absorption between π and σ polarization channels as a function of sample angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The vertical dashed line indicates the experimental configuration we used for charge order measurements while the horizontal dotted line denotes a unit ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' POLARIZATION DEPENDENCE OF FLUORESCENCE The polarization dependence of dd excitations in cuprates and nickelates has been widely discussed [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Here, we focus on fluorescence to show how the orbital information can be extracted by comparing the RIXS intensity in different polarization channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Since we are measuring at the Ni L edge, the RIXS signal can only arise from either Ni orbitals or Ni states hybridized with other orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For the fluorescence features, the photon emission process is quite similar, with electrons from the crystalline environment surrounding the Ni site deexciting to fill the core hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Hence, the main intensity difference between these two polarization channels comes from the photon absorption process, the cross-section of which can be simulated in the dipole approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S3 presents the x-ray dipole absorption intensity ratio between π and σ polarization channels as a function of sample angle θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For the experimental configuration we used (vertical dashed line), the biggest contribution to the π over σ polarization intensity ratio is the Ni 3d3z2−r2 orbitals while 3dx2−y2 and 3dxy contribute equally to σ over π polarization intensity ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Since the 3dx2−y2 orbitals dominate near the Fermi level and are expected to show stronger hybridization with oxygen orbitals [5], 3dxy orbitals are expected to play a less important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Moreover, the t2g states do not make any significant contribution to the unoccupied states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Thus, we focus on Ni eg orbitals during the discussion in the main text, which are the subject of most of the debates over the appropriate theoretical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' MINIMAL CONTRIBUTION OF SPIN ORDER TO THE REXS SIGNAL In La4Ni3O8, spin order takes place concomitantly with the charge order and shares the same Q∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Hence we need to invoke cross-section considerations to separate the possible contribution of charge and spin order [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With π incident x-ray polarization, charge order contributes to the measured signal in the π-π′ scattering channel while the spin order is responsible for the π-σ′ channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The resonant elastic x-ray scattering (REXS) intensity ratio between these channels can be estimated by (ki · kf)2/(ϵi × ϵ′ f · M)2 = cos2 2Θ/ sin2 θ ≈ 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8, where ki (kf) is the initial (final) x-ray wavevector, ϵi (ϵ′ f) is the initial (final) x-ray polarization, and M is the spin direction, which is parallel to the c-axis in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Based on this formula we can see that the REXS signal with π incident polarization is dominantly of charge order origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Regarding the spin order contribution with σ incident x-ray polarization, we can compare the peak intensity with grazing-in and grazing-out conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Since the charge order composes the σ-σ′ channel, its intensity is expected to be the same in these two geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' For spin order signal that is only observable in the σ-π′ channel, the intensity 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='35 (H, H) (r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=') 0 2 4 6 8 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) grazing-in 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='33 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='34 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='35 grazing-out FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Comparison of the superlattice peak intensity with grazing-in and grazing-out conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The scattering angle 2Θ was fixed to 153◦ and the data were collected in σ polarization channel at 40 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The solid lines are guides to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Both peaks are found to have essentially the same intensity, which confirms that the peak arises from charge, rather than spin, order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' TABLE S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Full list of parameters used for the ED calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The on-site orbital energies, hopping integrals and Coulomb interactions are kept the same as those used in the O K-edge calculations [7], and Vpdπ = −Vpdσ/2, Vppπ = −Vppσ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The potential difference, ∆ϵd, only applies to the Ni 3d orbitals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Note that the crystal field splitting that is instead used in a Ni atomic model is a combination of point charge potential and orbital hybridization, which can be estimated through ligand field theory [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The resulting effective crystal field splitting gives 10Dq = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='971, ∆eg = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='041, ∆t2g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='342 eV, which are of a similar energy scale as the dd excitations observed in the RIXS measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' ζi and ζn are spin-orbit coupling parameters of the Ni 3d electrons for the initial and intermediate states, respectively, and ζc is the spin-orbit coupling strength for the Ni 2p core electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The core-hole lifetime is set to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' All parameters are in units of eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' On-site orbital energies Hopping integrals ϵdx2−y2 ϵd3z2−r2 ϵdxy ϵdxz/yz ϵpσ ϵpπ/pz Vpdσ Vppσ 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 Spin-orbit coupling On-site Coulomb interactions ζi ζn ζc F 0 dd F 2 dd F 4 dd F 0 pp F 2 pp 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='083 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='102 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='507 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='58 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='89 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='31 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='3 5 Inter-site Coulomb interactions Core-hole potential F 0 dp F 2 dp G1 dp G3 dp F 0 dp F 2 dp G1 dp G3 dp 1 0 0 0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='869 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='405 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='051 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='304 ratio between grazing-in and grazing-out conditions is (kf, grazing−in ·M)2/(kf, grazing−out ·M)2 ≈ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6, indicating that the spin order signal should be strongly suppressed with grazing-out condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Figure S4 shows the Q dependence of the superlattice peak with both conditions, which are comparable with each other, proving that the superlattice peak observed with σ incident x-ray polarization is also dominantly of charge order origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' CHARGE ORDER IN ED CALCULATIONS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' We use cluster ED to study the charge order in the low-valence nickelate La4Ni3O8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The full list of the parameters used is presented in Table S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The validity of our cluster model and parameters has been verified by calculating the RIXS energy maps and confirming that they capture the main features of the measurements as shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In the calculations, we include all the Ni 3d and O 2p orbitals, which leads to a large Hilbert space and correspondingly only a limited number of states can be solved for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Fortunately, the accessible energy range covers the dd excitations so that we can make a direct comparison with the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The calculated results are broadened using a Gaussian profile with a full width at half maximum of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='3 eV and are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' 3 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To fully explore the charge order character in the ED calculations, we need to cover a large incident energy range but only the ground state is needed to calculate the REXS signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Thus, we only include the Ni 3dx2−y2 and O 2pσ 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 Hole occupation (a) Ni1 Ni1L Ni2 Ni2L 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 d (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='75 Charge disproportionation (b) (Ni2+Ni2L)-(Ni1+Ni1L) 10 12 14 16 18 Incident energy (eV) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) (c) REXS@QCO, -pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='1 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='2 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='3 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='4 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='7 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 d=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='9 d=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' The emergence of charge order by introducing the potential difference term ∆ϵd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (a) Hole occupations of different sites as a function of ∆ϵd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Ni1L stands for the ligand orbitals for Ni1 (the surrounding four oxygens).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Correspondingly, one oxygen is shared by Ni1L and Ni2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (b) Charge disproportionation defined as the hole occupation difference of Ni1+Ni1L and Ni2+Ni2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' (c) Calculated REXS signals at QCO with different ∆ϵd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' All the calculations are performed with ∆ = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' orbitals during the calculations of the charge order, which dominate the ground state, so that a tractable basis size is realized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' To trigger charge order in the Ni3O10 cluster, we introduce a potential difference ∆ϵd as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In a microscopic model like we use here, the onsite energy shift and charge occupation are intrinsically coupled, which is different from a phenomenological model where these two factors can be tuned independently [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S5, when ∆ϵd is zero, the hole occupations on different Ni sites are almost the same while the hole occupations of ligand orbitals are slightly imbalanced since Ni2 shares oxygens with both Ni1 and Ni3, leading to a small charge disproportionation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With increasing ∆ϵd, the charge imbalances on both the Ni and ligand orbitals are enhanced with the former much more prominent, indicating that most of the spatial charge modulation resides on the Ni sites, leading to a Ni site-centered charge order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Correspondingly, a charge-order peak emerge in the REXS calculations, the intensity of which increases with increasing charge disproportionation while the lineshape only evolves by a little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' After testing the effect of ∆ϵd, here we compare results with different charge-transfer energy ∆ in addition to the calculated results presented in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S6, in the charge-transfer regime (∆ ≪ Udd), a sharp resonant peak is obtained, resembling the experimental observations in cuprates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' With increasing ∆, the REXS lineshape evolves correspondingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' In the Mott-Hubbard limit (∆ ≫ Udd), the charge order peak becomes broader and shows multiple peak features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Compared with the data presented in the main text, we conclude that a charge-transfer energy with an intermediate strength (∆ ≈ Udd) matches the experimental results the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' [1] Junjie Zhang, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Brookes, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' de Luca, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Di Castro, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Grioni, 6 10 12 14 16 18 Incident energy (eV) 0 1 2 3 4 5 Intensity (arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' units) REXS@QCO, -pol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' =2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 =3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 =4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 =5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='6 =6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='7 =7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 =8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='9 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' S6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' Calculated REXS signals at the charge order wavevector QCO with different charge-transfer energy ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content=' All the calculations are performed with ∆ϵd = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='8 eV and U = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0tE2T4oBgHgl3EQf4wij/content/2301.04184v1.pdf'} +page_content='5 eV.' metadata={'source': 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[math.PR] 28 Jan 2023 +Normal approximation of subgraph counts in +the random-connection model +Qingwei Liu∗ +Nicolas Privault† +Division of Mathematical Sciences +School of Physical and Mathematical Sciences +Nanyang Technological University +21 Nanyang Link, Singapore 637371 +January 31, 2023 +Abstract +This paper derives normal approximation results for subgraph counts written as +multiparameter stochastic integrals in a random-connection model based on a Pois- +son point process. By combinatorial arguments we express the cumulants of general +subgraph counts using sums over connected partition diagrams, after cancellation of +terms obtained by M¨obius inversion. Using the Statuleviˇcius condition, we deduce con- +vergence rates in the Kolmogorov distance by studying the growth of subgraph count +cumulants as the intensity of the underlying Poisson point process tends to infinity. +Our analysis covers general subgraphs in the dilute and full random graph regimes, +and tree-like subgraphs in the sparse random graph regime. +Keywords: Random-connection model, subgraph count, normal approximation, Kolmogorov +distance, cumulant method, Poisson point process, random graphs. +Mathematics Subject Classification: 60F05, 60D05, 05C80, 60G55. +1 +Introduction +This paper treats the asymptotic behavior of random subgraph counts in the random- +connection model (RCM), which is used to model physical systems in e.g. wireless networks, +complex networks, and statistical mechanics. Our approach relies on the study of cumulant +growth rates as the intensity of the underlying Poisson point process tends to infinity. +The distributional approximation of subgraph counts has attracted significant interest +in the random graph literature. +In [Ruc88], conditions for the asymptotic normality of +∗qingwei.liu@ntu.edu.sg +†nprivault@ntu.edu.sg +1 + +renormalized subgraph counts have been obtained in the Erd˝os-R´enyi random graph model +[ER59, Gil59]. Those results have been made more precise in [BKR89] by the derivation +of convergence rates in the Wasserstein distance via Stein’s method. They have also been +strengthened in [KRT17] using the Kolmogorov distance in the case of triangle counts, and +in [PS20] in the case of general subgraphs G. The case of triangles has also been treated in +[R¨ol22] by the Stein-Tikhomirov method, which has been extended to general subgraphs in +[ER21]. In [Kho08], the counts of line (X-model) and cycles (Y -model) in discrete Erd˝os- +R´enyi models have been analyzed via the asymptotic behavior of their cumulants. In compar- +ison with [Kho08], we derive Kolmogorov convergence rates and our results are not restricted +to line and cycle graphs, as they cover more general subgraphs. +The random connection-model is a natural generalization of the Erd˝os-R´enyi random +graph in which vertices are randomly located and can be connected with position-dependent +probabilities. Studying the random-connection model and obtaining normal approximation +error bounds is more difficult due to the additional layer of complexity coming from the +randomness of vertex locations. In [LNS21], a central limit theorem and Kolmogorov con- +vergence rates have been presented for the number of components isomorphic to a given +finite connected graph in the random-connection model, together with a study of first mo- +ments and covariances. Recently, a Central Limit Theorem has been derived in [CT22] for +the counts of induced subgraphs in the random-connection model under certain stabilization +and moment conditions. +In this paper, we derive normal approximation rates under a relatively mild condition +on the connection function of the random-connection model, by deriving growth rates of +cumulants written as sums over connected partitions. To the best of our knowledge, this is +the first time that the normal approximation of subgraph counts with convergence rates is +established in the random-connection model. Furthermore, various random graph regimes +are discussed. +A number of probabilistic conclusions can be derived from the behavior of cumulants +of random variables using the Statuleviˇcius condition, including convergence rates in the +Kolmogorov distance and moderate deviation principles, see [SS91], [DE13], [DJS22]. In +[GT18a, GT18b], this method has been used to derive concentration inequalities, normal +approximation with error bounds, and moderate deviation principles for random polytopes. +Given µ a finite diffuse measure on Rd, we consider a random-connection model based on +an underlying Poisson point process Ξ on Rd with intensity of the form λµ(dx), in which any +2 + +two vertices x, y in Ξ are connected with the probability Hλ(x, y) := cλH(x, y) ∈ [0, 1], where +Hλ is the connection function of the model. Here, we investigate the limiting behavior of the +count NG of a given subgraph G as the intensity λ of the underlying Poisson point process +on Rd tends to infinity. To this end, we use the combinatorics of the cumulants κn(NG) +based on moment expressions obtained in [Pri19] for multiparameter stochastic integrals in +the random-connection model. +Using partition diagrams and dependency graph arguments, we start by showing in +Proposition 3.3 that the (virtual) cumulants of a random functional admitting a certain con- +nectedness factorization property (3.1) can be expressed as sums over connected partition +diagrams, generalizing Lemma 2 in [MM91]. A related result has been obtained in [Jan19] +in the particular case of two-parameter Poisson stochastic integrals, in relation to cluster +expansions for Gibbs point processes in statistical mechanics. In Proposition 4.3, we apply +Proposition 3.3 to express the cumulants of multiparameter stochastic integrals, for which +this factorization property can be checked from the moment formulas for multiparameter +stochastic integrals computed in Proposition 4.1. +Such expressions allow us to determine the dominant terms in the growth of cumulants +as the intensity λ of the underlying point process tends to infinity, by estimating the counts +of vertices and edges in connected partition diagrams as in [Kho08]. We work under a mild +condition (6.3) which is satisfied by e.g. any translation-invariant continuous connection +function H : Rd × Rd → [0, 1] non vanishing at 0, such as the Rayleigh connection function +given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0. +For our analysis of cumulant behavior we identify the leading terms in the sum (5.3) +over connected partition diagrams. When G is a connected graph with r := |V (G)| vertices, +satisfying Assumption 6.1 in the dilute regime (6.1) where λ−1 ≪ cλ ≤ 1, the dominant +terms correspond to connected partition diagrams with the highest number of blocks, as +found in [Pri22] in the case of k-hop counting in the one-dimensional random-connection +model. In Theorem 6.1 this yields the cumulant bounds +(n − 1)!cn|E(G)| +λ +(K1λ)1+(r−1)n ≤ κn(NG) ≤ n!r−1cn|E(G)| +λ +(K2λ)1+(r−1)n, +λ > 0, +for some constants K1, K2 > 0 independent of λ, n ≥ 1. From the Statuleviˇcius condition +(A.1) below, see [RSS78, DJS22], we deduce the Kolmogorov distance bound +sup +x∈R +��P +� � +NG ≤ x +� +− P(Z ≤ x) +�� ≤ +C +λ1/(4r−6) , +λ → ∞, +3 + +see Corollary 7.1, and a moderate deviation principle by Theorem 1.1 of [DE13]. +In the sparse regime (6.2) where cλ ≤ λ−α for some α ≥ 1, the maximal rate λα−(α−1)r is +attained for G a tree-like graph, and in Theorem 6.2 we obtain the cumulant bounds +(K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλα−(α−1)r, +λ > 0, +if G is a tree, and +(K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλr−α|E(G)|, +λ > 0, +if G is a not a tree, such as e.g. +a cycle graph. +As a consequence of the Statuleviˇcius +condition (A.1), when G is a tree we find the Kolmogorov distance bound +sup +x∈R +��P +� � +NG ≤ x +� +− P(Z ≤ x) +�� ≤ Cλ−(α−(α−1)r)/(4r−6), +λ → ∞, +provided that 1 ≤ α < r/(r − 1), see Corollary 7.2. +Convergence rates in the Kolmogorov distances may be improved into classical Berry- +Esseen rates when the connection function H(x, y) is {0, 1}-valued, e.g. +in disk models +as in [Pri22], by representing subgraph counts as multiple Poisson stochastic integrals and +using the fourth moment theorem for U-statistics and sums of multiple stochastic integrals +Corollary 4.10 in [ET14], see also Theorem 3 in [LRR16] or Theorem 6.3 in [PS22] for +Hoeffding decompositions. In the general case where H(x, y) is [0, 1]-valued this method no +longer applies, this is why we rely on the Statuleviˇcius condition which in turn may yield +suboptimal convergence rates. +The paper is organized as follows. Sections 2 and 3 introduce the preliminary frame- +work and notations on connected partition diagrams and combinatorics of virtual cumulants +that will be used for the expression of cumulants of multiparameter stochastic integrals in +Section 4 and for subgraph counts in Section 5. Those expressions are applied in Section 6 +to derive cumulant growth rates in the random-connection model, with application to Kol- +mogorov rates in subgraph counting via the Statuleviˇcius condition in Section 7. +2 +Set partitions and diagram connectivity +Given η a finite set, we denote by Π(η) the collection of its set partitions, and we let |σ| +denote the number of blocks in any partition σ ∈ Π(η). Given ρ, σ two set partitions, we +4 + +say that σ is coarser than ρ, or that ρ is finer than σ, and we write ρ ⪯ σ, if every block +in σ is a combination of blocks in ρ. We also denote by ρ ∨ σ the finest partition which is +coarser than ρ and σ, and by ρ∧σ the coarsest partition that is finer than ρ and σ. We let �0 +be the finest partition, which is made of a single element in each block, and we let �1 be the +coarsest (one-block) partition. In general, given any graph G we denote by V (G) the set of +its vertices, and by E(G) the set of its edges. +Our study of cumulants and moments of functionals of random fields relies on partition +diagrams, see [MM91, Kho08, PT11] and references therein for additional background. In +what follows we let [n] := {1, 2, . . . , n} for n ≥ 1. +Definition 2.1 Let n, r ≥ 1. +1. Given η ⊂ [n] we let Π(η × [r]) denote the set of all partitions of the set +η × [r] := +� +(k, l) : k ∈ η, l = 1, . . . , r +� +. +2. We also let πη := (πi)i∈η ∈ Π(η × [r]) denote the partition made of the |η| blocks of size r +given by +πk := {(k, 1), . . . , (k, r)}, +k ∈ η. +Next, we introduce the definition of partition diagrams. +Definition 2.2 Let n, r ≥ 1. Given η ⊂ [n] and ρ ∈ Π(η × [r]) a partition of η × [r], we +denote by Γ(ρ, πη) the diagram, or graphical representation of the partition ρ, constructed +by: +1. arranging the elements of η × [r] into an array of |η| rows and r columns, and +2. adding edges connecting neighbors within a same block in ρ. +In addition, we say that the partition diagram Γ(ρ, π) is connected when ρ ∨ πη = �1. +For example, taking η := {2, 3, 5, 8, 10}, given the partitions +ρ = +� +{(2, 1), (3, 1), (3, 2), (3, 3)}, {(2, 2), (2, 3), (2, 4), (3, 4)}, {(5, 1)}, {(5, 2), (8, 2)}, +{(5, 3)}, {(5, 4), (8, 3)}, {(8, 1), (10, 1)}, {(8, 4)}, {(10, 2), (10, 3), (10, 4)} +� +and +σ = +� +{(2, 1), (3, 1)}, {(2, 2)}, {(2, 3), (3, 4)}, {(2, 4)}, {(3, 2), (5, 2), (8, 2)}, +5 + +{(3, 3), (5, 4), (8, 3), (10, 2)}, {(5, 1)}, {(5, 3)}, {(8, 1), (10, 1)}, {(8, 4)}, {(10, 3)}, {(10, 4)} +� +, +of η × [4], Figure 1−a) presents an example of a non-connected partition diagram Γ(ρ, π), +and Figure 1−b) presents an example of a connected partition diagram Γ(σ, π), +2 +3 +5 +8 +10 +1 +2 +3 +4 +(a) Non-connected partition diagram Γ(ρ, π). +2 +3 +5 +8 +10 +1 +2 +3 +4 +(b) Connected partition diagram Γ(σ, π). +Figure 1: Two examples of partition diagrams with η = {2, 3, 5, 8, 10}, n = 10, r = 4. +Note that the above notion of connected partition diagram is distinct from that of irreducible +partition, see, e.g., [BOR85]. +Definition 2.3 Let n ≥ 1, G a connected graph with |V (G)| = r ≥ 1 vertices, and consider +G1, . . . , Gn copies of G respectively built on π1, . . . , πn. Let also ρ ∈ Π(η × [r]) be a partition +of η × [r]. +1. We let �ρG be the multigraph constructed on the blocks of ρ by adding an edge between two +blocks ρ1, ρ2 of the partition ρ whenever there exists (k, l1) ∈ ρ1 and (k, l2) ∈ ρ2 such that +(l1, l2) is an edge in Gk. +2. We let ρG be the graph constructed on the blocks of ρ by removing redundant edges in �ρG, +so that at most one edge remains between any two blocks ρ1, ρ2 ∈ ρ. +Figure 2-b) presents an illustration of the multigraph �ρG and graph ρG on the blocks of ρ +when G is the line graph {(1, 2), (2, 4), (3, 4)} on {1, 2, 3, 4}. +6 + +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and multigraph �ρG in blue. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(ρ, π) and graph ρG in red. +Figure 2: Diagram and graphs G, ρG, �ρG with n = 5, r = 4. +Definition 2.4 Let n, r ≥ 1, and let ρ ∈ Π([n] × [r]) be a partition of [n] × [r]. +1. For b ⊂ [n], we let ρb ⊂ ρ be defined as +ρb := {c ∈ ρ : c ⊂ b × [r]}. +2. Given η ⊂ [n] we split any partition ρ of η × [r] into the equivalence classes deduced from +the connected components of the diagram ρG, as +ρ = +� +b×[r]∈ρ∨π +b⊂[n] +ρb, +(2.1) +As an example, in Figure 3-a), when b = {1, 2} we have +ρ{1,2} = +� +{(1, 1), (2, 1), (2, 2), (2, 3)}, {(1, 2), (1, 3), (1, 4), (2, 4)} +� +, +and the partition (2.1) is illustrated in Figure 3-b) with b1 = {1, 2} and b2 = {3, 4, 5}. +7 + +1 +2 +3 +4 +5 +1 +2 +3 +4 +ρ{1,2} +(a) Diagram Γ(ρ, π) and block ρ{1,2}. +1 +2 +3 +4 +5 +1 +2 +3 +4 +ρb1 +ρb2 +(b) Splitting {ρb1, ρb2} of ρ according to ρG. +Figure 3: Splitting of the partition ρ with ρ ∨ π = {π1 ∪ π2, π3 ∪ π4 ∪ π5} and n = 5, r = 4. +Definition 2.5 Let n, r ≥ 1. Given σ ∈ Π([n]) a partition of [n], we let Πσ([n]×[r]) denote +the set of partitions ρ of [n] × [r] such that +ρ ∨ π = {b × [r] : b ∈ σ}, +and we partition Π([n] × [r]) as +Π([n] × [r]) = +� +σ∈Π([n]) +Πσ([n] × [r]). +(2.2) +We note that given η ⊂ [n], the set Π�1(η × [r]) consists of the partitions ρ of η × [r] for +which the diagram ρG is connected, as in Figure 4. In what follows, we also will use non-flat +partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, see Chapter 4 of [PT11] and Figure 4. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and multigraph �ρG in blue. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(ρ, π) and graph ρG in red. +Figure 4: Connected non-flat partition diagram with G a cycle graph and n = 5, r = 4. +8 + +Lemma 2.6 a) Let n, r ≥ 1. The cardinality of the set +C(n, r) := {ρ ∈ Π�1([n] × [r]) : ρ ∧ π = �0} +of connected non-flat partition diagrams on [n] × [r] satisfies +|C(n, r)| ≤ n!r−1rn−1r!n−1, +n ≥ 1. +(2.3) +b) Let n ≥ 1 and r ≥ 2. The cardinality of the set +Mn := {ρ ∈ C(n, r) : |ρ| = 1 + (r − 1)n} +of maximal connected non-flat partition diagrams on [n] × [r] satisfies +((r − 1)r)n−1(n − 1)! ≤ |Mn| ≤ ((r − 1)r)n−1n!2, +n ≥ 1. +(2.4) +Proof. +a) We have |C(1, r)| = 1. Given a connected partition diagram Γ(ρ, π) in C(n+1, r), +we construct a connected undirected graph �ρ on [n + 1] as in Figure 5-a), and note that +�ρ contains a spanning tree ρ, see e.g. Theorem 4.2.3 in [BR12], as shown in Figure 5-b). +In addition, the tree ρ has at most r leaves, because after removing any of root of ρ, the +remaining partition can be reconnected using no more than r vertices from the root. Then, +starting for any leaf in the tree ρ, ρ must be made from a connected partition diagram +in C(n, r), completed by a choice of at most (n + 1)r−1r! allocations of r − 1 vertices into +existing or new blocks. Indeed, note that at least one out of r vertices in the leaf is used for +an existing connection. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and graph �ρ. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(ρ, π) and spanning tree ρ. +Figure 5: Example of graph �ρ and its spanning tree subgraph. +9 + +This yields the induction inequality +|C(n + 1, r)| ≤ r(n + 1)r−1r!|C(n, r)|, +from which we conclude to (2.3). +b) Proceeding similarly to part (a), we have |M1| = 1 and the recursion +r × (1 + (r − 1)n) × |Mn| ≤ |Mn+1| ≤ (n + 1)r × (1 + (r − 1)n) × |Mn|, +n ≥ 1, +which yields +((r − 1)r)n−1 +n−1 +� +i=1 +� +i + +1 +r − 1 +� +≤ |Mn| ≤ n!((r − 1)r)n−1 +n−1 +� +i=1 +� +i + +1 +r − 1 +� +, +n ≥ 1, +from which (2.4) follows. +□ +3 +Virtual cumulants +The following definition uses the concept of independence of a virtual field with respect to +graph connectedness, see Relation (17) in [MM91, p. 34]. +Definition 3.1 Let n, r ≥ 1. We say that a mapping F defined on partitions of [n] × [r] +admits the connectedness factorization property if it decomposes according to the partition +(2.1) as +F(ρ) = +� +b×[r]∈ρ∨π +F(ρb), +ρ ∈ Π([n] × [r]). +(3.1) +In what follows, given F a mapping defined on the partitions of [n] × [r], we will use the +M¨obius transform �F of F, defined as +�F(η) := +� +ρ∈Π(η×[r]) +F(ρ), +η ⊂ [n], +with �F(∅) := 0, see [Rot64] and § 2.5 of [PT11]. We refer to [MM91, p. 33] for the following +definition. +Definition 3.2 Let n, r ≥ 1. The virtual cumulant G of a mapping F on � +η⊂[n] Π(η × [r]) +is defined by letting CF(η) := �F(η) when |η| = 1, and then recursively by +CF(η) := �F(η) − +� +σ∈Π(η) +|σ|≥2 +� +b∈σ +CF(b), +η ⊂ [n], +|η| ≥ 2. +(3.2) +10 + +In the particular case r = 1, we note that when (X1, . . . , Xn) is a sequence of random +variables, letting +F(ρ) := E +�� +b∈ρ +� +i∈b +Xi +� += E +� n +� +i=1 +Xi +� +, +Relation (3.2) shows that +CF(η) = +� +σ∈Π[η] +(−1)|σ|−1(|σ| − 1)! +� +b∈σ +F({b}) = +� +σ∈Π[η] +(−1)|σ|−1(|σ| − 1)! +� +b∈ρ +E +�� +i∈b +Xi +� +, +coincides with the actual joint cumulant of (Xi)t∈η, η ⊂ [n]. +The following proposition is an extension of the classical Lemma 2 in [MM91, p. 34], see +also Lemma 3.1 in [Kho08]. +Proposition 3.3 Let n, r ≥ 1. Let F be a mapping defined on � +η⊂[n] Π(η × [r]) and admit- +ting the connectedness factorization property (3.1). Then, for η ⊂ [n] with η ̸= ∅, the virtual +cumulant of F is given by the sum +CF(η) = +� +σ∈Π�1(η×[r]) +(connected) +F(σ) +(3.3) +over connected partition diagrams on η × [r]. +Proof. +The claim is true when |η| = 1. Assume that it is true for all η ⊂ [n] for some n ≥ 1, +and let η be such that |η| = n + 1. By (2.2) and (3.1), we have +�F(η) += +� +ρ∈Π(η×[r]) +F(ρ) += +� +σ∈Π(η) +� +ρ∈Πσ(η×[r]) +F(ρ) += +� +σ∈Π(η) +� +ρ∈Πσ(η×[r]) +� +b∈σ +F(ρb) += +� +σ∈Π(η) +� +b∈σ +� +ρ∈Π�1(b×[r]) +(connected) +F(ρ) += +� +ρ∈Π�1(η×[r]) +(connected) +F(ρ) + +� +σ∈Π(η) +|σ|≥2 +� +b∈σ +CF(b), +where the last equality follows from the induction hypothesis (3.3) when |η| ≤ n. The proof +is completed by subtracting the last term on both sides. +□ +11 + +In the particular case r = 1, we note that when (X1, . . . , Xn) is a sequence of independent +random variables, the functional +F(ρ) := E +�� +b∈ρ +� +i∈b +Xi +� += +� +i∈[n] +E[Xi] +satisfies the connectedness factorization property (3.1), and Proposition 3.3 recovers the +vanishing of the joint cumulants of (Xi)i∈η when |η| ≥ 2, as the set Π�1(η × [1]) of connected +partition diagrams on η × [1] is empty in this case. +4 +Cumulants of multiparameter stochastic integrals +Consider a Poisson point process Ξ on Rd, d ≥ 1, with intensity measure Λ on Rd, constructed +on the space +Ω = +� +ω = {xi}i∈I ⊂ Rd : #(A ∩ ω) < ∞ for all compact A ∈ B(Rd) +� +of locally finite configurations on Rd, whose elements ω ∈ Ω are identified with the Radon +point measures ω = +� +x∈ω +ǫx, where ǫx denotes the Dirac measure at x ∈ Rd. +By [LP18, +Corollary 6.5], almost every element ω of Ω can be represented as ω = {Vi}1≤i≤N, where +(Vi)i≥1 is a random sequence in Rd and a N ∪ {∞}-valued random variable N. +In this section, using sums over partitions we express the moments of the multiparameter +stochastic integral +� +V1,...,Vr∈Ξ +uG(V1, . . . , Vr) = +� +(Rd)r uG(x1, . . . , xr)ω(dx1) · · · ω(dxr), +(4.1) +where uG(x1, . . . , xr) is a measurable process of the form +uG(x1, . . . , xr) := +� +(i,j)∈E(G) +vi,j(xi, xj), +and vi,j(x, y), (i, j) ∈ E(G), are random processes v(x, y) independent of the underlying +Poisson point process Ξ. The next proposition is a consequence of Proposition 2 in [Pri19], +which relies on Proposition 3.1 of [Pri12] and Lemma 2.1 of [BRSW17]. +Proposition 4.1 Let n ≥ 1 and r ≥ 2. The n-th moment of the multiparameter stochastic +integral (4.1) is given by the summation +� +ρ∈Π([n]×[r]) +� +(Rd)|ρ| E + + +n +� +k=1 +� +(i,j)∈E(Gk) +v +� +xρ +k,i, xρ +k,j +� + + +� +η∈V (ρG) +Λ(dxη), +(4.2) +12 + +where we let xρ +k,l := xη whenever (k, l) ∈ η, for ρ ∈ Π([n] × [r]) and η ∈ ρ. +The next proposition rewrites the product in (4.2) as a product on the edges of the graph ρG +similarly to Proposition 4 of [Pri19] when v(x, y) vanishes on the diagonal, and it generalizes +Proposition 2.4 of [Jan19] from two-parameter Poisson stochastic integrals to multiparameter +integrals of higher orders. +Proposition 4.2 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag- +onals, i.e. v(x, x) = 0, x ∈ Rd. Then, the n-th moment of the multiparameter stochastic +integral (4.1) is given by the summation +� +ρ∈Π([n]×[r]) +ρ∧π=�0 +(non−flat) +� +(Rd)|ρ| +� +(η1,η2)∈E(ρG) +E +� +v(xη1, xη2)m(η1,η2)� +� +η∈V (ρG) +Λ(dxη), +over connected non-flat diagrams, where m(η1, η2) represents the multiplicity of the edge +(η1, η2) in the multigraph �ρG. +The next proposition is a consequence of Propositions 3.3 and 4.2, and it also extends +Proposition 2.5 of [Jan19] from the two-parameter case to the multiparameter case. Note +that in our setting, the two-parameter case only applies to the edge counting. +Proposition 4.3 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag- +onals, i.e. v(x, x) = 0, x ∈ Rd. Then, the n-th cumulant of the multiparameter stochastic +integral (4.1) is given by the summation +� +ρ∈Π�1([n]×[r]) +ρ∧π=�0 +(non−flat connected) +� +(Rd)|ρ| +� +(η1,η2)∈E(ρG) +E +� +v(xη1, xη2)m(η1,η2)� +� +η∈V (ρG) +Λ(dxη) +(4.3) +over connected non-flat partition diagrams. +Proof. +The functional +F(ρ) := +� +ρ∈Π([n]×[r]) +ρ∧π=�0 +(non−flat) +� +(Rd)|ρ| +� +(η1,η2)∈E(ρG) +E +� +v(xη1, xη2)m(η1,η2)� +� +η∈V (ρG) +Λ(dxη) +satisfies the connectedness factorization property (3.1), as for σ = b × [r] ∈ ρ ∨ π and +σ′ = b′ ×[r] ∈ ρ∨π with b ̸= b′, the variables (xη)η∈ρb are distinct from the variables (xη)η∈ρb′ +in the above integration. Hence, (4.3) follows from Proposition 3.3. +□ +13 + +5 +Cumulants of subgraph counts +Let H : Rd × Rd → [0, 1] denote a measurable connection function such that +0 < +� +Rd H(x, y)Λ(dx) < ∞, +for all y ∈ R. Given ω ∈ Ω, for any x, y ∈ ω with x ̸= y, an edge connecting x and y +is added with probability H(x, y), independently of the other pairs, and in this case we +write x ↔ y. The resulting random graph, together with the point process Ξ, is called the +random-connection model and denoted by GH(Ξ). +In the case where the connection function H is given by H(x, y) := 1{∥x−y∥≤R} for some +R > 0, the resulting graph is completely determined by the geometric of the underlying +point process Ξ, and is called a random geometric graph, which is included as a special case +in this paper. +Given G a connected graph with |V (G)| = r vertices, we denote NG the count of sub- +graphs isomorphic to G in the random-connection model GH(Ξ), which can be represented +as the multiparameter stochastic integral +NG := +� +V1,...,Vr∈Ξ +� +(i,j)∈E(G) +1{Vi↔Vj} = +� +(Rd)r +� +(i,j)∈E(G) +1{xi↔xj} ω(dx1) · · ·ω(dxr), +up to division by the number of automorphisms of G. Here, we have 1{Vi↔Vj} = 1 or 0 +depending whether Vi and Vj are connected or not by an edge in GH(Ξ), with +1{x↔x} = 0, +x ∈ Rd. +(5.1) +The following result is a direct consequence of Proposition 4.3 by taking v(x, y) := 1{x↔y} +in (4.3) and by using non-flat partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, to take into +account condition (5.1). +Proposition 5.1 Let n ≥ 1 and r ≥ 2. The moments and cumulants of NG are given by +the summation +E[(NG)n] = +� +ρ∈Π([n]×[r]) +ρ∧π=�0 +(non−flat) +� +(Rd)|ρ| +� +� +(η1,η2)∈E(ρG) +H(xη1, xη2) +� +� +η∈V (ρG) +Λ(dxη), +(5.2) +over non-flat partition diagrams, and by the summation +κn(NG) = +� +ρ∈Π�1([n]×[r]) +ρ∧π=�0 +(non−flat connected) +� +(Rd)|ρ| +� +� +(η1,η2)∈E(ρG) +H(xη1, xη2) +� +� +η∈V (ρG) +Λ(dxη), +(5.3) +14 + +over connected non-flat partition diagrams. +Proof. +Relations (5.2)-(5.3) are consequence of Proposition 4.3, after taking vi,j(xi, xj) := +1{xi↔xj}, (i, j) ∈ E(G). The summations are restricted to non-flat partition diagrams due +to condition (5.1) as in Section 2 of [Pri19]. +□ +6 +Asymptotic growth of subgraph count cumulants +We assume that the intensity measure of the Poisson point process Ξ on Rd has the form +Λλ(dx) = λµ(dx), +λ > 0, +where µ is a finite diffuse measure on Rd. We investigate the asymptotic behaviour of the +cumulants κn(NG) as the intensity λ tends to infinity, as a consequence of the partition +diagram representation of cumulant. For this, we consider the subgraph count in GH(Ξ) +obtained by replacing H(x, y) with Hλ(x, y) := cλH(x, y), in which case every term in (5.3) +contributes a factor c|E(ρG)| +λ +λ|V (ρG)|. +In what follows, given two positive functions f and g on (1, ∞) we write f(λ) ≪ g(λ) if +limλ→∞ g(λ)/f(λ) = ∞, and we consider the following regimes. +• Dilute regime: for some constant K > 0 we have +1 +λ ≪ cλ ≤ K, +λ → ∞. +(6.1) +• Sparse regime: for some constants K > 0 and α ≥ 1 we have +cλ ≤ K +λα, +λ → ∞. +(6.2) +In case cλ = K for all λ > 0 we also say that we are in the full random graph regime, and +in the sequel we take K = 1 for simplicity. +Assumption 6.1 Let r ≥ 2. There exist two constants c, C > 0 such that for any connected +non-flat partition diagram Γ(ρ, π), ρ ∈ Π�1([n] × [r]), n ≥ 1, we have +c|E(ρG)|C|V (ρG)| ≤ +� +Rd · · · +� +Rd +� +� +(i,j)∈E(ρG) +H(xi, xj) +� +� +k∈V (ρG) +µ(dxk). +(6.3) +15 + +We note that (6.3) is satisfied by e.g. any translation-invariant continuous kernel function +H : Rd×Rd → [0, 1] non vanishing at 0, including the standard Rayleigh connection function +given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0. Indeed, for those kernels there +exists c > 0 and a Borel set B ⊂ Rd such that µ(B) > 0 and +H(x, y) = H(x − y, 0) ≥ c1B(x)1B(y), +x, y ∈ Rd, +hence +c|E(ρG)|(µ(B))|V (ρG)| += +c|E(ρG)| +� +B +· · · +� +B +� +k∈V (ρG) +µ(dxk) +≤ +� +Rd · · · +� +Rd +� +� +(i,j)∈E(ρG) +H(xi, xj) +� +� +k∈V (ρG) +µ(dxk). +In what follows, we consider the centered and normalized subgraph count cumulants defined +as +� +NG := NG − κ1(NG) +� +κ2(NG) +, +n ≥ 1. +The following result shows that for n ≥ 3 the normalized cumulant κn( �NG) tends to zero +in (6.5), hence � +NG converges in distribution to the normal distribution by Theorem 1 in +[Jan88]. +Theorem 6.1 (Dilute regime) Let r ≥ 2 and consider G a connected graph with |V (G)| = +r vertices, satisfying Assumption 6.1 in the dilute regime (6.1). We have the cumulant bounds +(n − 1)!cn|E(G)| +λ +(K1λ)1+(r−1)n ≤ κn(NG) ≤ n!r−1cn|E(G)| +λ +(K2λ)1+(r−1)n +(6.4) +for some constants K1, K2 > 0 independent of λ, n ≥ 1, and +��κn +� �NG +��� ≤ n!r−1(Kλ)−(n/2−1), +λ ≥ 1, +n ≥ 2, +(6.5) +where K > 0 is a constant independent of λ > 0 and n ≥ 1. +Proof. +We identify the leading terms in the sum (5.3) over connected partition diagrams, +knowing that every vertex in ρG contributes a factor λ, and that every edge contributes a +factor cλ, therefore every summand in (5.3) contributes a factor c|E(ρG)| +λ +λ|V (ρG)|. +Modifying ρ ∈ Π�1([n] × [r]) by splitting a block in two means adding a vertex to ρG, and +therefore a adding factor λ to the corresponding term in (5.3). At the same time, this entails +16 + +no loss of edge but possibly the addition of an edge to ρG, which results into an additional +factor cλ with λcλ ≫ 1 by (6.1). Hence, the leading terms in (5.3) are those associated with +the connected partition diagrams Γ(ρ, π) having the highest block count, i.e. which have +1 + (r − 1)n blocks, see Figure 6 for a sample of such partition diagram. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and graph �ρG in blue. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(ρ, π) and graph ρG in red. +Figure 6: Example of maximal connected partition diagram with n = 5 and r = 4. +We note that any maximal partition ρ satisfies |E(ρG)| = n × |E(G)|, as can be checked in +Figure 6. Therefore, by (2.3)-(2.4), (5.3) and (6.3), we obtain +cn|E(G)|C1+(r−1)ncn|E(G)| +λ +((r − 1)r)n−1(n − 1)!λ1+(r−1)n +≤ +λ1+(r−1)ncn|E(G)| +λ +� +ρ∈Mn +� +(Rd)1+(r−1)n +� +� +(η1,η2)∈E(ρG) +Hλ(xη1, xη2) +� +� +η∈V (ρG) +µ(dxη), +≤ +κn(NG) +≤ +n!r−1rn−1r!n−1(µ(Rd))1+(r−1)ncn|E(G)| +λ +λ1+(r−1)n, +which yields (6.4). Regarding (6.5), we have, for n ≥ 2, +��κn( �NG) +�� ≤ +n!r−1cn|E(G)| +λ +(K2λ)1+(r−1)n +� +(2 − 1)!c2|E(G)| +λ +(K1λ)1+2(r−1)�n/2 = K2 +�(K2/K1)r−1 +√K1 +�n +n!r−1λ−(n/2−1). +□ +The following result yields a positive cumulant growth of order α − (α − 1)r > 0 in (6.6) for +trees in the sparse regime with α ∈ [1, r/(r − 1)), while in the case of non-tree graphs such +as cycle graphs the growth rate r − α|E(G)| ≤ (1 − α)r ≤ 0 is negative or zero in (6.8) and +(6.10). In addition, the normalized cumulant κn( �NG) tends to zero for n ≥ 3 in (6.7) only +17 + +when G is a tree, in which case � +NG converges in distribution to the normal distribution by +Theorem 1 in [Jan88]. We note that when α = 1, (6.7) is consistent with (6.5). +Theorem 6.2 (Sparse regime) Let G be a connected graph with |V (G)| = r vertices, +r ≥ 2, satisfying Assumption 6.1 in the sparse regime (6.2). +a) If G is a tree, i.e. |E(G)| = r − 1, we have the cumulant bounds +(K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλα−(α−1)r, +(6.6) +for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and +��κn( � +NG) +�� ≤ (K3)nn!r−1λ−(α−(α−1)r)(n/2−1), +λ ≥ 1, +n ≥ 2, +(6.7) +where K3 := (K2/K1)r−1. +b) If G is not a tree, i.e. |E(G)| ≥ r, we have the cumulant bounds +(K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλr−α|E(G)|, +(6.8) +for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and +��κn( � +NG) +�� ≤ n!r−1(K3)nλ(α|E(G)|−r)(n/2−1), +λ ≥ 1, +n ≥ 2, +(6.9) +for some K3 > 0. +c) If G is a cycle, i.e. |E(G)| = r, we have the cumulant bounds +(K1)rλ−(α−1)r ≤ κn(NG) ≤ n!r−1(K2)(r−1)nλ−(α−1)r, +(6.10) +for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and +��κn( � +NG) +�� ≤ n!r−1(K3)nλ(α−1)(n/2−1)r, +λ ≥ 1, +n ≥ 2, +(6.11) +for some K3 > 0. +Proof. +In the sparse regime (6.2), every edge in the graph ρG contributes a power λ−α and +every vertex contributes a power λ, hence every term in (5.3) contributes a power +λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|+(|V (ρG)|−|E(ρG)|−1)α ≤ λα−(α−1)|V (ρG)| +(6.12) +18 + +since |V (ρG)| − |E(ρG)| − 1 ≤ 0. In addition, for any connected partition diagram Γ(ρ, π) +with ρ ∈ Π�1([n] × [r]), we have +r ≤ |V (ρG)| ≤ 1 + (r − 1)n. +a) When G is a tree and the graph ρG is also a tree, i.e. |V (ρG)| − |E(ρG)| − 1 = 0, the +maximal order λα−(α−1)|V (ρG)| is attained in (6.12), see Figure 7 for an example. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and multigraph �ρG in blue. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) and graph ρG in red. +Figure 7: Example of connected partition diagram with ρG a tree and n = 5, r = 4. +In this case, the corresponding term in (5.3) contributes a power +λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|, +λ ≥ 1. +In this case, since |V (ρG)| ≥ r and α ≥ 1, the optimal rate λα−(α−1)r is attained by the +partition diagrams Γ(ρ, π) such that |V (ρG)| = r, as illustrated in Figure 8. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and multigraph �ρG in blue. +1 +2 +3 +4 +5 +1 +2 +3 +4 +(b) Diagram Γ(ρ, π) and graph ρG in red. +Figure 8: Tree diagram ρG with G a tree with |V (ρG)| = r and n = 5, r = 4. +19 + +We conclude to (6.6) using Lemma 2.6 as in the proof of Theorem 6.1, by upper bounding +the count of connected partitions from (2.3). Regarding (6.7), we have +��κn( � +NG) +�� +≤ +n!r−1K(r−1)n +2 +((K1)2(r−1)λα−(α−1)r)n/2λα−(α−1)r += +�K2 +K1 +�(r−1)n +n!r−1λ−(α−(α−1)r)(n/2−1), +n ≥ 2. +b) When G is not a tree it contains at least one cycle, and for any partition ρ ∈ Π�1([n] × [r]) +the same holds for the graph ρG. In this case, the highest order contribution in (5.3) is +attained by connected non-flat partition diagrams Γ(ρ, π), ρ ∈ Π�1([n] × [r]), such that ρG +has |V (ρG)| = r vertices, and their contribution is given by a power of order λr−α|E(G)|. An +example of such partition diagram ρ is given in Figure 9, with G a cycle. +1 +2 +3 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) and multigraph �ρG in blue. +1 +2 +3 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) and graph ρG in red. +Figure 9: Cycle diagram ρG with G a cycle graph and n = 5, r = 4. +Indeed, in order to remain non flat, the partition diagram ρ can only be modified into a +partition diagram σ by splitting a block of ρG in two, which entails the addition of a number +q of edges, q ≥ 1, resulting into an additional factor λ1−qα ≤ 1 that may only lower the order +of the contribution, see Figures 10-13 for an example where G is a graph with one cycle. +1 +2 +3 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) with order λ4−4α. +1 +2 +3 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) with order λ5−5α = λ4−4αλ−(α−1). +Figure 10: Splitting of a vertex with addition of one edge and n = 3, r = 4. +20 + +1 +2 +3 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) with order λ4−4α. +1 +2 +3 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α. +Figure 11: Splitting of a vertex with addition of three edges and n = 3, r = 4. +1 +2 +3 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) with order λ4−4α. +1 +2 +3 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α. +Figure 12: Splitting of a vertex with addition of two edges and n = 3, r = 4. +1 +2 +3 +1 +2 +3 +4 +(a) Diagram Γ(ρ, π) with order λ4−4α. +1 +2 +3 +1 +2 +3 +4 +(b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α. +Figure 13: Splitting of a vertex with addition of two edges and n = 3, r = 4. +When G is a triangle with n = 2 and r = 3, the above procedure can be reversed by first +merging a vertex and then gluing edges, see Figure 14, which results into “overlapping” all +copies of the graph G. +21 + +(a) Merging one vertex. +(b) Gluing one edge. +(c) Gluing three edges. +Figure 14: Diagram patterns with G a triangle and n = 2, r = 3. +As in part-(b) above, we lower bound κn(NG) using a single partition, and we upper bound +using the total count of connected non-flat partition diagrams using Lemma 2.6-b) to obtain +(6.8). Regarding (6.9), we have +��κn( � +NG) +�� ≤ n!r−1(K2)(r−1)nλr−α|E(G)| +((K1)rλr−α|E(G)|)n/2 += n!r−1(K2)(r−1)n +(K1)nr/2 λ−(r−α|E(G)|)(n/2−1), +n ≥ 2. +c) is a direct consequence of part b) above. +□ +7 +Asymptotic normality of subgraph counts +The cumulant bound (6.5) shows that the centered and normalized subgraph count � +NG +satisfies the Statuleviˇcius condition (A.1) below, see [RSS78, DJS22], with γ := r − 2. As a +consequence, we have the following result, in which the Berry-Esseen rate is obtained when +r = 2. +Corollary 7.1 (Dilute regime) Let G be a connected graph with |V (G)| = r vertices, +r ≥ 2, satisfying Assumption 6.1 in the dilute regime (6.1). +We have the Kolmogorov +distance bound +sup +x∈R +��P +� � +NG ≤ x +� +− P(Z ≤ x) +�� ≤ Cλ−1/(4r−6), +(7.1) +with rate 1/(4r − 6) as λ tends to infinity, where C > 0 depends only on H and G. +In addition, by Theorem 1.1 of [DE13], �NG satisfies a moderate deviation principle with +speed a2 +λ = o(λ1/(2r−3)) and rate function x2/2, see Lemma A.1-iii) in appendix. The cu- +mulant bounds (6.7), (6.9), (6.11) show that the centered and normalized subgraph count +� +NG satisfies the Statuleviˇcius condition (A.1) below, see [RSS78, DJS22], with γ := r − 2. +As a consequence, we have the following result, in which (7.2) is consistent with (7.1) when +α = 1. +22 + +Corollary 7.2 (Sparse regime) Let G be a tree with |V (G)| = r ≥ 2 vertices, satisfying +Assumption 6.1 in the sparse regime (6.2) with α ∈ [1, r/(r − 1)). We have the Kolmogorov +distance bound +sup +x∈R +��P +� � +NG ≤ x +� +− P(Z ≤ x) +�� ≤ Cλ−(α−(α−1)r)/(4r−6), +(7.2) +as λ tends to infinity, where C > 0 depends only on H and G. +In addition, by Theorem 1.1 of [DE13], �NG satisfies a moderate deviation principle with +speed a2 +λ = o(λ(α−(α−1)r)/(2r−3)) and rate function x2/2, see Lemma A.1-iii) in appendix. +Remark 7.3 We note that up to division by 2r − 3, the rate in (7.2) is consistent with +the rate (α − (α − 1)r)/2 obtained for the counting of trees in the Erd˝os-R´enyi graph, cf. +Corollary 4.10 of [PS20]. +Remark 7.4 Since (α|E(G)| − r)(n/2 − 1) ≥ (α − 1)(n/2 − 1)r ≥ 0, no significant Kol- +mogorov bounds are derived from (6.9) and (6.11) for cycle and other non-tree graphs in the +sparse regime, which is consistent with Corollaries 4.8-4.9 of [PS20]. +A +Appendix +The following results are summarized from the “main lemmas” in Chapter 2 of [SS91] and +[DE13], and are tailored to our RCM applications. We let Φ denote the cumulative distri- +bution function of the standard normal distribution. +Lemma A.1 Let (Xλ)λ≥1 be a family of random variables with mean zero and unit variance +for all λ > 0. Suppose that for all λ ≥ 1, all moments of the random variable Xλ exist and +that the cumulants of Xλ satisfy +|κj(Xλ)| ≤ (j!)1+γ +(∆λ)j−2, +j ≥ 3, +(A.1) +where γ ≥ 0 is a constant not depending on λ, while ∆λ ∈ (0, ∞) may depend on λ. Then, +the following assertions hold. +i) (Kolmogorov bound). One has +sup +x∈R +|P(Xλ ≤ x) − Φ(x)| ≤ +C +(∆λ)1/(1+2γ) , +(A.2) +for some constant C only depending on γ, see [SS91, Corollary 2.1] and [DJS22, Theo- +rem 2.4]. +23 + +ii) (Concentration inequality). For any x ≥ 0 and sufficiently large λ, +P(|Xλ| ≥ x) ≤ 2 exp +� +−1 +4 min +� +x2 +21/(1+γ), (x∆λ)1/(1+γ) +�� +. +(A.3) +See the corollary to [SS91, Lemma 2.4]. +iii) (Moderate deviation principle). Let (aλ)λ>0 be a sequence of real numbers tending to +infinity, and such that +lim +λ→∞ +aλ +(∆λ)1/(1+2γ) = 0. +Then, (a−1 +λ Xλ)λ>0 satisfies a moderate deviation principle with speed a2 +λ and rate func- +tion x2/2, see [DE13, Theorem 1.1]. +iv) (Normal approximation with Cram´er corrections). There exists a constant c > 0 such +that for all λ ≥ 1 and x ∈ (0, c(∆λ)1/(1+2γ)) we have +P(Xλ ≥ x) +1 − Φ(x) += +� +1 + O +� +x + 1 +(∆λ)1/(1+2γ) +�� +exp +�˜L(x) +� +, +P(Xλ ≤ −x) +Φ(−x) += +� +1 + O +� +x + 1 +(∆λ)1/(1+2γ) +�� +exp +�˜L(−x) +� +, +where ˜L(x) is related to the Cram´er-Petrov series, see [SS91, Lemma 2.3]. +References +[BKR89] +A.D. Barbour, M. Karo´nski, and A. Ruci´nski. A central limit theorem for decomposable random +variables with applications to random graphs. J. Combin. Theory Ser. B, 47(2):125–145, 1989. +[BOR85] +E.A. Bender, A.M. Odlyzko, and L.B. Richmond. The asymptotic number of irreducible parti- +tions. European J. Combin., 6(1):1–6, 1985. +[BR12] +R. 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On connected diagrams and cumulants of Erd˝os-R´enyi matrix models. Comm. +Math. Phys., 282:209–238, 2008. +[KRT17] +K. Krokowski, A. Reichenbachs, and C. Th¨ale. Discrete Malliavin-Stein method: Berry-Esseen +bounds for random graphs and percolation. Ann. Probab., 45(2):1071–1109, 2017. +[LNS21] +G. Last, F. Nestmann, and M. Schulte. The random connection model and functions of edge- +marked Poisson processes: second order properties and normal approximation. +Ann. Appl. +Probab., 31(1):128–168, 2021. +[LP18] +G. Last and M.D. Penrose. Lectures on the Poisson process, volume 7 of Institute of Mathematical +Statistics Textbooks. Cambridge University Press, Cambridge, 2018. +[LRR16] +R. Lachi`eze-Rey and M. Reitzner. U-statistics in stochastic geometry. In G. Peccati and M. Re- +itzner, editors, Stochastic Analysis for Poisson Point Processes: Malliavin Calculus, Wiener-Itˆo +Chaos Expansions and Stochastic Geometry, volume 7 of Bocconi & Springer Series, pages 229– +253. 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Wiener Chaos: Moments, Cumulants and Diagrams: A survey with +Computer Implementation. Bocconi & Springer Series. Springer, 2011. +[R¨ol22] +A. R¨ollin. +Kolmogorov bounds for the normal approximation of the number of triangles in +the Erd˝os-R´enyi random graph. +Probability in the Engineering and Informational Sciences, +36(3):747–773, 2022. +[Rot64] +G.-C. Rota. On the foundations of combinatorial theory. I. Theory of M¨obius functions. Z. +Wahrscheinlichkeitstheorie und Verw. Gebiete, 2:340–368, 1964. +[RSS78] +R. Rudzkis, L. Saulis, and V.A. Statuljaviˇcus. A general lemma on probabilities of large devia- +tions. Litovsk. Mat. Sb., 18(2):99–116, 217, 1978. +25 + +[Ruc88] +A. Ruci´nski. +When are small subgraphs of a random graph normally distributed? +Probab. +Theory Related Fields, 78:1–10, 1988. +[SS91] +L. Saulis and V.A. Statuleviˇcius. Limit theorems for large deviations, volume 73 of Mathematics +and its Applications (Soviet Series). Kluwer Academic Publishers Group, Dordrecht, 1991. +26 + diff --git a/3dFLT4oBgHgl3EQfry9C/content/tmp_files/load_file.txt b/3dFLT4oBgHgl3EQfry9C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..346e8c95e5db33a40ab7cfafb31d2797f6a58ff0 --- /dev/null +++ b/3dFLT4oBgHgl3EQfry9C/content/tmp_files/load_file.txt @@ -0,0 +1,854 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf,len=853 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='12145v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='PR] 28 Jan 2023 Normal approximation of subgraph counts in the random-connection model Qingwei Liu∗ Nicolas Privault† Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 21 Nanyang Link, Singapore 637371 January 31, 2023 Abstract This paper derives normal approximation results for subgraph counts written as multiparameter stochastic integrals in a random-connection model based on a Pois- son point process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' By combinatorial arguments we express the cumulants of general subgraph counts using sums over connected partition diagrams, after cancellation of terms obtained by M¨obius inversion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Using the Statuleviˇcius condition, we deduce con- vergence rates in the Kolmogorov distance by studying the growth of subgraph count cumulants as the intensity of the underlying Poisson point process tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Our analysis covers general subgraphs in the dilute and full random graph regimes, and tree-like subgraphs in the sparse random graph regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Keywords: Random-connection model, subgraph count, normal approximation, Kolmogorov distance, cumulant method, Poisson point process, random graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Mathematics Subject Classification: 60F05, 60D05, 05C80, 60G55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 Introduction This paper treats the asymptotic behavior of random subgraph counts in the random- connection model (RCM), which is used to model physical systems in e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' wireless networks, complex networks, and statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Our approach relies on the study of cumulant growth rates as the intensity of the underlying Poisson point process tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The distributional approximation of subgraph counts has attracted significant interest in the random graph literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In [Ruc88], conditions for the asymptotic normality of ∗qingwei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='liu@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='sg †nprivault@ntu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='sg 1 renormalized subgraph counts have been obtained in the Erd˝os-R´enyi random graph model [ER59, Gil59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Those results have been made more precise in [BKR89] by the derivation of convergence rates in the Wasserstein distance via Stein’s method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' They have also been strengthened in [KRT17] using the Kolmogorov distance in the case of triangle counts, and in [PS20] in the case of general subgraphs G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The case of triangles has also been treated in [R¨ol22] by the Stein-Tikhomirov method, which has been extended to general subgraphs in [ER21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In [Kho08], the counts of line (X-model) and cycles (Y -model) in discrete Erd˝os- R´enyi models have been analyzed via the asymptotic behavior of their cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In compar- ison with [Kho08], we derive Kolmogorov convergence rates and our results are not restricted to line and cycle graphs, as they cover more general subgraphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The random connection-model is a natural generalization of the Erd˝os-R´enyi random graph in which vertices are randomly located and can be connected with position-dependent probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Studying the random-connection model and obtaining normal approximation error bounds is more difficult due to the additional layer of complexity coming from the randomness of vertex locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In [LNS21], a central limit theorem and Kolmogorov con- vergence rates have been presented for the number of components isomorphic to a given finite connected graph in the random-connection model, together with a study of first mo- ments and covariances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Recently, a Central Limit Theorem has been derived in [CT22] for the counts of induced subgraphs in the random-connection model under certain stabilization and moment conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In this paper, we derive normal approximation rates under a relatively mild condition on the connection function of the random-connection model, by deriving growth rates of cumulants written as sums over connected partitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' To the best of our knowledge, this is the first time that the normal approximation of subgraph counts with convergence rates is established in the random-connection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Furthermore, various random graph regimes are discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A number of probabilistic conclusions can be derived from the behavior of cumulants of random variables using the Statuleviˇcius condition, including convergence rates in the Kolmogorov distance and moderate deviation principles, see [SS91], [DE13], [DJS22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In [GT18a, GT18b], this method has been used to derive concentration inequalities, normal approximation with error bounds, and moderate deviation principles for random polytopes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given µ a finite diffuse measure on Rd, we consider a random-connection model based on an underlying Poisson point process Ξ on Rd with intensity of the form λµ(dx), in which any 2 two vertices x, y in Ξ are connected with the probability Hλ(x, y) := cλH(x, y) ∈ [0, 1], where Hλ is the connection function of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Here, we investigate the limiting behavior of the count NG of a given subgraph G as the intensity λ of the underlying Poisson point process on Rd tends to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' To this end, we use the combinatorics of the cumulants κn(NG) based on moment expressions obtained in [Pri19] for multiparameter stochastic integrals in the random-connection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Using partition diagrams and dependency graph arguments, we start by showing in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 that the (virtual) cumulants of a random functional admitting a certain con- nectedness factorization property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) can be expressed as sums over connected partition diagrams, generalizing Lemma 2 in [MM91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A related result has been obtained in [Jan19] in the particular case of two-parameter Poisson stochastic integrals, in relation to cluster expansions for Gibbs point processes in statistical mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3, we apply Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 to express the cumulants of multiparameter stochastic integrals, for which this factorization property can be checked from the moment formulas for multiparameter stochastic integrals computed in Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Such expressions allow us to determine the dominant terms in the growth of cumulants as the intensity λ of the underlying point process tends to infinity, by estimating the counts of vertices and edges in connected partition diagrams as in [Kho08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We work under a mild condition (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) which is satisfied by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' any translation-invariant continuous connection function H : Rd × Rd → [0, 1] non vanishing at 0, such as the Rayleigh connection function given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' For our analysis of cumulant behavior we identify the leading terms in the sum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) over connected partition diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' When G is a connected graph with r := |V (G)| vertices, satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in the dilute regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) where λ−1 ≪ cλ ≤ 1, the dominant terms correspond to connected partition diagrams with the highest number of blocks, as found in [Pri22] in the case of k-hop counting in the one-dimensional random-connection model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 this yields the cumulant bounds (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='cn|E(G)| λ (K1λ)1+(r−1)n ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1cn|E(G)| λ (K2λ)1+(r−1)n, λ > 0, for some constants K1, K2 > 0 independent of λ, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' From the Statuleviˇcius condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) below, see [RSS78, DJS22], we deduce the Kolmogorov distance bound sup x∈R ��P � � NG ≤ x � − P(Z ≤ x) �� ≤ C λ1/(4r−6) , λ → ∞, 3 see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1, and a moderate deviation principle by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 of [DE13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In the sparse regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) where cλ ≤ λ−α for some α ≥ 1, the maximal rate λα−(α−1)r is attained for G a tree-like graph, and in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 we obtain the cumulant bounds (K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλα−(α−1)r, λ > 0, if G is a tree, and (K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλr−α|E(G)|, λ > 0, if G is a not a tree, such as e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' a cycle graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' As a consequence of the Statuleviˇcius condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1), when G is a tree we find the Kolmogorov distance bound sup x∈R ��P � � NG ≤ x � − P(Z ≤ x) �� ≤ Cλ−(α−(α−1)r)/(4r−6), λ → ∞, provided that 1 ≤ α < r/(r − 1), see Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Convergence rates in the Kolmogorov distances may be improved into classical Berry- Esseen rates when the connection function H(x, y) is {0, 1}-valued, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' in disk models as in [Pri22], by representing subgraph counts as multiple Poisson stochastic integrals and using the fourth moment theorem for U-statistics and sums of multiple stochastic integrals Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='10 in [ET14], see also Theorem 3 in [LRR16] or Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 in [PS22] for Hoeffding decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In the general case where H(x, y) is [0, 1]-valued this method no longer applies, this is why we rely on the Statuleviˇcius condition which in turn may yield suboptimal convergence rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Sections 2 and 3 introduce the preliminary frame- work and notations on connected partition diagrams and combinatorics of virtual cumulants that will be used for the expression of cumulants of multiparameter stochastic integrals in Section 4 and for subgraph counts in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Those expressions are applied in Section 6 to derive cumulant growth rates in the random-connection model, with application to Kol- mogorov rates in subgraph counting via the Statuleviˇcius condition in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2 Set partitions and diagram connectivity Given η a finite set, we denote by Π(η) the collection of its set partitions, and we let |σ| denote the number of blocks in any partition σ ∈ Π(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given ρ, σ two set partitions, we 4 say that σ is coarser than ρ, or that ρ is finer than σ, and we write ρ ⪯ σ, if every block in σ is a combination of blocks in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We also denote by ρ ∨ σ the finest partition which is coarser than ρ and σ, and by ρ∧σ the coarsest partition that is finer than ρ and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We let �0 be the finest partition, which is made of a single element in each block, and we let �1 be the coarsest (one-block) partition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In general, given any graph G we denote by V (G) the set of its vertices, and by E(G) the set of its edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Our study of cumulants and moments of functionals of random fields relies on partition diagrams, see [MM91, Kho08, PT11] and references therein for additional background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In what follows we let [n] := {1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , n} for n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given η ⊂ [n] we let Π(η × [r]) denote the set of all partitions of the set η × [r] := � (k, l) : k ∈ η, l = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , r � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We also let πη := (πi)i∈η ∈ Π(η × [r]) denote the partition made of the |η| blocks of size r given by πk := {(k, 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , (k, r)}, k ∈ η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Next, we introduce the definition of partition diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given η ⊂ [n] and ρ ∈ Π(η × [r]) a partition of η × [r], we denote by Γ(ρ, πη) the diagram, or graphical representation of the partition ρ, constructed by: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' arranging the elements of η × [r] into an array of |η| rows and r columns, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' adding edges connecting neighbors within a same block in ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, we say that the partition diagram Γ(ρ, π) is connected when ρ ∨ πη = �1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' For example,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' taking η := {2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 10},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' given the partitions ρ = � {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)} � and σ = � {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 5 {(3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 3)},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' {(10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4)} � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' of η × [4],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 1−a) presents an example of a non-connected partition diagram Γ(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' and Figure 1−b) presents an example of a connected partition diagram Γ(σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' π),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2 3 5 8 10 1 2 3 4 (a) Non-connected partition diagram Γ(ρ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2 3 5 8 10 1 2 3 4 (b) Connected partition diagram Γ(σ, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 1: Two examples of partition diagrams with η = {2, 3, 5, 8, 10}, n = 10, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Note that the above notion of connected partition diagram is distinct from that of irreducible partition, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', [BOR85].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 Let n ≥ 1, G a connected graph with |V (G)| = r ≥ 1 vertices, and consider G1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , Gn copies of G respectively built on π1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , πn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Let also ρ ∈ Π(η × [r]) be a partition of η × [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We let �ρG be the multigraph constructed on the blocks of ρ by adding an edge between two blocks ρ1, ρ2 of the partition ρ whenever there exists (k, l1) ∈ ρ1 and (k, l2) ∈ ρ2 such that (l1, l2) is an edge in Gk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We let ρG be the graph constructed on the blocks of ρ by removing redundant edges in �ρG, so that at most one edge remains between any two blocks ρ1, ρ2 ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 2-b) presents an illustration of the multigraph �ρG and graph ρG on the blocks of ρ when G is the line graph {(1, 2), (2, 4), (3, 4)} on {1, 2, 3, 4}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 6 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and multigraph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(ρ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 2: Diagram and graphs G, ρG, �ρG with n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4 Let n, r ≥ 1, and let ρ ∈ Π([n] × [r]) be a partition of [n] × [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' For b ⊂ [n], we let ρb ⊂ ρ be defined as ρb := {c ∈ ρ : c ⊂ b × [r]}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given η ⊂ [n] we split any partition ρ of η × [r] into the equivalence classes deduced from the connected components of the diagram ρG, as ρ = � b×[r]∈ρ∨π b⊂[n] ρb, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) As an example, in Figure 3-a), when b = {1, 2} we have ρ{1,2} = � {(1, 1), (2, 1), (2, 2), (2, 3)}, {(1, 2), (1, 3), (1, 4), (2, 4)} � , and the partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) is illustrated in Figure 3-b) with b1 = {1, 2} and b2 = {3, 4, 5}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 7 1 2 3 4 5 1 2 3 4 ρ{1,2} (a) Diagram Γ(ρ, π) and block ρ{1,2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 ρb1 ρb2 (b) Splitting {ρb1, ρb2} of ρ according to ρG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 3: Splitting of the partition ρ with ρ ∨ π = {π1 ∪ π2, π3 ∪ π4 ∪ π5} and n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given σ ∈ Π([n]) a partition of [n], we let Πσ([n]×[r]) denote the set of partitions ρ of [n] × [r] such that ρ ∨ π = {b × [r] : b ∈ σ}, and we partition Π([n] × [r]) as Π([n] × [r]) = � σ∈Π([n]) Πσ([n] × [r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) We note that given η ⊂ [n], the set Π�1(η × [r]) consists of the partitions ρ of η × [r] for which the diagram ρG is connected, as in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In what follows, we also will use non-flat partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, see Chapter 4 of [PT11] and Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and multigraph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(ρ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 4: Connected non-flat partition diagram with G a cycle graph and n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 8 Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6 a) Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The cardinality of the set C(n, r) := {ρ ∈ Π�1([n] × [r]) : ρ ∧ π = �0} of connected non-flat partition diagrams on [n] × [r] satisfies |C(n, r)| ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1rn−1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='n−1, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) b) Let n ≥ 1 and r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The cardinality of the set Mn := {ρ ∈ C(n, r) : |ρ| = 1 + (r − 1)n} of maximal connected non-flat partition diagrams on [n] × [r] satisfies ((r − 1)r)n−1(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' ≤ |Mn| ≤ ((r − 1)r)n−1n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2, n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' a) We have |C(1, r)| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given a connected partition diagram Γ(ρ, π) in C(n+1, r), we construct a connected undirected graph �ρ on [n + 1] as in Figure 5-a), and note that �ρ contains a spanning tree ρ, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 in [BR12], as shown in Figure 5-b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, the tree ρ has at most r leaves, because after removing any of root of ρ, the remaining partition can be reconnected using no more than r vertices from the root.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, starting for any leaf in the tree ρ, ρ must be made from a connected partition diagram in C(n, r), completed by a choice of at most (n + 1)r−1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' allocations of r − 1 vertices into existing or new blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Indeed, note that at least one out of r vertices in the leaf is used for an existing connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and graph �ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(ρ, π) and spanning tree ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 5: Example of graph �ρ and its spanning tree subgraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 9 This yields the induction inequality |C(n + 1, r)| ≤ r(n + 1)r−1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='|C(n, r)|, from which we conclude to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' b) Proceeding similarly to part (a), we have |M1| = 1 and the recursion r × (1 + (r − 1)n) × |Mn| ≤ |Mn+1| ≤ (n + 1)r × (1 + (r − 1)n) × |Mn|, n ≥ 1, which yields ((r − 1)r)n−1 n−1 � i=1 � i + 1 r − 1 � ≤ |Mn| ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' ((r − 1)r)n−1 n−1 � i=1 � i + 1 r − 1 � , n ≥ 1, from which (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ 3 Virtual cumulants The following definition uses the concept of independence of a virtual field with respect to graph connectedness, see Relation (17) in [MM91, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We say that a mapping F defined on partitions of [n] × [r] admits the connectedness factorization property if it decomposes according to the partition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) as F(ρ) = � b×[r]∈ρ∨π F(ρb), ρ ∈ Π([n] × [r]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) In what follows, given F a mapping defined on the partitions of [n] × [r], we will use the M¨obius transform �F of F, defined as �F(η) := � ρ∈Π(η×[r]) F(ρ), η ⊂ [n], with �F(∅) := 0, see [Rot64] and § 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5 of [PT11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We refer to [MM91, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 33] for the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The virtual cumulant G of a mapping F on � η⊂[n] Π(η × [r]) is defined by letting CF(η) := �F(η) when |η| = 1, and then recursively by CF(η) := �F(η) − � σ∈Π(η) |σ|≥2 � b∈σ CF(b), η ⊂ [n], |η| ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) 10 In the particular case r = 1, we note that when (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , Xn) is a sequence of random variables, letting F(ρ) := E �� b∈ρ � i∈b Xi � = E � n � i=1 Xi � , Relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) shows that CF(η) = � σ∈Π[η] (−1)|σ|−1(|σ| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' � b∈σ F({b}) = � σ∈Π[η] (−1)|σ|−1(|σ| − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' � b∈ρ E �� i∈b Xi � , coincides with the actual joint cumulant of (Xi)t∈η, η ⊂ [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The following proposition is an extension of the classical Lemma 2 in [MM91, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 34], see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in [Kho08].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 Let n, r ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Let F be a mapping defined on � η⊂[n] Π(η × [r]) and admit- ting the connectedness factorization property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, for η ⊂ [n] with η ̸= ∅, the virtual cumulant of F is given by the sum CF(η) = � σ∈Π�1(η×[r]) (connected) F(σ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) over connected partition diagrams on η × [r].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The claim is true when |η| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Assume that it is true for all η ⊂ [n] for some n ≥ 1, and let η be such that |η| = n + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1), we have �F(η) = � ρ∈Π(η×[r]) F(ρ) = � σ∈Π(η) � ρ∈Πσ(η×[r]) F(ρ) = � σ∈Π(η) � ρ∈Πσ(η×[r]) � b∈σ F(ρb) = � σ∈Π(η) � b∈σ � ρ∈Π�1(b×[r]) (connected) F(ρ) = � ρ∈Π�1(η×[r]) (connected) F(ρ) + � σ∈Π(η) |σ|≥2 � b∈σ CF(b), where the last equality follows from the induction hypothesis (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) when |η| ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The proof is completed by subtracting the last term on both sides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ 11 In the particular case r = 1, we note that when (X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , Xn) is a sequence of independent random variables, the functional F(ρ) := E �� b∈ρ � i∈b Xi � = � i∈[n] E[Xi] satisfies the connectedness factorization property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1), and Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 recovers the vanishing of the joint cumulants of (Xi)i∈η when |η| ≥ 2, as the set Π�1(η × [1]) of connected partition diagrams on η × [1] is empty in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 4 Cumulants of multiparameter stochastic integrals Consider a Poisson point process Ξ on Rd, d ≥ 1, with intensity measure Λ on Rd, constructed on the space Ω = � ω = {xi}i∈I ⊂ Rd : #(A ∩ ω) < ∞ for all compact A ∈ B(Rd) � of locally finite configurations on Rd, whose elements ω ∈ Ω are identified with the Radon point measures ω = � x∈ω ǫx, where ǫx denotes the Dirac measure at x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' By [LP18, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5], almost every element ω of Ω can be represented as ω = {Vi}1≤i≤N, where (Vi)i≥1 is a random sequence in Rd and a N ∪ {∞}-valued random variable N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In this section, using sums over partitions we express the moments of the multiparameter stochastic integral � V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=',Vr∈Ξ uG(V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , Vr) = � (Rd)r uG(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , xr)ω(dx1) · · · ω(dxr), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) where uG(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , xr) is a measurable process of the form uG(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' , xr) := � (i,j)∈E(G) vi,j(xi, xj), and vi,j(x, y), (i, j) ∈ E(G), are random processes v(x, y) independent of the underlying Poisson point process Ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The next proposition is a consequence of Proposition 2 in [Pri19], which relies on Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 of [Pri12] and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 of [BRSW17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let n ≥ 1 and r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The n-th moment of the multiparameter stochastic integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) is given by the summation � ρ∈Π([n]×[r]) � (Rd)|ρ| E \uf8ee \uf8f0 n � k=1 � (i,j)∈E(Gk) v � xρ k,i, xρ k,j � \uf8f9 \uf8fb � η∈V (ρG) Λ(dxη), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) 12 where we let xρ k,l := xη whenever (k, l) ∈ η, for ρ ∈ Π([n] × [r]) and η ∈ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The next proposition rewrites the product in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) as a product on the edges of the graph ρG similarly to Proposition 4 of [Pri19] when v(x, y) vanishes on the diagonal, and it generalizes Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4 of [Jan19] from two-parameter Poisson stochastic integrals to multiparameter integrals of higher orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag- onals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' v(x, x) = 0, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, the n-th moment of the multiparameter stochastic integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) is given by the summation � ρ∈Π([n]×[r]) ρ∧π=�0 (non−flat) � (Rd)|ρ| � (η1,η2)∈E(ρG) E � v(xη1, xη2)m(η1,η2)� � η∈V (ρG) Λ(dxη), over connected non-flat diagrams, where m(η1, η2) represents the multiplicity of the edge (η1, η2) in the multigraph �ρG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The next proposition is a consequence of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2, and it also extends Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5 of [Jan19] from the two-parameter case to the multiparameter case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Note that in our setting, the two-parameter case only applies to the edge counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 Let n ≥ 1, r ≥ 2, and assume that the process v(x, y) vanishes on diag- onals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' v(x, x) = 0, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, the n-th cumulant of the multiparameter stochastic integral (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) is given by the summation � ρ∈Π�1([n]×[r]) ρ∧π=�0 (non−flat connected) � (Rd)|ρ| � (η1,η2)∈E(ρG) E � v(xη1, xη2)m(η1,η2)� � η∈V (ρG) Λ(dxη) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) over connected non-flat partition diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The functional F(ρ) := � ρ∈Π([n]×[r]) ρ∧π=�0 (non−flat) � (Rd)|ρ| � (η1,η2)∈E(ρG) E � v(xη1, xη2)m(η1,η2)� � η∈V (ρG) Λ(dxη) satisfies the connectedness factorization property (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1), as for σ = b × [r] ∈ ρ ∨ π and σ′ = b′ ×[r] ∈ ρ∨π with b ̸= b′, the variables (xη)η∈ρb are distinct from the variables (xη)η∈ρb′ in the above integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) follows from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ 13 5 Cumulants of subgraph counts Let H : Rd × Rd → [0, 1] denote a measurable connection function such that 0 < � Rd H(x, y)Λ(dx) < ∞, for all y ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given ω ∈ Ω, for any x, y ∈ ω with x ̸= y, an edge connecting x and y is added with probability H(x, y), independently of the other pairs, and in this case we write x ↔ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The resulting random graph, together with the point process Ξ, is called the random-connection model and denoted by GH(Ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In the case where the connection function H is given by H(x, y) := 1{∥x−y∥≤R} for some R > 0, the resulting graph is completely determined by the geometric of the underlying point process Ξ, and is called a random geometric graph, which is included as a special case in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Given G a connected graph with |V (G)| = r vertices, we denote NG the count of sub- graphs isomorphic to G in the random-connection model GH(Ξ), which can be represented as the multiparameter stochastic integral NG := � V1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=',Vr∈Ξ � (i,j)∈E(G) 1{Vi↔Vj} = � (Rd)r � (i,j)∈E(G) 1{xi↔xj} ω(dx1) · · ·ω(dxr), up to division by the number of automorphisms of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Here, we have 1{Vi↔Vj} = 1 or 0 depending whether Vi and Vj are connected or not by an edge in GH(Ξ), with 1{x↔x} = 0, x ∈ Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) The following result is a direct consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 by taking v(x, y) := 1{x↔y} in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) and by using non-flat partition diagrams Γ(ρ, π) such that ρ ∧ π = �0, to take into account condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let n ≥ 1 and r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The moments and cumulants of NG are given by the summation E[(NG)n] = � ρ∈Π([n]×[r]) ρ∧π=�0 (non−flat) � (Rd)|ρ| � � (η1,η2)∈E(ρG) H(xη1, xη2) � � η∈V (ρG) Λ(dxη), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) over non-flat partition diagrams, and by the summation κn(NG) = � ρ∈Π�1([n]×[r]) ρ∧π=�0 (non−flat connected) � (Rd)|ρ| � � (η1,η2)∈E(ρG) H(xη1, xη2) � � η∈V (ρG) Λ(dxη), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) 14 over connected non-flat partition diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Relations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2)-(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) are consequence of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3, after taking vi,j(xi, xj) := 1{xi↔xj}, (i, j) ∈ E(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The summations are restricted to non-flat partition diagrams due to condition (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) as in Section 2 of [Pri19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ 6 Asymptotic growth of subgraph count cumulants We assume that the intensity measure of the Poisson point process Ξ on Rd has the form Λλ(dx) = λµ(dx), λ > 0, where µ is a finite diffuse measure on Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We investigate the asymptotic behaviour of the cumulants κn(NG) as the intensity λ tends to infinity, as a consequence of the partition diagram representation of cumulant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' For this, we consider the subgraph count in GH(Ξ) obtained by replacing H(x, y) with Hλ(x, y) := cλH(x, y), in which case every term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) contributes a factor c|E(ρG)| λ λ|V (ρG)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In what follows, given two positive functions f and g on (1, ∞) we write f(λ) ≪ g(λ) if limλ→∞ g(λ)/f(λ) = ∞, and we consider the following regimes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Dilute regime: for some constant K > 0 we have 1 λ ≪ cλ ≤ K, λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) Sparse regime: for some constants K > 0 and α ≥ 1 we have cλ ≤ K λα, λ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) In case cλ = K for all λ > 0 we also say that we are in the full random graph regime, and in the sequel we take K = 1 for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let r ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' There exist two constants c, C > 0 such that for any connected non-flat partition diagram Γ(ρ, π), ρ ∈ Π�1([n] × [r]), n ≥ 1, we have c|E(ρG)|C|V (ρG)| ≤ � Rd · · · � Rd � � (i,j)∈E(ρG) H(xi, xj) � � k∈V (ρG) µ(dxk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) 15 We note that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) is satisfied by e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' any translation-invariant continuous kernel function H : Rd×Rd → [0, 1] non vanishing at 0, including the standard Rayleigh connection function given by H(x, y) = e−β∥x−y∥2, x, y ∈ Rd, for some β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Indeed, for those kernels there exists c > 0 and a Borel set B ⊂ Rd such that µ(B) > 0 and H(x, y) = H(x − y, 0) ≥ c1B(x)1B(y), x, y ∈ Rd, hence c|E(ρG)|(µ(B))|V (ρG)| = c|E(ρG)| � B · · � B � k∈V (ρG) µ(dxk) ≤ � Rd · · · � Rd � � (i,j)∈E(ρG) H(xi, xj) � � k∈V (ρG) µ(dxk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In what follows, we consider the centered and normalized subgraph count cumulants defined as � NG := NG − κ1(NG) � κ2(NG) , n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The following result shows that for n ≥ 3 the normalized cumulant κn( �NG) tends to zero in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5), hence � NG converges in distribution to the normal distribution by Theorem 1 in [Jan88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 (Dilute regime) Let r ≥ 2 and consider G a connected graph with |V (G)| = r vertices, satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in the dilute regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We have the cumulant bounds (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='cn|E(G)| λ (K1λ)1+(r−1)n ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1cn|E(G)| λ (K2λ)1+(r−1)n (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4) for some constants K1, K2 > 0 independent of λ, n ≥ 1, and ��κn � �NG ��� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(Kλ)−(n/2−1), λ ≥ 1, n ≥ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5) where K > 0 is a constant independent of λ > 0 and n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We identify the leading terms in the sum (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) over connected partition diagrams, knowing that every vertex in ρG contributes a factor λ, and that every edge contributes a factor cλ, therefore every summand in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) contributes a factor c|E(ρG)| λ λ|V (ρG)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Modifying ρ ∈ Π�1([n] × [r]) by splitting a block in two means adding a vertex to ρG, and therefore a adding factor λ to the corresponding term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' At the same time, this entails 16 no loss of edge but possibly the addition of an edge to ρG, which results into an additional factor cλ with λcλ ≫ 1 by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Hence, the leading terms in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) are those associated with the connected partition diagrams Γ(ρ, π) having the highest block count, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' which have 1 + (r − 1)n blocks, see Figure 6 for a sample of such partition diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and graph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(ρ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 6: Example of maximal connected partition diagram with n = 5 and r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We note that any maximal partition ρ satisfies |E(ρG)| = n × |E(G)|, as can be checked in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Therefore, by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3), we obtain cn|E(G)|C1+(r−1)ncn|E(G)| λ ((r − 1)r)n−1(n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='λ1+(r−1)n ≤ λ1+(r−1)ncn|E(G)| λ � ρ∈Mn � (Rd)1+(r−1)n � � (η1,η2)∈E(ρG) Hλ(xη1, xη2) � � η∈V (ρG) µ(dxη), ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1rn−1r!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='n−1(µ(Rd))1+(r−1)ncn|E(G)| λ λ1+(r−1)n, which yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Regarding (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5), we have, for n ≥ 2, ��κn( �NG) �� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1cn|E(G)| λ (K2λ)1+(r−1)n � (2 − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='c2|E(G)| λ (K1λ)1+2(r−1)�n/2 = K2 �(K2/K1)r−1 √K1 �n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1λ−(n/2−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ The following result yields a positive cumulant growth of order α − (α − 1)r > 0 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6) for trees in the sparse regime with α ∈ [1, r/(r − 1)), while in the case of non-tree graphs such as cycle graphs the growth rate r − α|E(G)| ≤ (1 − α)r ≤ 0 is negative or zero in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='8) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, the normalized cumulant κn( �NG) tends to zero for n ≥ 3 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='7) only 17 when G is a tree, in which case � NG converges in distribution to the normal distribution by Theorem 1 in [Jan88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We note that when α = 1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='7) is consistent with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 (Sparse regime) Let G be a connected graph with |V (G)| = r vertices, r ≥ 2, satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in the sparse regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' a) If G is a tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' |E(G)| = r − 1, we have the cumulant bounds (K1)(r−1)nλα−(α−1)r ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλα−(α−1)r, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6) for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and ��κn( � NG) �� ≤ (K3)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1λ−(α−(α−1)r)(n/2−1), λ ≥ 1, n ≥ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='7) where K3 := (K2/K1)r−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' b) If G is not a tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' |E(G)| ≥ r, we have the cumulant bounds (K1)rλr−α|E(G)| ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλr−α|E(G)|, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='8) for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and ��κn( � NG) �� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K3)nλ(α|E(G)|−r)(n/2−1), λ ≥ 1, n ≥ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='9) for some K3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' c) If G is a cycle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' |E(G)| = r, we have the cumulant bounds (K1)rλ−(α−1)r ≤ κn(NG) ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλ−(α−1)r, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='10) for some constants K1 > 0, K2 > 1 independent of λ, n ≥ 1, and ��κn( � NG) �� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K3)nλ(α−1)(n/2−1)r, λ ≥ 1, n ≥ 2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='11) for some K3 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In the sparse regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2), every edge in the graph ρG contributes a power λ−α and every vertex contributes a power λ, hence every term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) contributes a power λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|+(|V (ρG)|−|E(ρG)|−1)α ≤ λα−(α−1)|V (ρG)| (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='12) 18 since |V (ρG)| − |E(ρG)| − 1 ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, for any connected partition diagram Γ(ρ, π) with ρ ∈ Π�1([n] × [r]), we have r ≤ |V (ρG)| ≤ 1 + (r − 1)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' a) When G is a tree and the graph ρG is also a tree, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' |V (ρG)| − |E(ρG)| − 1 = 0, the maximal order λα−(α−1)|V (ρG)| is attained in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='12), see Figure 7 for an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and multigraph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(σ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 7: Example of connected partition diagram with ρG a tree and n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In this case, the corresponding term in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) contributes a power λ|V (ρG)|−α|E(ρG)| = λα−(α−1)|V (ρG)|, λ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In this case, since |V (ρG)| ≥ r and α ≥ 1, the optimal rate λα−(α−1)r is attained by the partition diagrams Γ(ρ, π) such that |V (ρG)| = r, as illustrated in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (a) Diagram Γ(ρ, π) and multigraph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 4 5 1 2 3 4 (b) Diagram Γ(ρ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 8: Tree diagram ρG with G a tree with |V (ρG)| = r and n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 19 We conclude to (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6) using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6 as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1, by upper bounding the count of connected partitions from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Regarding (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='7), we have ��κn( � NG) �� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1K(r−1)n 2 ((K1)2(r−1)λα−(α−1)r)n/2λα−(α−1)r = �K2 K1 �(r−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1λ−(α−(α−1)r)(n/2−1), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' b) When G is not a tree it contains at least one cycle, and for any partition ρ ∈ Π�1([n] × [r]) the same holds for the graph ρG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In this case, the highest order contribution in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) is attained by connected non-flat partition diagrams Γ(ρ, π), ρ ∈ Π�1([n] × [r]), such that ρG has |V (ρG)| = r vertices, and their contribution is given by a power of order λr−α|E(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' An example of such partition diagram ρ is given in Figure 9, with G a cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (a) Diagram Γ(ρ, π) and multigraph �ρG in blue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (b) Diagram Γ(σ, π) and graph ρG in red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 9: Cycle diagram ρG with G a cycle graph and n = 5, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Indeed, in order to remain non flat, the partition diagram ρ can only be modified into a partition diagram σ by splitting a block of ρG in two, which entails the addition of a number q of edges, q ≥ 1, resulting into an additional factor λ1−qα ≤ 1 that may only lower the order of the contribution, see Figures 10-13 for an example where G is a graph with one cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (a) Diagram Γ(ρ, π) with order λ4−4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (b) Diagram Γ(σ, π) with order λ5−5α = λ4−4αλ−(α−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 10: Splitting of a vertex with addition of one edge and n = 3, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 20 1 2 3 1 2 3 4 (a) Diagram Γ(ρ, π) with order λ4−4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 11: Splitting of a vertex with addition of three edges and n = 3, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (a) Diagram Γ(ρ, π) with order λ4−4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 12: Splitting of a vertex with addition of two edges and n = 3, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (a) Diagram Γ(ρ, π) with order λ4−4α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 1 2 3 1 2 3 4 (b) Diagram Γ(σ, π) with order λ5−6α = λ4−4αλ1−2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 13: Splitting of a vertex with addition of two edges and n = 3, r = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' When G is a triangle with n = 2 and r = 3, the above procedure can be reversed by first merging a vertex and then gluing edges, see Figure 14, which results into “overlapping” all copies of the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 21 (a) Merging one vertex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (b) Gluing one edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (c) Gluing three edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Figure 14: Diagram patterns with G a triangle and n = 2, r = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' As in part-(b) above, we lower bound κn(NG) using a single partition, and we upper bound using the total count of connected non-flat partition diagrams using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='6-b) to obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Regarding (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='9), we have ��κn( � NG) �� ≤ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)nλr−α|E(G)| ((K1)rλr−α|E(G)|)n/2 = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='r−1(K2)(r−1)n (K1)nr/2 λ−(r−α|E(G)|)(n/2−1), n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' c) is a direct consequence of part b) above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' □ 7 Asymptotic normality of subgraph counts The cumulant bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='5) shows that the centered and normalized subgraph count � NG satisfies the Statuleviˇcius condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) below, see [RSS78, DJS22], with γ := r − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' As a consequence, we have the following result, in which the Berry-Esseen rate is obtained when r = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 (Dilute regime) Let G be a connected graph with |V (G)| = r vertices, r ≥ 2, satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in the dilute regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We have the Kolmogorov distance bound sup x∈R ��P � � NG ≤ x � − P(Z ≤ x) �� ≤ Cλ−1/(4r−6), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) with rate 1/(4r − 6) as λ tends to infinity, where C > 0 depends only on H and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 of [DE13], �NG satisfies a moderate deviation principle with speed a2 λ = o(λ1/(2r−3)) and rate function x2/2, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1-iii) in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The cu- mulant bounds (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='7), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='9), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='11) show that the centered and normalized subgraph count � NG satisfies the Statuleviˇcius condition (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) below, see [RSS78, DJS22], with γ := r − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' As a consequence, we have the following result, in which (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) is consistent with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) when α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 22 Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2 (Sparse regime) Let G be a tree with |V (G)| = r ≥ 2 vertices, satisfying Assumption 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 in the sparse regime (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) with α ∈ [1, r/(r − 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We have the Kolmogorov distance bound sup x∈R ��P � � NG ≤ x � − P(Z ≤ x) �� ≤ Cλ−(α−(α−1)r)/(4r−6), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) as λ tends to infinity, where C > 0 depends only on H and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In addition, by Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 of [DE13], �NG satisfies a moderate deviation principle with speed a2 λ = o(λ(α−(α−1)r)/(2r−3)) and rate function x2/2, see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1-iii) in appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3 We note that up to division by 2r − 3, the rate in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) is consistent with the rate (α − (α − 1)r)/2 obtained for the counting of trees in the Erd˝os-R´enyi graph, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='10 of [PS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4 Since (α|E(G)| − r)(n/2 − 1) ≥ (α − 1)(n/2 − 1)r ≥ 0, no significant Kol- mogorov bounds are derived from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='11) for cycle and other non-tree graphs in the sparse regime, which is consistent with Corollaries 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='8-4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='9 of [PS20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A Appendix The following results are summarized from the “main lemmas” in Chapter 2 of [SS91] and [DE13], and are tailored to our RCM applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' We let Φ denote the cumulative distri- bution function of the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1 Let (Xλ)λ≥1 be a family of random variables with mean zero and unit variance for all λ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Suppose that for all λ ≥ 1, all moments of the random variable Xλ exist and that the cumulants of Xλ satisfy |κj(Xλ)| ≤ (j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' )1+γ (∆λ)j−2, j ≥ 3, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1) where γ ≥ 0 is a constant not depending on λ, while ∆λ ∈ (0, ∞) may depend on λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, the following assertions hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' i) (Kolmogorov bound).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' One has sup x∈R |P(Xλ ≤ x) − Φ(x)| ≤ C (∆λ)1/(1+2γ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='2) for some constant C only depending on γ, see [SS91, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1] and [DJS22, Theo- rem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 23 ii) (Concentration inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' For any x ≥ 0 and sufficiently large λ, P(|Xλ| ≥ x) ≤ 2 exp � −1 4 min � x2 21/(1+γ), (x∆λ)1/(1+γ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3) See the corollary to [SS91, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' iii) (Moderate deviation principle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Let (aλ)λ>0 be a sequence of real numbers tending to infinity, and such that lim λ→∞ aλ (∆λ)1/(1+2γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Then, (a−1 λ Xλ)λ>0 satisfies a moderate deviation principle with speed a2 λ and rate func- tion x2/2, see [DE13, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' iv) (Normal approximation with Cram´er corrections).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' There exists a constant c > 0 such that for all λ ≥ 1 and x ∈ (0, c(∆λ)1/(1+2γ)) we have P(Xλ ≥ x) 1 − Φ(x) = � 1 + O � x + 1 (∆λ)1/(1+2γ) �� exp �˜L(x) � , P(Xλ ≤ −x) Φ(−x) = � 1 + O � x + 1 (∆λ)1/(1+2γ) �� exp �˜L(−x) � , where ˜L(x) is related to the Cram´er-Petrov series, see [SS91, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' References [BKR89] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Barbour, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Karo´nski, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Ruci´nski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A central limit theorem for decomposable random variables with applications to random graphs.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Odlyzko, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Richmond.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The asymptotic number of irreducible parti- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' European J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Combin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', 6(1):1–6, 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' [BR12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Balakrishnan and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Ranganathan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A textbook of graph theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Universitext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Springer, New York, second edition, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' [BRSW17] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Bogdan, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Rosi´nski, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Serafin, and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Wojciechowski.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' L´evy systems and moment formulas for mixed Poisson integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' In Stochastic analysis and related topics, volume 72 of Progr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', pages 139–164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Birkh¨auser/Springer, Cham, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' [CT22] V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Can and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Trinh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Random connection models in the thermodynamic regime: central limit theorems for add-one cost stabilizing functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Electron.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', 27:1–40, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' [DE13] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' D¨oring and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Eichelsbacher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Moderate deviations via cumulants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Theoret.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', 26:360– 385, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' [DJS22] H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' D¨oring, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Jansen, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Schubert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' The method of cumulants for the normal approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Probab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Surv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=', 19:185–270, 2022.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Saulis, and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Statuljaviˇcus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' A general lemma on probabilities of large devia- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Litovsk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Mat.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' Kluwer Academic Publishers Group, Dordrecht, 1991.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} +page_content=' 26' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/3dFLT4oBgHgl3EQfry9C/content/2301.12145v1.pdf'} diff --git a/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf b/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..823108f76541a074de271ad3b8782519f6660ffd --- /dev/null +++ b/4dAzT4oBgHgl3EQfffz3/content/2301.01455v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:ed3d2fc7428b661dc615feae5e7505c74ac577aec516c89c34ab6b019fa1c63d +size 2453675 diff --git a/4dAzT4oBgHgl3EQfffz3/vector_store/index.faiss b/4dAzT4oBgHgl3EQfffz3/vector_store/index.faiss new file mode 100644 index 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b/5NE0T4oBgHgl3EQfegCm/content/tmp_files/2301.02392v1.pdf.txt @@ -0,0 +1,1592 @@ +arXiv:2301.02392v1 [physics.plasm-ph] 6 Jan 2023 +Moment-Fourier approach to ion parallel fluid closures and +transport for a toroidally confined plasma +Jeong-Young Ji,∗ Eric D. Held, and J. Andrew Spencer +Department of Physics, Utah State University, Logan, Utah 84322, USA +Yong-Su Na +Department of Nuclear Engineering, +Seoul National University, Seoul 08826, South Korea +Abstract +A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic +field. Expanding a distribution function in general moments a set of ordinary differential equa- +tions are obtained. +Successively expanding the moments and magnetic-field involved quantities +in Fourier series, a set of linear algebraic equations is obtained. The set of full (Maxwellian and +non-Maxwellian) moment equations is solved to express the density, temperature, and flow veloc- +ity perturbations in terms of radial gradients of equilibrium pressure and temperature. Closure +relations that connect parallel heat flux density and viscosity to the radial gradients and parallel +gradients of temperature and flow velocity, are also obtained by solving the non-Maxwellian mo- +ment equations. The closure relations combined with the linearized fluid equations reproduce the +same solution obtained directly from the full moment equations. The method can be generalized +to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general +magnetic field. +∗ j.ji@usu.edu +1 + +I. +INTRODUCTION +For magnetically confined plasmas, neoclassical transport theory describes particle, heat, +and momentum transport of a steady-state plasma due to Coulomb collisions in an inhomo- +geneous magnetic field [1–7]. The neoclassical transport is obtained by solving the first order +drift kinetic equation [8, 9] assuming a zeroth order background distribution (see Ref. [10, 11] +for reviews). Due to difficulty in treating the integro-differential collision operator in veloc- +ity space, modified collision operators have been adopted for analytical work. Numerical +work may adopt the Landau (Fokker-Planck) collision operator with desired accuracy by in- +creasing velocity space resolution. Numerous transport codes have been developed to solve +the continuum drift kinetic equation with a modified [12, 13] or an exact Landau collision +operator [14–19]. +For describing a macroscopic state of a tokamak plasma, the fluid variables are of primary +importance and solving fluid equations instead of the kinetic equation may be sufficient. Due +to significantly lower dimensionality of position space compared to phase space, numerically +solving fluid equations has a great advantage over solving the kinetic equation [20–24]. The +key issue is to obtain proper closures to capture desired physics effects. Even though the +heat flux density is derived in neoclassical transport theory, it cannot serve as one of closures +for the temperature equation because it is derived from the fluid equations, and hence, +expressed in terms of the zeroth-order density and temperature instead of the (first-order) +fluid variables whose evolution equations are to be closed. That is, the heat flux derived +from the divergence free condition plays no role for the divergence term in the temperature +equation. +In this work, we introduce an analytic method to solve the drift kinetic equation to obtain +closures and transport. For a magnetized plasma, the parallel moment equations are derived +in Ref. [25]. One advantage of the moment approach is the availability of the exact collisional +moments of the linearized Landau operator [26]. The moment-based collision operator can +be utilized for the linear and nonlinear gyrokinetic Coulomb collision operator [27–29]. For +slab geometry where the magnetic field strength does not change along a magnetic field +line, the drift-kinetic equation can be converted to a linear system of ordinary differential +equations with constant coefficients. This linear system can be analytically solved for the +2 + +parallel moments using the eigenvector method [30]. +On the other hand, for an inhomogeneous magnetic field of a tokamak, the drift kinetic +equations becomes a linear system of ordinary differential equations with varying coefficients. +This means that the eigenvector method used in the integral closure [30] does not work. For +a system of linear differential equations with varying coefficients, we can Fourier-expand +the varying coefficients and moments to build a system of linear algebraic equations. While +truncation both in the moments and Fourier modes is inevitable, the solution of the truncated +system is equivalent to that of the drift kinetic equation when convergence is achieved by +increasing the number of moments and Fourier modes. The solution moments can then be +used to construct the distribution function that is the solution of the drift kinetic equation. +Therefore the moment solution can be used for benchmarking numerous fluid and kinetic +codes. +In Sec. II, we present the parallel moment equations which are equivalent to the first order +drift kinetic equation. In Sec. III, we use the Fourier expansion to solve the general moment +equations for fluid quantities in Fourier series. +The convergent solution is presented as +the numbers of moments and Fourier modes increase. In Sec. IV, we derive closures and +incorporate them into fluid equations to reproduce the fluid quantities. In Sec.V, we conclude +and discuss possible extensions of the work to more general plasmas. +II. +DRIFT KINETIC EQUATION AND MOMENT EQUATIONS +In standard neoclassical transport theory (see Ref. [11] for a general review), drift kinetic +equations are solved for ion and electron transport. An analytic solution can be obtained +for an axisymmetric magnetic field +B = I∇ζ + ∇ζ × ∇ψ +(1) +where 2πψ is the poloidal flux, 2πI/µ0 is the poloidal current, µ0 is the magnetic perme- +ability, and ζ is the toroidal angle. +For simplicity, we assume a circular magnetic field +B = +B0 +1 + ǫ cos θ +(2) +3 + +where θ is the poloidal angle, B0 is a constant reference field, ǫ = r/R0 is the inverse aspect +ratio, and R0 and r respectively are the major and minor radii of a circular-shape flux +surface. +For ion transport, the ion-electron collisions are often ignored and the reduced ion drift +kinetic equation for the first-order distribution function f1 becomes +v∥∂∥(f1 − F) = C(f1) +(3) +with +F = −Iv∥ +Ω +df0 +dψ = −Iv∥ +Ω +�d ln p0 +dψ ++ +� +s2 − 5 +2 +� d ln T0 +dψ +� +f0 +(4) +and +f0(ψ, w) = +n0(ψ) +[2πmT0(ψ)]3/2e−w/T0(ψ) = +n0 +π3/2v3 +0 +e−s2 +(5) +in the (ψ, θ, w = mv2/2, µ = mv2 +⊥/2B) coordinates, where ∂∥ = b·∇ = (B/B)·∇, v∥ = b·v, +Ω = qB/m, v0 = +� +2T0/m, and s = v/v0. Note that flux surfaces can be labeled by the +lowest-order density n0, temperature T0, or pressure p0 = n0T0. The collision operator is a +Landau operator linearized with respect to a static Maxwellian distribution function f0, +C(f1) = C(f1, f0) + C(f0, f1). +(6) +One difficulty of solving the kinetic equation (3) is in treating the collision operator, an +integro-differential operator in velocity space. In standard analytical neoclassical theory, +the Landau operator is often approximated as the Lorentz pitch-angle scattering operator +with an additional momentum restoring term for an analytical treatment. In the moment +approach, the linearized collision operator can be analytically calculated and explicitly rep- +resented by a matrix of collision coefficients. In this work, we solve a system of parallel +moment equations introduced in Ref. [25, 26]. The moment equations can also be derived +from the drift kinetic equation as shown below. +In the moment method of this work, a gyro-averaged distribution function f1 is expanded +as +f1 = f0 +� +l,k +ˆP lk ˆ +Mlk +(7) +4 + +with orthonormal polynomials +ˆP lk = +1 +√¯σlk +P lk = +1 +√¯σlk +slPl(v∥/v)L(l+1/2) +k +(s2), +where Pl is a Legendre polynomial, L(l+1/2) +k +is an associated Laguerre (Sonine) polynomial, +and the normalization constants are +¯σlk = ¯σlλlk, ¯σl = +1 +2l + 1, λlk = (l + k + 1/2)! +k!(1/2)! +. +(8) +Several lowest order moments of f1 are: +ˆ +M00 = n1/n0 (density), +ˆ +M01 = − +� +3/2T1/T0 +(temperature), +ˆ +M10 = +√ +2u/v0 (parallel flow velocity u = V1∥), +ˆ +M11 = − +� +4/5h∥/v0p0 +(parallel heat flux density), and ˆ +M20 = +� +3/4π∥/p0 (parallel viscosity), where p0 = n0T0. +The neoclassical thermodynamic drive term can also be expanded as +v∥∂∥F =v0∂∥ ln B +B/B0 +f0 +�� +2 ˆP 00 − 2 +� +2 +3 +ˆP 01 + 1 +√ +3 +ˆP 20 +� +ˆp0,ψ ++ +� +−5 +� +2 +3 +ˆP 01 + 2 +� +10 +3 +ˆP 02 + 1 +√ +3 +ˆP 20 − +� +7 +6 +ˆP 21 +� +ˆT0,ψ +� +, +(9) +where +ˆp0,ψ = +I +qv0B0n0 +dp0 +dψ , +(10) +ˆT0,ψ = +I +qv0B0 +dT0 +dψ . +(11) +Taking the ˆP jp moment of Eq. (3) yields +� +lk +ψjp,lk∂∥ ˆ +Mlk + ψjp,lk +B +(∂∥ ln B) ˆ +Mlk = 1 +λC +cjp,lk ˆ +Mlk + ∂∥ ln B +B/B0 +� +gjp +p ˆp0,ψ + gjp +T ˆT0,ψ +� +, (12) +where λC = v0τii (the ion mean free path). Note that eliminating (j, p) = (0, 0), (0, 1), and +(1,0) moment equations from Eq. (12) yields a set of closure moment equations, similar to +the closure moment equations in slab geometry Ref. [31]. The constant coefficients ψjp,lk, +ψjp,lk +B +, and cjp,lk are defined by +� +d3vv∥ ˆP jp ˆP lkf0 = n0v0ψjp,lk, +(13) +� +d3vv∥ ˆP jp(∂∥ ˆP lk)f0 = n0v0(∂∥ ln B)ψjp,lk +B +, +(14) +5 + +� +d3v ˆP jpC(f0 ˆP lk) = n0 +τii +cjp,lk = n0 +τii +δjlcj +pk. +(15) +The nonvanishing gjp in Eq. (9) are +g0,0 +p += 2, g0,1 +p += −2 +� +2 +3, g2,0 +p += 1 +√ +3 +(16) +and +g0,1 +T += −5 +� +2 +3, g0,2 +T += 2 +� +10 +3 , g2,0 +T += 1 +√ +3 +, g2,1 +T += − +� +7 +6. +(17) +Noting that ψjp,lk = δj,j±1ψj± +lk , ψjp,j+1,k +B += −(j + 2)ψjp,j+1,k/2, and ψjp,j−1,k +B += (j − +1)ψjp,j−1,k/2 (see Ref. [25]) and defining +∂j+ +∥ += ∂∥ − j + 2 +2 +∂∥ ln B, +∂j− +∥ += ∂∥ + j − 1 +2 +∂∥ ln B, +(18) +we can combine the ψ and ψB terms to rewrite Eq. (12) as +� +k +ψj− +pk ∂j− +∥ +ˆ +Mj−1,k + +� +k +ψj+ +pk ∂j+ +∥ +ˆ +Mj+1,k = 1 +λC +� +k +cj +pk ˆ +Mjk + ∂∥ ln B +B/B0 +� +gjp +p ˆpψ + gjp +T ˆTψ +� +. (19) +Although Eq. (12) for j = 0, 1, · · · , L − 1 and k = 0, 1, · · · , K − 1 is a truncated system, +there exist L and K such that the solution does not change when increasing the number of +moments higher than L and K. In other words, there exists a convergent solution of Eq. (12) +which can be considered as a solution of Eq. (3). Therefore Eq. (12) for the truncated set +of moments is quantitatively equivalent to Eq. (3). +III. +FOURIER METHOD OF SOLVING MOMENT EQUATIONS +In the axisymmetric magnetic field (1), physical quantities on a flux surface depends on θ +only. Using ∂∥ = (B · ∇θ/B)∂/∂θ = (Bθ/B)∂θ and dividing Eq. (12) by Bθ/B yields a +system of ordinary differential equations +� +lk +ψjp,lk∂θ ˆ +Mlk + ψjp,lk +B +(∂θ ln B) ˆ +Mlk = +B +BθλC +cjp,lk ˆ +Mlk + ∂θ ln B +B/B0 +� +gjp +p ˆpψ + gjp +t ˆTψ +� +. (20) +6 + +Since the coefficient ∂θ ln B is θ-dependent, the eigenvector method used in deriving integral +closures [30] does not work. Instead, we adopt the Fourier method to convert the system of +differential equations to a system of algebraic equations. Note that Eq. (20) forms a linear +system of ordinary differential equations for the parallel moments +ˆ +Mlk and the Fourier +expansion of coefficients, moments, and drive terms will convert the differential system to a +linear algebraic system. +In the Fourier method, all physical quantities are expanded in Fourier series. For A = ˆ +Mlk(θ) +and ∂θ ln B/(B/B0), +A(θ) = A(0)+A(1−) sin θ+A(1+) cos θ+A(2−) sin 2θ+A(2+) cos 2θ+· · · = +� +m +A(m)ϕ(m), (21) +with Fourier modes +ϕ(0) = 1, ϕ(1) = ϕ(1−) = sin θ, ϕ(2) = ϕ(1+) = cos θ, · · · , +ϕ(2n−1) = ϕ(n−) = sin nθ, ϕ(2n) = ϕ(n+) = cos nθ, · · · +(22) +where the Fourier index is denoted in the parentheses. The Fourier coefficient for A(θ) can +be obtained by +A(m) = +1 +σ(m) +� +dθϕ(m)A(θ), +(23) +where σ(0) = 2π and σ(m) = π for m > 0. The derivative ∂θ and the θ-dependent coefficients +in Eq. (20) become matrices in Fourier representation. For O = ∂θ, ∂θ ln B, and B/BθλC, +the Fourier matrix elements O(i,j) are obtained by +O(i,j) = +1 +σ(i) +� +dθϕ(i)Oϕ(j), +(24) +and the Fourier representation of O ˆ +Mlk becomes +� +O ˆ +Mlk� +(i) = +1 +σ(i) +� +dθϕ(i)O +� +j +ˆ +Mlk +(j)ϕ(j) = +� +j +O(i,j) ˆ +Mlk +(j). +(25) +Then the (m)th Fourier component of Eq. (20) becomes a system of algebraic equations +ψjp,lk (∂θ)(m,n) ˆ +Mlk +(n) + ψjp,lk +B +(∂θ ln B)(m,n) ˆ +Mlk +(n) = +cjp,lk +� +B +BθλC +� +(m,n) +ˆ +Mlk +(n) + +�∂θ ln B +B/B0 +� +(m) +� +gjp +p ˆp0,ψ + gjp +T ˆT0,ψ +� +, +(26) +7 + +-0.05 +0 +0.05 +-0.05 +0 +0.05 +-3 +-2 +-1 +0 +1 +2 +3 +0.3 +0.4 +0.5 +0.6 +0.7 +Figure 1. First-order density, temperature, and parallel flow velocity for ǫ = 0.1, K0 = 100, nF = 4, +and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160 +(cyan, dashed). The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ. +where summation over l, k, and n is implied. The system of algebraic equations can be +written in matrix form, +�ψ∂θ� +� +ˆ +M +� ++�ψB∂θ ln B� +� +ˆ +M +� += +� +cB/BθλC +� � +ˆ +M +� ++ +� +(gpˆp0 + gT ˆT0)(B0/B)(∂θ ln B) +� +, (27) +where �ψ∂θ� = [ψ] ⊗ (∂θ)F, �ψB∂θ ln B� = [ψB] ⊗ (∂θ ln B)F, and +� +cB/BθλC +� += [c] ⊗ +� +B/BθλC +� +F with ⊗ denoting a tensor product of two matrices. The ith row and jth column +of a Fourier matrix (O)F is O(i,j), and the dimension of the linear system is N = LKF = +(the number of Legendre polynomials)(the number of Laguerre polynomials)(the number of +Fourier modes). +8 + +-0.05 +0 +0.05 +-0.05 +0 +0.05 +-3 +-2 +-1 +0 +1 +2 +3 +0.2 +0.4 +0.6 +0.8 +Figure 2. First-order density, temperature, and parallel flow velocity for ǫ = 0.1, K0 = 100, LK = +40 × 80, and for nF = 1 (red, dotted), 2 (green, dash-dotted), 4 (blue solid), and 7 (cyan, dashed). +The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ. +The solution +� +ˆ +M +� +can be obtained by inverting or singular-value-decomposing the matrix, +� +ˆ +M +� += +� +�ψ∂θ� + �ψB∂θ ln B� − +� +cB/BθλC +��−1 +ns +� +(gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂θ ln B) +� +, (28) +where the subscript ‘ns’ denotes the nonsingular part of the matrix. It is found that elimi- +nating n(0) and T(0) components makes the matrix nonsingular [see also remarks in relation +to Eqs. (48) and (50)]. Then the Fourier components of the first order fluid quantities can +9 + +-0.05 +0 +0.05 +-0.05 +0 +0.05 +-3 +-2 +-1 +0 +1 +2 +3 +0.3 +0.4 +0.5 +0.6 +0.7 +Figure 3. First-order density, temperature, and parallel flow velocity for ǫ = 0.3, K0 = 100, nF = 4, +and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160 +(cyan, dashed). +be read from the solution +� +ˆ +M +� +, +N = ˆp0,ψNp0 + ˆT0,ψNT0, +T = ˆp0,ψTp0 + ˆT0,ψTT0, +(29) +U = ˆp0,ψUp0 + ˆT0,ψUT0, +where N = (ˆn)F = (n1/n0)F, T = ( ˆT)F = (T1/T0)F, U = (ˆu)F = (u/v0)F, Nα, Tβ, and Uβ +(β = p0, T0) are column vectors of Fourier components. With the Fourier components, the +first-order fluid quantities can be constructed from Eq. (21). For example, the density due +to ˆp0,ψ and ˆT0,ψ, respectively, are ˆn = � +m Np0 +(m)ϕ(m)ˆp0,ψ and ˆn = � +m NT0 +(m)ϕ(m) ˆT0,ψ, where +Nβ +(m) is the (m)th Fourier component of the column vector Nβ. +10 + +-0.05 +0 +0.05 +-0.05 +0 +0.05 +-3 +-2 +-1 +0 +1 +2 +3 +0.2 +0.4 +0.6 +0.8 +Figure 4. First-order density, temperature, and parallel flow velocity for ǫ = 0.3, K0 = 100, N = +LK = 40 × 80, and for nF = 1 (red, dotted), 5 (green, dash-dotted), 9 (blue solid), and 13 (cyan, +dashed). The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ. +The inverse collisionality of the system is characterized by a Knudsen number, the ratio of +the mean free path to the gradient scale length. Defining a basic Knudsen number for a +tokamak K0 = B/BθλC, the effective Knudsen number would be roughly K0∂θ ln B ∼ mK0 +where m is the typical Fourier mode of the system. +Although the solution (28) can be +obtained for an arbitrary axisymmetric magnetic field, circular magnetic fields [see Eq. (2)] +are considered in this work. For the circular magnetic field (2), the basic Knudsen number +is given by K0 ∼ λC/qR0 where q is the safety factor and the Fourier mode m is determined +by the inverse aspect ratio ǫ = r/R0. In general, the effective Knudsen number increases as +λC and ǫ increase. +11 + +Figure 5. The first-order distribution function f1 at θ = −π/3 in the s⊥-s∥ plane for ǫ = 0.3 +and K0 = 100. The white dashed lines indicate the passing/trapped boundary. The ratio f1/f0 is +plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b). +Figure 6. The first-order distribution function f1 at θ = π/3 on the s⊥-s∥ plane for ǫ = 0.3 and +K0 = 100. The white dashed lines indicate the passing/trapped boundary. The ratio f1/f0 is plotted +in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b). +12 + +0.2 +0.1 +0 +-0.1 +-0.2 +0.1 +0.05 +0 +-0.05 +-0.13 +3 +0.15 +2.5 +2.5 +0.1 +2 +0.05 +2 +1.5 +0 +1.5 +1 +0.05 +1 +-0.1 +0.5 +0.5 +-0.15 +0 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +s1 +(c) fi/ fo due to dpo /db and dTo /dab for To,b = po,b +(d) fi/ fo due to dpo /db and dTo/db for To, = 0.3po,b +3 +3 +2.5 +0.05 +2.5 +2 +2 +0 +1.5 +1.5 +1 +1 +-0.05 +0.5 +0.5 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +s1(a) fi/ fo due to dpo/db +(b) fi/ fo due to dTo/db0.2 +0.1 +0 +-0.1 +-0.2 +0.1 +0.05 +0 +-0.05 +-0.13 +3 +0.15 +2.5 +2.5 +0.1 +2 +0.05 +2 +1.5 +0 +1.5 +1 +0.05 +1 +-0.1 +0.5 +0.5 +-0.15 +0 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +s1 +(c) fi/fo due to dpo /db and dTo /dab for To,b = po,b +(d) fi/ fo due to dpo /db and dTo/db for To, = 0.3po,b +3 +3 +2.5 +2.5 +0.05 +2 +2 +1.5 +1.5 +0 +1 +1 +0.5 +-0.05 +0.5 +0 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +s1Figure 7. The first-order distribution function f1 at s = 0.7 on the θ-µ plane for ǫ = 0.3 and +K0 = 100. +The white dashed line indicates the passing/trapped boundary. +The ratio f1/f0 is +plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b). +The solution responding to the radial pressure gradient dp0/dψ shows that Np0 = 0, Tp0 = 0, +and Up0 = −(1, 0, ǫ, · · · )T = −(B0/B)F. This means that the ˆp0,ψ drive contributes only to +the flow velocity as ˆu = −ˆp0,ψB0/B + γuB/B0, consistent with the continuity equation +∇ · (n0V1) = 0. Here γu is an integration constant that can be determined by temperature +and flow velocity equations. It turns out that γu is proportional to ˆT0,ψ as verified from the +solution and as discussed in Sec. IV. +For the solution responding to the radial temperature gradient dT0/dψ, the density, tem- +perature, and parallel flow velocity are shown in Fig. 1 in the case of ǫ = 0.1, K0 = 100, and +nF = 4 (F = 2nF + 1 = 9). A convergence study increases the number of moments to show +that the LK = 40 × 80 moment solution converges and can be considered practically exact. +Note that the polynomials ˆP lk in Eq. (7) form a complete set. The necessary number of +moments for convergence increases as K0 increases. A convergence study that increases the +13 + +0.2 +0.15 +0.1 +0.05 +0 +0.01 +0 +-0.01 +0.02 +-0.03 +-0.04 +-0.05 +-0.06bm /odm on ann of/lf (e) +@n /on o ann of/lc (a) +0 +0.6 +0.6 +-0.02 + To/ Bol +0.5 +0.5 +0.04 +0.4 +0.4 +[units of +-0.06 +JOS +0.3 +[units +-0.08 +0.3 +0.2 +0.1 +0.2 +0.1 +-0.12 +0.1 +0 +0.14 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +0 +0 +(c) fi/ fo due to dpo /dab and dTo/db for To,b = po,b +(d) fi/ fo due to dpo/dab and dTo/db for To, = 0.3po,b +0.1 +0.6 +0.6 +0.08 +0.5 +0.5 +0.06 +0.4 +0.4 +0.04 +JO +JO +units +[units +0.3 +0.02 +0.3 +0.2 +0 +0.2 +0.1 +0.02 +0.1 +-0.04 +0 +0 +-3 +-2 +-1 +0 +1 +2 +3 +-3 +-2 +-1 +0 +1 +2 +3 +0 +0number of Fourier modes from 1 to 7 (see Figure 2) shows that the nF = 4 mode solution +converges and may be considered to be very accurate. The necessary number of Fourier +modes for convergence increases as ǫ increases. +Figures 3 and 4 show the density, temperature, and parallel flow velocity for ǫ = 0.3, a +larger inverse aspect ratio, and K0 = 100. The LK = 40 × 80 moment solution, while not as +accurate as in the ǫ = 0.1 case, is still very accurate for practical use, and the LK = 80×160 +solution is expected to be accurate. This is because ǫ = 0.3 requires more Fourier modes +than ǫ = 0.1 for an accurate expansion of the magnetic field. Higher Fourier modes make the +effective Knudsen number larger. The necessary number of Fourier modes for convergence +is nF = 13. +The moment solution can be used to construct the distribution function that is a solution +of the kinetic equation (3). Since all fluid quantities relevant to physical observables involve +several lowest order of moments, the reconstruction of the distribution function from the +moments may be redundant. Nevertheless, the distribution function itself is important for +understanding the kinetic behavior of a plasma. In the moment expansion, the high order +moments near truncation of the moment expansion could be inaccurate and may adversely +affect the convergence of the distribution function. However we find that those moments +near truncation are several orders smaller than the fluid moments, making the truncation +errors ignorable once the convergence is achieved. Figures 5 and 6 show the distribution +functions constructed from the moment solution on the s⊥-s∥ plane at θ = −π/3 and π/3, +respectively. Figure 7 shows the distribution function at s = 0.7 on the θ-µ plane. +IV. +FLUID EQUATIONS AND CLOSURES +In neoclassical transport theory, one solves Eq. (3) to express f1 in terms of f0 (or F) and take +moments of the solution f1 to express u in terms of dp0/dψ and dT0/dψ. These expressions +can be directly obtained by solving Eq. (12). In this section we derive closure relations +that can be used for closing and advancing (nonlinear) fluid equations for density, flow +velocity, and temperature. They can also be incorporated into linearized fluid equations to +reproduce the expressions of n1, T1 and u that are obtained in Sec. III. Although the closures +14 + +are represented in the Fourier basis, the formalism developed here can be applied to any +basis such as a finite element basis or finite difference basis in numerical methods. +The linearized fluid equations for n1, u, and T1 can be obtained from the original fluid +equations with n = n0 + n1, T = T0 + T1, V = ub + b × ∇p0/n0qB, h = h∥b + 5p0b × +∇T0/2qB, and π = (3π∥/2)(bb − b2I/3) where b = B/B. +They are equivalent to the +{P 00, mv0P 10, −T0P 01} moments of Eq. (3) and can be read from Eq. (20) for (j, p) = (0, 0), +(1, 0), and (0, 1): +∂0+ +θ ˆu = 2ˆp0,ψ +∂θ ln B +B/B0 +, +(30) +∂0+ +θ ˆu + ∂0+ +θ ˆh = (2ˆp0,ψ + 5 ˆT0,ψ)∂θ ln B +B/B0 +, +(31) +∂1− +θ ˆn + ∂1− +θ +ˆT + ∂1+ +θ ˆπ = 0, +(32) +where ˆu = u/v0, ˆh = h∥/v0p0, ˆπ = π∥/p0, and ∂l± +θ +is defined by Eq. (18) with ∂∥ replaced by +∂θ. For this fluid system to be closed, closure quantities ˆh and ˆπ should relate to first-order +(ˆn, ˆu, and ˆT) and equilibrium (ˆp0,ψ and ˆT0,ψ) fluid quantities. +In order to obtain the closure relations, the rows corresponding to fluid equations need to +be removed from Eq. (20). Then the corresponding columns appear as drives (sources) [gθ] +in the system: +[ψ′] +� +∂θ ˆ +M′� ++[ψ′ +B] (∂θ ln B) +� +ˆ +M′� += +B +BθλC +[c′] +� +ˆ +M′� ++[gθ]+ ∂θ ln B +B/B0 +�� +g′ +p +� +ˆp0,ψ + [g′ +T] ˆT0,ψ +� +, +(33) +where ′ denotes the removal of fluid columns and rows. For example, +� +ˆ +M′� +is a column vec- +tor ( ˆ +M0,2, · · · ˆ +M0,K+1, ˆ +M1,1, · · · , ˆ +M1,K, ˆ +M2,0, · · · , ˆ +M2,K−1, · · · , ˆ +ML−1,0, · · · , ˆ +ML−1,K−1). The +nonvanishing elements of [gθ] are +g1,1 +θ += +√ +5 +2 ∂θ ˆT, +(34) +g2,0 +θ += − +√ +3 +2 Wθ, Wθ = 4 +3∂2− +∥ ˆu. +(35) +From Fourier representation of Eq. (33), +�ψ′∂θ� +� +ˆ +M′� ++�ψ′ +B∂θ ln B� +� +ˆ +M′� += +� +cB/BθλC +� � +ˆ +M′� ++�gθ�+ +� +(g′ +pˆp0 + g′ +T ˆT0)(B0/B)(∂θ ln B) +� +, +15 + +(36) +the solution can be obtained, +� +ˆ +M′� += +� +�ψ′∂θ� + �ψ′ +B∂θ ln B� − +� +cB/BθλC +��−1 � +gθ + (gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂∥ ln B) +� +. +(37) +Fourier components of closures ˆh = − +√ +5 ˆ +M1,1/2 and ˆπ = 2 ˆ +M2,0/ +√ +3 can be read from the +solution and expressed in terms of ˆp0,ψ, and ˆT0,ψ, ˆT, and ˆu: +H = ˆp0,ψHp0 + ˆT0,ψHT0 + KhhDT + KhπW, +(38) +S = ˆp0,ψSp0 + ˆT0,ψST0 + KπhDT + KππW, +(39) +where H = (ˆh)F, S = (ˆπ)F, and W = (Wθ)F = (4/3)D2−U ≡ DWU, Hβ, and Sβ (β = p0, T0) +are column vectors, and D = (∂θ)F , Dl± = (∂l± +θ )F, and Kαβ (α, β = h, π) are matrices. Here a +column vector Hβ and Sβ connects the closures h∥ and π∥ to a radial gradient of zeroth-order +pressure (β = p0) or temperature (β = T0), and a matrix Kαβ connects closures α = h and +π to a parallel gradient of first-order temperature (β = h) or parallel flow velocity (β = π). +The closures in the position space can be constructed from the solution vector, for example, +ˆh(θ) = � +i ϕ(i){Hp0 +(i)ˆp0,ψ +HT0 +(i) ˆT0,ψ +� +j[Khh +(i,j)(DT)(j) +Khπ +(i,j)W(j)]ϕ(j)}, where Hβ +(i) is the (i)th +Fourier component of the column vector Hβ and Kαβ +(i,j) is the (i)th row and (j)th column +of the matrix Kαβ. Figures 8 and 9, respectively, show the parallel heat flux density and +viscosity due to ˆp0,ψ, ˆT0,ψ, and several Fourier modes of ∂θ ˆT and Wθ. As the Fourier mode +of the thermodynamic drives increases, the contribution to the closure quantity decreases. +By combining closure relations with the time-independent, linear fluid equations, we can +reproduce the fluid variables of Sec. III. Using (B0/B)∂θ ln B = −∂θ(B0/B) and eliminating +Eq. (30) from Eq. (31), we write the Fourier representation of Eqs. (30)-(32), +D0+U = −2ˆp0,ψDB−1, +(40) +D0+H = −5 ˆT0,ψDB−1, +(41) +DN + DT + D1+S = 0, +(42) +16 + +-2 +-1.5 +-1 +-0.5 +0 +-4 +-2 +0 +2 +-0.2 +-0.1 +0 +0.1 +0.2 +-2 +0 +2 +4 +-0.5 +0 +0.5 +1 +Figure 8. +Parallel heat flux density due to (a) dp0/dψ and dT0/dψ, (b) (∂θ ˆT)(m+) cos mθ, (c) +(∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ. +The dimensionless heat flux, +h∥/v0p0, is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and +(e) (Wθ)(m−). +17 + +-0.02 +-0.01 +0 +0.01 +0.02 +-0.1 +-0.05 +0 +0.05 +0.1 +-1 +0 +1 +2 +-80 +-60 +-40 +-20 +0 +-0.2 +0 +0.2 +Figure 9. +Parallel viscosity due to (a) dp0/dψ and dT0/dψ, +(b) (∂θ ˆT)(m+) cos mθ, +(c) +(∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ. The dimensionless viscosity π∥/p0 +is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and (e) +(Wθ)(m−). +18 + +where B−1 = (B0/B)F. Then we combine with closures (38) and (39) to write +L + + + + + +N +T +U + + + + + = Rp0 ˆp0,ψ + RT0 ˆT0,ψ. +(43) +where +L = + + + + + +0 +0 +D0+ +0 +D0+KhhD +D0+KhπDW +D D + D1+KπhD D1+KππDW + + + + + , +(44) +Rp0 = − + + + + + +2DB−1 +D0+Hp0 +D1+Sp0 + + + + + , RT0 = − + + + + + +0 +5DB−1 +D1+ST0 + + + + + . +(45) +Using the singular value decomposition, we can invert the nonsingular part of L and obtain +the solution vector (N, T, U) in terms of ˆp0,ψ and ˆT0,ψ. +The solution vector reproduces +Eq. (29) with the column vector (Nβ, Tβ, Uβ) = (L−1 +ns ) Rβ for β = p0 and T0. +Now we discuss how to obtain the parallel flow velocity and heat flux density when not +using the singular value decomposition but instead, analytically calculating the integration +constants. From Eqs. (40) and (41), we have +U = −ˆp0,ψB−1 + γuB, +(46) +H = −5 +2 +ˆT0,ψB−1 + γhB, +(47) +where γu and γh are expansion coefficients for the null space of D0+ (D0+B = 0), and +B = (B/B0)F. Combining Eq. (38) with (47), we have +DT = γuFu + γhFh + ˆp0,ψFp + ˆT0,ψFT, +(48) +where +Fu = −Khh,−1KhπDWB, +Fh = Khh,−1B, +Fp = −Khh,−1 � +Hp0 − KhπDWB−1 +� +, +FT = −Khh,−1 +� +HT0 + 5 +2B−1 +� +, +(49) +19 + +Combining Eq. (39) with Eq. (42) and using Eqs. (46) and (48), we have +DN + DT = γuGu + γhGh + ˆp0,ψGp + ˆT0,ψGT +(50) +where +Gu = −D1+ � +KπhFu + KππDWB +� +, +Gh = −D1+KπhFh, +Gp = −D1+ � +Sp0 + KπhFp − KππDWB−1 +� +, +GT = −D1+ � +ST0 + KπhFT� +. +(51) +The temperature and density can be obtained by inverting the nonsingular part of D in +Eqs. (48) and (50). The null space of D is spanned by [ϕ(0)]F, which corresponds to the +constant term in the Fourier series. Since the lowest-order density (n0) and temperature +(T0) are constant, we set n(0) = 0 and T(0) = 0 without loss of generality. From the first row +corresponding to the constant (0) Fourier mode, +0 = γuFu +(0) + γhFh +(0) + ˆp0,ψFp +(0) + ˆT0,ψFT +(0), +(52) +0 = γuGu +(0) + γhGh +(0) + ˆp0,ψGp +(0) + ˆT0,ψGT +(0), +(53) +we can determine the integration constants γu and γh, + + γu +γh + + = − + + Fu +(0) Fh +(0) +Gu +(0) Gh +(0) + + +−1  + Fp +(0) FT +(0) +Gp +(0) GT +(0) + + + + ˆp0,ψ +ˆT0,ψ + + . +(54) +Then Eqs. (46) and (47) with the constants obtained in Eq. (54) agree with the corresponding +column vectors of the solution (28). Note that the heat flux obtained here is not a closure +and satisfies ∇ · h = 0. +Before concluding this section, a few remarks are in order. First, Eqs. (40) and (41) are +equivalent to ∇ · (n0V1) = 0 and ∇ · h = 0. Inserting the lowest order solutions V1⊥ = +(1/qB2)B × ∇p0 and h⊥ = (5p0/2qB2)B × ∇T0 obtained from ∇p0 − n0qV1 × B/m = +0 and (5/2)p0∇T0 − qh × B = 0, one can derive ˆu = −ˆp0,ψB0/B + γuB/B0 and ˆh = +−5 ˆT0,ψB0/2B + γhB/B0 where γu and γh are integration constants. Second, Fp and Gp +vanish when ion-electron collisions are ignored. By setting f1 = g + F, Eq. (3) becomes +v∥∂∥g = C(g) + C(F). Note that the ˆp0,ψ term in C(F) = C(F, f 0) + C(f 0, F) vanishes due +20 + +to momentum conservation and does not affect g. Therefore the term ˆp0,ψ contributes only +to the flow velocity moment of f1 and hence Fp in Eq. (48) and Gp in Eq. (50) must vanish. +Third, in the closure calculation, the ˆp0,ψ drive appears in Wθ of the viscosity equation and +affects closure quantities. However, the ˆp0,ψ term in V1∥ of Wθ exactly cancels the ˆp0,ψ term +in V1⊥ of Wθ making n1 and T1 independent of the ˆp0,ψ drive. Fourth, for an electron-ion +plasma (a, b) = (e, i) and (i, e), the ˆpa0,ψ and ˆpb0,ψ drives do not vanish in the collision +operator C(Fa, f 0 +b ) + C(f 0 +a, Fb) for the ga equation and do affect ga unless V1a∥ = V1b∥. +V. +CONCLUSION AND FUTURE WORK +We have demonstrated how to solve the drift kinetic equation using the general moment +equations to obtain transport and closure relations. +Using the moment-Fourier method +developed here, one can directly solve a full set of parallel moment equations equivalent to +the drift kinetic equation for fluid variables (density, flow velocity, and temperature) and/or +fluxes (particle flux, electric current, heat flux, etc.). The solution moments can be used to +construct the distribution function that is the solution of the drift kinetic equation. One can +also solve the non-Maxwellian moment equations to express parallel closures in terms of fluid +variables. The closures can be combined with linearized fluid equations to reproduce the +fluid variables and/or fluxes obtained from the full set of parallel moment equations. More +importantly, the closures can be utilized to advance a system of fluid equations in numerical +simulations with nonlinear terms kept when nonlinear effects are significant. Note that the +drift kinetic equation yields only linearized fluid equations by nature, e.g. Eqs. (30)-(32), +and hence cannot capture the nonlinear effects. +While the formalism developed here is only applied in the case of a single component plasma +in a circular axisymmetric magnetic field, it can be generalized to a multi-component plasma +in a tokamak with arbitrarily shaped nested flux surfaces. As long as the magnetic field +is Fourier-expandable, the moment-Fourier approach developed here is applicable. For a +multi-component plasma, the collisional heating and friction terms, respectively, will modify +Eqs. (31) and (32). The collision terms introduce couplings of temperatures and flow veloc- +ities between unlike species and, as a result, the dp0/dψ term will affect all other fluid and +closure moments as remarked at the end of Sec. IV. Although ion-electron collisions in the +21 + +ion theory are ignored based on the small-mass-ratio approximation in the existing theories +(including this work), the momentum and energy conservations require those terms in the +ion fluid equations. These effects can be investigated by solving coupled moment equations +with the Fourier method. The transport and closure relations for an electron-ion plasma +will be presented in the near future. +The moment-Fourier method developed here is applicable to a plasma with an arbitrary +Knudsen number in a general magnetic field, as long as convergence can be achieved by +increasing the number of moments and Fourier modes. +In the high-collisionality limit, +B/BθλC ≪ 1, the closure coefficients Kαβ in Eqs. (38) and (39) reproduce the corresponding +Braginskii closure coefficient [32, 33]. In the small inverse aspect ratio limit, ǫ ≪ 1, the Kαβ +reproduce the corresponding integral closure [31]. In principle, the moment-Fourier solutions +are practically exact once convergence is achieved. The necessary numbers of moments and +Fourier modes, respectively, increase as the Knudsen number and the inverse aspect ratio +increase. In practice, the moment approach is limited by the accuracy of the inverse matrix +in Eqs. (28) and (37). For low collisionality nFK0 ≳ 104, the required matrix dimension +for convergence is LKF ≳ 106, and the inverse matrix becomes inaccurate due to a large +condition number, even with the exact null space eliminated in the case of Eq. (28). For low +collisionality, the drift kinetic equation may be solved numerically. However, in the collision- +less limit, we find that the drift kinetic equation should be solved analytically for accurate +closure and transport relations. The results in the collisionless limit will be presented in the +near future, too. It is also notable that the finite element basis used in Refs. [18] and [19] +makes the convergence faster than the Legendre polynomial basis. +Since the computational effort to calculate the convergent closures is tremendous when +the effective collisionality is low, it may be impractical to compute the closures during a +fluid simulation. For practical applications, we plan to develop explicit formulas of closures +which can be expressed in terms of magnetic field parameters, ǫ for a circular geometry +or Fourier components for a general magnetic field. The explicit expressions of closures +can be developed for practical values of ǫ ≲ 0.4 (at the edge of the ITER tokamak) and +nFK0 ≲ 104 (at the core of ITER). Once the closures have been obtained for the magnetic field +parameters, they can be conveniently used without time-consuming moment calculations. +Furthermore, calculating γu in Eq. (46) will be performed for general ǫ and collisionality of +22 + +interest for a quantitative analysis of convergence depending on the number of moments and +Fourier modes. +DATA AVAILABILITY STATEMENT +The data that support the findings of this study are available upon request from the authors. +ACKNOWLEDGMENTS +The research was supported by the U.S. DOE under Grant Nos. DE-SC0022048 and DE- +FG02-04ER54746 and by National R&D Program through the National Research Foundation +of Korea (NRF) funded by Ministry of Science and ICT (2021M3F7A1084419). +[1] A. A. Galeev and R. Z. Sagdeev, Soviet Physics JETP 26, 233 (1968). +[2] M. N. Rosenbluth, R. D. Hazeltine, and F. L. Hinton, Phys. Fluids 15, 116 (1972). +[3] R. D. Hazeltine, F. L. Hinton, and M. N. 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Plasmas 22, 062114 (2015). +25 + diff --git a/5NE0T4oBgHgl3EQfegCm/content/tmp_files/load_file.txt b/5NE0T4oBgHgl3EQfegCm/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..74d465e8990a7698712196db086cd4698837c0cb --- /dev/null +++ b/5NE0T4oBgHgl3EQfegCm/content/tmp_files/load_file.txt @@ -0,0 +1,736 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf,len=735 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02392v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='plasm-ph] 6 Jan 2023 Moment-Fourier approach to ion parallel fluid closures and transport for a toroidally confined plasma Jeong-Young Ji,∗ Eric D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Held, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Andrew Spencer Department of Physics, Utah State University, Logan, Utah 84322, USA Yong-Su Na Department of Nuclear Engineering, Seoul National University, Seoul 08826, South Korea Abstract A general method of solving the drift kinetic equation is developed for an axisymmetric magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Expanding a distribution function in general moments a set of ordinary differential equa- tions are obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Successively expanding the moments and magnetic-field involved quantities in Fourier series, a set of linear algebraic equations is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The set of full (Maxwellian and non-Maxwellian) moment equations is solved to express the density, temperature, and flow veloc- ity perturbations in terms of radial gradients of equilibrium pressure and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Closure relations that connect parallel heat flux density and viscosity to the radial gradients and parallel gradients of temperature and flow velocity, are also obtained by solving the non-Maxwellian mo- ment equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The closure relations combined with the linearized fluid equations reproduce the same solution obtained directly from the full moment equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The method can be generalized to derive closures and transport for an electron-ion plasma and a multi-ion plasma in a general magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' ∗ j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='ji@usu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='edu 1 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' INTRODUCTION For magnetically confined plasmas, neoclassical transport theory describes particle, heat, and momentum transport of a steady-state plasma due to Coulomb collisions in an inhomo- geneous magnetic field [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The neoclassical transport is obtained by solving the first order drift kinetic equation [8, 9] assuming a zeroth order background distribution (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [10, 11] for reviews).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Due to difficulty in treating the integro-differential collision operator in veloc- ity space, modified collision operators have been adopted for analytical work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Numerical work may adopt the Landau (Fokker-Planck) collision operator with desired accuracy by in- creasing velocity space resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Numerous transport codes have been developed to solve the continuum drift kinetic equation with a modified [12, 13] or an exact Landau collision operator [14–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For describing a macroscopic state of a tokamak plasma, the fluid variables are of primary importance and solving fluid equations instead of the kinetic equation may be sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Due to significantly lower dimensionality of position space compared to phase space, numerically solving fluid equations has a great advantage over solving the kinetic equation [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The key issue is to obtain proper closures to capture desired physics effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Even though the heat flux density is derived in neoclassical transport theory, it cannot serve as one of closures for the temperature equation because it is derived from the fluid equations, and hence, expressed in terms of the zeroth-order density and temperature instead of the (first-order) fluid variables whose evolution equations are to be closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' That is, the heat flux derived from the divergence free condition plays no role for the divergence term in the temperature equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In this work, we introduce an analytic method to solve the drift kinetic equation to obtain closures and transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For a magnetized plasma, the parallel moment equations are derived in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' One advantage of the moment approach is the availability of the exact collisional moments of the linearized Landau operator [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The moment-based collision operator can be utilized for the linear and nonlinear gyrokinetic Coulomb collision operator [27–29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For slab geometry where the magnetic field strength does not change along a magnetic field line, the drift-kinetic equation can be converted to a linear system of ordinary differential equations with constant coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' This linear system can be analytically solved for the 2 parallel moments using the eigenvector method [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' On the other hand, for an inhomogeneous magnetic field of a tokamak, the drift kinetic equations becomes a linear system of ordinary differential equations with varying coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' This means that the eigenvector method used in the integral closure [30] does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For a system of linear differential equations with varying coefficients, we can Fourier-expand the varying coefficients and moments to build a system of linear algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' While truncation both in the moments and Fourier modes is inevitable, the solution of the truncated system is equivalent to that of the drift kinetic equation when convergence is achieved by increasing the number of moments and Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The solution moments can then be used to construct the distribution function that is the solution of the drift kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Therefore the moment solution can be used for benchmarking numerous fluid and kinetic codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' II, we present the parallel moment equations which are equivalent to the first order drift kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' III, we use the Fourier expansion to solve the general moment equations for fluid quantities in Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The convergent solution is presented as the numbers of moments and Fourier modes increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' IV, we derive closures and incorporate them into fluid equations to reproduce the fluid quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='V, we conclude and discuss possible extensions of the work to more general plasmas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' DRIFT KINETIC EQUATION AND MOMENT EQUATIONS In standard neoclassical transport theory (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [11] for a general review), drift kinetic equations are solved for ion and electron transport.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' An analytic solution can be obtained for an axisymmetric magnetic field B = I∇ζ + ∇ζ × ∇ψ (1) where 2πψ is the poloidal flux, 2πI/µ0 is the poloidal current, µ0 is the magnetic perme- ability, and ζ is the toroidal angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For simplicity, we assume a circular magnetic field B = B0 1 + ǫ cos θ (2) 3 where θ is the poloidal angle, B0 is a constant reference field, ǫ = r/R0 is the inverse aspect ratio, and R0 and r respectively are the major and minor radii of a circular-shape flux surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For ion transport, the ion-electron collisions are often ignored and the reduced ion drift kinetic equation for the first-order distribution function f1 becomes v∥∂∥(f1 − F) = C(f1) (3) with F = −Iv∥ Ω df0 dψ = −Iv∥ Ω �d ln p0 dψ + � s2 − 5 2 � d ln T0 dψ � f0 (4) and f0(ψ, w) = n0(ψ) [2πmT0(ψ)]3/2e−w/T0(ψ) = n0 π3/2v3 0 e−s2 (5) in the (ψ, θ, w = mv2/2, µ = mv2 ⊥/2B) coordinates, where ∂∥ = b·∇ = (B/B)·∇, v∥ = b·v, Ω = qB/m, v0 = � 2T0/m, and s = v/v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that flux surfaces can be labeled by the lowest-order density n0, temperature T0, or pressure p0 = n0T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The collision operator is a Landau operator linearized with respect to a static Maxwellian distribution function f0, C(f1) = C(f1, f0) + C(f0, f1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (6) One difficulty of solving the kinetic equation (3) is in treating the collision operator, an integro-differential operator in velocity space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In standard analytical neoclassical theory, the Landau operator is often approximated as the Lorentz pitch-angle scattering operator with an additional momentum restoring term for an analytical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the moment approach, the linearized collision operator can be analytically calculated and explicitly rep- resented by a matrix of collision coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In this work, we solve a system of parallel moment equations introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [25, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The moment equations can also be derived from the drift kinetic equation as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the moment method of this work, a gyro-averaged distribution function f1 is expanded as f1 = f0 � l,k ˆP lk ˆ Mlk (7) 4 with orthonormal polynomials ˆP lk = 1 √¯σlk P lk = 1 √¯σlk slPl(v∥/v)L(l+1/2) k (s2), where Pl is a Legendre polynomial, L(l+1/2) k is an associated Laguerre (Sonine) polynomial, and the normalization constants are ¯σlk = ¯σlλlk, ¯σl = 1 2l + 1, λlk = (l + k + 1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (1/2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (8) Several lowest order moments of f1 are: ˆ M00 = n1/n0 (density), ˆ M01 = − � 3/2T1/T0 (temperature), ˆ M10 = √ 2u/v0 (parallel flow velocity u = V1∥), ˆ M11 = − � 4/5h∥/v0p0 (parallel heat flux density), and ˆ M20 = � 3/4π∥/p0 (parallel viscosity), where p0 = n0T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The neoclassical thermodynamic drive term can also be expanded as v∥∂∥F =v0∂∥ ln B B/B0 f0 �� 2 ˆP 00 − 2 � 2 3 ˆP 01 + 1 √ 3 ˆP 20 � ˆp0,ψ + � −5 � 2 3 ˆP 01 + 2 � 10 3 ˆP 02 + 1 √ 3 ˆP 20 − � 7 6 ˆP 21 � ˆT0,ψ � , (9) where ˆp0,ψ = I qv0B0n0 dp0 dψ , (10) ˆT0,ψ = I qv0B0 dT0 dψ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (11) Taking the ˆP jp moment of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3) yields � lk ψjp,lk∂∥ ˆ Mlk + ψjp,lk B (∂∥ ln B) ˆ Mlk = 1 λC cjp,lk ˆ Mlk + ∂∥ ln B B/B0 � gjp p ˆp0,ψ + gjp T ˆT0,ψ � , (12) where λC = v0τii (the ion mean free path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that eliminating (j, p) = (0, 0), (0, 1), and (1,0) moment equations from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) yields a set of closure moment equations, similar to the closure moment equations in slab geometry Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The constant coefficients ψjp,lk, ψjp,lk B , and cjp,lk are defined by � d3vv∥ ˆP jp ˆP lkf0 = n0v0ψjp,lk, (13) � d3vv∥ ˆP jp(∂∥ ˆP lk)f0 = n0v0(∂∥ ln B)ψjp,lk B , (14) 5 � d3v ˆP jpC(f0 ˆP lk) = n0 τii cjp,lk = n0 τii δjlcj pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (15) The nonvanishing gjp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (9) are g0,0 p = 2, g0,1 p = −2 � 2 3, g2,0 p = 1 √ 3 (16) and g0,1 T = −5 � 2 3, g0,2 T = 2 � 10 3 , g2,0 T = 1 √ 3 , g2,1 T = − � 7 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (17) Noting that ψjp,lk = δj,j±1ψj± lk , ψjp,j+1,k B = −(j + 2)ψjp,j+1,k/2, and ψjp,j−1,k B = (j − 1)ψjp,j−1,k/2 (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [25]) and defining ∂j+ ∥ = ∂∥ − j + 2 2 ∂∥ ln B, ∂j− ∥ = ∂∥ + j − 1 2 ∂∥ ln B, (18) we can combine the ψ and ψB terms to rewrite Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) as � k ψj− pk ∂j− ∥ ˆ Mj−1,k + � k ψj+ pk ∂j+ ∥ ˆ Mj+1,k = 1 λC � k cj pk ˆ Mjk + ∂∥ ln B B/B0 � gjp p ˆpψ + gjp T ˆTψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (19) Although Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) for j = 0, 1, · · · , L − 1 and k = 0, 1, · · · , K − 1 is a truncated system, there exist L and K such that the solution does not change when increasing the number of moments higher than L and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In other words, there exists a convergent solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) which can be considered as a solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Therefore Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) for the truncated set of moments is quantitatively equivalent to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' FOURIER METHOD OF SOLVING MOMENT EQUATIONS In the axisymmetric magnetic field (1), physical quantities on a flux surface depends on θ only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Using ∂∥ = (B · ∇θ/B)∂/∂θ = (Bθ/B)∂θ and dividing Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12) by Bθ/B yields a system of ordinary differential equations � lk ψjp,lk∂θ ˆ Mlk + ψjp,lk B (∂θ ln B) ˆ Mlk = B BθλC cjp,lk ˆ Mlk + ∂θ ln B B/B0 � gjp p ˆpψ + gjp t ˆTψ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20) 6 Since the coefficient ∂θ ln B is θ-dependent, the eigenvector method used in deriving integral closures [30] does not work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Instead, we adopt the Fourier method to convert the system of differential equations to a system of algebraic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20) forms a linear system of ordinary differential equations for the parallel moments ˆ Mlk and the Fourier expansion of coefficients, moments, and drive terms will convert the differential system to a linear algebraic system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the Fourier method, all physical quantities are expanded in Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For A = ˆ Mlk(θ) and ∂θ ln B/(B/B0), A(θ) = A(0)+A(1−) sin θ+A(1+) cos θ+A(2−) sin 2θ+A(2+) cos 2θ+· · · = � m A(m)ϕ(m), (21) with Fourier modes ϕ(0) = 1, ϕ(1) = ϕ(1−) = sin θ, ϕ(2) = ϕ(1+) = cos θ, · · · , ϕ(2n−1) = ϕ(n−) = sin nθ, ϕ(2n) = ϕ(n+) = cos nθ, · · · (22) where the Fourier index is denoted in the parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The Fourier coefficient for A(θ) can be obtained by A(m) = 1 σ(m) � dθϕ(m)A(θ), (23) where σ(0) = 2π and σ(m) = π for m > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The derivative ∂θ and the θ-dependent coefficients in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20) become matrices in Fourier representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For O = ∂θ, ∂θ ln B, and B/BθλC, the Fourier matrix elements O(i,j) are obtained by O(i,j) = 1 σ(i) � dθϕ(i)Oϕ(j), (24) and the Fourier representation of O ˆ Mlk becomes � O ˆ Mlk� (i) = 1 σ(i) � dθϕ(i)O � j ˆ Mlk (j)ϕ(j) = � j O(i,j) ˆ Mlk (j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (25) Then the (m)th Fourier component of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20) becomes a system of algebraic equations ψjp,lk (∂θ)(m,n) ˆ Mlk (n) + ψjp,lk B (∂θ ln B)(m,n) ˆ Mlk (n) = cjp,lk � B BθλC � (m,n) ˆ Mlk (n) + �∂θ ln B B/B0 � (m) � gjp p ˆp0,ψ + gjp T ˆT0,ψ � , (26) 7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' First-order density, temperature, and parallel flow velocity for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1, K0 = 100, nF = 4, and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160 (cyan, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' where summation over l, k, and n is implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The system of algebraic equations can be written in matrix form, �ψ∂θ� � ˆ M � +�ψB∂θ ln B� � ˆ M � = � cB/BθλC � � ˆ M � + � (gpˆp0 + gT ˆT0)(B0/B)(∂θ ln B) � , (27) where �ψ∂θ� = [ψ] ⊗ (∂θ)F, �ψB∂θ ln B� = [ψB] ⊗ (∂θ ln B)F, and � cB/BθλC � = [c] ⊗ � B/BθλC � F with ⊗ denoting a tensor product of two matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ith row and jth column of a Fourier matrix (O)F is O(i,j), and the dimension of the linear system is N = LKF = (the number of Legendre polynomials)(the number of Laguerre polynomials)(the number of Fourier modes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='8 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' First-order density, temperature, and parallel flow velocity for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1, K0 = 100, LK = 40 × 80, and for nF = 1 (red, dotted), 2 (green, dash-dotted), 4 (blue solid), and 7 (cyan, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The solution � ˆ M � can be obtained by inverting or singular-value-decomposing the matrix, � ˆ M � = � �ψ∂θ� + �ψB∂θ ln B� − � cB/BθλC ��−1 ns � (gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂θ ln B) � , (28) where the subscript ‘ns’ denotes the nonsingular part of the matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' It is found that elimi- nating n(0) and T(0) components makes the matrix nonsingular [see also remarks in relation to Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (48) and (50)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Then the Fourier components of the first order fluid quantities can 9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' First-order density, temperature, and parallel flow velocity for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3, K0 = 100, nF = 4, and for LK = 10×20 (red, dotted), 20×40 (green, dash-dotted), 40×80 (blue solid), and 80×160 (cyan, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' be read from the solution � ˆ M � , N = ˆp0,ψNp0 + ˆT0,ψNT0, T = ˆp0,ψTp0 + ˆT0,ψTT0, (29) U = ˆp0,ψUp0 + ˆT0,ψUT0, where N = (ˆn)F = (n1/n0)F, T = ( ˆT)F = (T1/T0)F, U = (ˆu)F = (u/v0)F, Nα, Tβ, and Uβ (β = p0, T0) are column vectors of Fourier components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' With the Fourier components, the first-order fluid quantities can be constructed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For example, the density due to ˆp0,ψ and ˆT0,ψ, respectively, are ˆn = � m Np0 (m)ϕ(m)ˆp0,ψ and ˆn = � m NT0 (m)ϕ(m) ˆT0,ψ, where Nβ (m) is the (m)th Fourier component of the column vector Nβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='8 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' First-order density, temperature, and parallel flow velocity for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3, K0 = 100, N = LK = 40 × 80, and for nF = 1 (red, dotted), 5 (green, dash-dotted), 9 (blue solid), and 13 (cyan, dashed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratios n1/n0, T1/T0, and u/v0 are plotted in units of ˆT0,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The inverse collisionality of the system is characterized by a Knudsen number, the ratio of the mean free path to the gradient scale length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Defining a basic Knudsen number for a tokamak K0 = B/BθλC, the effective Knudsen number would be roughly K0∂θ ln B ∼ mK0 where m is the typical Fourier mode of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Although the solution (28) can be obtained for an arbitrary axisymmetric magnetic field, circular magnetic fields [see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (2)] are considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For the circular magnetic field (2), the basic Knudsen number is given by K0 ∼ λC/qR0 where q is the safety factor and the Fourier mode m is determined by the inverse aspect ratio ǫ = r/R0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In general, the effective Knudsen number increases as λC and ǫ increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 11 Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The first-order distribution function f1 at θ = −π/3 in the s⊥-s∥ plane for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 and K0 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The white dashed lines indicate the passing/trapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratio f1/f0 is plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The first-order distribution function f1 at θ = π/3 on the s⊥-s∥ plane for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 and K0 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The white dashed lines indicate the passing/trapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratio f1/f0 is plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='13 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='15 0 0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 s1 (c) fi/ fo due to dpo /db and dTo /dab for To,b = po,b (d) fi/ fo due to dpo /db and dTo/db for To, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3po,b 3 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 2 2 0 1.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0 0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 s1Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The first-order distribution function f1 at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='7 on the θ-µ plane for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 and K0 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The white dashed line indicates the passing/trapped boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The ratio f1/f0 is plotted in units of ˆp0,ψ in (a), (c), and (d) and in units of ˆT0,ψ in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The solution responding to the radial pressure gradient dp0/dψ shows that Np0 = 0, Tp0 = 0, and Up0 = −(1, 0, ǫ, · · · )T = −(B0/B)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' This means that the ˆp0,ψ drive contributes only to the flow velocity as ˆu = −ˆp0,ψB0/B + γuB/B0, consistent with the continuity equation ∇ · (n0V1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Here γu is an integration constant that can be determined by temperature and flow velocity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' It turns out that γu is proportional to ˆT0,ψ as verified from the solution and as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For the solution responding to the radial temperature gradient dT0/dψ, the density, tem- perature, and parallel flow velocity are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 1 in the case of ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1, K0 = 100, and nF = 4 (F = 2nF + 1 = 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' A convergence study increases the number of moments to show that the LK = 40 × 80 moment solution converges and can be considered practically exact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that the polynomials ˆP lk in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (7) form a complete set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The necessary number of moments for convergence increases as K0 increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' A convergence study that increases the 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='06bm /odm on ann of/lf (e) @n /on o ann of/lc (a) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 To/ Bol 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 [units of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='06 JOS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 [units 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='14 0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 0 0 (c) fi/ fo due to dpo /dab and dTo/db for To,b = po,b (d) fi/ fo due to dpo/dab and dTo/db for To, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3po,b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='6 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='04 0 0 3 2 1 0 1 2 3 3 2 1 0 1 2 3 0 0number of Fourier modes from 1 to 7 (see Figure 2) shows that the nF = 4 mode solution converges and may be considered to be very accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The necessary number of Fourier modes for convergence increases as ǫ increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Figures 3 and 4 show the density, temperature, and parallel flow velocity for ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3, a larger inverse aspect ratio, and K0 = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The LK = 40 × 80 moment solution, while not as accurate as in the ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 case, is still very accurate for practical use, and the LK = 80×160 solution is expected to be accurate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' This is because ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='3 requires more Fourier modes than ǫ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 for an accurate expansion of the magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Higher Fourier modes make the effective Knudsen number larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The necessary number of Fourier modes for convergence is nF = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The moment solution can be used to construct the distribution function that is a solution of the kinetic equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Since all fluid quantities relevant to physical observables involve several lowest order of moments, the reconstruction of the distribution function from the moments may be redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Nevertheless, the distribution function itself is important for understanding the kinetic behavior of a plasma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the moment expansion, the high order moments near truncation of the moment expansion could be inaccurate and may adversely affect the convergence of the distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' However we find that those moments near truncation are several orders smaller than the fluid moments, making the truncation errors ignorable once the convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Figures 5 and 6 show the distribution functions constructed from the moment solution on the s⊥-s∥ plane at θ = −π/3 and π/3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Figure 7 shows the distribution function at s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='7 on the θ-µ plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' FLUID EQUATIONS AND CLOSURES In neoclassical transport theory, one solves Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3) to express f1 in terms of f0 (or F) and take moments of the solution f1 to express u in terms of dp0/dψ and dT0/dψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' These expressions can be directly obtained by solving Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In this section we derive closure relations that can be used for closing and advancing (nonlinear) fluid equations for density, flow velocity, and temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' They can also be incorporated into linearized fluid equations to reproduce the expressions of n1, T1 and u that are obtained in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Although the closures 14 are represented in the Fourier basis, the formalism developed here can be applied to any basis such as a finite element basis or finite difference basis in numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The linearized fluid equations for n1, u, and T1 can be obtained from the original fluid equations with n = n0 + n1, T = T0 + T1, V = ub + b × ∇p0/n0qB, h = h∥b + 5p0b × ∇T0/2qB, and π = (3π∥/2)(bb − b2I/3) where b = B/B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' They are equivalent to the {P 00, mv0P 10, −T0P 01} moments of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3) and can be read from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20) for (j, p) = (0, 0), (1, 0), and (0, 1): ∂0+ θ ˆu = 2ˆp0,ψ ∂θ ln B B/B0 , (30) ∂0+ θ ˆu + ∂0+ θ ˆh = (2ˆp0,ψ + 5 ˆT0,ψ)∂θ ln B B/B0 , (31) ∂1− θ ˆn + ∂1− θ ˆT + ∂1+ θ ˆπ = 0, (32) where ˆu = u/v0, ˆh = h∥/v0p0, ˆπ = π∥/p0, and ∂l± θ is defined by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (18) with ∂∥ replaced by ∂θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For this fluid system to be closed, closure quantities ˆh and ˆπ should relate to first-order (ˆn, ˆu, and ˆT) and equilibrium (ˆp0,ψ and ˆT0,ψ) fluid quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In order to obtain the closure relations, the rows corresponding to fluid equations need to be removed from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Then the corresponding columns appear as drives (sources) [gθ] in the system: [ψ′] � ∂θ ˆ M′� +[ψ′ B] (∂θ ln B) � ˆ M′� = B BθλC [c′] � ˆ M′� +[gθ]+ ∂θ ln B B/B0 �� g′ p � ˆp0,ψ + [g′ T] ˆT0,ψ � , (33) where ′ denotes the removal of fluid columns and rows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For example, � ˆ M′� is a column vec- tor ( ˆ M0,2, · · · ˆ M0,K+1, ˆ M1,1, · · · , ˆ M1,K, ˆ M2,0, · · · , ˆ M2,K−1, · · · , ˆ ML−1,0, · · · , ˆ ML−1,K−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The nonvanishing elements of [gθ] are g1,1 θ = √ 5 2 ∂θ ˆT, (34) g2,0 θ = − √ 3 2 Wθ, Wθ = 4 3∂2− ∥ ˆu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (35) From Fourier representation of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (33), �ψ′∂θ� � ˆ M′� +�ψ′ B∂θ ln B� � ˆ M′� = � cB/BθλC � � ˆ M′� +�gθ�+ � (g′ pˆp0 + g′ T ˆT0)(B0/B)(∂θ ln B) � , 15 (36) the solution can be obtained, � ˆ M′� = � �ψ′∂θ� + �ψ′ B∂θ ln B� − � cB/BθλC ��−1 � gθ + (gpˆp0,ψ + gT ˆT0,ψ)(B0/B)(∂∥ ln B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (37) Fourier components of closures ˆh = − √ 5 ˆ M1,1/2 and ˆπ = 2 ˆ M2,0/ √ 3 can be read from the solution and expressed in terms of ˆp0,ψ, and ˆT0,ψ, ˆT, and ˆu: H = ˆp0,ψHp0 + ˆT0,ψHT0 + KhhDT + KhπW, (38) S = ˆp0,ψSp0 + ˆT0,ψST0 + KπhDT + KππW, (39) where H = (ˆh)F, S = (ˆπ)F, and W = (Wθ)F = (4/3)D2−U ≡ DWU, Hβ, and Sβ (β = p0, T0) are column vectors, and D = (∂θ)F , Dl± = (∂l± θ )F, and Kαβ (α, β = h, π) are matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Here a column vector Hβ and Sβ connects the closures h∥ and π∥ to a radial gradient of zeroth-order pressure (β = p0) or temperature (β = T0), and a matrix Kαβ connects closures α = h and π to a parallel gradient of first-order temperature (β = h) or parallel flow velocity (β = π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The closures in the position space can be constructed from the solution vector, for example, ˆh(θ) = � i ϕ(i){Hp0 (i)ˆp0,ψ +HT0 (i) ˆT0,ψ +� j[Khh (i,j)(DT)(j) +Khπ (i,j)W(j)]ϕ(j)}, where Hβ (i) is the (i)th Fourier component of the column vector Hβ and Kαβ (i,j) is the (i)th row and (j)th column of the matrix Kαβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Figures 8 and 9, respectively, show the parallel heat flux density and viscosity due to ˆp0,ψ, ˆT0,ψ, and several Fourier modes of ∂θ ˆT and Wθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' As the Fourier mode of the thermodynamic drives increases, the contribution to the closure quantity decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' By combining closure relations with the time-independent, linear fluid equations, we can reproduce the fluid variables of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Using (B0/B)∂θ ln B = −∂θ(B0/B) and eliminating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (30) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (31), we write the Fourier representation of Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (30)-(32), D0+U = −2ˆp0,ψDB−1, (40) D0+H = −5 ˆT0,ψDB−1, (41) DN + DT + D1+S = 0, (42) 16 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0 4 2 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 2 0 2 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='5 1 Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Parallel heat flux density due to (a) dp0/dψ and dT0/dψ, (b) (∂θ ˆT)(m+) cos mθ, (c) (∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The dimensionless heat flux, h∥/v0p0, is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and (e) (Wθ)(m−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 17 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='1 1 0 1 2 80 60 40 20 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='2 Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Parallel viscosity due to (a) dp0/dψ and dT0/dψ, (b) (∂θ ˆT)(m+) cos mθ, (c) (∂θ ˆT)(m−) sin mθ, (d) (Wθ)(m+) cos mθ, and (e) (Wθ)(m−) sin mθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The dimensionless viscosity π∥/p0 is plotted in units of (a) ˆp0,ψ and ˆT0,ψ, (b) (∂θ ˆT)(m+), (c) (∂θ ˆT)(m−), (d) (Wθ)(m+), and (e) (Wθ)(m−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' 18 where B−1 = (B0/B)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Then we combine with closures (38) and (39) to write L \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed N T U \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = Rp0 ˆp0,ψ + RT0 ˆT0,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (43) where L = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 0 D0+ 0 D0+KhhD D0+KhπDW D D + D1+KπhD D1+KππDW \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , (44) Rp0 = − \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 2DB−1 D0+Hp0 D1+Sp0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 , RT0 = − \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed 0 5DB−1 D1+ST0 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (45) Using the singular value decomposition, we can invert the nonsingular part of L and obtain the solution vector (N, T, U) in terms of ˆp0,ψ and ˆT0,ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The solution vector reproduces Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (29) with the column vector (Nβ, Tβ, Uβ) = (L−1 ns ) Rβ for β = p0 and T0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Now we discuss how to obtain the parallel flow velocity and heat flux density when not using the singular value decomposition but instead, analytically calculating the integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' From Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (40) and (41), we have U = −ˆp0,ψB−1 + γuB, (46) H = −5 2 ˆT0,ψB−1 + γhB, (47) where γu and γh are expansion coefficients for the null space of D0+ (D0+B = 0), and B = (B/B0)F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (38) with (47), we have DT = γuFu + γhFh + ˆp0,ψFp + ˆT0,ψFT, (48) where Fu = −Khh,−1KhπDWB, Fh = Khh,−1B, Fp = −Khh,−1 � Hp0 − KhπDWB−1 � , FT = −Khh,−1 � HT0 + 5 2B−1 � , (49) 19 Combining Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (39) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (42) and using Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (46) and (48), we have DN + DT = γuGu + γhGh + ˆp0,ψGp + ˆT0,ψGT (50) where Gu = −D1+ � KπhFu + KππDWB � , Gh = −D1+KπhFh, Gp = −D1+ � Sp0 + KπhFp − KππDWB−1 � , GT = −D1+ � ST0 + KπhFT� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (51) The temperature and density can be obtained by inverting the nonsingular part of D in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (48) and (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The null space of D is spanned by [ϕ(0)]F, which corresponds to the constant term in the Fourier series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Since the lowest-order density (n0) and temperature (T0) are constant, we set n(0) = 0 and T(0) = 0 without loss of generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' From the first row corresponding to the constant (0) Fourier mode, 0 = γuFu (0) + γhFh (0) + ˆp0,ψFp (0) + ˆT0,ψFT (0), (52) 0 = γuGu (0) + γhGh (0) + ˆp0,ψGp (0) + ˆT0,ψGT (0), (53) we can determine the integration constants γu and γh, \uf8eb \uf8ed γu γh \uf8f6 \uf8f8 = − \uf8eb \uf8ed Fu (0) Fh (0) Gu (0) Gh (0) \uf8f6 \uf8f8 −1 \uf8eb \uf8ed Fp (0) FT (0) Gp (0) GT (0) \uf8f6 \uf8f8 \uf8eb \uf8ed ˆp0,ψ ˆT0,ψ \uf8f6 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (54) Then Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (46) and (47) with the constants obtained in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (54) agree with the corresponding column vectors of the solution (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that the heat flux obtained here is not a closure and satisfies ∇ · h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Before concluding this section, a few remarks are in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' First, Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (40) and (41) are equivalent to ∇ · (n0V1) = 0 and ∇ · h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Inserting the lowest order solutions V1⊥ = (1/qB2)B × ∇p0 and h⊥ = (5p0/2qB2)B × ∇T0 obtained from ∇p0 − n0qV1 × B/m = 0 and (5/2)p0∇T0 − qh × B = 0, one can derive ˆu = −ˆp0,ψB0/B + γuB/B0 and ˆh = −5 ˆT0,ψB0/2B + γhB/B0 where γu and γh are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Second, Fp and Gp vanish when ion-electron collisions are ignored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' By setting f1 = g + F, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (3) becomes v∥∂∥g = C(g) + C(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that the ˆp0,ψ term in C(F) = C(F, f 0) + C(f 0, F) vanishes due 20 to momentum conservation and does not affect g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Therefore the term ˆp0,ψ contributes only to the flow velocity moment of f1 and hence Fp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (48) and Gp in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (50) must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Third, in the closure calculation, the ˆp0,ψ drive appears in Wθ of the viscosity equation and affects closure quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' However, the ˆp0,ψ term in V1∥ of Wθ exactly cancels the ˆp0,ψ term in V1⊥ of Wθ making n1 and T1 independent of the ˆp0,ψ drive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Fourth, for an electron-ion plasma (a, b) = (e, i) and (i, e), the ˆpa0,ψ and ˆpb0,ψ drives do not vanish in the collision operator C(Fa, f 0 b ) + C(f 0 a, Fb) for the ga equation and do affect ga unless V1a∥ = V1b∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' CONCLUSION AND FUTURE WORK We have demonstrated how to solve the drift kinetic equation using the general moment equations to obtain transport and closure relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Using the moment-Fourier method developed here, one can directly solve a full set of parallel moment equations equivalent to the drift kinetic equation for fluid variables (density, flow velocity, and temperature) and/or fluxes (particle flux, electric current, heat flux, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The solution moments can be used to construct the distribution function that is the solution of the drift kinetic equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' One can also solve the non-Maxwellian moment equations to express parallel closures in terms of fluid variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The closures can be combined with linearized fluid equations to reproduce the fluid variables and/or fluxes obtained from the full set of parallel moment equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' More importantly, the closures can be utilized to advance a system of fluid equations in numerical simulations with nonlinear terms kept when nonlinear effects are significant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Note that the drift kinetic equation yields only linearized fluid equations by nature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (30)-(32), and hence cannot capture the nonlinear effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' While the formalism developed here is only applied in the case of a single component plasma in a circular axisymmetric magnetic field, it can be generalized to a multi-component plasma in a tokamak with arbitrarily shaped nested flux surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' As long as the magnetic field is Fourier-expandable, the moment-Fourier approach developed here is applicable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For a multi-component plasma, the collisional heating and friction terms, respectively, will modify Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (31) and (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The collision terms introduce couplings of temperatures and flow veloc- ities between unlike species and, as a result, the dp0/dψ term will affect all other fluid and closure moments as remarked at the end of Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Although ion-electron collisions in the 21 ion theory are ignored based on the small-mass-ratio approximation in the existing theories (including this work), the momentum and energy conservations require those terms in the ion fluid equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' These effects can be investigated by solving coupled moment equations with the Fourier method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The transport and closure relations for an electron-ion plasma will be presented in the near future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The moment-Fourier method developed here is applicable to a plasma with an arbitrary Knudsen number in a general magnetic field, as long as convergence can be achieved by increasing the number of moments and Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the high-collisionality limit, B/BθλC ≪ 1, the closure coefficients Kαβ in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (38) and (39) reproduce the corresponding Braginskii closure coefficient [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In the small inverse aspect ratio limit, ǫ ≪ 1, the Kαβ reproduce the corresponding integral closure [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In principle, the moment-Fourier solutions are practically exact once convergence is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The necessary numbers of moments and Fourier modes, respectively, increase as the Knudsen number and the inverse aspect ratio increase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' In practice, the moment approach is limited by the accuracy of the inverse matrix in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (28) and (37).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For low collisionality nFK0 ≳ 104, the required matrix dimension for convergence is LKF ≳ 106, and the inverse matrix becomes inaccurate due to a large condition number, even with the exact null space eliminated in the case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For low collisionality, the drift kinetic equation may be solved numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' However, in the collision- less limit, we find that the drift kinetic equation should be solved analytically for accurate closure and transport relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The results in the collisionless limit will be presented in the near future, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' It is also notable that the finite element basis used in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [18] and [19] makes the convergence faster than the Legendre polynomial basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Since the computational effort to calculate the convergent closures is tremendous when the effective collisionality is low, it may be impractical to compute the closures during a fluid simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' For practical applications, we plan to develop explicit formulas of closures which can be expressed in terms of magnetic field parameters, ǫ for a circular geometry or Fourier components for a general magnetic field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' The explicit expressions of closures can be developed for practical values of ǫ ≲ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='4 (at the edge of the ITER tokamak) and nFK0 ≲ 104 (at the core of ITER).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Once the closures have been obtained for the magnetic field parameters, they can be conveniently used without time-consuming moment calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' Furthermore, calculating γu in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' (46) will be performed for general ǫ and collisionality of 22 interest for a quantitative analysis of convergence depending on the number of moments and Fourier modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT The data that support the findings of this study are available upon request from the authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' ACKNOWLEDGMENTS The research was supported by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' DOE under Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' DE-SC0022048 and DE- FG02-04ER54746 and by National R&D Program through the National Research Foundation of Korea (NRF) funded by Ministry of Science and ICT (2021M3F7A1084419).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/5NE0T4oBgHgl3EQfegCm/content/2301.02392v1.pdf'} +page_content=' [1] A.' metadata={'source': 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He +State Key Laboratory of Advanced Optical Communication Systems and Networks +Shanghai Jiao Tong University +Shanghai 200240, China +{Wenjia Zhang}wenjia.zhang@sjtu.edu.cn +ABSTRACT +To tackle challenging combinatorial optimization problems, analog computing machines based on +the nature-inspired Ising model are attracting increasing attentions in order to disruptively overcome +the impending limitations on conventional electronic computers. Photonic spatial Ising machine has +become an unique and primitive solution with all-to-all connections to solve large-scale Max-cut +problems. However, spin configuration and flipping requires two independent sets of spatial light +modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering +difficulty of optical alignment and coupling. We report a novel quadrature photonic spatial-Euler +Ising machine to realize large-scale and flexible spin-interaction configuration and spin-flip in a +single spatial light modulator, and develop a noise enhancement approach by adding digital white +noise onto detected optical signals. We experimentally show that such proposal accelerates solving +(un)weighted, (non)fully connected, 20736-node Max-cut problems, which offers obvious advantages +over simulation and heuristic algorithm results in digital computers. +1 +Introduction +Complex systems related research has progressed at a rapid pace due to high-throughput data acquisition techniques +[1, 2, 3]. Contrarily, comprehensive processing and optimization of big data with complex structures and correlations is +a prerequisite for the vast applications and spectacular advancement in bioinformatics [4, 5], pharmaceutical medicine +[6, 7], finance [8, 9], cryptography [10, 11], and artificial intelligence (AI) [12, 13]. Therefore, powerful mathematical +models and hardware processors are critically utilised to analyse high-dimensional data sets and complex systems. The +Ising model, depicting Markov chains of interacting binary units, is a typical model used to study complex systems +[14, 15]. Various artificial Ising machines developed based on this model accelerate conventional electronic computers +in performing optimization tasks involving non-deterministic polynomial time (NP)-hard problems and combinatorial +optimisation tasks, such as the Max-cut, protein folding, number partition and travelling salesman problem(TSP) +[16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36]. +Among these Ising solutions, the photonic Ising machine, by leveraging light interference to emulate spin interaction in +ferromagnets, offers substantial benefits of high connectivity and speed in ground state search [37]. Recently, there +propose various innovative photonic constructions for Ising model, such as optical coherent Ising machines (CIM) +[19, 24, 20, 21, 22, 23], photonic recurrent Ising sampler (PRIS) [34, 35], and spatial photonic Ising machines (SPIM) +[28, 29, 30, 31, 32, 33, 38]. These proposals, originated from Ising model given by H = − � + Jl,kxlxk where +Jl,k is the interaction between spins and spin binary state xl ∈ {1, −1}, are designed to search for ground state of +Ising model with the minimum Hamiltonian by either iterative sampling or directly evolving the ensemble energy +regarding the established mapping of a particular combinatorial problem. Although the coherent Ising machines +performs comparably to the quantum annealing, it lacks the advantages of parallel processing in optical computing since +it requires an extremely long fiber cavity to simulate spins through temporal multiplexing [19]. Chip-level photonic +Ising samplers are embedded with specialised heuristic method to provide sample solutions to the ground state of Ising +∗ +arXiv:2301.04651v1 [cs.ET] 11 Jan 2023 + +Figure 1: Architecture of the quadrature photonic spatial-Euler Ising machine. (a)The schematic and principle +of Euler-SIM. (b) Images of the initial and target intensity. The white bar corresponds to the length of 20 µm. (c) Initial +and final phase masks encoding on SLM in one experiment. +models, but currently fail to scale up [34] and heuristic algorithms are difficult to converge into optimum point for +a large-scale problem. In contrast, spatial photonic Ising machines encoding the spins as a phase matrix in spatial +light modulators (SLMs), can implement spin scales up to tens of thousands [28, 31]. This approach, using spatial +Fourier transformation as basic building block, can be expressed by H = − � + εlεkxlxk, which indicates that the +interaction coefficient Jl,k is set by the amplitude modulation εl and εk. This scheme is compatible with an Ising model +with fully connected interactions (or an equivalent quadratic unconstrained binary optimization (QUBO) problem) due +to its high connectivity and scalability [31]. +However, the proposed Ising machine still need external spatial amplitude modulator and thereby spin configuration +and flipping will require two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation, +which will lead to tremendous engineering difficulty of optical alignment and coupling [31]. In our previous work, we +proposed quadrature spatial Ising machine to provide flexibility for interaction configuration by introducing spatial +spins interference with quadrature phase design [32, 39]. However, the proposed Ising machine still need external +spatial amplitude modulator and thereby spin configuration and flipping will require two independent sets of spatial +light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of +optical alignment and coupling. +In this paper, we propose a novel quadrature photonic spatial-Euler Ising machine (Euler-SIM) where intensity +modulation is performed based on Euler’s Formula by extending quadrature phase configuration. To estimate the +performance of Euler-SIM, we conduct experiments and simulations on the Max-cut problem with over 20000 nodes. +The max cut value in experiment is improved by 32% over simulation results and 34% over Sahni-Gonzales (SG) +algorithm with a hundredfold speedup. The results demonstrate the superiority of our structure in terms of result +yield and speed of solving NP-hard problems beyond the traditional von Neumann processor. Furthermore, we also +investigate noise enhancement approach through experiments, finding that up to 8% performance enhancement by +adding external Gaussian white noise on the detected optical amplitude. +2 +Principle of quadrature photonic spatial-Euler Ising machine +Fig.1(a) shows the architecture design of Euler-SIM. An extended coherent light source shines on the SLM screen. +The phase mask of SLM is configured by four parts to encode both the interaction coefficients and the spin states. In +this case, a spin with amplitude information will consist of four parts ei(φl−αl),ei(θl−βl),ei(φl+αl),ei(θl+βl). On the +one hand, the spin state xl is encoded by the modulated phase φl ∈ {0, π}, and the corresponding yl is encoded by +the quadrature phase θl ∈ { π +2 , 3π +2 }. There satisfies a specific transformation relation between y and x determined by +the interaction matrix, y = Ax [32]. On the other hand, arbitrary amplitudes scaled down to the range (−1, 1) can be +converted into phase. According to the corollary to Euler’s Formula, the cosine functions can be interpreted as weighted +sums of the exponential functions,as +cos αl = ℜ(eiαl) = eiαl + e−iαl +2 +(1) +2 + +Initial I(u,v) +Initial Phase Mask +ei(t-βi) +20μm +ei(pi+an) +ntyi +Target I(u,v) + Final Phase Mask +H= +(EIEXiXkNnyiyk) +,k +1x13 +15 +20μm +feedbackFigure 2: Experimental and simulation results of Max-cut problem. (a) Graph division of a 100-node Max-cut +problem obtained by Euler-SIM. (b) Graph division of a 100-node Max-cut problem obtained with the SG algorithm. +(c) Experimental searching for max cut value of 20736 nodes. (d) Simulated searching for max cut value of 20736 +nodes. (e) Experimental and simulation results for Max-cut problems with graph densities of [0.5, 1.0]. +Thus, phase and amplitude information can then be encoded simultaneously according to extra phases αl, as +εlxl = 1 +2[ei(φl−αl) + ei(φl+αl)] +(2) +The modulated wave is passed through a lens to achieve spatial Fourier Transform and result in a superimposed effect at +the centre intensity +I(0, 0) = (xT ε + yT η)(εT x + ηT y) +(3) +which followed an opposite trend to the corresponding Hamiltonian +H = − +� + +(εlεkxlxk ± ηlηkylyk) +(4) +Therefore, we can search for the ground state of Ising model by maximising the central light intensity during the +experiment. +Based on the above architecture design, we construct the experimental setup. An incident beam with λ = 632.8nm is +injected into the beam expander with a rectangular aperture to produce a 12.5 × 7.1 mm2 rectangular light spot, which +completely covers the phase-only reflective SLM (HOLOEYE LETO-3-CFS-127) plane to activate all the 1920×1080 +pixels with a pixel pitch of 6.4 µm. Here, a full-coverage spot is essential to maximise pixel utilisation while minimizing +modulation-related pixel alignment issues. In the rear, a lens (focal length f = 150 mm) is arranged to perform a +two-dimensional Fourier transform on the beam with modulated wavefront. Finally, we probe the field intensity by the +charge-coupled device (CCD) camera on the back focal plane of the lens. The 2758×2208 sensor in the CCD (QSI 600) +has 800 kHz read rate and 16 bit digital resolution, providing extremely high resolution. The intensity images are loaded +into the central processing unit (CPU) for subsequent calculations in the electrical domain and feed back. Fig.1(b) +3 + +(a) +(b) +(c) +×104 +×108 +.285 +2 +1.8 +3.28 +1.6 + Distance +Euclidean Distance +1.4 +3.275 +Cut Value +1.2 +Value +Euclidean J +3.27 +1 +0.8 +3.265 +0.6 +0.4 +3.26 +0.2 +Euler-SIM +SG algorithm +3.255 +0 +(100 nodes) +(100 nodes) +1 +11 +21 +31 +41 +51 +61 +71 +81 +91 +Iteration +(d) +(e) +×108 +×108 +×108 +5 +1.2 +4.5 +2 +4 +Hamiltonian +3.5 +0.8 +1.5 +Hamiltonian +Cut Value + Value, +３ +2.5 +Cut +2 + Experimental results +0.4 +1.5 + Simulated results +0.5 +1 +0.2 + SG algorithm +0.5 +0 +0 +0 +1 +11 +21 +31 +41 +51 +61 +71 +81 +91 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Iteration +Graph DensityFigure 3: Experimental and simulation results of Max-cut problem with added digital noise. (a) Initial image +acquired by CCD. (b) Gaussian white noise matrices with noise level of 0.1. (c) Polluted image for computation. (d) +Simulated results of Max-cut problems with added digital noise of 0.02 0.08. The black dashed line represents the +unnoised results as a reference line. (e) Experimental results for Max-cut problems with added digital noise. The ideal +noise levels related to different graph densities are marked at the top of the orange bars. +exhibits the images of the initial detected intensityI and target intensityIT , which is focused by a uniform beam without +any modulation. Here, we calculate the Euclidean distance ∥IT − I∥2 as a cost function of the simulated annealing (SA) +algorithm, thus generating a new phase mask to refresh SLM screen. And the initial and final phase masks describing +the spin states are illustrated in Fig.1(c). This procedure is continuously cycled to govern the Hamiltonian evolution +until the system stabilises to the ground state. +3 +Experiments and Discussion +3.1 +Experimental performances and numerical simulations +The Max-cut problem, requiring to find the cut of the given graph into two subsets with the maximum value of their +connecting weighted edges, can be formulated into an equivalent Ising model without local fields [40]. An unweighted +and all-to-all connection max-cut problem make it easier by assuming Jl,k taking values of ±1, whereas many NP +problems can only be converted into weighted sparse max-cut problems for solution [41]. Our proposed scheme +perfectly implements the mapping of the latter. For each cut, the cut value is denoted as +W = 1 +2 +� + +wl,k(1 − xlxk) +(5) +where wl,k is the weight between the l-th vertex and the k-th vertex. The related Hamiltonian we use is H = +� + wl,kxlxk and the weight can be expressed as +wl,k = cos αl cos αk ± cos βl cos βk +(6) +Thus, we can maximise the cut value by looking for the minimum Hamiltonian. +Given that it is too complex to be solved precisely with a large scale problem as the exact solvers generally fail with +1000 vertexes [34, 42]. Before the experiments we need to perform a reference calculation on the conventional electrical +computing platform. Usually, the Goemans-Williamson SDP (GW-SDP) algorithm is one of the most popular methods +to solve the Max-cut problem with a guarantee of solution quality. However, it fails to solve large-scale problems owing +to the inordinately long time consumption [43]. Therefore, for large instances, we choose to employ another classical +greedy heuristic algorithm called the Sahni-Gonzales (SG) method, which is known to find approximate solutions +to large Max-cut problems in polynomial time, comparable to the GW-SDP [24, 36]. Using this method, a set of +4 + +(a) +(b) +(c) +(e) +×108 +1.85 +0.8 +0.08 +0.6 +1.8 + Spontaneous Noise Only +0.4 +0.07 +0.2 +1.75 + Added Digital Noise +(d) +×108 +1.4 +1.7 +-0.02 +0.04 +-0 +1.35 +lue +1.65 +-0.03 +0.04 +Val +1.3 +Value +0.06 +0.08 +0.06 +1.25 +1.55 +0.05 +0.07 +1.2 +1.5 +1.15 +1.45 +1.1 +1.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +0.5 +0.6 +0.7 +0.8 +0.9 +1 +Graph Density +Graph DensityTable 1: Performance comparison between Euler-SIM and other Ising machines for solving Max-cut problems. +Ising machine +Implementation +Problem Type +Problem Scale +Time to Resolution +8-FPGA SB machine [44] +Easy +All-to-all,Weighted +16,384-node +1.2 ms +PRIS [34] +Easy +All-to-all,Unweighted +100-node +63 ns per-step +CIM with DOPO [24] +Very hard +All-to-all,Unweighted +100000-node +785 µs +CIM with OEPO [45] +Hard +Sparse,Unweighted +56-node +4.5 µs +D-wave 2000Q [46, 36] +Very hard +Sparse,Weighted +2500-node +> 104 s (for 55 nodes) +Euler-SIM +Easy +All-to-all,Weighted +20000-node +325 s +20736-node Max-cut problems with varies graph densities is implemented on CPUs (Intel i9-13900K, 5.8 GHz) to +derive the max cut values, consuming 11 hours on average. +The division of 20736 points is too convoluted to be plotted. We present the division of the 100-node Max-cut problem +solved by the Euler-SIM in Fig. 2 (a) and the SG algorithm in Fig. 2 (b), respectively. Fig. 2 (c) plots the results of five +experiments on the weighted Max-cut problem with 20736 fully connected nodes. During the 100 iterations, the cut +value increases as the Euclidean distance decreases and stabilizes at around 1.759 × 108, with a 122 times speedup +compared to the SG algorithm. +Furthermore, we carried out five simulations of the same problem in MATLAB to approximate the operation of the +photonic Ising machine. As shown in Fig. 2 (d), the cut value remarkably increases to 1.076 × 108 as the Hamiltonian +of the Ising model converges rapidly. An interesting finding is that the simulation results are inferior to the experimental +results. Finally, we extended our experiments and simulations for the Max-cut problem with graph densities of 0.5-1.0 +compared with the SG algorithm. The results are shown as Fig. 2 (e), which statistically demonstrates that our +Euler-SIM offers compelling advantages for handling large-scale Max-cut problem that outweighs electronic computers, +in comparison with both simulation results and SG algorithm. The experimental max cut values exceed the SG algorithm +by an average of 34% and achieve a maximum of 49% with graph density of 1.0, which precisely captures the inherent +advantage of fully connected systems. Additionally, the experimental results routinely outperform the simulated results +by roughly 32%. The reasons for this occurrence will be discussed in the next section. +3.2 +Noise enhancement approach +The detection susceptible to noise may cause some uncertainty in experiments and we speculate that it is the discrepancy +that makes it easier to jump out of the local optimum and fit better with the SA algorithm, resulting in a better solution. +In fact, several related works have reported that noise-accelerated or noise-enhanced photonic Ising machines can be +used to solve large-scale combinatorial optimization problems [29, 35]. Considering that the spontaneous noise of the +system is challenging to gauge, we develop a noise enhancement approach by adding digital white noise onto detected +optical signals. More specifically, we generate a group of white noise matrices with different variances and add to +the CCD acquired images (after normalization) separately to provide a group of polluted images for computation and +feedback, as shown in Fig. 3(a)-(c). +Since large noise can blur the image and prevent the algorithm from converging, we pre-simulate to clarify a suitable +range of noise levels, defined as variance. The noise level is eventually lowered to less than 0.1, ensuring the correct +execution of the algorithm to obtain a feasible solution. And the simulated results in Fig. 3(d) show that Gaussian noise +with a noise level of 0.02 to 0.03 may enhance the outcomes, which will be more striking for higher graph densities, up +to 2.7%. In subsequent experiments shown as Fig. 3(e), we find that the added digital noise do improve the experimental +results with an average increase of 2.9%. Similar to the numerical simulation results, a more significant improvement +still appears in the larger graph density, up to 8.6%. Note that the ideal noise level fluctuates with the graph density +rather than in a fixed threshold, which is different from numerical simulations. +3.3 +Discussion on the Euler-SIM performance +We also compare the performance of the Euler-SIM in solving Max-Cut problems to other Ising machines in Table +1. We evaluated relevant metrics, such as implementation, problem type and scale, time to resolution (or speed) and +obtained the following conclusions: +1. Efficiently solving large-scale Max-cut problems. Compared to most solutions [44, 34, 45, 46], we comfortably +solve Max-cut problems with size over 20,000, approaching the highest reported record so far [24]. In fact, we +take the adjacent 10×10 pixels as an operation unit for the same encoding to ensure the consistency of the +5 + +Ising system in our experiments, thus do not maximise the use of all pixel points. With further optimisation of +the alignment and detection capabilities, it is feasible to scale up the problem hundredfold. +2. Flexible mapping of (non)fully connected Max-cut problems with arbitrary amplitude. Considering experi- +mental setups, many schemes prefer to demonstrate the process of solving the benchmark sparse unweighted +Max-cut problem [34, 24, 45]. Obviously, being unweighted reduces the complexity, and fully connected +problems are of more practical value and harder to implement than sparse ones [44]. As a result, many designs +take great efforts to achieve fully connection. Quantum annealers sacrifice scale, and CIMs also address this +deficiency by various schemes [15]. And achieving weighted is even harder. However, it is where SIM excel +and our design further magnifies this advantage by liberal switching between fully and non-fully connected, +weighted and unweighted problems. +3. Simple and cost-effective experimental construction. Large power consumption and high costs are required +by quantum annealers because of the cryogenic environment. Even CIMs impose rigorous experimental +requirements. Fiber oscillators of tens of kilometres are applied to keep optical loss and optical gain within +thresholds and thus guarantee spin-to-spin coupling, bringing fairly large roundtrip loss [15]. In contrast, our +approach based on a simple SLM is superior in terms of experimental cost and manoeuvrability. +Despite the fact that different Ising machines demonstrate their respective attractions in tackling Max-cut problems, such +as ultra-large scale [24], ultra-high speed [34], high stability [45], and arbitrary Max-cut problem mapping [44, 34], our +design, by adopting a more economical experimental architecture, achieves the magnitude adjacent to the largest scale +and free mapping of (non)fully connected, (un)weighted Max-cut problems, which has greater practical implications for +solving NP-hard problems. Although the design leaves much to be desired in terms of computational speed, which +is constrained by the optoelectronic transmission of data and the refresh frequency of the SLM, it still exhibits speed +advantages over electrical computation and even quantum annealing. +4 +Conclusion +In summary, our proposed Euler-SIM utilises Euler’s Formula to achieve amplitude-phase integrated modulation +and solves the Max-cut problem with 20,736 nodes. The experimental results present around 32% max cut value +improvement over simulations and 34% over SG algorithm running on the electronic computer, validating better +optimization performance and fast speed of the optical computing paradigm. Additionally, it is also a noise-friendly +Ising machine that not only exhibits a large tolerance for system noise, but even rationalizes noise as a potential boost +to system performance. 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Light: +Science & Applications, 11(1):1–10, 2022. +[46] D-Wave Systems Inc. +Hybrid solvers for quadratic optimization. +https://www.dwavesys.com/media/ +soxph512/hybrid-solvers-for-quadratic-optimization.pdf, April 2022. +8 + diff --git a/8NE3T4oBgHgl3EQfqQrl/content/tmp_files/load_file.txt b/8NE3T4oBgHgl3EQfqQrl/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b40f9ba9c2f196643706dcfb5f0c069cca9b21dd --- /dev/null +++ b/8NE3T4oBgHgl3EQfqQrl/content/tmp_files/load_file.txt @@ -0,0 +1,397 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf,len=396 +page_content='PHOTONIC SPATIAL-EULER ISING MACHINE FOR SOLVING 20000-NODE MAX-CUT PROBLEM ∗ Xin Ye, Wenjia Zhang, Shaomeng Wang, Xiaoxuan Yang, Zuyuan He State Key Laboratory of Advanced Optical Communication Systems and Networks Shanghai Jiao Tong University Shanghai 200240, China {Wenjia Zhang}wenjia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='zhang@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='cn ABSTRACT To tackle challenging combinatorial optimization problems, analog computing machines based on the nature-inspired Ising model are attracting increasing attentions in order to disruptively overcome the impending limitations on conventional electronic computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Photonic spatial Ising machine has become an unique and primitive solution with all-to-all connections to solve large-scale Max-cut problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' However, spin configuration and flipping requires two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of optical alignment and coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' We report a novel quadrature photonic spatial-Euler Ising machine to realize large-scale and flexible spin-interaction configuration and spin-flip in a single spatial light modulator, and develop a noise enhancement approach by adding digital white noise onto detected optical signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' We experimentally show that such proposal accelerates solving (un)weighted, (non)fully connected, 20736-node Max-cut problems, which offers obvious advantages over simulation and heuristic algorithm results in digital computers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' 1 Introduction Complex systems related research has progressed at a rapid pace due to high-throughput data acquisition techniques [1, 2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Contrarily, comprehensive processing and optimization of big data with complex structures and correlations is a prerequisite for the vast applications and spectacular advancement in bioinformatics [4, 5], pharmaceutical medicine [6, 7], finance [8, 9], cryptography [10, 11], and artificial intelligence (AI) [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Therefore, powerful mathematical models and hardware processors are critically utilised to analyse high-dimensional data sets and complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' The Ising model, depicting Markov chains of interacting binary units, is a typical model used to study complex systems [14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Various artificial Ising machines developed based on this model accelerate conventional electronic computers in performing optimization tasks involving non-deterministic polynomial time (NP)-hard problems and combinatorial optimisation tasks, such as the Max-cut, protein folding, number partition and travelling salesman problem(TSP) [16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Among these Ising solutions, the photonic Ising machine, by leveraging light interference to emulate spin interaction in ferromagnets, offers substantial benefits of high connectivity and speed in ground state search [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Recently, there propose various innovative photonic constructions for Ising model, such as optical coherent Ising machines (CIM) [19, 24, 20, 21, 22, 23], photonic recurrent Ising sampler (PRIS) [34, 35], and spatial photonic Ising machines (SPIM) [28, 29, 30, 31, 32, 33, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' These proposals, originated from Ising model given by H = − � Jl,kxlxk where Jl,k is the interaction between spins and spin binary state xl ∈ {1, −1}, are designed to search for ground state of Ising model with the minimum Hamiltonian by either iterative sampling or directly evolving the ensemble energy regarding the established mapping of a particular combinatorial problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Although the coherent Ising machines performs comparably to the quantum annealing, it lacks the advantages of parallel processing in optical computing since it requires an extremely long fiber cavity to simulate spins through temporal multiplexing [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Chip-level photonic Ising samplers are embedded with specialised heuristic method to provide sample solutions to the ground state of Ising ∗ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='04651v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='ET] 11 Jan 2023 Figure 1: Architecture of the quadrature photonic spatial-Euler Ising machine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (a)The schematic and principle of Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (b) Images of the initial and target intensity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' The white bar corresponds to the length of 20 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (c) Initial and final phase masks encoding on SLM in one experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' models, but currently fail to scale up [34] and heuristic algorithms are difficult to converge into optimum point for a large-scale problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' In contrast, spatial photonic Ising machines encoding the spins as a phase matrix in spatial light modulators (SLMs), can implement spin scales up to tens of thousands [28, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' This approach, using spatial Fourier transformation as basic building block, can be expressed by H = − � εlεkxlxk, which indicates that the interaction coefficient Jl,k is set by the amplitude modulation εl and εk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' This scheme is compatible with an Ising model with fully connected interactions (or an equivalent quadratic unconstrained binary optimization (QUBO) problem) due to its high connectivity and scalability [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' However, the proposed Ising machine still need external spatial amplitude modulator and thereby spin configuration and flipping will require two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of optical alignment and coupling [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' In our previous work, we proposed quadrature spatial Ising machine to provide flexibility for interaction configuration by introducing spatial spins interference with quadrature phase design [32, 39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' However, the proposed Ising machine still need external spatial amplitude modulator and thereby spin configuration and flipping will require two independent sets of spatial light modulators (SLMs) for amplitude and phase modulation, which will lead to tremendous engineering difficulty of optical alignment and coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' In this paper, we propose a novel quadrature photonic spatial-Euler Ising machine (Euler-SIM) where intensity modulation is performed based on Euler’s Formula by extending quadrature phase configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' To estimate the performance of Euler-SIM, we conduct experiments and simulations on the Max-cut problem with over 20000 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' The max cut value in experiment is improved by 32% over simulation results and 34% over Sahni-Gonzales (SG) algorithm with a hundredfold speedup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' The results demonstrate the superiority of our structure in terms of result yield and speed of solving NP-hard problems beyond the traditional von Neumann processor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Furthermore, we also investigate noise enhancement approach through experiments, finding that up to 8% performance enhancement by adding external Gaussian white noise on the detected optical amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' 2 Principle of quadrature photonic spatial-Euler Ising machine Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='1(a) shows the architecture design of Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' An extended coherent light source shines on the SLM screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' The phase mask of SLM is configured by four parts to encode both the interaction coefficients and the spin states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' In this case, a spin with amplitude information will consist of four parts ei(φl−αl),ei(θl−βl),ei(φl+αl),ei(θl+βl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' On the one hand, the spin state xl is encoded by the modulated phase φl ∈ {0, π}, and the corresponding yl is encoded by the quadrature phase θl ∈ { π 2 , 3π 2 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' There satisfies a specific transformation relation between y and x determined by the interaction matrix, y = Ax [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' On the other hand, arbitrary amplitudes scaled down to the range (−1, 1) can be converted into phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' According to the corollary to Euler’s Formula, the cosine functions can be interpreted as weighted sums of the exponential functions,as cos αl = ℜ(eiαl) = eiαl + e−iαl 2 (1) 2 Initial I(u,v) Initial Phase Mask ei(t-βi) 20μm ei(pi+an) ntyi Target I(u,v) Final Phase Mask H= (EIEXiXkNnyiyk) ,k 1x13 15 20μm feedbackFigure 2: Experimental and simulation results of Max-cut problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (a) Graph division of a 100-node Max-cut problem obtained by Euler-SIM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (b) Graph division of a 100-node Max-cut problem obtained with the SG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (c) Experimental searching for max cut value of 20736 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (d) Simulated searching for max cut value of 20736 nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' (e) Experimental and simulation results for Max-cut problems with graph densities of [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content='0].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' Thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' phase and amplitude information can then be encoded simultaneously according to extra phases αl,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' as εlxl = 1 2[ei(φl−αl) + ei(φl+αl)] (2) The modulated wave is passed through a lens to achieve spatial Fourier Transform and result in a superimposed effect at the centre intensity I(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8NE3T4oBgHgl3EQfqQrl/content/2301.04651v1.pdf'} +page_content=' 0) = (xT ε + yT η)(εT x + ηT y) (3) which followed an opposite trend to the corresponding Hamiltonian H = − � 0 and 1 ≤ 푟 ≤ 푛, we say that +푝 ≡ 푟 (mod 푛), or that 푟 = mod(푝, 푛), if 푝 = 푞푛 + 푟, for some integer 푞. For a set 푃 ⊆ {0, 1}푛, +we use the notation +1푃(x) = + + +1, if x ∈ 푃, +0, otherwise, +and we write 1푃 as the indicator function (1푃(x) : x ∈ {0, 1}푛). For a given length 푛, we also +use the notation 푊푖 := {x ∈ {0, 1}푛 : 푤(x) = 푖}. +B. Block Codes and Constrained Sequences +We recall the following definitions of block codes and linear codes over F2 (see, for example, +Chapters 1 and 2 of [11]). +Definition II.1. An (푛, 푀) block code C over F2 is a nonempty subset of F푛 +2, with |C| = 푀. +The rate of the block code C is given by +rate(C) := log2 푀 +푛 +. +Definition II.2. The minimum distance 푑(C) of a block code C is the minimum distance between +January 13, 2023 +DRAFT + +JANUARY 2023 +6 +any two distinct codewords of C, i.e., +푑(C) = +min +c1,c2∈C: c1≠c2 푑(c1, c2). +An (푛, 푀) block code with minimum distance 푑 will be called an (푛, 푀, 푑) block code. +Moreover, given a sequence of block codes {C(푛)}푛≥1, if it holds that rate(C(푛)) +푛→∞ +−−−−→ 푅, for +some 푅 ∈ [0, 1], then we say that {C(푛)}푛≥1 is of rate 푅. +Definition II.3. An [푛, 푘] linear code C over F2 is an (푛, 2푘) block code that is a subspace of +F푛 +2. +A constraint is represented by a set A ⊆ {0, 1}푛 of binary words. We call the sequences in A +as constrained sequences, and refer to a block code C, all of whose codewords lie in A, as a +“constrained code”. We refer the reader to [1] for a detailed exposition on constrained sequences +and coding in the presence of constraints. Note that we make no further assumption about the +constrained system (such as it being finite-type, almost-finite-type, irreducible, etc.). Given such +a collection of constrained sequences of blocklength 푛, the noiseless capacity (see Chapter 3 of +[1]) of the constraint is defined as +퐶0(A) := lim +푛→∞ +log2 |A| +푛 +, +where the limit in the expression above exists, by subadditivity arguments. +Further, for a given blocklength 푛, we use the notation 퐴(푛, 푑; A) to denote the size of the +largest constrained code, of minimum distance at least 푑, such that all of its codewords lie in +A. More formally, +퐴(푛, 푑; A) := +max +C⊆A: 푑(C)≥푑 |C|. +For the case where A = {0, 1}푛, we write 퐴(푛, 푑; A) as simply 퐴(푛, 푑). +C. Fourier Expansion of Boolean Functions +Consider functions 푓 : {0, 1}푛 → R, mapping x = (푥1, . . . , 푥푛) ∈ {0, 1}푛 to 푓 (x) ∈ R. If the +range of 푓 is {0, 1}, then 푓 is called a Boolean function. Now, given any function 푓 : {0, 1}푛 → R +January 13, 2023 +DRAFT + +JANUARY 2023 +7 +and a vector s = (푠1, . . . , 푠푛) ∈ {0, 1}푛, we define the Fourier coefficient of 푓 at s as +�푓 (s) := 1 +2푛 +� +x∈{0,1}푛 +푓 (x) · (−1)x·s. +The function �푓 is known as the Fourier transform (sometimes called the Hadamard transform) +of 푓 . Moreover, the functions (휒s : s ∈ {0, 1}푛), where 휒s(x) := (−1)x·s, form a basis for the +vector space 푉 of functions 푓 : {0, 1}푛 → R. If we define an inner product ⟨·, ·⟩ over the vector +space 푉, as follows: +⟨ 푓 , 푔⟩ := 1 +2푛 +� +x∈{0,1}푛 +푓 (x)푔(x), +for functions 푓 , 푔 ∈ 푉, we also have that the basis functions (휒s : s ∈ {0, 1}푛) are orthonormal, +in that +⟨휒s, 휒s′⟩ = + + +1, if s = s′, +0, otherwise. +The above discussion leads us to the following well-known theorem: +Theorem II.1 (Theorem 1.1 in [21]). Every function 푓 : {0, 1}푛 → R can be uniquely expressed +as its Fourier expansion +푓 (x) = +� +s∈{0,1}푛 +�푓 (s) · (−1)x·s. +For more details on the Fourier analysis of Boolean functions, we refer the reader to [21]. In +our paper, we shall make use of Plancherel’s Theorem from Fourier analysis, which is recalled +below, without proof. +Theorem II.2 (Plancherel’s Theorem). For any 푓 , 푔 ∈ {0, 1}푛 → R, we have that +⟨ 푓 , 푔⟩ = +� +s∈{0,1}푛 +�푓 (s)�푔(s). +We also recall the operation of convolution of two functions 푓 , 푔 : {0, 1}푛 → R, defined as +푓 ★ 푔(x) = 1 +2푛 +� +z∈{0,1}푛 +푓 (z) · 푔(x + z), +January 13, 2023 +DRAFT + +JANUARY 2023 +8 +where the ‘+’ operation in x + z above is over vectors in F푛 +2. +It is well-known (see [21]) that the Fourier transform +� +푓 ★ 푔(s) = �푓 (s) · �푔(s), +for any s ∈ {0, 1}푛. +III. MAIN RESULTS +A. Counting Constrained Codewords in Linear Codes +First, we work towards characterizing the number of constrained codewords in an arbitrary +linear code. Consider an [푛, 푘] linear code C. Suppose that we are interested in computing the +number of codewords c ∈ C each of which satisfies a certain property, which we call a constraint. +Let A ⊆ {0, 1}푛 denote the set of length-푛 words that respect the constraint. We let 푁(C; A) +denote the number of such constrained codewords in C. We can then write +푁(C; A) = +� +c∈C +1A(c). +(1) +Observe that the summation in (1) is over a set of size 2푘, which could be quite large, especially +when 푘 > 푛/2. Our interest is in obtaining insight into the summation above, by employing a +simple trick from the Fourier expansions of Boolean functions. +Our first observation is summarized below as a theorem. For a linear code C, we denote its +dual code by C⊥. +Theorem III.1. Given a linear code C of blocklength 푛 and a set A ⊆ {0, 1}푛, we have that +푁(C; A) = |C| · +� +s∈C⊥ +� +1A(s). +Proof. The proof is a straightforward application of Plancherel’s Theorem. Observe that +푁(C; A) = +� +c∈C +1A(c) += +� +x∈{0,1}푛 +1A(x) · 1C(x) +January 13, 2023 +DRAFT + +JANUARY 2023 +9 += 2푛 · +� +s∈{0,1}푛 +� +1A(s) · � +1C(s). +(2) +Now, we claim that +� +1C(s) = + + +|C| +2푛 , if s ∈ C⊥, +0, otherwise. +(3) +To prove the above claim, recall that +� +1C(s) = 1 +2푛 +� +x∈{0,1}푛 +1C(x) · (−1)x·s. +If s ∈ C⊥, it holds that x · s = 0, for all x ∈ C. Hence, in this case, we have that � +1C(s) = |C| +2푛 . +If s ∉ C⊥, this means that there is some x★ ∈ C such that x★ · s = 1. Further, it is true that +0 · s = 0, with 0 ∈ C, since C is a linear code. Hence, to any c ∈ C such that c · s = 0, we +can uniquely map the codeword c + x★ (where the summation is over F푛 +2), with the property that +�c + x★� · s = 1. Since this map is bijective, we obtain that the number of codewords x ∈ C such +that (−1)x·s = 1, equals the number of codewords x ∈ C such that (−1)x·s = −1, for s ∉ C⊥. +Therefore, we get that in this case, � +1C(s) = 0. +Plugging (3) back in (2), we see that +푁(C; A) = |C| · +� +s∈C⊥ +� +1A(s). +□ +Theorem III.1 provides an alternative approach to addressing our problem of counting con- +strained codewords in linear codes. In particular, note that if C had large dimension, i.e., if +푘 > 푛/2, then, it is computationally less intensive to calculate the number of constrained code- +words using Theorem III.1, provided we knew the Fourier coefficients � +1A(s), since dim (C⊥) = +푛 − 푘 < 푛/2, in this case. Additionally, if the structure of the Fourier coefficients is simple to +handle, we could also use Theorem III.1 to construct linear codes that have a large (or small) +number of constrained codewords, or to obtain estimates of the number of constrained codewords +in a fixed linear code. In the rest of the paper, we shall study various examples of constraints, +January 13, 2023 +DRAFT + +JANUARY 2023 +10 +which in turn correspond to sets A, whose Fourier coefficients � +1A(s) are computable either +analytically or numerically. +In Appendices A and B, we provide a Fourier-analytic perspective on the problem of calcu- +lating the weight distribution of constrained sequences in the ambient space F푛 +2 and in linear +codes C. Appendix C discusses the connection between Theorem III.1 and the well-known +MacWilliams’ identities for linear codes [22]. +In Section IV, we shall look at specific examples of constraints and apply Theorem III.1 above. +In particular, as recurring motifs, we shall consider the [2푚−1, 2푚−1−푚] binary Hamming code, +for 푚 ≥ 1 and the binary Reed-Muller codes. Since the constraints we work with are sensitive +to the ordering of coordinates of the code, in the sense that a permutation of the coordinates can +transform a codeword that does not satisfy the constraint into one that does, we shall first fix a +canonical ordering of coordinates for the codes that we analyze. For the binary Hamming code, +we assume that a parity-check matrix 퐻Ham is such that 퐻Ham[푖] = B푚(푖), for 1 ≤ 푖 ≤ 2푚 − 1. +Note that the Reed-Muller (RM) family of codes are codes of large blocklength, which are +known to achieve the capacities of BMS channels under bit-MAP decoding [8] (see also [23]). +RM codes are hence linear codes that offer the maximum resilience to symmetric, stochas- +tic noise, for a given rate. For 푚 ≥ 1 and 푟 ≤ 푚, the 푟th-order binary Reed-Muller code +RM(푚, 푟) is the set of binary vectors obtained as evaluations of multilinear Boolean polynomials +푓 (푥1, . . . , 푥푚), in the variables 푥1, . . . , 푥푚, of maximum degree 푟, on points of the unit hypercube +(see Chapter 13 of [24] for more information on Reed-Muller codes). We use the convention +that the coordinates of RM(푚, 푟) are written as binary 푚-tuples that are ordered according +to the standard lexicographic ordering, i.e., the 푖th coordinate from the start is the 푚-tuple +B푚(푖 − 1), for 1 ≤ 푖 ≤ 2푚. We thus have that the blocklength of RM(푚, 푟) is 푛 = 2푚 and +dim(RM(푚, 푟)) = �푟 +푖=0 +�푚 +푖 +� =: � 푚 +≤푟 +�. +B. A Linear Program for Constrained Systems +In this section, we consider the problem of upper bounding the sizes of constrained codes +with a prescribed minimum distance. In particular, we present a linear program (LP) to upper +bound 퐴(푛, 푑; A), for any A ⊆ {0, 1}푛. This LP is based on Delsarte’s linear programming +January 13, 2023 +DRAFT + +JANUARY 2023 +11 +approach [18] to bounding from above the value of 퐴(푛, 푑), for 푛 ≥ 1 and 1 ≤ 푑 ≤ 푛. We first +recall Delsarte’s LP2, which we call Del(푛, 푑). Given an LP L, we denote by val(L) its optimal +value, and for any feasible solution 푓 of L, we denote the value of the objective function of L +evaluated at 푓 as valL( 푓 ). The subscript will be omitted when the LP being referred to is clear +from the context. We remark here that the LPs in this paper can return non-integral optimal +values, and that integer upper bounds on the sizes of codes can be obtained by suitable rounding +of real numbers. +Del(푛, 푑) +maximize +푓 : {0,1}푛→R +� +x∈{0,1}푛 +푓 (x) +(Obj) +subject to: +푓 (x) ≥ 0, ∀ x ∈ {0, 1}푛, +(C1) +�푓 (s) ≥ 0, ∀ s ∈ {0, 1}푛, +(C2) +푓 (x) = 0, if 1 ≤ 푤(x) ≤ 푑 − 1, +(C3) +푓 (0푛) = 1. +(C4) +Now, for any block code C of blocklength 푛 and minimum distance at least 푑, let 1C denote its +indicator function. Let us define3 +푓C = 2푛 +|C|1C ★ 1C. +We claim that 푓C is a feasible solution for Del(푛, 푑), with the objective function (Obj) evaluating +to |C|. Indeed, observe that (C1) is trivially satisfied, by the definition of the convolution operator. +Further, since � +푓C = 2푛 +|C| · � +1C +2, it holds that (C2) is satisfied as well. Next, note that since C is such +2The version of Delsarte’s LP that is most often used in papers in coding theory, such as in [25], is obtained after symmetrizing +Del(푛, 푑). In particular, the common version of Delsarte’s LP is Del/푆푛 (푛, 푑) (see the remark following Theorem III.2), where +푆푛 is the symmetry group on 푛 elements. +3Note that when C is a linear code (or a subspace of F푛 +2), for all x, z ∈ {0, 1}푛, we have that 1C(z)1C (x+z) = 1C(z)1C (x), +and hence 푓C evaluates to simply 1C. +January 13, 2023 +DRAFT + +JANUARY 2023 +12 +that 푑(C) ≥ 푑, it holds that 1C(x + z) = 0, for all z ∈ C and any x such that 1 ≤ 푤(x) ≤ 푑 − 1. +In other words, we have that 1C(z) · 1C(x + z) = 0, for all z ∈ {0, 1}푛, if 1 ≤ 푤(x) ≤ 푑 − 1. +Hence, (C3) is also true. Finally, observe that +푓C(0푛) = 1 +|C| +� +z∈{0,1}푛 +1C(z) · 1C(z) += 1 +|C| +� +z∈{0,1}푛 +1C(z) = 1, +thereby satisfying (C4) also. Now, it holds that the objective value of 푓C, which is � +x∈{0,1}푛 푓C(x), +is given by +� +x∈{0,1}푛 +푓C(x) = 1 +|C| +� +x∈{0,1}푛 +� +z∈{0,1}푛 +1C(z) · 1C(x + z) += 1 +|C| +� +z∈{0,1}푛 +1C(z) +� +x∈{0,1}푛 +1C(x + z) += |C|2 +|C| = |C|. +(4) +Hence, it holds that the optimal value of Del(푛, 푑), is an upper bound on |C|, for all block +codes C of blocklength 푛 and minimum distance at least 푑, and therefore also an upper bound +on 퐴(푛, 푑). We refer the reader to [18], [19], [25]–[28] and the references therein for a more +detailed treatment of linear programming-based upper bounds on the sizes of block codes and +linear codes, and for the derivation of analytical upper bounds via the dual LP or using modern +Fourier-theoretic or expander graph-based arguments. +Our LP, which we call Del(푛, 푑; A), is but a small modification of Del(푛, 푑), to take into +account the fact that all codewords of the code of minimum distance at least 푑, whose size we +are attempting to bound, must also lie in the set A ⊆ F푛 +2. +January 13, 2023 +DRAFT + +JANUARY 2023 +13 +Del(푛, 푑; A) +maximize +푓 : {0,1}푛→R +� +x∈{0,1}푛 +푓 (x) +(Obj′) +subject to: +푓 (x) ≥ 0, ∀ x ∈ {0, 1}푛, +(D1) +�푓 (s) ≥ 0, ∀ s ∈ {0, 1}푛, +(D2) +푓 (x) = 0, if 1 ≤ 푤(x) ≤ 푑 − 1, +(D3) +푓 (0푛) ≤ val(Del(푛, 푑)), +(D4) +푓 (x) ≤ 2푛 · (1A ★ 1A)(x), ∀ x ∈ {0, 1}푛. (D5) +Like in the case with Del(푛, 푑), we produce a feasible solution for Del(푛, 푑; A), and claim that +val(Del(푛, 푑; A)) is at least (퐴(푛, 푑; A))2 (note the difference with Del(푛, 푑), whose value is +at least 퐴(푛, 푑)). To this end, let CA be any length-푛 constrained code, with 푑(CA) ≥ 푑, such +that all codewords in CA lie in A. Observe that we can write CA as C ∩ A, for some block +(not necessarily constrained) code C, with 푑(C) ≥ 푑. Thus, an upper bound on +max +C: 푑(C)≥푑 |C ∩A| +serves as an upper bound on (and in fact, equals) 퐴(푛, 푑; A). +Let 1C be the indicator function of a block code C as above, and let 1A be the indicator +function of the constraint. We define +푓C,A = 2푛 · (1C1A ★ 1C1A), +and claim that 푓C,A is a feasible solution for Del(푛, 푑; A), with the objective function (Obj′) +evaluating to |C ∩ A|2. To see this, note that the LP constraints (D1)–(D3) are satisfied for the +same reasons as why 푓C satisfied (C1)–(C3) in Del(푛, 푑). Furthermore, +푓C,A(0푛) = +� +z∈{0,1}푛 +1C∩A(z) +January 13, 2023 +DRAFT + +JANUARY 2023 +14 += |C ∩ A| ≤ |C| ≤ val(Del(푛, 푑)), +since C is a block code of distance at least 푑. Hence, (D4) is satisfied by 푓C,A. Finally, observe +that for any x ∈ {0, 1}푛, +푓C,A(x) = +� +z∈{0,1}푛 +1C1A(z) · 1C(x + z)1A(x + z) +≤ +� +z∈{0,1}푛 +1A(z) · 1A(x + z) = 2푛 · (1A ★ 1A)(x), +showing that (D5) also holds. Now, note that val( 푓C,A), which is � +x∈{0,1}푛 푓C,A(x), equals +|C ∩ A|2, by calculations as in (4). Hence, we have that val(Del(푛, 푑; A)) ≥ |C ∩ A|2, for any +C with minimum distance at least 푑. Therefore, it holds that val(Del(푛, 푑; A)) ≥ (퐴(푛, 푑; A))2, +or, that 퐴(푛, 푑; A) ≤ (val(Del(푛, 푑; A)))1/2. +We now discuss a couple of observations about Del(푛, 푑; A). Firstly, note that by (D5), we +have that for any feasible solution 푓 of Del(푛, 푑; A), it holds that the objective value +� +x∈{0,1}푛 +푓 (x) ≤ +� +x∈{0,1}푛 +� +z∈{0,1}푛 +1A(z) · 1A(x + z) += +� +z∈{0,1}푛 +1A(z) +� +x∈{0,1}푛 +1A(x + z) = |A|2. +Therefore, we have that our upper bound that is (val(Del(푛, 푑; A)))1/2 is less than or equal to +|A|, which is the total number of constrained sequences of length 푛. +Second, from the proof of Proposition 1 in [28], we obtain that for any feasible solution 푓 of +Del(푛, 푑; A), for all x ∈ {0, 1}푛, it holds that 푓 (x) ≤ 푓 (0푛) ≤ val(Del(푛, 푑)). Hence, we have +that the objective value of 푓 , for any feasible 푓 , obeys +� +x∈{0,1}푛 +푓 (x) ≤ +�� 푛 +≥ 푑 +� ++ 1 +� +· val(Del(푛, 푑)), +since 푓 (x) = 0, if 1 ≤ wt(x) ≤ 푑 − 1, where we define � 푛 +≥푑 +� := �푛 +푖=푑 +�푛 +푖 +�. Therefore, we have +that (val(Del(푛, 푑; A)))1/2 ≤ +�� 푛 +≥푑 +� + 1 +�1/2 +· (val(Del(푛, 푑)))1/2. Moreover, for most constraints +that we ran either Del(푛, 푑; A) or the equivalent symmetrized LP Del/퐺 A (푛, 푑; A) (see Section +January 13, 2023 +DRAFT + +JANUARY 2023 +15 +III-C) on, the value of the LP came out to be strictly less than val(Del(푛, 푑)) (see also Corollary +III.1). +Next, we derive an upper bound on the optimal value val(Del(푛, 푑; A)) of our LP, in the +following lemma that is essentially a formulation of the dual LP. While we do not apply Lemma +III.1 elsewhere in this paper, we believe that it will serve useful in the derivation of asymptotic +(as the blocklength goes to infinity) upper bounds on the rate-distance tradeoff for constrained +codes. We abbreviate val(Del(푛, 푑)) as 푣 and val(Del(푛, 푑; A)) as 푣A. +Lemma III.1. Let 훽 : {0, 1}푛 → R be a function that satisfies �훽(s) ≥ 0, for all s ∈ {0, 1}푛, and +� +x 훽(x) = 1. Then, +푣A ≤ 2푛 · + +훽(0푛) · min{푣, |A|} + 2푛 · +� +x: 푤(x)≥푑 +훽(x) · (1A ★ 1A)(x) + +. +Proof. Consider any function 휆 : {0, 1}푛 → [0, ∞). Now, observe that for any feasible solution +푓 of Del(푛, 푑; A), we have that +휆(0푛) · �푓 (0푛) +(푎) +≤ +� +x∈{0,1}푛 +휆(x) · �푓 (x) +(푏)= 2푛 · +� +s∈{0,1}푛 +�휆(s) · 푓 (s) +2푛 +(푐)= +� +s: s = 0푛 or +푤(s)≥푑 +�휆(s) · 푓 (s) +(푑) +≤ �휆(0푛) · min{푣, |A|} + 2푛 · +� +s: 푤(s)≥푑 +�휆(s) · (1A ★ 1A)(s), +where (a) holds since 휆, �푓 ≥ 0, (b) holds by an application of Plancherel’s Theorem, along with +the fact that � +( �푓 ) = 2−푛 · 푓 . Next, (c) is true since 푓 satisfies (D3) and (d) holds since 푓 satisfies +(D4) and (D5), and since 2푛 · (1A ★ 1A)(0푛) = |A|. +Moreover, we have that val( 푓 ) = � +x∈{0,1}푛 푓 (x) = 2푛 · �푓 (0푛), and that 휆(0푛) = � +s∈{0,1}푛 �휆(s), +by the definition of the Fourier transform. Putting everything together, we obtain that for any +January 13, 2023 +DRAFT + +JANUARY 2023 +16 +feasible 푓 of Del(푛, 푑; A), it holds that +val( 푓 ) ≤ +2푛 +� +s �휆(s) +· + +�휆(0푛) · min{푣, |A|} + 2푛 · +� +s: 푤(s)≥푑 +�휆(s) · (1A ★ 1A)(s) + +. +Finally, by substituting 훽 as �휆 and ensuring that � +x �훽(x) = 1, we obtain the thesis of the +lemma. +□ +The following simple corollary then follows from Lemma III.1. +Corollary III.1. If |A| ≤ 2푛/2, it holds that 푣A ≤ 푣 + 1. +Proof. Note that in Lemma III.1 above, we can pick 훽(x) = 2−푛, for all x ∈ {0, 1}푛, so that +�훽(s) = 2−2푛, if s = 0푛 and �훽(s) = 0, for s ≠ 0푛. Then, we obtain that +푣A ≤ min{푣, |A|} + 2−푛 · +� +x: 푤(x)≥푑 +� +z +1A(z) · 1A(x + z) +≤ min{푣, |A|} + |A|2 +2푛 . +Hence, if |A| ≤ 2푛/2, we get that 푣A ≤ 푣 + 1. +□ +C. Symmetrizing Del(푛, 푑; A) +The linear program Del(푛, 푑; A) discussed in Section III-B, for a fixed A ⊆ F푛 +2, suffers from +the drawback that the variables, which are precisely the values ( 푓 (x) : x ∈ {0, 1}푛), are 2푛 in +number, i.e., exponentially large in the blocklength. The number of LP constraints, similarly, +are exponentially large in 푛. It would therefore be of interest to check if the size of the linear +program Del(푛, 푑; A), which is the sum of the number of variables and the number of LP +constraints, can be reduced, using symmetries present in the formulation. +Our exposition in this section on symmetrizing Del(푛, 푑; A), follows that in [28]. Let 푆푛 +denote the symmetry group on 푛 elements, which is the set of all permutations 휎 : [푛] → [푛]. +Note that given a length-푛 vector x = (푥1, . . . , 푥푛) ∈ {0, 1}푛, a permutation 휎 ∈ 푆푛 acts on x as +follows: +휎 · x = (푥휎(1), 푥휎(2), . . . , 푥휎(푛)). +January 13, 2023 +DRAFT + +JANUARY 2023 +17 +The permutation 휎 also acts on functions 푓 : {0, 1}푛 → R via the mapping (휎◦ 푓 )(x) = 푓 (휎·x), +for x ∈ {0, 1}푛. Now, given any set A ⊆ F푛 +2, we define the “symmetry group” of the constraint +represented by A to be the set of all permutations 휋 ∈ 푆푛 that leave the indicator function 1A +invariant. In other words, the symmetry group 퐺A of the constraint represented by A is the set +of all permutations 휋 ∈ 푆푛 such that 1A = 휋 ◦ 1A. +Given a group 퐺 ⊆ 푆푛 of permutations, which acts on the vectors x ∈ {0, 1}푛, we say that +Del(푛, 푑; A) is 퐺-invariant, if for all 휎 ∈ 퐺, it holds that if 푓 : {0, 1}푛 → R is a feasible +solution to Del(푛, 푑; A), then so is 휎 ◦ 푓 , with val( 푓 ) = val(휎 ◦ 푓 ). In what follows, we shall +prove that Del(푛, 푑; A) is, in fact, 퐺A-invariant. Let 휋 ∈ 퐺A be a permutation in the symmetry +group of A and let 푓 be some feasible solution to Del(푛, 푑; A). +(D1) It is clear that if 푓 (x) ≥ 0, then 푓 (휋 · x) ≥ 0, for all x ∈ {0, 1}푛. +(D2) The fact that if �푓 (s) ≥ 0, then it holds that � +휋 ◦ 푓 (s) ≥ 0, for all s ≥ 0, follows from the +simple lemma below: +Lemma III.2. For any function 푓 : {0, 1}푛 → R and for any permutation 휎 ∈ 푆푛, it holds +that +� +휎 ◦ 푓 (s) = (휎 ◦ �푓 )(s), +for all s ∈ {0, 1}푛. +Proof. Observe that +� +휎 ◦ 푓 (s) = +� +x∈{0,1}푛 +푓 (휎 · x) · (−1)x·s += +� +x∈{0,1}푛 +푓 (휎 · x) · (−1)(휎·x)·(휎·s) += +� +x∈{0,1}푛 +푓 (x) · (−1)x·(휎·s) = (휎 ◦ �푓 )(s). +□ +(D3) Since any permutation in 퐺A also lies in 푆푛 and hence preserves the weights of vectors in +{0, 1}푛, it holds that (휋 ◦ 푓 )(x) = 0, for all x ∈ {0, 1}푛 such that 1 ≤ 푤(x) ≤ 푑 − 1. +(D4) This constraint is also satisfied by 휋 ◦ 푓 , since 휋(0푛) = 0푛, for all 휋 ∈ 퐺A. +January 13, 2023 +DRAFT + +JANUARY 2023 +18 +(D5) Observe that for any 휋 ∈ 퐺A, +2푛 · (1A ★ 1A)(휋 · x) = +� +z∈{0,1}푛 +1A(z) · 1A(휋 · x + z) += +� +z∈{0,1}푛 +1A(휋 · z) · 1A(휋 · x + 휋 · z) += +� +z∈{0,1}푛 +1A(휋 · z) · 1A(휋 · (x + z)) += +� +z∈{0,1}푛 +1A(z) · 1A(x + z) = 2푛 · (1A ★ 1A)(x). +Hence, since for all x ∈ {0, 1}푛, we have that +휋 ◦ 푓 (x) ≤ 2푛 · (1A ★ 1A)(휋 · x) += 2푛 · (1A ★ 1A)(x), +we have that (D5) is also satisfied by 휋 ◦ 푓 . +(Obj′) It is clear that � +x 푓 (x) = � +x 푓 (휋 · x), and hence that val( 푓 ) = val(휋 ◦ 푓 ). +From the preceding discussion, we see that given a feasible solution 푓 to Del(푛, 푑; A), we can +construct the function +푓 := +1 +|퐺A| +� +휋∈퐺 A +휋 ◦ 푓 , +such that 푓 is also a feasible solution to the LP (by linearity), with val( 푓 ) = val( 푓 ). Observe, +in addition, that 푓 is such that 휋 ◦ 푓 = 푓 , for all 휋 ∈ 퐺A. Hence, it follows that in order to +arrive at an optimal solution to Del(푛, 푑; A), one can restrict oneself to searching among feasible +solutions 푓 that are constant on each orbit 푂 in {0, 1}푛/퐺A. Such functions 푓 can be expressed +as +푓 (x) = +� +푂∈{0,1}푛/퐺 A +푎푂 · 1푂(x), +(5) +where 푎푂 ∈ R, for all 푂 ∈ {0, 1}푛/퐺A. Before we work on symmetrizing the constraints of +Del(푛, 푑; A), we introduce some notation. For an orbit 푂 ∈ {0, 1}푛/퐺A, we denote by |푂| the +number of elements in the orbit and by x푂 (or s푂) a representative element of the orbit. Further, +January 13, 2023 +DRAFT + +JANUARY 2023 +19 +for a given element x ∈ {0, 1}푛, we define 푂(x) to be the orbit in which x lies. We shall now +formulate the constraints (D1)–(D5) and the objective function (Obj′) in Del(푛, 푑; A) based on +(5). +(D1′) The fact that 푓 (x) ≥ 0 for all x implies that 푎푂 ≥ 0, for all 푂 ∈ {0, 1}푛/퐺A. +(D2′) By the linearity of the Fourier transform operation, we obtain that +�푓 (s) = +� +푂∈{0,1}푛/퐺 A +푎푂 · � +1푂(s) ≥ 0, +for all s ∈ {0, 1}푛. +In fact, note that since 퐺A ⊆ 푆푛, it can be argued using Lemma III.2 that the above +inequality only needs to hold for orbit representatives s푂 ∈ {0, 1}푛, of 푂 ∈ {0, 1}푛/퐺A. +Indeed, we have that for any 휋 ∈ 퐺A, +�푓 (휋 · s) = � +휋 ◦ 푓 (s) += �푓 (s), +where the first equality holds by Lemma III.2 and the second holds since 휋 ◦ 푓 = 푓 , for +functions 푓 as in (5). +(D3′) The constraint (D3) can be written as +푎푂 = 0, for all 푂 such that 1 ≤ 푤(x푂) ≤ 푑 − 1, +where x푂 ∈ {0, 1}푛 is a representative element of the orbit 푂 ∈ {0, 1}푛/퐺A. +(D4′) The constraint (D4) becomes +푎푂(0푛) ≤ val(Del(푛, 푑)), +where 푂(0푛) is the orbit that contains the all-zeros word 0푛. +(D5′) Similarly, the constraint (D5) reduces to +푎푂 ≤ 2푛 · (1A ★ 1A)(x푂), +January 13, 2023 +DRAFT + +JANUARY 2023 +20 +where, again, x푂 is some representative element of the orbit 푂 ∈ {0, 1}푛/퐺A. +(Obj′′) From (5), we see that the new objective function simply becomes +maximize +푎푂∈R +� +푂∈{0,1}푛/퐺 A +|푂| · 푎푂. +We call the symmetrized version of Del(푛, 푑; A) as Del/퐺 A (푛, 푑; A), which is given below. +Del/퐺 A (푛, 푑; A) +maximize +{푎푂∈R: 푂∈{0,1}푛/퐺 A} +� +푂 +|푂| · 푎푂 +(Obj′′) +subject to: +푎푂 ≥ 0, ∀ 푂 ∈ {0, 1}푛/퐺A, +(D1′) +� +푂∈{0,1}푛/퐺 A +푎푂 · � +1푂(s ˜푂) ≥ 0, ∀ orbit rep. s ˜푂 ∈ {0, 1}푛, (D2′) +푎푂 = 0, if 1 ≤ 푤(x푂) ≤ 푑 − 1, +(D3′) +푎푂(0푛) ≤ val(Del(푛, 푑)), +(D4′) +푎푂 ≤ 2푛 · (1A ★ 1A)(x푂), ∀ 푂 ∈ {0, 1}푛/퐺A. +(D5′) +The preceding discussion can then be summarized as a theorem. +Theorem III.2. The LPs Del(푛, 푑; A) and Del/퐺 A (푛, 푑; A) are equivalent in that +val(Del(푛, 푑; A)) = val(Del/퐺 A (푛, 푑; A)). +Remark. All the above arguments remain valid if we use a subgroup 퐻 of the symmetry group +퐺A as well. For the special case when A = {0, 1}푛, it holds that 퐺A = 푆푛, and we then recover +the more common version of Delsarte’s LP that is 푀LP(푛, 푑) in [25]. It is this version that we +use for evaluating the right-hand side of constraints (D4) and (D4′) in our numerical examples. +Observe that in the symmetrized LP Del/퐺 A (푛, 푑; A), the number of variables is the number +January 13, 2023 +DRAFT + +JANUARY 2023 +21 +푁A of orbits 푂 ∈ {0, 1}푛/퐺A and the number of constraints is at most 4푁A + 1. Hence, if the +constraint is such that the number of orbits 푁A induced by its symmetry group is small (as a +function of the blocklength 푛), then the size of the symmetrized LP is small. In the subsections +that follow, we shall explicitly write down Del/퐺 A (푛, 푑; A), for select constraints (or sets A), +and provide numerical results obtained by running Del/퐺 A (푛, 푑; A) on those constraints. +IV. EXAMPLES +We now take up specific examples of constrained sequences and apply Theorem III.1 and the +LP discussed in Section III. +A. 2-Charge Constraint +In this subsection, we work with a special kind of a spectral null constraint [29], [30]. This +constraint that we shall study is the so-called 2-charge constraint (see Section 1.5.4 in [1]), +whose sequences have a spectral null at zero frequency (such a constraint is also called a DC- +free constraint). The 2-charge constraint admits only sequences y ∈ {−1, +1}푛, whose running +sum �푟 +푖=1 푦푖, for any 1 ≤ 푟 ≤ 푛, obeys 0 ≤ �푟 +푖=1 푦푖 ≤ 2. The graph in Figure 1 below represents +the constraint, in that all sequences y ∈ {−1, +1}푛 that are 2-charge constrained can be read off +the labels of edges of this graph. The nodes (or states) of the graph represent the values that +�푟 +푖=1 푦푖 can take, for any 1 ≤ 푟 ≤ 푛. We take the initial state to be the 0 state. +0 +1 +2 ++1 ++1 +−1 +−1 +Fig. 1: State transition graph representing the 2-charge constraint +To any sequence x ∈ {0, 1}푛, we map (in a one-one manner) the sequence y = ((−1)푥1, . . . , (−1)푥푛) ∈ +{−1, +1}푛. We let 푆(푛) +2 +denote the set of sequences x ∈ {0, 1}푛 such that y = ((−1)푥1, (−1)푥2, . . . , (−1)푥푛) +is 2-charge constrained. We drop the superscript ‘(푛)’ when clear from the context. The set of +constrained sequences of interest to us, hence, is A = 푆2. Figure 2 shows a state transition graph +for sequences in the set 푆2. +January 13, 2023 +DRAFT + +JANUARY 2023 +22 +0 +1 +2 +0 +0 +1 +1 +Fig. 2: State transition graph for sequences in the set 푆2. +Let the labelled, directed graph 퐺 = (푉, 퐸, L) represent the state transition graph above, +with 푉 = {0, 1, 2} being the set of states, 퐸 ⊆ 푉 × 푉 being the set of directed edges, and +L : 퐸 → {0, 1} being the labelling function that assigns to each edge a label that is 0 or +1. For example, L((1, 2)) = 0 and L((1, 0)) = 1. We assume that the initial state is 푣0 = 0. +Since labels of paths in the state transition graph (beginning at state 0) correspond to binary +sequences x ∈ 푆2, we denote by 푥푖 the label of the 푖th edge in the path. Observe that 푥1 = 0, by +our choice of initial state. Further, for a given path in the graph, we let 푣푖 denote the 푖th state, +which is the terminal state of the 푖th edge. Hence, given such a state transition graph, where the +labels of distinct outgoing edges from a state are different, one can define a transition function +휙 : 푉 × {0, 1} → 푉, with the property that 푣푖 = 휙(푣푖−1, 푥푖), for 1 ≤ 푖 ≤ 푛. +Now, observe that for any x ∈ 푆2, it holds that the state 푣2푖−1, for any 1 ≤ 푖 ≤ 푛, equals 1. +Owing to this fact, the label of the 푗th edge, 푥 푗, in any path in the graph 퐺 in Figure 2, can be +either 0 or 1, when 푗 = 2푖, and is fixed to be exactly one of 0 or 1, when 푗 = 2푖 + 1, based on +the label of the ( 푗 − 1)th edge, for 1 ≤ 푗 ≤ 푛. In particular, it holds that 푥2푖 + 푥2푖+1 = 1 (over the +reals), for all 1 ≤ 푖 ≤ +� 푛 +2 +� +, if 푛 is odd, and for all 1 ≤ 푖 ≤ 푛 +2 − 1, if 푛 is even, with 푥1 fixed to be +0 in both cases. From this observation, we see that for blocklength 푛, it holds that |푆2| = 2⌊ 푛 +2⌋. +We now state a lemma that completely determines the Fourier transform of 1푆2. But before +we do so, we need some more notation: we define the set of vectors B = +� +b0, b1, . . . , b⌈ 푛 +2⌉−1 +� +, +where b0 = 10푛−1 and for 1 ≤ 푖 ≤ +�푛 +2 +� +− 1, the vector b푖 is such that 푏푖,푗 = 1, for 푗 ∈ {2푖, 2푖 + 1}, +and 푏푖,푗 = 0, otherwise. For example, when 푛 = 5, we have that B = {10000, 01100, 00011}. Let +푉B = span(B). +Lemma IV.1. For a = +� +푎0, 푎1, . . . , 푎⌈ 푛 +2⌉−1 +� +∈ {0, 1}⌈ 푛 +2⌉, consider s = +⌈ 푛 +2⌉−1 +� +푖=0 +푎푖 · b푖 (where the +January 13, 2023 +DRAFT + +JANUARY 2023 +23 +summation is over F푛 +2). It holds that +� +1푆2 (s) = 2⌊ 푛 +2⌋−푛 · (−1)푤(a)−푎0. +Further, for s ∉ 푉B, we have that � +1푆2 (s) = 0. +Proof. First, we note that for any s ∈ {0, 1}푛, +� +1푆2(s) = 1 +2푛 +� +x∈{0,1}푛 +1푆2(x) · (−1)x·s += 2−푛 · (#{x ∈ 푆2 : 푤s(x) is even} − #{x ∈ 푆2 : 푤s(x) is odd}) . +(6) +Now, for s = b0, note that since all words x ∈ 푆2 have 푥1 = 0, we have that #{x ∈ 푆2 : +푤s(x) is even} = |푆2| = 2⌊ 푛 +2⌋, and #{x ∈ 푆2 : 푤s(x) is odd} = 0. Plugging back in (6), we get +that � +1푆2(b0) = 2⌊ 푛 +2⌋−푛. +Further, recall that since 푣2푖−1 = 1, for any 1 ≤ 푖 ≤ 푛, we have that 푥2푖 + 푥2푖+1 = 1 (over F2). +Hence, we see that for s = b푗, for 1 ≤ 푗 ≤ +�푛 +2 +� +−1, it is true that #{x ∈ 푆2 : 푤s(x) is odd} = |푆2| = +2⌊ 푛 +2⌋ and #{x ∈ 푆2 : 푤s(x) is even} = 0. Substituting in (6), we get that � +1푆2(b푗) = −2⌊ 푛 +2⌋−푛 for +all 1 ≤ 푗 ≤ +�푛 +2 +� +− 1. Furthermore, we claim that � +1푆2(0푛) = 2⌊ 푛 +2⌋−푛. To see this, note that +� +1푆2(0푛) = 1 +2푛 +� +x∈{0,1}푛 +1푆2(x) · (−1)x·0푛 += 1 +2푛 +� +x∈푆2 +1 = |푆2| +2푛 = 2⌊ 푛 +2⌋−푛. +Now, suppose that for some s1, s2 ∈ 푉B, it holds that � +1푆2(s1) = (−1)푖1 · 2⌊ 푛 +2⌋−푛 and � +1푆2(s2) = +(−1)푖2 · 2⌊ 푛 +2⌋−푛, for some 푖1, 푖2 ∈ {0, 1}. From the arguments above, we can deduce that for all +x ∈ 푆2, it holds that 푤s1(x) is even, if 푖1 = 0, and for all x ∈ 푆2, we have that 푤s1(x) is odd, if +푖1 = 1. Similar arguments hold for 푤s2(x) as well, for all x ∈ 푆2. Hence, it can be checked that if +푖1 = 푖2, it holds that 푤s1+s2(x) is even, and hence, � +1푆2(s1 + s2) = 2⌊ 푛 +2⌋−푛 = (−1)푖1+푖2 · 2⌊ 푛 +2⌋−푛, and +if 푖1 ≠ 푖2, it holds that 푤s1+s2(x) is odd, and hence, � +1푆2(s1 + s2) = −2⌊ 푛 +2⌋−푛 = (−1)푖1+푖2 · 2⌊ 푛 +2⌋−푛. +By applying this fact iteratively, and using the expressions for the Fourier coefficients � +1푆2(b푗), +for 0 ≤ 푗 ≤ +�푛 +2 +� +− 1, we obtain the first part of the lemma. +January 13, 2023 +DRAFT + +JANUARY 2023 +24 +To show that � +1푆2 (s) = 0 for s ∉ 푉B, we use Plancherel’s Theorem again. Note that +|푆2| +2푛 +(푎)= 1 +2푛 +� +x∈{0,1}푛 +1푆2(x) +(푏)= 1 +2푛 +� +x∈{0,1}푛 +12 +푆2(x) +(푐)= +� +s∈{0,1}푛 +� +� +1푆2(s) +�2 +(푑)= +� +s∈푉B +� +� +1푆2(s) +�2 ++ +� +s∉푉B +� +� +1푆2(s) +�2 +, +where (b) holds since 1푆2 is a boolean function, and (c) holds by Plancherel’s Theorem. Now, +by equating the left side of equality (a) and the right side of equality (d), we see that +|푆2| +2푛 = +� +s∈푉B +� +� +1푆2(s) +�2 ++ +� +s∉푉B +� +� +1푆2(s) +�2 +. +(7) +However, from the first part of the lemma, we get that +� +s∈푉B +� +� +1푆2(s) +�2 += |푉B| · 22·(⌊ 푛 +2⌋−푛) +(푒)= 2⌈ 푛 +2⌉ · 22·(⌊ 푛 +2⌋−푛) += + + +2− 푛 +2 , if 푛 is even, +2−( 푛+1 +2 ), if 푛 is odd += |푆2| +2푛 , +where equality (e) follows from the fact that |푉B| = 2⌈ 푛 +2⌉, since 푉B = span(B) and the vectors +in B are linearly independent. Hence, plugging back in (7), we obtain that � +s∉푉B +� +� +1푆2(s) +�2 += 0, +implying that � +1푆2(s) = 0, for all s ∉ 푉B. +□ +Lemma IV.1 informs the construction of linear codes C that have a large number of codewords +c ∈ 푆2. In particular, note that from Theorem III.1, we have that +푁(C; 푆2) = |C| · +� +s∈C⊥ +� +1푆2(s) +January 13, 2023 +DRAFT + +JANUARY 2023 +25 += |C| · +� +s∈C⊥∩푉B +� +1푆2(s), +(8) +where, for s ∈ C⊥ ∩ 푉B, with s = �⌈ 푛 +2⌉−1 +푖=0 +푎푖 · b푖, for some a = +� +푎0, 푎1, . . . , 푎⌈ 푛 +2⌉−1 +� +∈ {0, 1}⌈ 푛 +2⌉, +we have that � +1푆2(s) = 2⌊ 푛 +2⌋−푛 · +� +(−1) +�⌈ 푛 +2⌉−1 +푗=1 +푎 푗 +� +. Now, suppose that C is such that C⊥ does not +satisfy the criterion (C) below: +(C) +For all s ∈ C⊥ ∩ 푉B, it holds that � +1푆2(s) ≥ 0. +If (C) does not hold, then, it implies that for some s★ ∈ C⊥ ∩ 푉B, it holds that � +1푆2(s★) < 0. +Hence, following the reasoning in the proof of Lemma IV.1, since C⊥ ∩푉B is a vector space, we +have that via the map s ↦→ s+s★, the number of elements s ∈ C⊥∩푉B such that � +1푆2(s) < 0 equals +the number of elements s ∈ C⊥ ∩푉B such that � +1푆2(s) > 0. Furthermore, since +��� � +1푆2(s) +��� = 2⌊ 푛 +2⌋−푛, +for all s ∈ C⊥ ∩ 푉B, we get from (8) that 푁(C; 푆2) = 0, in this case. +Hence, in order to construct linear codes C such that 푁(C, 푆2) > 0, we require that criterion +(C) is indeed satisfied by the dual code C⊥ of C, with � +1푆2(s★) > 0, for some s★ ∈ C⊥. With +this instruction in mind, we can construct linear codes C such that its dual code C⊥ contains +푡 linearly independent vectors (s1, . . . , s푡) with � +1푆2(s푖) > 0, for all 1 ≤ 푖 ≤ 푡, and no vectors +s ∈ 푉B with � +1푆2(s) < 0. In such a case, we obtain that +푁(C; 푆2) = |C| · 2푡+⌊ 푛 +2⌋−푛. +From the structure of 푉B, we see that the largest number of vectors s ∈ {0, 1}푛 such that +� +1푆2(s) > 0, equals |푉B| +2 += 2⌈ 푛 +2⌉−1. Hence, the largest number of linearly independent vectors 푡 as +above, is +�푛 +2 +� +− 1. The discussion above is summarized below as a lemma. +Lemma IV.2. For any linear code C of blocklength 푛 ≥ 1, the following are true: +1) If criterion (C) is not satisfied, then, 푁(C, 푆2) = 0. +2) If criterion (C) is satisfied and there exist 푡푛 ∈ +� +1 : +�푛 +2 +� +− 1 +� +linearly independent vectors +�s1, . . . , s푡푛 +� in C⊥ with � +1푆2(s푖) > 0, for all 1 ≤ 푖 ≤ 푡푛, then, 푁(C; 푆2) = |C| · 2푡푛+⌊ 푛 +2⌋−푛. +We thus understand that given a linear code whose dual code satisfies item 2 of Lemma IV.2, +January 13, 2023 +DRAFT + +JANUARY 2023 +26 +the rate of the largest constrained subcode, C2, of C, all of whose codewords are in 푆2, obeys +rate (C2) = log2 푁(C; 푆2) +푛 += +log2 +� +|C| · 2푡푛+⌊ 푛 +2⌋−푛� +푛 += log2 (|C|) +푛 ++ 푡푛 + +� 푛 +2 +� +− 푛 +푛 +. +In particular, given a sequence of linear codes +� +C(푛)� +푛≥1 satisfying item 2 of Lemma IV.2, if +it holds that rate(C(푛)) +푛→∞ +−−−−→ 푅 ∈ (0, 1), then, the rate of their largest constrained subcodes +� +C(푛) +2 +� +푛≥1, all of whose codewords are in 푆2, obeys +lim inf +푛→∞ rate +� +C(푛) +2 +� += 푅 − 1 +2 + lim inf +푛→∞ +푡푛 +푛 . +(9) +From [31]4, we observe that for the constraint identified by the set 푆2, there exist cosets of the +linear codes +� +C(푛)� +푛≥1 with rate(C(푛)) +푛→∞ +−−−−→ 푅, the rate of the constrained subcodes of which (in +the limit as the blocklength goes to infinity) is at least 푅 − 1 +2. From (9), since 푡푛 ∈ +� +1 : +�푛 +2 +� +− 1 +� +, +we see that we can construct a sequence of linear codes whose constrained subcodes are of rate +larger than or equal to the coset-averaging lower bound in [31]. In other words, it is possible +to achieve the coset-averaging rate lower bound (and potentially more) by using the linear code +itself, instead of one of its cosets. +Specifically, suppose that we choose 푡푛 = +�푛 +2 +� +− 푝푛, for some positive integer 푝푛 such that +lim푛→∞ +푝푛 +푛 = 0, thereby making dim�C⊥ +푛 +� ≥ ⌈ 푛 +2⌉−푝푛 +푛 +, where C⊥ +푛 is the dual code of C푛. Note +that this implies that 1 − 푅 = lim푛→∞ rate (C⊥) ≥ 1 +2, and hence that 푅 ∈ (0, 1 +2]. In this case, +by plugging into (9), we obtain that the rate of the largest constrained subcodes +� +C(푛) +2 +� +푛≥1 of +� +C(푛)� +푛≥1 is +lim +푛→∞ rate +� +C(푛) +2 +� += 푅 − 1 +2 + lim +푛→∞ +푡푛 +푛 += 푅. +4Such a result is also attributed to Elias and Bassalygo. +January 13, 2023 +DRAFT + +JANUARY 2023 +27 +In other words, in the case where 푡푛 = +�푛 +2 +� +− 푝푛, for 푝푛 > 0 as above, the asymptotic rate of the +codewords that lie in 푆2 equals the asymptotic rate 푅 ∈ (0, 1 +2] of the code itself. +Next, we shall make use of Theorem III.1 to compute the number of codewords of specific +linear codes C, which lie in 푆2. We first consider the binary single parity-check code. +Corollary IV.1. For C being the [푛, 푛 − 1] single parity-check code, we have that +푁(C; 푆2) = + + +2⌊ 푛 +2⌋−1, if 푛 is even, +2⌊ 푛 +2⌋, if 푛 = 4푧 + 1, for some non-negative integer 푧, +0, otherwise. +Proof. Observe that if C is the [푛, 푛 − 1] single parity-check code, then C⊥ is the [푛, 1] binary +repetition code. Consider the case when 푛 is even. We shall plug in the Fourier coefficients given +in Lemma IV.1 in (8). Note that in this case, C⊥ ∩ 푉B = 0푛. Hence, +푁(C; 푆2) = |C| · � +1푆2(0푛) += 2푛−1 · 2⌊ 푛 +2⌋−푛 = 2⌊ 푛 +2⌋−1, +where the second equality applies Lemma IV.1. +Similarly, when 푛 = 4푧 +1, for some non-negative integer 푧, it can be verified that the all-ones +vector 1푛 equals �⌈ 푛 +2⌉−1 +푖=0 +b푖 = �⌈ 푛 +2⌉−1 +푖=0 +푎푖 · b푖, where 푎푖 = 1, for all 0 ≤ 푖 ≤ +�푛 +2 +� +− 1. From Lemma +IV.1, it can be checked that since 푛 = 4푧 +1, we have that � +1푆2(1푛) = 2⌊ 푛 +2⌋−푛. Hence, in this case, +we get that +푁(C; 푆2) = |C| · +� +� +1푆2(0푛) + � +1푆2(1푛) +� += 2⌊ 푛 +2⌋. +Finally, when 푛 = 4푧 +3, for some non-negative integer 푧, by arguments as before, we can check +that the all-ones vector 1푛 indeed belongs to 푉B, with � +1푆2(1푛) = −2⌊ 푛 +2⌋−푛. Hence, in this case, +we obtain from the discussion preceding Lemma IV.2 that 푁(C; 푆2) = 0. +□ +January 13, 2023 +DRAFT + +JANUARY 2023 +28 +In other words, the lemma above identifies the number of words in 푆2 that are of even weight, +for different values of the blocklength 푛. +Next, we shall apply our results to the [2푚 −1, 2푚 −1 − 푚] binary Hamming code, for 푚 ≥ 3. +We shall use the coordinate ordering discussed in Section III-A. +Corollary IV.2. For 푚 ≥ 3 and for C being the [2푚 − 1, 2푚 − 1 − 푚] Hamming code, we have +that 푁(C; 푆2) = 2 +� +2푚−1 +2 +� +−1. +Proof. The dual code C⊥ of the [2푚 − 1, 2푚 − 1 − 푚] Hamming code is the [2푚 − 1, 푚] simplex +code, all of whose non-zero codewords (i.e., codewords that are not equal to 02푚−1) are of weight +2푚−1. Further, a generator matrix of the simplex code under consideration is 퐻Ham. Now, let the +columns of 퐻Ham be indexed by 푚-tuples (푥1, . . . , 푥푚) ∈ {0, 1}푚 \ {0푚}, ordered in the standard +lexicographic order, i.e., the 푖th column of 퐺 is indexed as B푚(푖), for 1 ≤ 푖 ≤ 2푚 − 1. It is +well-known (see, for example, Section 1.10 of [32]) that the 푗th row 퐻Ham( 푗) is the evaluation +vector, over the 푚-tuples indexing the columns, of the monomial 푥 푗, for 1 ≤ 푗 ≤ 푚. We write +this row as Eval\0(푥 푗). +Consider the first 푚 − 1 rows of 퐻Ham, which are the evaluation vectors Eval\0(푥 푗), for +1 ≤ 푗 ≤ 푚 − 1. It can be checked that the Hamming weight, 2푚−1, of any of these rows is a +multiple of 4, when 푚 ≥ 3. Moreover, in any of these rows, if the entry corresponding to the +evaluation point (푥1, . . . , 푥푚−1, 0) equals 1, then so does the entry corresponding to the evaluation +point (푥1, . . . , 푥푚−1, 1). The above two facts imply that each of the first 푚 −1 rows of 퐻Ham can +be written as a linear combination of an even number of vectors bℓ ∈ B, for ℓ ∈ [1 : +�2푚−1 +2 +� +−1]. +Hence, from Lemma IV.1, it holds that the Fourier coefficient � +1푆2 +� +Eval\0(푥 푗) +� += 2 +� +2푚−1 +2 +� +−(2푚−1), +for all 1 ≤ 푗 ≤ 푚 − 1. Furthermore, observe that the above arguments also hold for any linear +combination of the first 푚 − 1 rows of 퐻Ham, i.e., it holds that � +1푆2 +� +Eval\0(s) +� += 2 +� +2푚−1 +2 +� +−(2푚−1), +where s = �푚 +푗=2 푐 푗 · Eval\0(푥 푗), for 푐 푗 ∈ {0, 1}, 푗 ∈ [2 : 푚]. +It can also be seen that since Eval\0(푥푚) ∉ 푉B, we have that � +1푆2 +� +Eval\0(푥푚) +� += 0, and +similarly, that � +1푆2 +� +Eval\0(s) +� += 0, where s = Eval(푥1) + �푚−1 +푗=1 푐 푗 · Eval(푥 푗), for 푐 푗 ∈ {0, 1}, +푗 ∈ [푚 − 1]. Putting everything together, we observe that for half of the codewords s ∈ C⊥, +January 13, 2023 +DRAFT + +JANUARY 2023 +29 +(푚, 푟) +푁(RM(푚, 푟); 푆2) +(4, 2) +16 +(4, 3) +128 +(5, 3) +2048 +(6, 4) +6.711 × 107 +(7, 5) +1.441 × 1017 +(8, 6) +1.329 × 1036 +TABLE I: Table of values of 푁(RM(푚, 푟); 푆2), for select parameters 푚 and 푟 +the Fourier coefficient � +1푆2(s) equals 2 +� +2푚−1 +2 +� +−(2푚−1), and for another half of the codewords, the +Fourier coefficient � +1푆2(s) equals zero. Applying (8), we get that +푁(C; 푆2) = |C| · +� +s∈C⊥ +� +1푆2(s) += 22푚−1−푚 · 2푚−1 · 2 +� +2푚−1 +2 +� +−(2푚−1) = 2 +� +2푚−1 +2 +� +−1, +where the second inequality holds since |C| = 22푚−1−푚 and |C⊥| = 2푚, and half the codewords +s ∈ C⊥ have nonzero Fourier coefficient � +1푆2(s). +□ +Note that in Corollary IV.2 and in Corollary IV.1, when 푛 is even, the number of constrained +codewords in the linear codes are half the total number of constrained codewords, 2⌊ 푛 +2⌋, of the +same blocklength 푛 as the codes under consideration. However, in the limit as the blocklength +goes to infinity, the rates of the subcodes of the single parity-check and Hamming codes that lie +in 푆2, equal the noiseless capacity 퐶0(푆2) of the constraint, which in turn equals 1 +2. +We then move on to counting constrained codewords in the Reed-Muller (RM) family of +codes. Using the structure of Fourier coefficients given in Lemma IV.1 and using the fact that +the dual code of RM(푚, 푟) is the code RM(푚, 푚 −푟 −1), for 푟 ≤ 푚 −1, we numerically calculate +the number of constrained codewords 푁(RM(푚, 푟); 푆2), for certain (large) values of 푚 and 푟. +Our results are documented in Table I. Note that the computational technique in Theorem III.1 +proves particularly useful when the rate of RM(푚, 푟) is larger than 1 +2, or equivalently, when +푟 > +� 푚 +2 +� +. +Next, we shall work towards obtaining bounds on the sizes of constrained codes that are +January 13, 2023 +DRAFT + +JANUARY 2023 +30 +subsets of 푆2, of minimum distance at least 푑. In other words, we are interested in formulating +the symmetrized LP Del/퐺푆2 (푛, 푑; 푆2). In what follows, we fix the blocklength 푛 to be odd. +Slight modifications of the construction of the symmetry group 퐺푆2 and the identification of the +orbits, below, yield Del/퐺푆2 (푛, 푑; 푆2), when 푛 is even. +Now, consider the following permutations, where 푛 is odd: +1) For even indices 푖 ∈ [푛], define 휋adj +푖 +: [푛] → [푛], such that 휋adj +푖 +(푖) = 푖 + 1, 휋adj +푖 +(푖 + 1) = 푖, +and 휋adj +푖 +( 푗) = 푗, for 푗 ∉ {푖, 푖 + 1}. +In words, 휋adj +푖,푗 swaps adjacent positions 푖 and 푖 + 1, for even 푖 ∈ [푛], and leaves other +positions unchanged. +2) For even indices 푖, 푗 ∈ [푛], define 휋swap +푖 +: [푛] → [푛], such that 휋swap +푖 +(푖) = 푗, 휋swap +푖 +(푖 + 1) = +푗 + 1, and 휋swap +푖 +( 푗) = 푖, 휋swap +푖 +( 푗 + 1) = 푖 + 1, with 휋swap +푖 +(푘) = 푘, for 푘 ∉ {푖, 푖 + 1, 푗, 푗 + 1}. +In words, 휋swap +푖,푗 +swaps 푖 and 푗, and 푖 + 1 and 푗 + 1, for 푖, 푗 being even, and leaves other +positions unchanged. +The discussion above on the sequences in 푆2 implies that the symmetry group 퐺푆2 of the +constraint is generated (via compositions) by {휋adj +푖 +: +푖 even} ∪ {휋adj +푖,푗 : +푖, 푗 even}. Further, +consider tuples 휶 ∈ {0, 1} × +� +0 : +� 푛 +2 +�� +× +� +0 : +� 푛 +2 +�� +of the form 휶 = (푏, 푡00, 푡11), with 푡00 + +푡11 ≤ +� 푛 +2 +� +. For a sequence x ∈ {0, 1}푛, we identify 푏 ∈ {0, 1} with 푥1, the integer 푡00 with +|{푖 : 푖 even and (푥푖, 푥푖+1) = (0, 0)}|, and the integer 푡11 with |{푖 : 푖 even and (푥푖, 푥푖+1) = (1, 1)}|. +Note that then |{푖 : 푖 even and (푥푖, 푥푖+1) = (0, 1) or (1, 0)}| = +� 푛 +2 +� +− 푡00 − 푡11. We thus have that +the orbits of the symmetry group of the constraint {0, 1}푛/퐺푆2 are in one-one correspondence +with tuples of the form 휶 = (푏, 푡00, 푡11). Observe that the number of orbits is hence bounded +above by 2 · +�푛 +2 +�2, and therefore the number of variables and the number of constraints in the +LP Del/퐺푆2 (푛, 푑; 푆2), are bounded above by a polynomial function of the blocklength 푛, unlike +the number of variables in Del(푛, 푑; 푆2), which equals 2푛. +If we use the notation 푆(휶) to denote the size of an orbit represented by 휶, the symmetrized +LP Del/퐺푆2 (푛, 푑; 푆2) then becomes: +January 13, 2023 +DRAFT + +JANUARY 2023 +31 +푑 +Del/푆2(푛, 푑; 푆2) +GenSph(푛, 푑; 푆2) +Del(푛, 푑) +2 +64 +64 +4096 +3 +45.255 +64 +512 +4 +45.255 +64 +292.571 +5 +22.627 +64 +64 +6 +17.889 +64 +40 +7 +5.657 +32 +8 +8 +4.619 +32 +5.333 +9 +2.828 +16 +3.333 +10 +2.619 +16 +2.857 +TABLE II: Table of values of optimal values of the symmetrized Del/푆2(푛, 푑; 푆2) LP, the +generalized sphere packing bound LP GenSph(푛, 푑; 푆2) in [17] and [15], and the Del(푛, 푑) +LP, for 푛 = 13 and varying values of 푑. +Del/퐺푆2 (푛, 푑; 푆2) +maximize +{푎휶∈R: 휶 is an orbit} +� +휶 +푆(휶) · 푎휶 +(Obj′′) +subject to: +푎휶 ≥ 0, ∀ orbits 휶, +(D1′) +� +휶 +푎휶 · � +1휶(s ˜휶) ≥ 0, ∀ orbit rep. s ˜휶 ∈ {0, 1}푛, (D2′) +푎휶 = 0, if 1 ≤ +�푛 +2 +� ++ 훼1 − 훼2 + 훼3 ≤ 푑 − 1, +(D3′) +푎(0,⌊ 푛 +2⌋,0) ≤ val(Del(푛, 푑)), +(D4′) +푎휶 ≤ 2푛 · (1A ★ 1A)(x휶), ∀ orbits 휶. +(D5′) +Table II shows numerical evaluations of Del/퐺푆2 (푛, 푑; 푆2), when 푛 = 13, for varying values of +푑. The table also includes comparisons with upper bounds via the generalized sphere packing +bound of [15] and [17] and with Del(푛, 푑). We observe that once again our LP provides tighter +upper bounds than those obtained by the sphere packing approach. +January 13, 2023 +DRAFT + +JANUARY 2023 +32 +B. Constant Subblock Composition Constraint +We now move on to studying the constant subblock-composition CSC푝 +푧 constraint, which +requires that each one of the 푝 “subblocks” of a binary sequence have a constant number, 푧, of 1s. +In particular, for any sequence x ∈ {0, 1}푛, we first partition the 푛 coordinates into 푝 subblocks, +with the ℓth subblock being the vector of symbols xℓ := +� +푥푖 ∈ {0, 1} : +(ℓ−1)푛 +푝 ++ 1 ≤ 푖 ≤ ℓ푛 +푝 +� +, for +1 ≤ ℓ ≤ 푝. We implicitly assume that 푝 divides 푛. Note that hence x = x1x2 . . . x푝. A binary +sequence x respects the CSC푝 +푧 constraint if 푤(xℓ) = 푧, for all 1 ≤ ℓ ≤ 푝. We let 퐶푝,(푛) +푧 +(or +simply, 퐶푝 +푧 ) denote the set of all CSC푝 +푧 -constrained sequences of length 푛. CSC푝 +푧 -constrained +sequences were introduced in [5] for simultaneous information and energy transfer from a +powered transmitter to an energy harversting receiver, while ensuring that the receiver battery +does not drain out during periods of low signal energy. The applications of such constrained +codes to visible light [33] and powerline communications [34] have also been investigated. +As before, we are interested in computing the Fourier coefficients of the function 1퐶 푝 +푧 : +{0, 1}푛 → {0, 1}. The lemma below provides these Fourier coefficients. +Lemma IV.3. For s ∈ {0, 1}푛 with s = s1s2 . . . s푝, we have that +2푛 · � +1퐶 푝 +푧 (s) = +푝 +� +ℓ=1 +퐾(푛/푝) +푧 +(푤(sℓ)), +where 퐾(푛/푝) +푖 +( 푗) = �푖 +푡=0(−1)푡 � 푗 +푡 +� �푛/푝−푗 +푖−푡 +� is the 푖th-Krawtchouk polynomial, for the length 푛/푝. +Proof. We have that +2푛 · � +1퐶 푝 +푧 (s) = +� +x∈{0,1}푛: x∈퐶 푝 +푧 +(−1)x·s += +� +x1∈{0,1}푛/푝: 푤(x)=푧 +. . . +� +x푝∈{0,1}푛/푝: 푤(x)=푧 +(−1)x·s1 . . . (−1)x·s푝 += +푝 +� +ℓ=1 +�� +� +� +xℓ∈{0,1}푛/푝: 푤(x)=푧 +(−1)x·sℓ�� +� +. +Now, by following a line of argument similar to that in the proof of Theorem A.1 in Appendix +A, we obtain that for any ℓ ∈ [푝], the value of the inner summand depends on sℓ only via its +January 13, 2023 +DRAFT + +JANUARY 2023 +33 +weight. In other words, it holds that for any ℓ ∈ [푝], +� +xℓ∈{0,1}푛/푝: 푤(x)=푧 +(−1)x·sℓ = +� +xℓ∈{0,1}푛/푝: 푤(x)=푧 +(−1)x·˜sℓ, +where +˜sℓ = (1, 1, . . . , 1 +���������������� +푤(sℓ) such +, 0, 0, . . . , 0). +By direct calculations, it holds that the sum in right-hand side of the expression above equals +퐾(푛/푝) +푧 +(푤(sℓ)). +□ +In what follows, we shall concern ourselves with the application of Lemma IV.4 and Theorem +III.1 to calculating the number of subblock constrained codewords in Reed-Muller (RM) codes +RM(푚, 푟), for select values of the number of subblocks 푝. +First, we recall an important prpoerty of RM codes, which is sometimes called the Plotkin +decomposition (see [24, Chap. 13] or the survey [35]): any length-2푚 codeword c ∈ RM(푚, 푟) can +be written as the concatenation c = (u | u+v), where u ∈ RM(푚 −1, 푟) and v ∈ RM(푚 −1, 푟 −1) +and the ‘+’ operation in u+v is over F2푚−1 +2 +. Observe that since RM(푚, 푡), for 1 ≤ 푡 ≤ 푚, consists +of evaluation vectors of Boolean polynomials of degree at most 푡, it holds that RM(푚−1, 푟−1) ⊂ +RM(푚 − 1, 푟). In what follows, we ensure that 푟 ≥ 1 and 푚 is large. +Assume, for simplicity, that 푝 = 2. We then have that for 0 ≤ 푧 ≤ 2푚−1, +푁 +� +RM(푚, 푟); 퐶2 +푧 +� += +� +c∈RM(푚,푟) +1퐶2푧 (x) += +� +u∈RM(푚−1,푟), +v∈RM(푚−1,푟−1) +1푊푧 (u) · 1푊푧 (u + v), +(10) +where the second equality uses the Plotkin decomposition and the fact that the set 푊푧 consists of +sequences of Hamming weight exactly 푧. Further, let u1, u2, . . . , u푀 be an enumeration of coset +representatives of distinct cosets of RM(푚−1, 푟 −1) in RM(푚−1, 푟), where 푀 = +|RM(푚−1,푟)| +|RM(푚−1,푟−1)| = +2(푚−1 +≤푟 )−( 푚−1 +≤푟−1) = 2(푚−1 +푟 ). In other words, u푖 is a representative of the coset u푖 + RM(푚 − 1, 푟 − 1), +with u푖 ∈ RM(푚 − 1, 푟), for 1 ≤ 푖 ≤ 푀, where the cosets u푗 + RM(푚 − 1, 푟 − 1), for different +January 13, 2023 +DRAFT + +JANUARY 2023 +34 +values of 푗, are disjoint. Let 퐴u(푦) be the weight enumerator of the coset u + RM(푚 − 1, 푟 − 1), +at the weight 0 ≤ 푦 ≤ 2푚−1, for u ∈ RM(푚 − 1, 푟). Then, from (10), we see that +푁 +� +RM(푚, 푟); 퐶2 +푧 +� += +� +u∈RM(푚−1,푟), +v∈RM(푚−1,푟−1) +1푊푧 (u) · 1푊푧 (u + v) += +� +u∈RM(푚−1,푟) +1푊푧 (u) · +� +v∈RM(푚−1,푟−1) +1푊푧 (u + v) += +� +u∈RM(푚−1,푟) +1푊푧 (u) · 퐴u(푧) +(푎)= +푀 +� +푖=1 +� +u∈u푖+RM(푚−1,푟−1) +1푊푧 (u) · 퐴u푖 (푧) += +푀 +� +푖=1 +�퐴u푖 (푧)�2 , +(11) +where equality (a) uses the fact that any u ∈ RM(푚 − 1, 푟) belongs to some coset u푖 + RM(푚 − +1, 푟 − 1). +While equality (11) provides a neat method to count the number of constrained codewords +푁 �RM(푚, 푟); 퐶2 +푧 +�, provided the coset weight enumerators 퐴u푖 (푧), 1 ≤ 푖 ≤ 푀, are known, observe +that in the summation in (11), we need to perform 푀 −1 = 2(푚−1 +푟 ) −1 additions. If 푟 is large, the +number of such additions can be fairly high. We show next that with the help of Theorem III.1 and +Lemma IV.4, it is possible to reduce the number of computations, when 푟 is large. Before we do +so, we recall the fact that for 푟 ≤ 푚−1, the dual code of RM(푚, 푟) is the code RM(푚, 푚−푟 −1). +We let 퐴u(푦) be the weight enumerator of the coset u + RM(푚 − 1, 푚 − 푟 − 2), at the weight +0 ≤ 푦 ≤ 2푚−1, for u ∈ RM(푚 − 1, 푚 − 푟 − 1). Further, we let u1, u2, . . . , u푀 be an enumeration +of coset representatives of distinct cosets of RM(푚 − 1, 푚 − 푟 − 2) in RM(푚 − 1, 푚 − 푟 − 1), +where 푀 = |RM(푚−1,푚−푟−1)| +|RM(푚−1,푚−푟−2)| = 2( 푚−1 +푚−푟−1). +Now, applying Theorem III.1, we see that +푁 +� +RM(푚, 푟); 퐶2 +푧 +� += +� +s1s2∈RM(푚,푚−푟−1) +퐾(푛/2) +푧 +(푤(s1)) · 퐾(푛/2) +푧 +(푤(s2)) +January 13, 2023 +DRAFT + +JANUARY 2023 +35 += +� +u∈RM(푚−1,푚−푟−1), +v∈RM(푚−1,푚−푟−2) +퐾(푛/2) +푧 +(푤(u)) · 퐾(푛/2) +푧 +(푤(u + v)) += +� +u∈RM(푚−1,푚−푟−1) +퐾(푛/2) +푧 +(푤(u)) · +� +v∈RM(푚−1,푚−푟−2) +퐾(푛/2) +푧 +(푤(u + v)) += +� +u∈RM(푚−1,푚−푟−1) +퐾(푛/2) +푧 +(푤(u)) · +2푚−1 +� +푗=0 +퐴u( 푗) · 퐾(푛/2) +푧 +( 푗) += +푀 +� +푖=1 +�� +� +2푚−1 +� +푗=0 +퐴u푖 ( 푗) · 퐾(푛/2) +푧 +( 푗)�� +� +2 +. +(12) +Now, observe that using equality (12), the number of computations required, in the form of +summations, assuming that the coset weight enumerators 퐴u푖 (·) are known, for all 1 ≤ 푖 ≤ +푀, is 2푚−1+푀 − 1 = 2푚−1+( 푚−1 +푚−푟−1) − 1. Clearly, since for large 푟 (and large 푚), we have that +푚 − 1 + � 푚−1 +푚−푟−1 +� < �푚−1 +푟 +�, we note the relative ease of calculating 푁 �RM(푚, 푟); 퐶2 +푧 +� via (12), +with the aid of Theorem III.1, as compared to using (11). We remark here that the analysis of +the number of codewords in RM(푚, 푟) that lie in 퐶푝 +푧 can be extended to values of 푝 that are +powers of 2, by iteratively applying the Plotkin decomposition. Finally, we note that in order +to compute the coset weight enumerators required in (11) and (12), one can use the recursive +algorithm provided in [36], which applies to RM codes, in addition to polar codes. +Next, we provide a more explicit form of Del/퐺 A (푛, 푑; A), when A = 퐶푝 +푧 , for a fixed +blocklength 푛 and parameters 푝 and 푧. From the description of the constraint, it can be checked +that the symmetry group 퐺퐶 푝 +푧 is generated (via compositions) by the following permutations: +1) For 1 ≤ ℓ ≤ 푝, and (ℓ−1)푛 +푝 ++ 1 ≤ 푗 ≤ ℓ푛 +푝 , define 휋perm,푗 +ℓ +: [푛] → [푛] such that 휋perm,푗 +ℓ +swaps +the indices (ℓ−1)푛 +푝 ++ 1 and 푗, and leaves the other indices in [푛] unchanged. +Note that for a fixed block indexed by 1 ≤ ℓ ≤ 푝, the collection of permutations {휋perm,푗 +ℓ +: +(ℓ−1)푛 +푝 ++ 1 ≤ 푗 ≤ ℓ푛 +푝 } generates a group isomorphic to the symmetric group 푆푛/푝, which +contains all permutations of the indices (ℓ−1)푛 +푝 ++ 1 ≤ 푖 ≤ ℓ푛 +푝 . +2) For 1 ≤ ℓ, ℓ′ ≤ 푝, define 휋exch +ℓ,ℓ′ : [푛] → [푛] such that 휋exch +ℓ,ℓ′ swaps the element (ℓ−1)푛 +푝 ++ 푗 +with (ℓ′−1)푛 +푝 ++ 푗, for all 1 ≤ 푗 ≤ 푛 +푝, and leaves the other indices in [푛] unchanged. +In other words, 휋exch +ℓ,ℓ′ exchanges entire blocks indexed by ℓ and ℓ′. +January 13, 2023 +DRAFT + +JANUARY 2023 +36 +From the description of the symmetry group 퐺퐶 푝 +푧 above, we arrive at the fact that the orbits of the +symmetry group are in one-one correspondence with unordered 푝-tuples 휶 ∈ +� +0 : 푛 +푝 +� 푝 +. Indeed, +a given sequence x ∈ {0, 1}푛 lies in the orbit 휶(x) = (훼1(x), . . . , 훼푝(x)), where wt(xℓ) = 훼휎(ℓ), +for 1 ≤ ℓ ≤ 푝 and some permutation 휎 ∈ 푆푝. Note hence that the number of orbits, and +therefore the sum of the number of variables and the number of constraints in Del/퐺퐶 푝 +푧 (푛, 푑; 퐶푝 +푧 ) +is bounded above by 푐 · +� +푛 +푝 +� 푝 +, for some constant 푐 > 0, which is only a polynomial function of +the blocklength. Further, for a given orbit 휶, we let x휶 be a representative element of the orbit. +In particular, we define x휶 to be the concatenation x휶,1x휶,2 . . . x휶,푝, with +x휶,ℓ = (1, 1, . . . , 1 +���������������� +훼1 such +, 0, 0, . . . , 0) +(13) +being of length 푛/푝, for 1 ≤ ℓ ≤ 푝. We thus obtain the following lemma: +Lemma IV.4. For given orbits 휶, ˜휶, with s ˜휶 being an orbit representative of ˜휶, it holds that +2푛 · � +1휶(s ˜휶) = +푝 +� +ℓ=1 +퐾(푛/푝) +훼ℓ +( ˜훼ℓ), +where for a given length 푚, 퐾(푚) +푖 +is the 푖th Krawtchouk polynomial, with 퐾(푚) +푖 +( 푗) = �푖 +푡=0(−1)푡 � 푗 +푡 +� �푚−푗 +푖−푡 +�. +The proof of the above lemma is similar to the proof of Lemma IV.4 and is hence omitted. +Again, using the notation 푆(휶) to denote the number of elements in an orbit 휶, the symmetrized +LP Del/퐺퐶 푝 +푧 (푛, 푑; 퐶푝 +푧 ) then becomes: +January 13, 2023 +DRAFT + +JANUARY 2023 +37 +푑 +Del/퐶2 +5 (푛, 푑; 퐶2 +5) +GenSph(푛, 푑; 퐶2 +5) +2 +441 +441 +3 +197.9899 +441 +4 +197.9899 +441 +5 +49.574 +147 +6 +35.0542 +147 +7 +11.3137 +73.5 +TABLE III: Table of values of optimal values of the Del/퐶2 +5 (푛, 푑; 퐶2 +5) LP, and the generalized +sphere packing bound LP GenSph(푛, 푑; 퐶2 +5), for (푛, 푝, 푧) = (14, 2, 5), and varying values of 푑. +Del/퐺퐶 푝 +푧 (푛, 푑; 퐶푝 +푧 ) +maximize +{푎휶∈R: 휶 is an orbit} +� +휶 +푆(휶) · 푎휶 +(Obj′′) +subject to: +푎휶 ≥ 0, ∀ orbits 휶, +(D1′) +� +휶 +푎휶 · � +1휶(s ˜휶) ≥ 0, ∀ orbit rep. s ˜휶 ∈ {0, 1}푛, (D2′) +푎휶 = 0, if 1 ≤ +푝 +� +푡=1 +훼푡 ≤ 푑 − 1, +(D3′) +푎0푝 ≤ val(Del(푛, 푑)), +(D4′) +푎휶 ≤ 2푛 · (1A ★ 1A)(x휶), ∀ orbits 휶. +(D5′) +Tables III, IV, and V show numerical evaluations of Del/퐺퐶 푝 +푧 (푛, 푑; 퐶푝 +푧 ), when 푛 = 14, 푛 = 15, +and 푛 = 18, respectively, for fixed parameters 푝 and 푧, and for varying values of 푑. In Tables III +and IV, we again compare with upper bounds via the generalized sphere packing bound of [15] +and [17]. Here too our LP provides tighter upper bounds than the generalized sphere packing +bounds. +January 13, 2023 +DRAFT + +JANUARY 2023 +38 +푑 +Del퐶3 +2 (푛, 푑; 퐶3 +2) +GenSph(푛, 푑; 퐶3 +2) +2 +1000 +1000 +3 +826.236 +1000 +4 +826.236 +1000 +5 +157.767 +333.333 +6 +110.851 +333.333 +7 +22.627 +166.667 +TABLE IV: Table of values of optimal values of the Del/퐶3 +2 (푛, 푑; 퐶3 +2) LP, and the generalized +sphere packing bound LP GenSph(푛, 푑; 퐶3 +2), for (푛, 푝, 푧) = (15, 3, 2), and varying values of 푑. +푑 +3 +4 +5 +6 +7 +8 +9 +Del/퐶2 +2 (푛, 푑; 퐶2 +2) +556.38 +556.38 +227.111 +165.247 +38.118 +28.540 +4.472 +TABLE V: Table of values of optimal values of the Del/퐶2 +2 (푛, 푑; 퐶2 +2) LP, for (푛, 푝, 푧) = (18, 2, 2), +and varying values of 푑. +C. Runlength-Limited (RLL) Constraints +In this subsection, we shall work with runlength-limited constraints on binary sequences. +Unlike in the previous subsections, where the Fourier coefficients of the indicator functions of +the constraints were explicitly (or analytically) computable, in the application of Theorem III.1, +for the constraints considered in this section, we shall provide recurrence relations for the Fourier +coefficients, which allow them to be efficiently computable, numerically. +We concern ourselves with the (푑, ∞)-runlength limited (RLL) constraint. This constraint +mandates that there be at least 푑 0s between every pair of successive 1s in the binary input +sequence, where 푑 ≥ 1. For example, when 푑 = 3, the sequence 10001000010 respects the (3, ∞)- +RLL constraint, but the sequence 10100010, does not. It can also be checked that the (1, ∞)- +RLL constraint is the same as a “no-consecutive-ones” constraint. The (푑, ∞)-RLL constraint +is a special case of the (푑, 푘)-RLL constraint, which admits only binary sequences in which +successive 1s are separated by at least 푑 0s, and the length of any run of 0s is at most 푘. +Such constraints find application in magnetic and optical recording systems, where the (푑, ∞)- +RLL constraint on the data sequence (with the bit 1 corresponding to a voltage peak of high +amplitude and the bit 0 corresponding to no peak) ensures that successive 1s are spaced far +January 13, 2023 +DRAFT + +JANUARY 2023 +39 +enough apart, so that there is little inter-symbol interference (ISI) between the voltage responses +corresponding to the magnetic transitions. Reference [37] contains many examples of (푑, 푘)- +RLL codes used in practice in magnetic storage and recording. More recently, (푑, 푘)-RLL input +constrained sequences have also been investigated for joint energy and information transfer +performance [6]. We let 푆푑 denote the set of (푑, ∞)-RLL constrained binary words of length 푛. +Now, for 푛 ≥ 1, and for s ∈ {0, 1}푛, let � +1푆푑 +(푛)(s) denote the Fourier coefficient at s, when the +blocklength is 푛. We then have that: +Lemma IV.5. For 푛 ≥ 푑 + 2 and for s = (푠1, . . . , 푠푛) ∈ {0, 1}푛, it holds that when 푠1 = 0, +� +1푆푑 +(푛)(s) = 2−1 · � +1푆푑 +(푛−1) �푠푛 +2 +� + 2−(푑+1) · � +1푆푑 +(푛−푑−1) �푠푛 +푑+2 +� , +and when 푠1 = 1, +� +1푆푑 +(푛)(s) = 2−1 · � +1푆푑 +(푛−1) �푠푛 +2 +� − 2−(푑+1) · � +1푆푑 +(푛−푑−1) �푠푛 +푑+2 +� . +Proof. To prove the first recurrence relation, we write +� +1푆푑 +(푛)(s) = 1 +2푛 · +� +x∈푆푑 +(−1)x·s += 2−푛 · +� +#{푥푛 ∈ 푆푑 : 푤s(푥푛) is even} − #{푥푛 ∈ 푆푑 : 푤s(푥푛) is odd} +� +. +(14) +Now, observe that +#{푥푛 ∈ 푆푑 : 푤s(푥푛) is even} = #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥1 = 0} + +#{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥1 = 1} +(푎)= #{푥푛 +2 ∈ 푆푑 : 푤푠푛 +2 (푥푛 +2) is even} + +#{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥(푑+1) +1 += 10푑} += #{푥푛 +2 ∈ 푆푑 : 푤푠푛 +2 (푥푛 +2) is even} + #{푥푛 +푑+2 ∈ 푆푑 : 푤푠푛 +푑+2(푥푛) is even}, +(15) +where (a) holds because 푠1 = 0 and from the fact that the (푑, ∞)-RLL constraint requires that +January 13, 2023 +DRAFT + +JANUARY 2023 +40 +푥푑+1 +2 += 0푑, if 푥1 = 1. Similarly, we obtain that +#{푥푛 ∈ 푆푑 : 푤s(푥푛) is odd} = #{푥푛 +2 ∈ 푆푑 : 푤푠푛 +2 (푥푛 +2) is odd} + #{푥푛 +푑+2 ∈ 푆푑 : 푤푠푛 +푑+2 (푥푛) is odd}. +(16) +Now, observe that +� +1푆푑 +(푛−1)(푠푛 +2) = 2−(푛−1) · +� +#{푥푛 +2 ∈ 푆푑 : 푤푠푛 +2 (푥푛 +2) is even} − #{푥푛 +2 ∈ 푆푑 : 푤푠푛 +2 (푥푛 +2) is odd} +� +(17) +and that +� +1푆푑 +(푛−푑−1)(푠푛 +푑+2) = 2−(푛−푑−1) · +� +#{푥푛 +푑+2 ∈ 푆푑 : 푤푠푛 +푑+2(푥푛 +푑+2) is even}− +#{푥푛 +푑+2 ∈ 푆푑 : 푤푠푛 +푑+2(푥푛 +푑+2) is odd} +� +. +(18) +Substituting (15) and (16) in (14) and using (17) and (18), we get the first recurrence relation. +The second recurrence relation is also proved by similar arguments. +□ +We shall now explain how Lemma IV.5 helps compute the Fourier coefficients for a given +(large) 푛, efficiently. First, we note that a direct computation of all the Fourier coefficients of +1푆푑 at blocklength 푛, can be accomplished by the fast Walsh-Hadamard transform (FWHT) +algorithm (see Exercise 1.12(b) in [21]), in time 푛 · 2푛. Now, let us assume that we pre-compute +and store the Fourier coefficients +� +� +1푆푑 +(푚)(s) : s ∈ {0, 1}푚� +, for 1 ≤ 푚 ≤ 푑 + 1. These Fourier +coefficients help initialize the recurrences in Lemma IV.5. Now, give a fixed (large) 푛, the Fourier +coefficients at which blocklength we intend computing, we shall calculate, using the recurrence +relations above, the Fourier coefficients at all blocklengths 푑 + 2 ≤ 푚 ≤ 푛, iteratively, beginning +at length 푑 + 2, and increasing 푚. Assuming that the additions and multiplications in Lemma +IV.5 take unit time, it can be seen that the time complexity of computing the Fourier coefficient +at length 푛 grows as �푛 +푑+2 2푖 < 2푛+1. This is much less than the time that is 2푛+log2 푛, taken by +the FWHT algorithm. +However, there still remains the issue of storage cost: at “level” 푚, one needs to store all 2푚 +Fourier coefficients in order to facilitate computation of the Fourier coefficients at blocklengths +January 13, 2023 +DRAFT + +JANUARY 2023 +41 +C +푁(C; 푆1) +RM(4, 2) +83 +RM(4, 3) +1292 +Ham3 +4 +Ham4 +101 +TABLE VI: Table of values of 푁(C; 푆1), for select codes C +푛 > 푚. Hence, assuming that the storage of a single Fourier coefficient takes up one unit of +space, we see that we require at least 2푛 units of memory in order to store the Fourier coefficients +at blocklength 푛. For 푛 ⪆ 20, for example, this storage cost becomes prohibitively expensive. +We now use the Fourier coefficients that are numerically computed using Lemma IV.5, to +calculate the number of (1, ∞)-RLL constrained codewords in select codes, by applying Theorem +III.1. These values are listed in Table VI. We denote the binary Hamming code of blocklength +2푡 −1 as Ham푡. Further, we assume that the coordinates of the Hamming and Reed-Muller codes +follow the orderings discussed in Section III-A. +Next, we obtain upper bounds on the sizes of (푑, ∞)-RLL constrained codes with a given +minimum distance. Since there is no apparent symmetry group of 푆푑, we shall first directly run +the Del(푛, 푑; 푆푑) LP. +Table VII (resp. Table VIII) shows comparisons between the upper bounds on 퐴(푛, 푑; 푆1) (resp. +퐴(푛, 푑; 푆2)), obtained using our Del(푛, 푑; 푆1) LP (resp. Del(푛, 푑; 푆2) LP) with the generalized +sphere packing bound of [15] and [17], when 푛 = 10, and for varying values of the minimum +distance 푑. We also compare these upper bounds with the optimal value of Del(푛, 푑), since this +is a trivial upper bound on 퐴(푛, 푑; A), for any A ⊆ {0, 1}푛. Note that, from the numerical trials, +for certain values of 푑, the generalized sphere-packing bound returns a value that is larger (and +hence worse) than the value of Del(푛, 푑), whereas the optimal value of our Del(푛, 푑; A) LP is +uniformly bounded above by Del(푛, 푑). +Next, we study upper bounds on the sizes of constrained codes for a given minimum distance, +for a slightly stronger variant of the (푑, ∞)-RLL constraint. In particular, we require that the +constraint is satisfied across the boundary when the constrained sequence is cyclically wrapped +January 13, 2023 +DRAFT + +JANUARY 2023 +42 +푑 +Del(푛, 푑; 푆1) +GenSph(푛, 푑; 푆1) +Del(푛, 푑) +2 +128.557 +144 +512 +3 +74.762 +111 +85.333 +4 +42.048 +111 +42.667 +5 +12 +63 +12 +6 +6 +63 +6 +7 +3.2 +26 +3.2 +TABLE VII: Table of values of optimal values of the Del(푛, 푑; 푆1) LP, the generalized sphere +packing bound LP GenSph(푛, 푑; 푆1) in [15] and [17], and the Del(푛, 푑) LP, for 푛 = 10 and +varying values of 푑. +푑 +Del(푛, 푑; 푆2) +GenSph(푛, 푑; 푆2) +Del(푛, 푑) +2 +49.578 +60 +512 +3 +32.075 +46.5 +85.333 +4 +21.721 +46.5 +42.667 +5 +7.856 +34 +12 +6 +4.899 +34 +6 +7 +2.529 +19 +3.2 +TABLE VIII: Table of values of optimal values of the Del(푛, 푑; 푆2) LP, the generalized sphere +packing bound LP GenSph(푛, 푑; 푆2), and the Del(푛, 푑) LP, for 푛 = 10 and varying values of 푑. +around. We let 푆tail,푑 denote this constraint that admits only those sequences x ∈ {0, 1}푛 such +that x ∈ 푆푑 and ℓ − 푠 ≤ 푛 − 푑 −1, where 푠 ∈ [푛] denotes the position of the first occurring 1 in x +and ℓ ∈ [푛] denotes the position of the last occurring 1. In the paragraphs that follow, we first +identify the symmetry group of the more broad class of tail-biting constraints, to which 푆tail,푑 +also belongs. +Let a tail-biting constraint of interest be represented by a special set A of constrained +sequences. In particular, A has the property that if a sequence x ∈ A, then 휋cyc,푖 · x also +lies in A, where for 1 ≤ 푖 ≤ 푛, 휋cyc,푖 shifts each bit in x by 푖 bits to the left, wrapping around +cyclically, if needed. More formally, 푥휋cyc,푖 ( 푗) = 푥mod( 푗+푖,푛), for 1 ≤ 푗 ≤ 푛. Clearly, we have that +the symmetry group of the constraint 퐺A contains the cyclic group 퐶푛, and it is hence possible +to symmetrize Del(푛, 푑; A) using 퐶푛. The orbits 휶 thus are in one-one correspondence with +(fixed) 2-ary necklaces of length 푛, with turnovers prohibited (see pg. 18 in [38] and sequence +A000031 of [39]). It is known that the number of such necklaces, and hence the number of +January 13, 2023 +DRAFT + +JANUARY 2023 +43 +푑 +Del/퐺푆tail +(1,∞) +(푛, 푑; 푆tail +(1,∞)) +GenSph(푛, 푑; 푆tail +(1,∞)) +Del(푛, 푑) +2 +480.676 +521 +4096 +3 +350.055 +448.5 +512 +4 +229.569 +448.5 +292.571 +5 +64 +316.727 +64 +6 +40 +316.727 +40 +7 +8 +169 +8 +8 +5.333 +169 +5.333 +9 +3.333 +73.667 +3.333 +TABLE IX: Table of values of optimal values of the Del/퐺푆tail,1 (푛, 푑; 푆tail,1) LP, the generalized +sphere packing bound LP GenSph(푛, 푑; 푆tail,1) in [17] and [15], and Del(푛, 푑), for 푛 = 13, and +varying values of 푑. +orbits, 푁cyc(푛), is such that lim푛→∞ +푁cyc(푛) +2푛/푛 += 1 (see [40]); in other words, the sum of the number +of variables and constraints in Del/퐺 A (푛, 푑; A), is bounded above by 푐 · 2푛 +푛 , for some constant +푐 > 0, and large enough 푛. Thus, we obtain a slight reduction in the size of the LP as compared +to Del(푛, 푑; A), which had 2푛 variables. +We then apply this symmetrization procedure to the tail-biting (1, ∞)-RLL constraint, 푆tail,1, +which admits only binary sequences x ∈ {0, 1}푛 with no consecutive ones and which are such +that (푥1, 푥푛) ≠ (1, 1). Similar to the approach in the previous two subsections, we can set up +a symmetrized LP Del/퐺푆tail,1 (푛, 푑; 푆tail,1), using the orbits of the cyclic group 퐶푛. Table IX +shows numerical evaluations of Del/퐺푆tail,1 (푛, 푑; 푆tail,1), when 푛 = 13, for varying values of 푑, +and comparisons with the generalized sphere packing bound. Again, our LP provides tighter +upper bounds than the generalized sphere packing bounds. Observe also that for certain values +of minimum distance 푑, the optimal value of our LP coincides with the optimal value of the +Delsarte LP Del(푛, 푑). +V. CONCLUSION +In this work, we took two approaches to the problem of estimating the sizes of binary error- +correcting constrained codes. First, motivated by the application of transmission of codes over +stochastic, symmetric, channel noise models—a problem for which explicit capacity-achieving +linear codes have been constructed, we consider the question of computing the sizes of con- +January 13, 2023 +DRAFT + +JANUARY 2023 +44 +strained subcodes of linear codes. Such constrained subcodes of capacity-achieving linear codes, +for example, are resilient to symmetric errors and erasures, in that their error probabilities using +the same decoding strategy as for the larger linear code, vanish as the blocklength of the code +goes to infinity. Our approach was to view the problem through a Fourier-analytic lens, thereby +transforming it into a counting problem in the space of the dual code. As part of our method, +we analyzed (analytically or numerically) the Fourier transform of the indicator function of the +constraint, and provided values of the number of constrained codewords in select linear codes +and algorithmic procedures for efficient counting, in the cases of certain constraints. For some +constraints, we also obtained insights into the construction of linear codes with a large number +of constrained codewords. +Next, we considered the scenario where the constrained codes were subjected to adversarial +bit-flip errors or erasures, with a combinatorial bound on the number of errors or erasures that +can be induced. We then proposed numerical upper bounds on the sizes of constrained codes +with a given resilience to such combinatorial errors and erasures (equivalently, with a prescribed +minimum Hamming distance), via an extension of Delsarte’s linear program (LP). We then +applied our LPs to different constraints, and observed that the optimal numerical values returned +by our LP are better than those provided by the generalized sphere packing bounds of Fazeli, +Vardy, and Yaakobi (2015). +There are many interesting directions for future work. One line of study would be to build +on the Fourier-theoretic techniques in this paper and study the asymptotics (in the limit as the +blocklength goes to infinity) of the rates of constrained subcodes of specific linear codes of a +given rate 푅 ∈ (0, 1). Similarly, one could try to use our dual LP formulation to derive asymptotic +upper bounds on the rate-distance tradeoff for constrained codes. This, for example, will help +us understand if the Gilbert-Varshamov lower bounds of Kolesnik and Krachkovsky (1991) and +Marcus and Roth (1992) are tight for any constrained system. Another direction of work could +study the extension of results here to codes with larger alphabet sizes. +VI. ACKNOWLEDGEMENTS +The authors would like to thank Prof. H. D. Pfister for useful discussions. +January 13, 2023 +DRAFT + +JANUARY 2023 +45 +APPENDIX +A. On the Weight Distribution of Constrained Sequences in F푛 +2 +Suppose that we are interested in computing the weight distribution of words in {0, 1}푛 that lie +in a (constrained) set A ⊆ {0, 1}푛. Let 푎푖,A denote the number of constrained words of weight +푖 ∈ [0 : 푛] and recall the definition of the 푖th-Krawtchouk polynomial 퐾(푛) +푖 +, for a given blocklength +푛, where 퐾(푛) +푖 +(푧) = �푖 +ℓ=0(−1)ℓ�푧 +ℓ +� �푛−푧 +푖−ℓ +�, and the notation 푊푖 = {x ∈ {0, 1}푛 : 푤(x) = 푖}. The +following theorem then holds true: +Theorem A.1. The weight distribution of sequences that lie in a set A ⊆ {0, 1}푛 obeys +푎푖,A = +푛 +� +푗=0 +퐾(푛) +푖 +( 푗) · +� +s: 푤(s)=푗 +� +1A(s), +푖 ∈ [0 : 푛]. +Proof. The proof is again a simple application of Plancherel’s Theorem. Observe that +푎푖,A = +� +x∈A +1푊푖(x) += +� +x∈{0,1}푛 +1푊푖 (x) · 1A(x) += 2푛 · +� +s∈{0,1}푛 +� +1푊푖 (s) · � +1A(s). +(19) +We now recall the well-known proof of the fact that 2푛 · � +1푊푖(s) = 퐾푖(푤(s)) (see Chapter 5 in +[24] for more details on Krawtchouk polynomials). Note that +2푛 · � +1푊푖(s) = +� +x∈{0,1}푛: 푤(x)=푖 +(−1)x·s. +Now, the summation on the right side depends only on the weight 푤(s), i.e., for any permutation +of coordinates 휋 : {0, 1}푛 → {0, 1}푛, it holds that +� +x∈{0,1}푛: 푤(x)=푖 +(−1)x·휋(s) = +� +x∈{0,1}푛: 푤(x)=푖 +(−1)휋(x)·휋(s) += +� +x∈{0,1}푛: 푤(x)=푖 +(−1)x·s. +January 13, 2023 +DRAFT + +JANUARY 2023 +46 +In other words, we have that � +1푊푖 (s) = � +1푊푖 (휋(s)). Hence, for s such that 푤(s) = 푗, it suffices that +we calculate 2푛 · � +1푊푖(s★), where s★ = (푠★ +1, . . . , 푠★ +푛) is such that 푠★ +1 = . . . = 푠★ +푗 = 1 and 푠★ +푗+1 = . . . = +푠★ +푛 = 0. By a direct computation, it can be checked that 2푛 · � +1푊푖 (s★) = �푖 +ℓ=0(−1)ℓ� 푗 +ℓ +� �푛−푗 +푖−ℓ +� = 2푛 · +� +1푊푖 (s). Plugging this back into (19) and simplifying, we obtain the expression in the theorem. +□ +As before, Theorem A.1 implies that if the Fourier coefficients � +1A(s) (or, the sum of Fourier +coefficients at a fixed weight � +s: 푤(s)=푗 � +1A(s)) were available to us, we can easily compute the +number of constrained words of a given weight. +B. Weight Distribution of Constrained Codewords in Linear Codes +In this section, we briefly describe another useful computation that is facilitated by knowledge +of the Fourier coefficients of the indicator function that a word belongs to a set of constrained +sequences. Before we do so, we recall another property of Fourier transforms: +Theorem A.2. Given functions 푓 , 푔 : {0, 1}푛 → R, the Fourier transform of 푓 · 푔 is the function +�푓 ⃝★ �푔, where +( �푓 ⃝★ �푔)(s) := +� +z∈{0,1}푛 +�푓 (z)�푔(z + s) = 2푛 · ( �푓 ★ �푔)(s), +s ∈ {0, 1}푛. +The calculation we wish to perform puts together the theses of Theorems III.1 and A.1. In +particular, we ask for the number of codewords of a linear code C, which lie in a fixed set A +and are of a given weight. We thus obtain Theorem A.3. For a given v ∈ {0, 1}푛, we use the +notation v + C to denote the coset of C to which v belongs. We let 푎푖,A(C) denote the number +of constrained codewords of weight 푖 ∈ [0 : 푛], which lie in the set A and in a linear code C. +Theorem A.3. Given a linear code C of blocklength 푛, we have that +푎푖,A(C) = |C| +2푛 +푛 +� +푗=0 +퐾(푛) +푖 +( 푗) +� +s: 푤(s)=푗 +� +z∈s+C⊥ +� +1A(z). +Proof. We observe that +푎푖,A(C) = +� +x∈{0,1}푛 +1푊푖 (x) · 1A(x) · 1C(x) +January 13, 2023 +DRAFT + +JANUARY 2023 +47 +=2푛 · +� +s∈{0,1}푛 +� +1푊푖(s) · � +1A · 1C(s), +where the last equality above uses Plancherel’s Theorem (see Section II-C). Further, by employing +Theorem A.2 to expand the last equality, we get that +푎푖,A(C) = 2푛 · +� +s∈{0,1}푛 +� +1푊푖 (s) · +� +� +1A ⃝★ � +1C +� +(s) +(푎)= |C| +2푛 · +� +s∈{0,1}푛 +퐾(푛) +푖 +(푤(s)) +� +z∈{0,1}푛 +� +1C⊥(z) · � +1A(s + z) += |C| +2푛 · +푛 +� +푗=0 +퐾(푛) +푖 +( 푗) +� +s: 푤(s)=푗 +� +z∈s+C⊥ +� +1A(z), +where equality (a) above uses the fact that 2푛 · � +1푊푖 (s) = 퐾(푛) +푖 +(푤(s)) and that � +1C(s) = |C| +2푛 · 1C⊥(s) +(see the proofs of Theorems A.1 and III.1). +□ +Theorem A.3 could prove useful in the following context: consider the transmission of code- +words of a linear code C over an input-constrained binary-input memoryless symmetric (BMS) +channel, which admits only binary constrained sequences that lie in the set A as inputs. Suppose +that the decoder being used is the maximum a-posteriori (MAP) (equivalently, the maximum like- +lihood (ML)) decoder of the linear code C. By calculating the weight distribution of constrained +codewords in a linear code C as above, it is possible to obtain an upper bound on the block +error probability (via a union bounding argument), when the MAP decoder is used (see Chapter +1 of [41]). +C. Obtaining MacWilliams’ Identities for Linear Codes Via Theorem III.1 +Consider the simple constraint that admits only sequences having a fixed weight 푖 ∈ [0 : 푛], +where 푛 is the blocklength of the code. Note that in this case, the set of constrained sequences +is A = 푊푖. By applying Theorem III.1 to this constraint, for a given linear code C, we obtain +the well-known MacWilliams’ identities [22] for linear codes. We use the notation 푎푖(C) for the +number of codewords of weight 푖 ∈ [0 : 푛] in C, which equals 푁(C;푊푖), following the notation +of Theorem III.1. +January 13, 2023 +DRAFT + +JANUARY 2023 +48 +Theorem A.4 (MacWilliams’ identities). It is true that +푎푖(C) = +1 +|C⊥| +푛 +� +푗=0 +퐾(푛) +푖 +( 푗) · 푎 푗(C⊥). +Proof. 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Urbanke, Modern Coding Theory. +Cambridge University Press, 2008. +January 13, 2023 +DRAFT + +This figure "BEC_bms.png" is available in "png"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "BMS.png" is available in "png"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "ConstDMC.PNG" is available in "PNG"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "DMC.PNG" is available in "PNG"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "dinf_new.png" is available in "png"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "dk_new.png" is available in "png"� format from: +http://arxiv.org/ps/2301.05098v1 + +This figure "no_consec.PNG" is available in "PNG"� format from: +http://arxiv.org/ps/2301.05098v1 + diff --git a/CdE4T4oBgHgl3EQfeQ2g/content/tmp_files/load_file.txt b/CdE4T4oBgHgl3EQfeQ2g/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..cb2338764d24a6aeb41b63bc45e55352f11f1603 --- /dev/null +++ b/CdE4T4oBgHgl3EQfeQ2g/content/tmp_files/load_file.txt @@ -0,0 +1,1296 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf,len=1295 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='05098v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='IT] 12 Jan 2023 JANUARY 2023 1 Estimating the Sizes of Binary Error-Correcting Constrained Codes V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Arvind Rameshwar, Student Member, IEEE, and Navin Kashyap, Senior Member, IEEE Abstract In this paper, we study binary constrained codes that are also resilient to bit-flip errors and erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In our first approach, we compute the sizes of constrained subcodes of linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Since there exist well-known linear codes that achieve vanishing probabilities of error over the binary symmetric channel (which causes bit-flip errors) and the binary erasure channel, constrained subcodes of such linear codes are also resilient to random bit-flip errors and erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We employ a simple identity from the Fourier analysis of Boolean functions, which transforms the problem of counting constrained codewords of linear codes to a question about the structure of the dual code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Via examples of constraints, we illustrate the utility of our method in providing explicit values or efficient algorithms for our counting problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our second approach is to obtain good upper bounds on the sizes of the largest constrained codes that can correct a fixed number of combinatorial errors or erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We accomplish this using an extension of Delsarte’s linear program (LP) to the setting of constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We observe that the numerical values of our LP-based upper bounds beat those obtained by using the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Index Terms Constrained coding, Fourier analysis, linear programming bounds This work was supported in part by a Qualcomm Innovation Fellowship India 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The work of V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Rameshwar was supported by a Prime Minister’s Research Fellowship, from the Ministry of Education, Govt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' of India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The authors are with the Department of Electrical Communication Engineering, Indian Institute of Science, Bengaluru 560012, India (e-mail: vrameshwar@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='in;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' nkashyap@iisc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 2 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' INTRODUCTION Constrained coding is a method that is employed in several domains such as magneto-optical recording (see, for example, [1] or [2]), DNA data storage [3], [4], and energy harvesting communication [5], [6], which allows the encoding of arbitrary user data sequences into only those sequences that respect a certain constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our interest in this paper is in constrained codes that are also resilient to symmetric errors and erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In our first approach, we consider the setting of the transmission of constrained codes over a noisy channel that induces errors stochastically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For example, if a code designer would like to write data onto a magnetic disk, then it is imperative that the data sequences satisfy a constraint, which is often imposed by the physical recording medium, and in addition, also be able to correct (possibly) random errors and erasures that are introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is in this context that we consider the transmission of constrained subcodes of binary linear codes: if the channel introducing errors or erasures is memoryless and symmetric, there are well-known binary linear codes that achieve the capacity or whose rates are very close to the capacity of the channel (see [7]–[9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, this means that constrained subcodes of such linear codes also enjoy vanishing error probabilities over such binary-input memoryless symmetric (BMS) channels (which include the binary symmetric channel (BSC) that introduces bit-flip errors and the binary erasure channel (BEC)), in the limit as the blocklength goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This observation can be useful, hence, for the construction of error-correcting constrained coding schemes of good rates over input-constrained BMS channels, without feedback [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We mention here that the problem of constructing capacity-achieving codes over input-constrained memoryless channels, which form a special class of finite-state channels (FSCs), is still wide open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' As part of our first approach, we are hence interested in the problem of determining the sizes of constrained subcodes of linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our approach to this counting problem makes use of a simple identity from the Fourier analysis of Boolean functions, namely, Plancherel’s Theorem, which transforms our counting problem to one in the space of the dual code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' An immediate advantage of this approach is that the dimension of the vector space over which we count, which is the minimum of the dimensions of the linear code and its dual, is always January 13, 2023 DRAFT JANUARY 2023 3 bounded above by half the blocklength of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We also show, using specific examples of constraints, that our approach can yield not just the values of the sizes of constrained subcodes of specific linear codes, but also interesting insights into the construction of linear codes with a presribed number of constrained codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' An important ingredient in our method is the Fourier transform of the indicator function that a word is constrained—an object that we believe is of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We show that in the cases of certain constraints, this Fourier transform is explicitly computable, and in some others, is efficiently calculable via a recursive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, in our second approach, we consider the situation when the constrained codewords we seek to transmit or store are subject to adversarial (or combinatorial) errors or erasures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' More precisely, we are interested in the scenario where there is an upper bound on the number of errors or erasures that the channel can introduce (with potential adversarial knowledge of the codeword as well), and we would like to recover our constrained codeword with zero probability of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is well-known (see, for example, [11]) that the minimum Hamming distance (or simply, minimum distance) of the constrained code determines the number of errors or erasures that it can tolerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, we seek to obtain good bounds on the sizes of constrained codes with a prescribed minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' There is extensive literature on the construction of and bounds for constrained codes with a certain minimum distance, and we refer the reader to Chapter 9 of [1] for references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, [12] provided a simple lower bound on the sizes of runlength-limited (RLL) constrained codes with a given minimum Hamming distance, by a coset-averaging argument for linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' These bounds were then improved upon by Kolesnik and Krachkovsky [13] and Marcus and Roth [14], via the solutions to certain constrained optimization problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Less was known in the case of upper bounds on constrained codes with a given minimum distance, until the works of Cullina and Kiyavash [15] (see also [16]) and Fazeli, Vardy, and Yaakobi [17], which provided a generalization of the well-known sphere packing bound for codes, to the setting of constrained codes1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The approach in [15] and [17] was based on finding the size of the largest matching, or 1While these papers were focussed on obtaining bounds on the sizes of codes for combinatorial error models, their techniques can be easily applied to determining upper bounds on the sizes of constrained codes with a given minimum distance as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 4 equivalently, the size of the smallest transversal, in a suitably defined hypergraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In this paper, we provide a different approach to deriving good upper bounds on the sizes of constrained codes with a given minimum Hamming distance, by modifying Delsarte’s well- known linear program (LP) [18] to the setting of constrained systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' While on a first pass, we propose an LP whose number of variables is exponential in the blocklength of the code, we show that for certain constraints, it is possible to “symmetrize” this LP to derive an equivalent LP with much smaller numbers of variables and constraints, which are sometimes only polynomial functions of the blocklength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We use our LPs to numerically calculate upper bounds on the sizes of the largest constrained codes with a prescribed minimum distance, for different constraints, and show that the values we obtain by our approach beat those obtained via the generalized sphere packing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We believe that further study of our LP with the aid of modern Fourier- analytic techniques (see [19], [20]) could provide insight into asymptotic upper bounds on the rate-distance tradeoff for constrained codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The remainder of the paper is organized as follows: Section II puts down notation and refreshes some background on binary codes and elementary Fourier analysis on the hypercube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Section III introduces the main theorem for counting constrained codewords in linear codes and our LPs for upper bounding the sizes of constrained codes with a given minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Section IV then applies the theorems and LPs discussed in Section III to different constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The paper is concluded in Section V with some directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' PRELIMINARIES A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Notation Given integers 푎, 푏, we use the notation [푎 : 푏] to denote the set {푎, 푎 + 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푏}, for 푎 ≤ 푏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We use [푛] as shorthand for [1 : 푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a vector a ∈ {0, 1}푛, we denote by 푤(a) its Hamming weight (or, simply, weight), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', 푤(a) is the number of ones present in a, and we use the notation supp(a) to denote the set of coordinates {푖 ∈ [푛] : 푎푖 = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We say that a length-푛 vector a is supported on 푆 ⊆ [푛], if supp(a) ⊆ 푆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given a pair of vectors a, b ∈ R푛, we define the Hamming distance (or simply, distance) 푑(a, b) between a and b as the number of coordinates January 13, 2023 DRAFT JANUARY 2023 5 in which they differ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', 푑(a, b) = |{푖 ∈ [푛] : 푎푖 ≠ 푏푖}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, given a length-푛 vector a, let 푎 푗 푖 denote the vector (푎푘 : 푖 ≤ 푘 ≤ 푗), for 푗 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given a length 푛, we let B푛(푖) denote the binary representation of 0 ≤ 푖 ≤ 2푛 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The subscript ‘푛’ will be dropped when evident from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for z ∈ {0, 1}푛, we define 푤z(a) to be the number of ones in a, in the positions in supp(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For 푟 ∈ {0, 1}, we define the vector 푟푚 to be the length-푚 vector all of whose symbols equal 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, given two binary vectors a = (푎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푎푛) and b = (푏1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푏푛), their dot product a · b equals �푛 푖=1 푎푖푏푖, and we denote their concatenation by ab or (a | b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For an 푚 × 푛 matrix 푀, we use the notation 푀[푖] to denote its 푖th column, for 푖 ∈ [푛], and 푀( 푗) to denote its 푗th row, for 푗 ∈ [푚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a given real number 푟 ∈ R, we define ⌊푟⌋ and ⌈푟⌉ to be, respectively, the largest integer less than or equal to 푟, and the smallest integer larger than or equal to 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for integers 푝, 푟, and 푛, with 푛 > 0 and 1 ≤ 푟 ≤ 푛, we say that 푝 ≡ 푟 (mod 푛), or that 푟 = mod(푝, 푛), if 푝 = 푞푛 + 푟, for some integer 푞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a set 푃 ⊆ {0, 1}푛, we use the notation 1푃(x) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 1, if x ∈ 푃, 0, otherwise, and we write 1푃 as the indicator function (1푃(x) : x ∈ {0, 1}푛).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a given length 푛, we also use the notation 푊푖 := {x ∈ {0, 1}푛 : 푤(x) = 푖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Block Codes and Constrained Sequences We recall the following definitions of block codes and linear codes over F2 (see, for example, Chapters 1 and 2 of [11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' An (푛, 푀) block code C over F2 is a nonempty subset of F푛 2, with |C| = 푀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The rate of the block code C is given by rate(C) := log2 푀 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The minimum distance 푑(C) of a block code C is the minimum distance between January 13, 2023 DRAFT JANUARY 2023 6 any two distinct codewords of C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', 푑(C) = min c1,c2∈C: c1≠c2 푑(c1, c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' An (푛, 푀) block code with minimum distance 푑 will be called an (푛, 푀, 푑) block code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Moreover, given a sequence of block codes {C(푛)}푛≥1, if it holds that rate(C(푛)) 푛→∞ −−−−→ 푅, for some 푅 ∈ [0, 1], then we say that {C(푛)}푛≥1 is of rate 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Definition II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' An [푛, 푘] linear code C over F2 is an (푛, 2푘) block code that is a subspace of F푛 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A constraint is represented by a set A ⊆ {0, 1}푛 of binary words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We call the sequences in A as constrained sequences, and refer to a block code C, all of whose codewords lie in A, as a “constrained code”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We refer the reader to [1] for a detailed exposition on constrained sequences and coding in the presence of constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that we make no further assumption about the constrained system (such as it being finite-type, almost-finite-type, irreducible, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given such a collection of constrained sequences of blocklength 푛, the noiseless capacity (see Chapter 3 of [1]) of the constraint is defined as 퐶0(A) := lim 푛→∞ log2 |A| 푛 , where the limit in the expression above exists, by subadditivity arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for a given blocklength 푛, we use the notation 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) to denote the size of the largest constrained code, of minimum distance at least 푑, such that all of its codewords lie in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' More formally, 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) := max C⊆A: 푑(C)≥푑 |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For the case where A = {0, 1}푛, we write 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) as simply 퐴(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Fourier Expansion of Boolean Functions Consider functions 푓 : {0, 1}푛 → R, mapping x = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푛) ∈ {0, 1}푛 to 푓 (x) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If the range of 푓 is {0, 1}, then 푓 is called a Boolean function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, given any function 푓 : {0, 1}푛 → R January 13, 2023 DRAFT JANUARY 2023 7 and a vector s = (푠1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푠푛) ∈ {0, 1}푛, we define the Fourier coefficient of 푓 at s as �푓 (s) := 1 2푛 � x∈{0,1}푛 푓 (x) · (−1)x·s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The function �푓 is known as the Fourier transform (sometimes called the Hadamard transform) of 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Moreover, the functions (휒s : s ∈ {0, 1}푛), where 휒s(x) := (−1)x·s, form a basis for the vector space 푉 of functions 푓 : {0, 1}푛 → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If we define an inner product ⟨·, ·⟩ over the vector space 푉, as follows: ⟨ 푓 , 푔⟩ := 1 2푛 � x∈{0,1}푛 푓 (x)푔(x), for functions 푓 , 푔 ∈ 푉, we also have that the basis functions (휒s : s ∈ {0, 1}푛) are orthonormal, in that ⟨휒s, 휒s′⟩ = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 1, if s = s′, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The above discussion leads us to the following well-known theorem: Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 (Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 in [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Every function 푓 : {0, 1}푛 → R can be uniquely expressed as its Fourier expansion 푓 (x) = � s∈{0,1}푛 �푓 (s) · (−1)x·s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For more details on the Fourier analysis of Boolean functions, we refer the reader to [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In our paper, we shall make use of Plancherel’s Theorem from Fourier analysis, which is recalled below, without proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Theorem II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 (Plancherel’s Theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For any 푓 , 푔 ∈ {0, 1}푛 → R, we have that ⟨ 푓 , 푔⟩ = � s∈{0,1}푛 �푓 (s)�푔(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We also recall the operation of convolution of two functions 푓 , 푔 : {0, 1}푛 → R, defined as 푓 ★ 푔(x) = 1 2푛 � z∈{0,1}푛 푓 (z) · 푔(x + z), January 13, 2023 DRAFT JANUARY 2023 8 where the ‘+’ operation in x + z above is over vectors in F푛 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is well-known (see [21]) that the Fourier transform � 푓 ★ 푔(s) = �푓 (s) · �푔(s), for any s ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' MAIN RESULTS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Counting Constrained Codewords in Linear Codes First, we work towards characterizing the number of constrained codewords in an arbitrary linear code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Consider an [푛, 푘] linear code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Suppose that we are interested in computing the number of codewords c ∈ C each of which satisfies a certain property, which we call a constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let A ⊆ {0, 1}푛 denote the set of length-푛 words that respect the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) denote the number of such constrained codewords in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We can then write 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) = � c∈C 1A(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (1) Observe that the summation in (1) is over a set of size 2푘, which could be quite large, especially when 푘 > 푛/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our interest is in obtaining insight into the summation above, by employing a simple trick from the Fourier expansions of Boolean functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our first observation is summarized below as a theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a linear code C, we denote its dual code by C⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given a linear code C of blocklength 푛 and a set A ⊆ {0, 1}푛, we have that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) = |C| · � s∈C⊥ � 1A(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The proof is a straightforward application of Plancherel’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) = � c∈C 1A(c) = � x∈{0,1}푛 1A(x) · 1C(x) January 13, 2023 DRAFT JANUARY 2023 9 = 2푛 · � s∈{0,1}푛 � 1A(s) · � 1C(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (2) Now, we claim that � 1C(s) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 |C| 2푛 , if s ∈ C⊥, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (3) To prove the above claim, recall that � 1C(s) = 1 2푛 � x∈{0,1}푛 1C(x) · (−1)x·s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If s ∈ C⊥, it holds that x · s = 0, for all x ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, in this case, we have that � 1C(s) = |C| 2푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If s ∉ C⊥, this means that there is some x★ ∈ C such that x★ · s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, it is true that 0 · s = 0, with 0 ∈ C, since C is a linear code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, to any c ∈ C such that c · s = 0, we can uniquely map the codeword c + x★ (where the summation is over F푛 2), with the property that �c + x★� · s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Since this map is bijective, we obtain that the number of codewords x ∈ C such that (−1)x·s = 1, equals the number of codewords x ∈ C such that (−1)x·s = −1, for s ∉ C⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Therefore, we get that in this case, � 1C(s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Plugging (3) back in (2), we see that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) = |C| · � s∈C⊥ � 1A(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 provides an alternative approach to addressing our problem of counting con- strained codewords in linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, note that if C had large dimension, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', if 푘 > 푛/2, then, it is computationally less intensive to calculate the number of constrained code- words using Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, provided we knew the Fourier coefficients � 1A(s), since dim (C⊥) = 푛 − 푘 < 푛/2, in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Additionally, if the structure of the Fourier coefficients is simple to handle, we could also use Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 to construct linear codes that have a large (or small) number of constrained codewords, or to obtain estimates of the number of constrained codewords in a fixed linear code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In the rest of the paper, we shall study various examples of constraints, January 13, 2023 DRAFT JANUARY 2023 10 which in turn correspond to sets A, whose Fourier coefficients � 1A(s) are computable either analytically or numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In Appendices A and B, we provide a Fourier-analytic perspective on the problem of calcu- lating the weight distribution of constrained sequences in the ambient space F푛 2 and in linear codes C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Appendix C discusses the connection between Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and the well-known MacWilliams’ identities for linear codes [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In Section IV, we shall look at specific examples of constraints and apply Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, as recurring motifs, we shall consider the [2푚−1, 2푚−1−푚] binary Hamming code, for 푚 ≥ 1 and the binary Reed-Muller codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Since the constraints we work with are sensitive to the ordering of coordinates of the code, in the sense that a permutation of the coordinates can transform a codeword that does not satisfy the constraint into one that does, we shall first fix a canonical ordering of coordinates for the codes that we analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For the binary Hamming code, we assume that a parity-check matrix 퐻Ham is such that 퐻Ham[푖] = B푚(푖), for 1 ≤ 푖 ≤ 2푚 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that the Reed-Muller (RM) family of codes are codes of large blocklength, which are known to achieve the capacities of BMS channels under bit-MAP decoding [8] (see also [23]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' RM codes are hence linear codes that offer the maximum resilience to symmetric, stochas- tic noise, for a given rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For 푚 ≥ 1 and 푟 ≤ 푚, the 푟th-order binary Reed-Muller code RM(푚, 푟) is the set of binary vectors obtained as evaluations of multilinear Boolean polynomials 푓 (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푚), in the variables 푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푚, of maximum degree 푟, on points of the unit hypercube (see Chapter 13 of [24] for more information on Reed-Muller codes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We use the convention that the coordinates of RM(푚, 푟) are written as binary 푚-tuples that are ordered according to the standard lexicographic ordering, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', the 푖th coordinate from the start is the 푚-tuple B푚(푖 − 1), for 1 ≤ 푖 ≤ 2푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We thus have that the blocklength of RM(푚, 푟) is 푛 = 2푚 and dim(RM(푚, 푟)) = �푟 푖=0 �푚 푖 � =: � 푚 ≤푟 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A Linear Program for Constrained Systems In this section, we consider the problem of upper bounding the sizes of constrained codes with a prescribed minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, we present a linear program (LP) to upper bound 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), for any A ⊆ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This LP is based on Delsarte’s linear programming January 13, 2023 DRAFT JANUARY 2023 11 approach [18] to bounding from above the value of 퐴(푛, 푑), for 푛 ≥ 1 and 1 ≤ 푑 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We first recall Delsarte’s LP2, which we call Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given an LP L, we denote by val(L) its optimal value, and for any feasible solution 푓 of L, we denote the value of the objective function of L evaluated at 푓 as valL( 푓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The subscript will be omitted when the LP being referred to is clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We remark here that the LPs in this paper can return non-integral optimal values, and that integer upper bounds on the sizes of codes can be obtained by suitable rounding of real numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Del(푛, 푑) maximize 푓 : {0,1}푛→R � x∈{0,1}푛 푓 (x) (Obj) subject to: 푓 (x) ≥ 0, ∀ x ∈ {0, 1}푛, (C1) �푓 (s) ≥ 0, ∀ s ∈ {0, 1}푛, (C2) 푓 (x) = 0, if 1 ≤ 푤(x) ≤ 푑 − 1, (C3) 푓 (0푛) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (C4) Now, for any block code C of blocklength 푛 and minimum distance at least 푑, let 1C denote its indicator function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let us define3 푓C = 2푛 |C|1C ★ 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We claim that 푓C is a feasible solution for Del(푛, 푑), with the objective function (Obj) evaluating to |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Indeed, observe that (C1) is trivially satisfied, by the definition of the convolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, since � 푓C = 2푛 |C| · � 1C 2, it holds that (C2) is satisfied as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, note that since C is such 2The version of Delsarte’s LP that is most often used in papers in coding theory, such as in [25], is obtained after symmetrizing Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, the common version of Delsarte’s LP is Del/푆푛 (푛, 푑) (see the remark following Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2), where 푆푛 is the symmetry group on 푛 elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 3Note that when C is a linear code (or a subspace of F푛 2), for all x, z ∈ {0, 1}푛, we have that 1C(z)1C (x+z) = 1C(z)1C (x), and hence 푓C evaluates to simply 1C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 12 that 푑(C) ≥ 푑, it holds that 1C(x + z) = 0, for all z ∈ C and any x such that 1 ≤ 푤(x) ≤ 푑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, we have that 1C(z) · 1C(x + z) = 0, for all z ∈ {0, 1}푛, if 1 ≤ 푤(x) ≤ 푑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, (C3) is also true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Finally, observe that 푓C(0푛) = 1 |C| � z∈{0,1}푛 1C(z) · 1C(z) = 1 |C| � z∈{0,1}푛 1C(z) = 1, thereby satisfying (C4) also.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, it holds that the objective value of 푓C, which is � x∈{0,1}푛 푓C(x), is given by � x∈{0,1}푛 푓C(x) = 1 |C| � x∈{0,1}푛 � z∈{0,1}푛 1C(z) · 1C(x + z) = 1 |C| � z∈{0,1}푛 1C(z) � x∈{0,1}푛 1C(x + z) = |C|2 |C| = |C|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (4) Hence, it holds that the optimal value of Del(푛, 푑), is an upper bound on |C|, for all block codes C of blocklength 푛 and minimum distance at least 푑, and therefore also an upper bound on 퐴(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We refer the reader to [18], [19], [25]–[28] and the references therein for a more detailed treatment of linear programming-based upper bounds on the sizes of block codes and linear codes, and for the derivation of analytical upper bounds via the dual LP or using modern Fourier-theoretic or expander graph-based arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our LP, which we call Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), is but a small modification of Del(푛, 푑), to take into account the fact that all codewords of the code of minimum distance at least 푑, whose size we are attempting to bound, must also lie in the set A ⊆ F푛 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 13 Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) maximize 푓 : {0,1}푛→R � x∈{0,1}푛 푓 (x) (Obj′) subject to: 푓 (x) ≥ 0, ∀ x ∈ {0, 1}푛, (D1) �푓 (s) ≥ 0, ∀ s ∈ {0, 1}푛, (D2) 푓 (x) = 0, if 1 ≤ 푤(x) ≤ 푑 − 1, (D3) 푓 (0푛) ≤ val(Del(푛, 푑)), (D4) 푓 (x) ≤ 2푛 · (1A ★ 1A)(x), ∀ x ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D5) Like in the case with Del(푛, 푑), we produce a feasible solution for Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), and claim that val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) is at least (퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A))2 (note the difference with Del(푛, 푑), whose value is at least 퐴(푛, 푑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' To this end, let CA be any length-푛 constrained code, with 푑(CA) ≥ 푑, such that all codewords in CA lie in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that we can write CA as C ∩ A, for some block (not necessarily constrained) code C, with 푑(C) ≥ 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Thus, an upper bound on max C: 푑(C)≥푑 |C ∩A| serves as an upper bound on (and in fact, equals) 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 1C be the indicator function of a block code C as above, and let 1A be the indicator function of the constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We define 푓C,A = 2푛 · (1C1A ★ 1C1A), and claim that 푓C,A is a feasible solution for Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), with the objective function (Obj′) evaluating to |C ∩ A|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' To see this, note that the LP constraints (D1)–(D3) are satisfied for the same reasons as why 푓C satisfied (C1)–(C3) in Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Furthermore, 푓C,A(0푛) = � z∈{0,1}푛 1C∩A(z) January 13, 2023 DRAFT JANUARY 2023 14 = |C ∩ A| ≤ |C| ≤ val(Del(푛, 푑)), since C is a block code of distance at least 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, (D4) is satisfied by 푓C,A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Finally, observe that for any x ∈ {0, 1}푛, 푓C,A(x) = � z∈{0,1}푛 1C1A(z) · 1C(x + z)1A(x + z) ≤ � z∈{0,1}푛 1A(z) · 1A(x + z) = 2푛 · (1A ★ 1A)(x), showing that (D5) also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, note that val( 푓C,A), which is � x∈{0,1}푛 푓C,A(x), equals |C ∩ A|2, by calculations as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, we have that val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) ≥ |C ∩ A|2, for any C with minimum distance at least 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Therefore, it holds that val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) ≥ (퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A))2, or, that 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) ≤ (val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We now discuss a couple of observations about Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Firstly, note that by (D5), we have that for any feasible solution 푓 of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), it holds that the objective value � x∈{0,1}푛 푓 (x) ≤ � x∈{0,1}푛 � z∈{0,1}푛 1A(z) · 1A(x + z) = � z∈{0,1}푛 1A(z) � x∈{0,1}푛 1A(x + z) = |A|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Therefore, we have that our upper bound that is (val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)))1/2 is less than or equal to |A|, which is the total number of constrained sequences of length 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Second, from the proof of Proposition 1 in [28], we obtain that for any feasible solution 푓 of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), for all x ∈ {0, 1}푛, it holds that 푓 (x) ≤ 푓 (0푛) ≤ val(Del(푛, 푑)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, we have that the objective value of 푓 , for any feasible 푓 , obeys � x∈{0,1}푛 푓 (x) ≤ �� 푛 ≥ 푑 � + 1 � val(Del(푛, 푑)), since 푓 (x) = 0, if 1 ≤ wt(x) ≤ 푑 − 1, where we define � 푛 ≥푑 � := �푛 푖=푑 �푛 푖 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Therefore, we have that (val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)))1/2 ≤ �� 푛 ≥푑 � + 1 �1/2 (val(Del(푛, 푑)))1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Moreover, for most constraints that we ran either Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) or the equivalent symmetrized LP Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) (see Section January 13, 2023 DRAFT JANUARY 2023 15 III-C) on, the value of the LP came out to be strictly less than val(Del(푛, 푑)) (see also Corollary III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we derive an upper bound on the optimal value val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) of our LP, in the following lemma that is essentially a formulation of the dual LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' While we do not apply Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 elsewhere in this paper, we believe that it will serve useful in the derivation of asymptotic (as the blocklength goes to infinity) upper bounds on the rate-distance tradeoff for constrained codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We abbreviate val(Del(푛, 푑)) as 푣 and val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) as 푣A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 훽 : {0, 1}푛 → R be a function that satisfies �훽(s) ≥ 0, for all s ∈ {0, 1}푛, and � x 훽(x) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Then, 푣A ≤ 2푛 · \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 훽(0푛) · min{푣, |A|} + 2푛 · � x: 푤(x)≥푑 훽(x) · (1A ★ 1A)(x) \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Consider any function 휆 : {0, 1}푛 → [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, observe that for any feasible solution 푓 of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), we have that 휆(0푛) · �푓 (0푛) (푎) ≤ � x∈{0,1}푛 휆(x) · �푓 (x) (푏)= 2푛 · � s∈{0,1}푛 �휆(s) · 푓 (s) 2푛 (푐)= � s: s = 0푛 or 푤(s)≥푑 �휆(s) · 푓 (s) (푑) ≤ �휆(0푛) · min{푣, |A|} + 2푛 · � s: 푤(s)≥푑 �휆(s) · (1A ★ 1A)(s), where (a) holds since 휆, �푓 ≥ 0, (b) holds by an application of Plancherel’s Theorem, along with the fact that � ( �푓 ) = 2−푛 · 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, (c) is true since 푓 satisfies (D3) and (d) holds since 푓 satisfies (D4) and (D5), and since 2푛 · (1A ★ 1A)(0푛) = |A|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Moreover, we have that val( 푓 ) = � x∈{0,1}푛 푓 (x) = 2푛 · �푓 (0푛), and that 휆(0푛) = � s∈{0,1}푛 �휆(s), by the definition of the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Putting everything together, we obtain that for any January 13, 2023 DRAFT JANUARY 2023 16 feasible 푓 of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), it holds that val( 푓 ) ≤ 2푛 � s �휆(s) \uf8ee\uf8ef\uf8ef\uf8ef\uf8ef\uf8f0 �휆(0푛) · min{푣, |A|} + 2푛 · � s: 푤(s)≥푑 �휆(s) · (1A ★ 1A)(s) \uf8f9\uf8fa\uf8fa\uf8fa\uf8fa\uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Finally, by substituting 훽 as �휆 and ensuring that � x �훽(x) = 1, we obtain the thesis of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ The following simple corollary then follows from Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Corollary III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If |A| ≤ 2푛/2, it holds that 푣A ≤ 푣 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that in Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 above, we can pick 훽(x) = 2−푛, for all x ∈ {0, 1}푛, so that �훽(s) = 2−2푛, if s = 0푛 and �훽(s) = 0, for s ≠ 0푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Then, we obtain that 푣A ≤ min{푣, |A|} + 2−푛 · � x: 푤(x)≥푑 � z 1A(z) · 1A(x + z) ≤ min{푣, |A|} + |A|2 2푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, if |A| ≤ 2푛/2, we get that 푣A ≤ 푣 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Symmetrizing Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) The linear program Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) discussed in Section III-B, for a fixed A ⊆ F푛 2, suffers from the drawback that the variables, which are precisely the values ( 푓 (x) : x ∈ {0, 1}푛), are 2푛 in number, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', exponentially large in the blocklength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The number of LP constraints, similarly, are exponentially large in 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It would therefore be of interest to check if the size of the linear program Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), which is the sum of the number of variables and the number of LP constraints, can be reduced, using symmetries present in the formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our exposition in this section on symmetrizing Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), follows that in [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 푆푛 denote the symmetry group on 푛 elements, which is the set of all permutations 휎 : [푛] → [푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that given a length-푛 vector x = (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푛) ∈ {0, 1}푛, a permutation 휎 ∈ 푆푛 acts on x as follows: 휎 · x = (푥휎(1), 푥휎(2), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥휎(푛)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 17 The permutation 휎 also acts on functions 푓 : {0, 1}푛 → R via the mapping (휎◦ 푓 )(x) = 푓 (휎·x), for x ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, given any set A ⊆ F푛 2, we define the “symmetry group” of the constraint represented by A to be the set of all permutations 휋 ∈ 푆푛 that leave the indicator function 1A invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, the symmetry group 퐺A of the constraint represented by A is the set of all permutations 휋 ∈ 푆푛 such that 1A = 휋 ◦ 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given a group 퐺 ⊆ 푆푛 of permutations, which acts on the vectors x ∈ {0, 1}푛, we say that Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) is 퐺-invariant, if for all 휎 ∈ 퐺, it holds that if 푓 : {0, 1}푛 → R is a feasible solution to Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), then so is 휎 ◦ 푓 , with val( 푓 ) = val(휎 ◦ 푓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In what follows, we shall prove that Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) is, in fact, 퐺A-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 휋 ∈ 퐺A be a permutation in the symmetry group of A and let 푓 be some feasible solution to Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D1) It is clear that if 푓 (x) ≥ 0, then 푓 (휋 · x) ≥ 0, for all x ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D2) The fact that if �푓 (s) ≥ 0, then it holds that � 휋 ◦ 푓 (s) ≥ 0, for all s ≥ 0, follows from the simple lemma below: Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For any function 푓 : {0, 1}푛 → R and for any permutation 휎 ∈ 푆푛, it holds that � 휎 ◦ 푓 (s) = (휎 ◦ �푓 )(s), for all s ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that � 휎 ◦ 푓 (s) = � x∈{0,1}푛 푓 (휎 · x) · (−1)x·s = � x∈{0,1}푛 푓 (휎 · x) · (−1)(휎·x)·(휎·s) = � x∈{0,1}푛 푓 (x) · (−1)x·(휎·s) = (휎 ◦ �푓 )(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ (D3) Since any permutation in 퐺A also lies in 푆푛 and hence preserves the weights of vectors in {0, 1}푛, it holds that (휋 ◦ 푓 )(x) = 0, for all x ∈ {0, 1}푛 such that 1 ≤ 푤(x) ≤ 푑 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D4) This constraint is also satisfied by 휋 ◦ 푓 , since 휋(0푛) = 0푛, for all 휋 ∈ 퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 18 (D5) Observe that for any 휋 ∈ 퐺A, 2푛 · (1A ★ 1A)(휋 · x) = � z∈{0,1}푛 1A(z) · 1A(휋 · x + z) = � z∈{0,1}푛 1A(휋 · z) · 1A(휋 · x + 휋 · z) = � z∈{0,1}푛 1A(휋 · z) · 1A(휋 · (x + z)) = � z∈{0,1}푛 1A(z) · 1A(x + z) = 2푛 · (1A ★ 1A)(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, since for all x ∈ {0, 1}푛, we have that 휋 ◦ 푓 (x) ≤ 2푛 · (1A ★ 1A)(휋 · x) = 2푛 · (1A ★ 1A)(x), we have that (D5) is also satisfied by 휋 ◦ 푓 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (Obj′) It is clear that � x 푓 (x) = � x 푓 (휋 · x), and hence that val( 푓 ) = val(휋 ◦ 푓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From the preceding discussion, we see that given a feasible solution 푓 to Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), we can construct the function 푓 := 1 |퐺A| � 휋∈퐺 A 휋 ◦ 푓 , such that 푓 is also a feasible solution to the LP (by linearity), with val( 푓 ) = val( 푓 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe, in addition, that 푓 is such that 휋 ◦ 푓 = 푓 , for all 휋 ∈ 퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, it follows that in order to arrive at an optimal solution to Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), one can restrict oneself to searching among feasible solutions 푓 that are constant on each orbit 푂 in {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Such functions 푓 can be expressed as 푓 (x) = � 푂∈{0,1}푛/퐺 A 푎푂 · 1푂(x), (5) where 푎푂 ∈ R, for all 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Before we work on symmetrizing the constraints of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), we introduce some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For an orbit 푂 ∈ {0, 1}푛/퐺A, we denote by |푂| the number of elements in the orbit and by x푂 (or s푂) a representative element of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, January 13, 2023 DRAFT JANUARY 2023 19 for a given element x ∈ {0, 1}푛, we define 푂(x) to be the orbit in which x lies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We shall now formulate the constraints (D1)–(D5) and the objective function (Obj′) in Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) based on (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D1′) The fact that 푓 (x) ≥ 0 for all x implies that 푎푂 ≥ 0, for all 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D2′) By the linearity of the Fourier transform operation, we obtain that �푓 (s) = � 푂∈{0,1}푛/퐺 A 푎푂 · � 1푂(s) ≥ 0, for all s ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In fact, note that since 퐺A ⊆ 푆푛, it can be argued using Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 that the above inequality only needs to hold for orbit representatives s푂 ∈ {0, 1}푛, of 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Indeed, we have that for any 휋 ∈ 퐺A, �푓 (휋 · s) = � 휋 ◦ 푓 (s) = �푓 (s), where the first equality holds by Lemma III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 and the second holds since 휋 ◦ 푓 = 푓 , for functions 푓 as in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D3′) The constraint (D3) can be written as 푎푂 = 0, for all 푂 such that 1 ≤ 푤(x푂) ≤ 푑 − 1, where x푂 ∈ {0, 1}푛 is a representative element of the orbit 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D4′) The constraint (D4) becomes 푎푂(0푛) ≤ val(Del(푛, 푑)), where 푂(0푛) is the orbit that contains the all-zeros word 0푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D5′) Similarly, the constraint (D5) reduces to 푎푂 ≤ 2푛 · (1A ★ 1A)(x푂), January 13, 2023 DRAFT JANUARY 2023 20 where, again, x푂 is some representative element of the orbit 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (Obj′′) From (5), we see that the new objective function simply becomes maximize 푎푂∈R � 푂∈{0,1}푛/퐺 A |푂| · 푎푂.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We call the symmetrized version of Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) as Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), which is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) maximize {푎푂∈R: 푂∈{0,1}푛/퐺 A} � 푂 |푂| · 푎푂 (Obj′′) subject to: 푎푂 ≥ 0, ∀ 푂 ∈ {0, 1}푛/퐺A, (D1′) � 푂∈{0,1}푛/퐺 A 푎푂 · � 1푂(s ˜푂) ≥ 0, ∀ orbit rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' s ˜푂 ∈ {0, 1}푛, (D2′) 푎푂 = 0, if 1 ≤ 푤(x푂) ≤ 푑 − 1, (D3′) 푎푂(0푛) ≤ val(Del(푛, 푑)), (D4′) 푎푂 ≤ 2푛 · (1A ★ 1A)(x푂), ∀ 푂 ∈ {0, 1}푛/퐺A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D5′) The preceding discussion can then be summarized as a theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The LPs Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) and Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) are equivalent in that val(Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)) = val(Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Remark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' All the above arguments remain valid if we use a subgroup 퐻 of the symmetry group 퐺A as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For the special case when A = {0, 1}푛, it holds that 퐺A = 푆푛, and we then recover the more common version of Delsarte’s LP that is 푀LP(푛, 푑) in [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is this version that we use for evaluating the right-hand side of constraints (D4) and (D4′) in our numerical examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that in the symmetrized LP Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), the number of variables is the number January 13, 2023 DRAFT JANUARY 2023 21 푁A of orbits 푂 ∈ {0, 1}푛/퐺A and the number of constraints is at most 4푁A + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, if the constraint is such that the number of orbits 푁A induced by its symmetry group is small (as a function of the blocklength 푛), then the size of the symmetrized LP is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In the subsections that follow, we shall explicitly write down Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), for select constraints (or sets A), and provide numerical results obtained by running Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) on those constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' EXAMPLES We now take up specific examples of constrained sequences and apply Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and the LP discussed in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 2-Charge Constraint In this subsection, we work with a special kind of a spectral null constraint [29], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This constraint that we shall study is the so-called 2-charge constraint (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4 in [1]), whose sequences have a spectral null at zero frequency (such a constraint is also called a DC- free constraint).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The 2-charge constraint admits only sequences y ∈ {−1, +1}푛, whose running sum �푟 푖=1 푦푖, for any 1 ≤ 푟 ≤ 푛, obeys 0 ≤ �푟 푖=1 푦푖 ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The graph in Figure 1 below represents the constraint, in that all sequences y ∈ {−1, +1}푛 that are 2-charge constrained can be read off the labels of edges of this graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The nodes (or states) of the graph represent the values that �푟 푖=1 푦푖 can take, for any 1 ≤ 푟 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We take the initial state to be the 0 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 0 1 2 +1 +1 −1 −1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 1: State transition graph representing the 2-charge constraint To any sequence x ∈ {0, 1}푛, we map (in a one-one manner) the sequence y = ((−1)푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , (−1)푥푛) ∈ {−1, +1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 푆(푛) 2 denote the set of sequences x ∈ {0, 1}푛 such that y = ((−1)푥1, (−1)푥2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , (−1)푥푛) is 2-charge constrained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We drop the superscript ‘(푛)’ when clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The set of constrained sequences of interest to us, hence, is A = 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Figure 2 shows a state transition graph for sequences in the set 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 22 0 1 2 0 0 1 1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 2: State transition graph for sequences in the set 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let the labelled, directed graph 퐺 = (푉, 퐸, L) represent the state transition graph above, with 푉 = {0, 1, 2} being the set of states, 퐸 ⊆ 푉 × 푉 being the set of directed edges, and L : 퐸 → {0, 1} being the labelling function that assigns to each edge a label that is 0 or 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For example, L((1, 2)) = 0 and L((1, 0)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We assume that the initial state is 푣0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Since labels of paths in the state transition graph (beginning at state 0) correspond to binary sequences x ∈ 푆2, we denote by 푥푖 the label of the 푖th edge in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that 푥1 = 0, by our choice of initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for a given path in the graph, we let 푣푖 denote the 푖th state, which is the terminal state of the 푖th edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, given such a state transition graph, where the labels of distinct outgoing edges from a state are different, one can define a transition function 휙 : 푉 × {0, 1} → 푉, with the property that 푣푖 = 휙(푣푖−1, 푥푖), for 1 ≤ 푖 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, observe that for any x ∈ 푆2, it holds that the state 푣2푖−1, for any 1 ≤ 푖 ≤ 푛, equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Owing to this fact, the label of the 푗th edge, 푥 푗, in any path in the graph 퐺 in Figure 2, can be either 0 or 1, when 푗 = 2푖, and is fixed to be exactly one of 0 or 1, when 푗 = 2푖 + 1, based on the label of the ( 푗 − 1)th edge, for 1 ≤ 푗 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, it holds that 푥2푖 + 푥2푖+1 = 1 (over the reals), for all 1 ≤ 푖 ≤ � 푛 2 � , if 푛 is odd, and for all 1 ≤ 푖 ≤ 푛 2 − 1, if 푛 is even, with 푥1 fixed to be 0 in both cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From this observation, we see that for blocklength 푛, it holds that |푆2| = 2⌊ 푛 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We now state a lemma that completely determines the Fourier transform of 1푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' But before we do so, we need some more notation: we define the set of vectors B = � b0, b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , b⌈ 푛 2⌉−1 � , where b0 = 10푛−1 and for 1 ≤ 푖 ≤ �푛 2 � − 1, the vector b푖 is such that 푏푖,푗 = 1, for 푗 ∈ {2푖, 2푖 + 1}, and 푏푖,푗 = 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For example, when 푛 = 5, we have that B = {10000, 01100, 00011}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 푉B = span(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a = � 푎0, 푎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푎⌈ 푛 2⌉−1 � ∈ {0, 1}⌈ 푛 2⌉, consider s = ⌈ 푛 2⌉−1 � 푖=0 푎푖 · b푖 (where the January 13, 2023 DRAFT JANUARY 2023 23 summation is over F푛 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It holds that � 1푆2 (s) = 2⌊ 푛 2⌋−푛 · (−1)푤(a)−푎0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for s ∉ 푉B, we have that � 1푆2 (s) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' First, we note that for any s ∈ {0, 1}푛, � 1푆2(s) = 1 2푛 � x∈{0,1}푛 1푆2(x) · (−1)x·s = 2−푛 · (#{x ∈ 푆2 : 푤s(x) is even} − #{x ∈ 푆2 : 푤s(x) is odd}) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (6) Now, for s = b0, note that since all words x ∈ 푆2 have 푥1 = 0, we have that #{x ∈ 푆2 : 푤s(x) is even} = |푆2| = 2⌊ 푛 2⌋, and #{x ∈ 푆2 : 푤s(x) is odd} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Plugging back in (6), we get that � 1푆2(b0) = 2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, recall that since 푣2푖−1 = 1, for any 1 ≤ 푖 ≤ 푛, we have that 푥2푖 + 푥2푖+1 = 1 (over F2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, we see that for s = b푗, for 1 ≤ 푗 ≤ �푛 2 � −1, it is true that #{x ∈ 푆2 : 푤s(x) is odd} = |푆2| = 2⌊ 푛 2⌋ and #{x ∈ 푆2 : 푤s(x) is even} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Substituting in (6), we get that � 1푆2(b푗) = −2⌊ 푛 2⌋−푛 for all 1 ≤ 푗 ≤ �푛 2 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Furthermore, we claim that � 1푆2(0푛) = 2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' To see this, note that � 1푆2(0푛) = 1 2푛 � x∈{0,1}푛 1푆2(x) · (−1)x·0푛 = 1 2푛 � x∈푆2 1 = |푆2| 2푛 = 2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, suppose that for some s1, s2 ∈ 푉B, it holds that � 1푆2(s1) = (−1)푖1 · 2⌊ 푛 2⌋−푛 and � 1푆2(s2) = (−1)푖2 · 2⌊ 푛 2⌋−푛, for some 푖1, 푖2 ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From the arguments above, we can deduce that for all x ∈ 푆2, it holds that 푤s1(x) is even, if 푖1 = 0, and for all x ∈ 푆2, we have that 푤s1(x) is odd, if 푖1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Similar arguments hold for 푤s2(x) as well, for all x ∈ 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, it can be checked that if 푖1 = 푖2, it holds that 푤s1+s2(x) is even, and hence, � 1푆2(s1 + s2) = 2⌊ 푛 2⌋−푛 = (−1)푖1+푖2 · 2⌊ 푛 2⌋−푛, and if 푖1 ≠ 푖2, it holds that 푤s1+s2(x) is odd, and hence, � 1푆2(s1 + s2) = −2⌊ 푛 2⌋−푛 = (−1)푖1+푖2 · 2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By applying this fact iteratively, and using the expressions for the Fourier coefficients � 1푆2(b푗), for 0 ≤ 푗 ≤ �푛 2 � − 1, we obtain the first part of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 24 To show that � 1푆2 (s) = 0 for s ∉ 푉B, we use Plancherel’s Theorem again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that |푆2| 2푛 (푎)= 1 2푛 � x∈{0,1}푛 1푆2(x) (푏)= 1 2푛 � x∈{0,1}푛 12 푆2(x) (푐)= � s∈{0,1}푛 � � 1푆2(s) �2 (푑)= � s∈푉B � � 1푆2(s) �2 + � s∉푉B � � 1푆2(s) �2 , where (b) holds since 1푆2 is a boolean function, and (c) holds by Plancherel’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, by equating the left side of equality (a) and the right side of equality (d), we see that |푆2| 2푛 = � s∈푉B � � 1푆2(s) �2 + � s∉푉B � � 1푆2(s) �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (7) However, from the first part of the lemma, we get that � s∈푉B � � 1푆2(s) �2 = |푉B| · 22·(⌊ 푛 2⌋−푛) (푒)= 2⌈ 푛 2⌉ · 22·(⌊ 푛 2⌋−푛) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f3 2− 푛 2 , if 푛 is even, 2−( 푛+1 2 ), if 푛 is odd = |푆2| 2푛 , where equality (e) follows from the fact that |푉B| = 2⌈ 푛 2⌉, since 푉B = span(B) and the vectors in B are linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, plugging back in (7), we obtain that � s∉푉B � � 1푆2(s) �2 = 0, implying that � 1푆2(s) = 0, for all s ∉ 푉B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 informs the construction of linear codes C that have a large number of codewords c ∈ 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, note that from Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, we have that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · � s∈C⊥ � 1푆2(s) January 13, 2023 DRAFT JANUARY 2023 25 = |C| · � s∈C⊥∩푉B � 1푆2(s), (8) where, for s ∈ C⊥ ∩ 푉B, with s = �⌈ 푛 2⌉−1 푖=0 푎푖 · b푖, for some a = � 푎0, 푎1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푎⌈ 푛 2⌉−1 � ∈ {0, 1}⌈ 푛 2⌉, we have that � 1푆2(s) = 2⌊ 푛 2⌋−푛 · � (−1) �⌈ 푛 2⌉−1 푗=1 푎 푗 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, suppose that C is such that C⊥ does not satisfy the criterion (C) below: (C) For all s ∈ C⊥ ∩ 푉B, it holds that � 1푆2(s) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If (C) does not hold, then, it implies that for some s★ ∈ C⊥ ∩ 푉B, it holds that � 1푆2(s★) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, following the reasoning in the proof of Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, since C⊥ ∩푉B is a vector space, we have that via the map s ↦→ s+s★, the number of elements s ∈ C⊥∩푉B such that � 1푆2(s) < 0 equals the number of elements s ∈ C⊥ ∩푉B such that � 1푆2(s) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Furthermore, since ��� � 1푆2(s) ��� = 2⌊ 푛 2⌋−푛, for all s ∈ C⊥ ∩ 푉B, we get from (8) that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = 0, in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, in order to construct linear codes C such that 푁(C, 푆2) > 0, we require that criterion (C) is indeed satisfied by the dual code C⊥ of C, with � 1푆2(s★) > 0, for some s★ ∈ C⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' With this instruction in mind, we can construct linear codes C such that its dual code C⊥ contains 푡 linearly independent vectors (s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , s푡) with � 1푆2(s푖) > 0, for all 1 ≤ 푖 ≤ 푡, and no vectors s ∈ 푉B with � 1푆2(s) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In such a case, we obtain that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · 2푡+⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From the structure of 푉B, we see that the largest number of vectors s ∈ {0, 1}푛 such that � 1푆2(s) > 0, equals |푉B| 2 = 2⌈ 푛 2⌉−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, the largest number of linearly independent vectors 푡 as above, is �푛 2 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The discussion above is summarized below as a lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For any linear code C of blocklength 푛 ≥ 1, the following are true: 1) If criterion (C) is not satisfied, then, 푁(C, 푆2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 2) If criterion (C) is satisfied and there exist 푡푛 ∈ � 1 : �푛 2 � − 1 � linearly independent vectors �s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , s푡푛 � in C⊥ with � 1푆2(s푖) > 0, for all 1 ≤ 푖 ≤ 푡푛, then, 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · 2푡푛+⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We thus understand that given a linear code whose dual code satisfies item 2 of Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2, January 13, 2023 DRAFT JANUARY 2023 26 the rate of the largest constrained subcode, C2, of C, all of whose codewords are in 푆2, obeys rate (C2) = log2 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) 푛 = log2 � |C| · 2푡푛+⌊ 푛 2⌋−푛� 푛 = log2 (|C|) 푛 + 푡푛 + � 푛 2 � − 푛 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, given a sequence of linear codes � C(푛)� 푛≥1 satisfying item 2 of Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2, if it holds that rate(C(푛)) 푛→∞ −−−−→ 푅 ∈ (0, 1), then, the rate of their largest constrained subcodes � C(푛) 2 � 푛≥1, all of whose codewords are in 푆2, obeys lim inf 푛→∞ rate � C(푛) 2 � = 푅 − 1 2 + lim inf 푛→∞ 푡푛 푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (9) From [31]4, we observe that for the constraint identified by the set 푆2, there exist cosets of the linear codes � C(푛)� 푛≥1 with rate(C(푛)) 푛→∞ −−−−→ 푅, the rate of the constrained subcodes of which (in the limit as the blocklength goes to infinity) is at least 푅 − 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From (9), since 푡푛 ∈ � 1 : �푛 2 � − 1 � , we see that we can construct a sequence of linear codes whose constrained subcodes are of rate larger than or equal to the coset-averaging lower bound in [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, it is possible to achieve the coset-averaging rate lower bound (and potentially more) by using the linear code itself, instead of one of its cosets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Specifically, suppose that we choose 푡푛 = �푛 2 � − 푝푛, for some positive integer 푝푛 such that lim푛→∞ 푝푛 푛 = 0, thereby making dim�C⊥ 푛 � ≥ ⌈ 푛 2⌉−푝푛 푛 , where C⊥ 푛 is the dual code of C푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that this implies that 1 − 푅 = lim푛→∞ rate (C⊥) ≥ 1 2, and hence that 푅 ∈ (0, 1 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In this case, by plugging into (9), we obtain that the rate of the largest constrained subcodes � C(푛) 2 � 푛≥1 of � C(푛)� 푛≥1 is lim 푛→∞ rate � C(푛) 2 � = 푅 − 1 2 + lim 푛→∞ 푡푛 푛 = 푅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 4Such a result is also attributed to Elias and Bassalygo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 27 In other words, in the case where 푡푛 = �푛 2 � − 푝푛, for 푝푛 > 0 as above, the asymptotic rate of the codewords that lie in 푆2 equals the asymptotic rate 푅 ∈ (0, 1 2] of the code itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we shall make use of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 to compute the number of codewords of specific linear codes C, which lie in 푆2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We first consider the binary single parity-check code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For C being the [푛, 푛 − 1] single parity-check code, we have that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = \uf8f1\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f2 \uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f4\uf8f3 2⌊ 푛 2⌋−1, if 푛 is even, 2⌊ 푛 2⌋, if 푛 = 4푧 + 1, for some non-negative integer 푧, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that if C is the [푛, 푛 − 1] single parity-check code, then C⊥ is the [푛, 1] binary repetition code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Consider the case when 푛 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We shall plug in the Fourier coefficients given in Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 in (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that in this case, C⊥ ∩ 푉B = 0푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · � 1푆2(0푛) = 2푛−1 · 2⌊ 푛 2⌋−푛 = 2⌊ 푛 2⌋−1, where the second equality applies Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Similarly, when 푛 = 4푧 +1, for some non-negative integer 푧, it can be verified that the all-ones vector 1푛 equals �⌈ 푛 2⌉−1 푖=0 b푖 = �⌈ 푛 2⌉−1 푖=0 푎푖 · b푖, where 푎푖 = 1, for all 0 ≤ 푖 ≤ �푛 2 � − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, it can be checked that since 푛 = 4푧 +1, we have that � 1푆2(1푛) = 2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, in this case, we get that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · � � 1푆2(0푛) + � 1푆2(1푛) � = 2⌊ 푛 2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Finally, when 푛 = 4푧 +3, for some non-negative integer 푧, by arguments as before, we can check that the all-ones vector 1푛 indeed belongs to 푉B, with � 1푆2(1푛) = −2⌊ 푛 2⌋−푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, in this case, we obtain from the discussion preceding Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ January 13, 2023 DRAFT JANUARY 2023 28 In other words, the lemma above identifies the number of words in 푆2 that are of even weight, for different values of the blocklength 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we shall apply our results to the [2푚 −1, 2푚 −1 − 푚] binary Hamming code, for 푚 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We shall use the coordinate ordering discussed in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For 푚 ≥ 3 and for C being the [2푚 − 1, 2푚 − 1 − 푚] Hamming code, we have that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = 2 � 2푚−1 2 � −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The dual code C⊥ of the [2푚 − 1, 2푚 − 1 − 푚] Hamming code is the [2푚 − 1, 푚] simplex code, all of whose non-zero codewords (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', codewords that are not equal to 02푚−1) are of weight 2푚−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, a generator matrix of the simplex code under consideration is 퐻Ham.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, let the columns of 퐻Ham be indexed by 푚-tuples (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푚) ∈ {0, 1}푚 \\ {0푚}, ordered in the standard lexicographic order, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', the 푖th column of 퐺 is indexed as B푚(푖), for 1 ≤ 푖 ≤ 2푚 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is well-known (see, for example, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='10 of [32]) that the 푗th row 퐻Ham( 푗) is the evaluation vector, over the 푚-tuples indexing the columns, of the monomial 푥 푗, for 1 ≤ 푗 ≤ 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We write this row as Eval\\0(푥 푗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Consider the first 푚 − 1 rows of 퐻Ham, which are the evaluation vectors Eval\\0(푥 푗), for 1 ≤ 푗 ≤ 푚 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It can be checked that the Hamming weight, 2푚−1, of any of these rows is a multiple of 4, when 푚 ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Moreover, in any of these rows, if the entry corresponding to the evaluation point (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푚−1, 0) equals 1, then so does the entry corresponding to the evaluation point (푥1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푥푚−1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The above two facts imply that each of the first 푚 −1 rows of 퐻Ham can be written as a linear combination of an even number of vectors bℓ ∈ B, for ℓ ∈ [1 : �2푚−1 2 � −1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, from Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, it holds that the Fourier coefficient � 1푆2 � Eval\\0(푥 푗) � = 2 � 2푚−1 2 � −(2푚−1), for all 1 ≤ 푗 ≤ 푚 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Furthermore, observe that the above arguments also hold for any linear combination of the first 푚 − 1 rows of 퐻Ham, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', it holds that � 1푆2 � Eval\\0(s) � = 2 � 2푚−1 2 � −(2푚−1), where s = �푚 푗=2 푐 푗 · Eval\\0(푥 푗), for 푐 푗 ∈ {0, 1}, 푗 ∈ [2 : 푚].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It can also be seen that since Eval\\0(푥푚) ∉ 푉B, we have that � 1푆2 � Eval\\0(푥푚) � = 0, and similarly, that � 1푆2 � Eval\\0(s) � = 0, where s = Eval(푥1) + �푚−1 푗=1 푐 푗 · Eval(푥 푗), for 푐 푗 ∈ {0, 1}, 푗 ∈ [푚 − 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Putting everything together, we observe that for half of the codewords s ∈ C⊥, January 13, 2023 DRAFT JANUARY 2023 29 (푚, 푟) 푁(RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) (4, 2) 16 (4, 3) 128 (5, 3) 2048 (6, 4) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='711 × 107 (7, 5) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='441 × 1017 (8, 6) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='329 × 1036 TABLE I: Table of values of 푁(RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), for select parameters 푚 and 푟 the Fourier coefficient � 1푆2(s) equals 2 � 2푚−1 2 � −(2푚−1), and for another half of the codewords, the Fourier coefficient � 1푆2(s) equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Applying (8), we get that 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) = |C| · � s∈C⊥ � 1푆2(s) = 22푚−1−푚 · 2푚−1 · 2 � 2푚−1 2 � −(2푚−1) = 2 � 2푚−1 2 � −1, where the second inequality holds since |C| = 22푚−1−푚 and |C⊥| = 2푚, and half the codewords s ∈ C⊥ have nonzero Fourier coefficient � 1푆2(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ Note that in Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 and in Corollary IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, when 푛 is even, the number of constrained codewords in the linear codes are half the total number of constrained codewords, 2⌊ 푛 2⌋, of the same blocklength 푛 as the codes under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' However, in the limit as the blocklength goes to infinity, the rates of the subcodes of the single parity-check and Hamming codes that lie in 푆2, equal the noiseless capacity 퐶0(푆2) of the constraint, which in turn equals 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then move on to counting constrained codewords in the Reed-Muller (RM) family of codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Using the structure of Fourier coefficients given in Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and using the fact that the dual code of RM(푚, 푟) is the code RM(푚, 푚 −푟 −1), for 푟 ≤ 푚 −1, we numerically calculate the number of constrained codewords 푁(RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), for certain (large) values of 푚 and 푟.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our results are documented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that the computational technique in Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 proves particularly useful when the rate of RM(푚, 푟) is larger than 1 2, or equivalently, when 푟 > � 푚 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we shall work towards obtaining bounds on the sizes of constrained codes that are January 13, 2023 DRAFT JANUARY 2023 30 subsets of 푆2, of minimum distance at least 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, we are interested in formulating the symmetrized LP Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In what follows, we fix the blocklength 푛 to be odd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Slight modifications of the construction of the symmetry group 퐺푆2 and the identification of the orbits, below, yield Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), when 푛 is even.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, consider the following permutations, where 푛 is odd: 1) For even indices 푖 ∈ [푛], define 휋adj 푖 : [푛] → [푛], such that 휋adj 푖 (푖) = 푖 + 1, 휋adj 푖 (푖 + 1) = 푖, and 휋adj 푖 ( 푗) = 푗, for 푗 ∉ {푖, 푖 + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In words, 휋adj 푖,푗 swaps adjacent positions 푖 and 푖 + 1, for even 푖 ∈ [푛], and leaves other positions unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 2) For even indices 푖, 푗 ∈ [푛], define 휋swap 푖 : [푛] → [푛], such that 휋swap 푖 (푖) = 푗, 휋swap 푖 (푖 + 1) = 푗 + 1, and 휋swap 푖 ( 푗) = 푖, 휋swap 푖 ( 푗 + 1) = 푖 + 1, with 휋swap 푖 (푘) = 푘, for 푘 ∉ {푖, 푖 + 1, 푗, 푗 + 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In words, 휋swap 푖,푗 swaps 푖 and 푗, and 푖 + 1 and 푗 + 1, for 푖, 푗 being even, and leaves other positions unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The discussion above on the sequences in 푆2 implies that the symmetry group 퐺푆2 of the constraint is generated (via compositions) by {휋adj 푖 : 푖 even} ∪ {휋adj 푖,푗 : 푖, 푗 even}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, consider tuples 휶 ∈ {0, 1} × � 0 : � 푛 2 �� × � 0 : � 푛 2 �� of the form 휶 = (푏, 푡00, 푡11), with 푡00 + 푡11 ≤ � 푛 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a sequence x ∈ {0, 1}푛, we identify 푏 ∈ {0, 1} with 푥1, the integer 푡00 with |{푖 : 푖 even and (푥푖, 푥푖+1) = (0, 0)}|, and the integer 푡11 with |{푖 : 푖 even and (푥푖, 푥푖+1) = (1, 1)}|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that then |{푖 : 푖 even and (푥푖, 푥푖+1) = (0, 1) or (1, 0)}| = � 푛 2 � − 푡00 − 푡11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We thus have that the orbits of the symmetry group of the constraint {0, 1}푛/퐺푆2 are in one-one correspondence with tuples of the form 휶 = (푏, 푡00, 푡11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that the number of orbits is hence bounded above by 2 · �푛 2 �2, and therefore the number of variables and the number of constraints in the LP Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), are bounded above by a polynomial function of the blocklength 푛, unlike the number of variables in Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), which equals 2푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If we use the notation 푆(휶) to denote the size of an orbit represented by 휶, the symmetrized LP Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) then becomes: January 13, 2023 DRAFT JANUARY 2023 31 푑 Del/푆2(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) Del(푛, 푑) 2 64 64 4096 3 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='255 64 512 4 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='255 64 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='571 5 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='627 64 64 6 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='889 64 40 7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='657 32 8 8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='619 32 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='828 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 10 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='619 16 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='857 TABLE II: Table of values of optimal values of the symmetrized Del/푆2(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) LP, the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) in [17] and [15], and the Del(푛, 푑) LP, for 푛 = 13 and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) maximize {푎휶∈R: 휶 is an orbit} � 휶 푆(휶) · 푎휶 (Obj′′) subject to: 푎휶 ≥ 0, ∀ orbits 휶, (D1′) � 휶 푎휶 · � 1휶(s ˜휶) ≥ 0, ∀ orbit rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' s ˜휶 ∈ {0, 1}푛, (D2′) 푎휶 = 0, if 1 ≤ �푛 2 � + 훼1 − 훼2 + 훼3 ≤ 푑 − 1, (D3′) 푎(0,⌊ 푛 2⌋,0) ≤ val(Del(푛, 푑)), (D4′) 푎휶 ≤ 2푛 · (1A ★ 1A)(x휶), ∀ orbits 휶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D5′) Table II shows numerical evaluations of Del/퐺푆2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), when 푛 = 13, for varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The table also includes comparisons with upper bounds via the generalized sphere packing bound of [15] and [17] and with Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We observe that once again our LP provides tighter upper bounds than those obtained by the sphere packing approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 32 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Constant Subblock Composition Constraint We now move on to studying the constant subblock-composition CSC푝 푧 constraint, which requires that each one of the 푝 “subblocks” of a binary sequence have a constant number, 푧, of 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, for any sequence x ∈ {0, 1}푛, we first partition the 푛 coordinates into 푝 subblocks, with the ℓth subblock being the vector of symbols xℓ := � 푥푖 ∈ {0, 1} : (ℓ−1)푛 푝 + 1 ≤ 푖 ≤ ℓ푛 푝 � , for 1 ≤ ℓ ≤ 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We implicitly assume that 푝 divides 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that hence x = x1x2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' x푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A binary sequence x respects the CSC푝 푧 constraint if 푤(xℓ) = 푧, for all 1 ≤ ℓ ≤ 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 퐶푝,(푛) 푧 (or simply, 퐶푝 푧 ) denote the set of all CSC푝 푧 -constrained sequences of length 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' CSC푝 푧 -constrained sequences were introduced in [5] for simultaneous information and energy transfer from a powered transmitter to an energy harversting receiver, while ensuring that the receiver battery does not drain out during periods of low signal energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The applications of such constrained codes to visible light [33] and powerline communications [34] have also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' As before, we are interested in computing the Fourier coefficients of the function 1퐶 푝 푧 : {0, 1}푛 → {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The lemma below provides these Fourier coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For s ∈ {0, 1}푛 with s = s1s2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' s푝, we have that 2푛 · � 1퐶 푝 푧 (s) = 푝 � ℓ=1 퐾(푛/푝) 푧 (푤(sℓ)), where 퐾(푛/푝) 푖 ( 푗) = �푖 푡=0(−1)푡 � 푗 푡 � �푛/푝−푗 푖−푡 � is the 푖th-Krawtchouk polynomial, for the length 푛/푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We have that 2푛 · � 1퐶 푝 푧 (s) = � x∈{0,1}푛: x∈퐶 푝 푧 (−1)x·s = � x1∈{0,1}푛/푝: 푤(x)=푧 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' � x푝∈{0,1}푛/푝: 푤(x)=푧 (−1)x·s1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (−1)x·s푝 = 푝 � ℓ=1 �� � � xℓ∈{0,1}푛/푝: 푤(x)=푧 (−1)x·sℓ�� � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, by following a line of argument similar to that in the proof of Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 in Appendix A, we obtain that for any ℓ ∈ [푝], the value of the inner summand depends on sℓ only via its January 13, 2023 DRAFT JANUARY 2023 33 weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, it holds that for any ℓ ∈ [푝], � xℓ∈{0,1}푛/푝: 푤(x)=푧 (−1)x·sℓ = � xℓ∈{0,1}푛/푝: 푤(x)=푧 (−1)x·˜sℓ, where ˜sℓ = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 1 ���������������� 푤(sℓ) such , 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By direct calculations, it holds that the sum in right-hand side of the expression above equals 퐾(푛/푝) 푧 (푤(sℓ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ In what follows, we shall concern ourselves with the application of Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4 and Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 to calculating the number of subblock constrained codewords in Reed-Muller (RM) codes RM(푚, 푟), for select values of the number of subblocks 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' First, we recall an important prpoerty of RM codes, which is sometimes called the Plotkin decomposition (see [24, Chap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 13] or the survey [35]): any length-2푚 codeword c ∈ RM(푚, 푟) can be written as the concatenation c = (u | u+v), where u ∈ RM(푚 −1, 푟) and v ∈ RM(푚 −1, 푟 −1) and the ‘+’ operation in u+v is over F2푚−1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that since RM(푚, 푡), for 1 ≤ 푡 ≤ 푚, consists of evaluation vectors of Boolean polynomials of degree at most 푡, it holds that RM(푚−1, 푟−1) ⊂ RM(푚 − 1, 푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In what follows, we ensure that 푟 ≥ 1 and 푚 is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Assume, for simplicity, that 푝 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then have that for 0 ≤ 푧 ≤ 2푚−1, 푁 � RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 푧 � = � c∈RM(푚,푟) 1퐶2푧 (x) = � u∈RM(푚−1,푟), v∈RM(푚−1,푟−1) 1푊푧 (u) · 1푊푧 (u + v), (10) where the second equality uses the Plotkin decomposition and the fact that the set 푊푧 consists of sequences of Hamming weight exactly 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, let u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , u푀 be an enumeration of coset representatives of distinct cosets of RM(푚−1, 푟 −1) in RM(푚−1, 푟), where 푀 = |RM(푚−1,푟)| |RM(푚−1,푟−1)| = 2(푚−1 ≤푟 )−( 푚−1 ≤푟−1) = 2(푚−1 푟 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, u푖 is a representative of the coset u푖 + RM(푚 − 1, 푟 − 1), with u푖 ∈ RM(푚 − 1, 푟), for 1 ≤ 푖 ≤ 푀, where the cosets u푗 + RM(푚 − 1, 푟 − 1), for different January 13, 2023 DRAFT JANUARY 2023 34 values of 푗, are disjoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 퐴u(푦) be the weight enumerator of the coset u + RM(푚 − 1, 푟 − 1), at the weight 0 ≤ 푦 ≤ 2푚−1, for u ∈ RM(푚 − 1, 푟).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Then, from (10), we see that 푁 � RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 푧 � = � u∈RM(푚−1,푟), v∈RM(푚−1,푟−1) 1푊푧 (u) · 1푊푧 (u + v) = � u∈RM(푚−1,푟) 1푊푧 (u) · � v∈RM(푚−1,푟−1) 1푊푧 (u + v) = � u∈RM(푚−1,푟) 1푊푧 (u) · 퐴u(푧) (푎)= 푀 � 푖=1 � u∈u푖+RM(푚−1,푟−1) 1푊푧 (u) · 퐴u푖 (푧) = 푀 � 푖=1 �퐴u푖 (푧)�2 , (11) where equality (a) uses the fact that any u ∈ RM(푚 − 1, 푟) belongs to some coset u푖 + RM(푚 − 1, 푟 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' While equality (11) provides a neat method to count the number of constrained codewords 푁 �RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 푧 �, provided the coset weight enumerators 퐴u푖 (푧), 1 ≤ 푖 ≤ 푀, are known, observe that in the summation in (11), we need to perform 푀 −1 = 2(푚−1 푟 ) −1 additions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' If 푟 is large, the number of such additions can be fairly high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We show next that with the help of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4, it is possible to reduce the number of computations, when 푟 is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Before we do so, we recall the fact that for 푟 ≤ 푚−1, the dual code of RM(푚, 푟) is the code RM(푚, 푚−푟 −1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 퐴u(푦) be the weight enumerator of the coset u + RM(푚 − 1, 푚 − 푟 − 2), at the weight 0 ≤ 푦 ≤ 2푚−1, for u ∈ RM(푚 − 1, 푚 − 푟 − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, we let u1, u2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , u푀 be an enumeration of coset representatives of distinct cosets of RM(푚 − 1, 푚 − 푟 − 2) in RM(푚 − 1, 푚 − 푟 − 1), where 푀 = |RM(푚−1,푚−푟−1)| |RM(푚−1,푚−푟−2)| = 2( 푚−1 푚−푟−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, applying Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, we see that 푁 � RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 푧 � = � s1s2∈RM(푚,푚−푟−1) 퐾(푛/2) 푧 (푤(s1)) · 퐾(푛/2) 푧 (푤(s2)) January 13, 2023 DRAFT JANUARY 2023 35 = � u∈RM(푚−1,푚−푟−1), v∈RM(푚−1,푚−푟−2) 퐾(푛/2) 푧 (푤(u)) · 퐾(푛/2) 푧 (푤(u + v)) = � u∈RM(푚−1,푚−푟−1) 퐾(푛/2) 푧 (푤(u)) · � v∈RM(푚−1,푚−푟−2) 퐾(푛/2) 푧 (푤(u + v)) = � u∈RM(푚−1,푚−푟−1) 퐾(푛/2) 푧 (푤(u)) · 2푚−1 � 푗=0 퐴u( 푗) · 퐾(푛/2) 푧 ( 푗) = 푀 � 푖=1 �� � 2푚−1 � 푗=0 퐴u푖 ( 푗) · 퐾(푛/2) 푧 ( 푗)�� � 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (12) Now, observe that using equality (12), the number of computations required, in the form of summations, assuming that the coset weight enumerators 퐴u푖 (·) are known, for all 1 ≤ 푖 ≤ 푀, is 2푚−1+푀 − 1 = 2푚−1+( 푚−1 푚−푟−1) − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Clearly, since for large 푟 (and large 푚), we have that 푚 − 1 + � 푚−1 푚−푟−1 � < �푚−1 푟 �, we note the relative ease of calculating 푁 �RM(푚, 푟);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 푧 � via (12), with the aid of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, as compared to using (11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We remark here that the analysis of the number of codewords in RM(푚, 푟) that lie in 퐶푝 푧 can be extended to values of 푝 that are powers of 2, by iteratively applying the Plotkin decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Finally, we note that in order to compute the coset weight enumerators required in (11) and (12), one can use the recursive algorithm provided in [36], which applies to RM codes, in addition to polar codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we provide a more explicit form of Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), when A = 퐶푝 푧 , for a fixed blocklength 푛 and parameters 푝 and 푧.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' From the description of the constraint, it can be checked that the symmetry group 퐺퐶 푝 푧 is generated (via compositions) by the following permutations: 1) For 1 ≤ ℓ ≤ 푝, and (ℓ−1)푛 푝 + 1 ≤ 푗 ≤ ℓ푛 푝 , define 휋perm,푗 ℓ : [푛] → [푛] such that 휋perm,푗 ℓ swaps the indices (ℓ−1)푛 푝 + 1 and 푗, and leaves the other indices in [푛] unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that for a fixed block indexed by 1 ≤ ℓ ≤ 푝, the collection of permutations {휋perm,푗 ℓ : (ℓ−1)푛 푝 + 1 ≤ 푗 ≤ ℓ푛 푝 } generates a group isomorphic to the symmetric group 푆푛/푝, which contains all permutations of the indices (ℓ−1)푛 푝 + 1 ≤ 푖 ≤ ℓ푛 푝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 2) For 1 ≤ ℓ, ℓ′ ≤ 푝, define 휋exch ℓ,ℓ′ : [푛] → [푛] such that 휋exch ℓ,ℓ′ swaps the element (ℓ−1)푛 푝 + 푗 with (ℓ′−1)푛 푝 + 푗, for all 1 ≤ 푗 ≤ 푛 푝, and leaves the other indices in [푛] unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In other words, 휋exch ℓ,ℓ′ exchanges entire blocks indexed by ℓ and ℓ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 36 From the description of the symmetry group 퐺퐶 푝 푧 above, we arrive at the fact that the orbits of the symmetry group are in one-one correspondence with unordered 푝-tuples 휶 ∈ � 0 : 푛 푝 � 푝 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Indeed, a given sequence x ∈ {0, 1}푛 lies in the orbit 휶(x) = (훼1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 훼푝(x)), where wt(xℓ) = 훼휎(ℓ), for 1 ≤ ℓ ≤ 푝 and some permutation 휎 ∈ 푆푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note hence that the number of orbits, and therefore the sum of the number of variables and the number of constraints in Del/퐺퐶 푝 푧 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶푝 푧 ) is bounded above by 푐 · � 푛 푝 � 푝 , for some constant 푐 > 0, which is only a polynomial function of the blocklength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, for a given orbit 휶, we let x휶 be a representative element of the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, we define x휶 to be the concatenation x휶,1x휶,2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' x휶,푝, with x휶,ℓ = (1, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 1 ���������������� 훼1 such , 0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 0) (13) being of length 푛/푝, for 1 ≤ ℓ ≤ 푝.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We thus obtain the following lemma: Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For given orbits 휶, ˜휶, with s ˜휶 being an orbit representative of ˜휶, it holds that 2푛 · � 1휶(s ˜휶) = 푝 � ℓ=1 퐾(푛/푝) 훼ℓ ( ˜훼ℓ), where for a given length 푚, 퐾(푚) 푖 is the 푖th Krawtchouk polynomial, with 퐾(푚) 푖 ( 푗) = �푖 푡=0(−1)푡 � 푗 푡 � �푚−푗 푖−푡 �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The proof of the above lemma is similar to the proof of Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4 and is hence omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Again, using the notation 푆(휶) to denote the number of elements in an orbit 휶, the symmetrized LP Del/퐺퐶 푝 푧 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶푝 푧 ) then becomes: January 13, 2023 DRAFT JANUARY 2023 37 푑 Del/퐶2 5 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 5) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 5) 2 441 441 3 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='9899 441 4 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='9899 441 5 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='574 147 6 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='0542 147 7 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3137 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 TABLE III: Table of values of optimal values of the Del/퐶2 5 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 5) LP, and the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 5), for (푛, 푝, 푧) = (14, 2, 5), and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Del/퐺퐶 푝 푧 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶푝 푧 ) maximize {푎휶∈R: 휶 is an orbit} � 휶 푆(휶) · 푎휶 (Obj′′) subject to: 푎휶 ≥ 0, ∀ orbits 휶, (D1′) � 휶 푎휶 · � 1휶(s ˜휶) ≥ 0, ∀ orbit rep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' s ˜휶 ∈ {0, 1}푛, (D2′) 푎휶 = 0, if 1 ≤ 푝 � 푡=1 훼푡 ≤ 푑 − 1, (D3′) 푎0푝 ≤ val(Del(푛, 푑)), (D4′) 푎휶 ≤ 2푛 · (1A ★ 1A)(x휶), ∀ orbits 휶.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (D5′) Tables III, IV, and V show numerical evaluations of Del/퐺퐶 푝 푧 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶푝 푧 ), when 푛 = 14, 푛 = 15, and 푛 = 18, respectively, for fixed parameters 푝 and 푧, and for varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In Tables III and IV, we again compare with upper bounds via the generalized sphere packing bound of [15] and [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Here too our LP provides tighter upper bounds than the generalized sphere packing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 38 푑 Del퐶3 2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶3 2) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶3 2) 2 1000 1000 3 826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='236 1000 4 826.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='236 1000 5 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='767 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 6 110.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='851 333.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 7 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='627 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='667 TABLE IV: Table of values of optimal values of the Del/퐶3 2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶3 2) LP, and the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶3 2), for (푛, 푝, 푧) = (15, 3, 2), and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푑 3 4 5 6 7 8 9 Del/퐶2 2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 2) 556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='38 556.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='38 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='111 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='247 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='118 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='540 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='472 TABLE V: Table of values of optimal values of the Del/퐶2 2 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐶2 2) LP, for (푛, 푝, 푧) = (18, 2, 2), and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Runlength-Limited (RLL) Constraints In this subsection, we shall work with runlength-limited constraints on binary sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Unlike in the previous subsections, where the Fourier coefficients of the indicator functions of the constraints were explicitly (or analytically) computable, in the application of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, for the constraints considered in this section, we shall provide recurrence relations for the Fourier coefficients, which allow them to be efficiently computable, numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We concern ourselves with the (푑, ∞)-runlength limited (RLL) constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This constraint mandates that there be at least 푑 0s between every pair of successive 1s in the binary input sequence, where 푑 ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For example, when 푑 = 3, the sequence 10001000010 respects the (3, ∞)- RLL constraint, but the sequence 10100010, does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It can also be checked that the (1, ∞)- RLL constraint is the same as a “no-consecutive-ones” constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The (푑, ∞)-RLL constraint is a special case of the (푑, 푘)-RLL constraint, which admits only binary sequences in which successive 1s are separated by at least 푑 0s, and the length of any run of 0s is at most 푘.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Such constraints find application in magnetic and optical recording systems, where the (푑, ∞)- RLL constraint on the data sequence (with the bit 1 corresponding to a voltage peak of high amplitude and the bit 0 corresponding to no peak) ensures that successive 1s are spaced far January 13, 2023 DRAFT JANUARY 2023 39 enough apart, so that there is little inter-symbol interference (ISI) between the voltage responses corresponding to the magnetic transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Reference [37] contains many examples of (푑, 푘)- RLL codes used in practice in magnetic storage and recording.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' More recently, (푑, 푘)-RLL input constrained sequences have also been investigated for joint energy and information transfer performance [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 푆푑 denote the set of (푑, ∞)-RLL constrained binary words of length 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, for 푛 ≥ 1, and for s ∈ {0, 1}푛, let � 1푆푑 (푛)(s) denote the Fourier coefficient at s, when the blocklength is 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then have that: Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For 푛 ≥ 푑 + 2 and for s = (푠1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푠푛) ∈ {0, 1}푛, it holds that when 푠1 = 0, � 1푆푑 (푛)(s) = 2−1 · � 1푆푑 (푛−1) �푠푛 2 � + 2−(푑+1) · � 1푆푑 (푛−푑−1) �푠푛 푑+2 � , and when 푠1 = 1, � 1푆푑 (푛)(s) = 2−1 · � 1푆푑 (푛−1) �푠푛 2 � − 2−(푑+1) · � 1푆푑 (푛−푑−1) �푠푛 푑+2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' To prove the first recurrence relation, we write � 1푆푑 (푛)(s) = 1 2푛 · � x∈푆푑 (−1)x·s = 2−푛 · � #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even} − #{푥푛 ∈ 푆푑 : 푤s(푥푛) is odd} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (14) Now, observe that #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even} = #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥1 = 0} + #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥1 = 1} (푎)= #{푥푛 2 ∈ 푆푑 : 푤푠푛 2 (푥푛 2) is even} + #{푥푛 ∈ 푆푑 : 푤s(푥푛) is even and 푥(푑+1) 1 = 10푑} = #{푥푛 2 ∈ 푆푑 : 푤푠푛 2 (푥푛 2) is even} + #{푥푛 푑+2 ∈ 푆푑 : 푤푠푛 푑+2(푥푛) is even}, (15) where (a) holds because 푠1 = 0 and from the fact that the (푑, ∞)-RLL constraint requires that January 13, 2023 DRAFT JANUARY 2023 40 푥푑+1 2 = 0푑, if 푥1 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Similarly, we obtain that #{푥푛 ∈ 푆푑 : 푤s(푥푛) is odd} = #{푥푛 2 ∈ 푆푑 : 푤푠푛 2 (푥푛 2) is odd} + #{푥푛 푑+2 ∈ 푆푑 : 푤푠푛 푑+2 (푥푛) is odd}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (16) Now, observe that � 1푆푑 (푛−1)(푠푛 2) = 2−(푛−1) · � #{푥푛 2 ∈ 푆푑 : 푤푠푛 2 (푥푛 2) is even} − #{푥푛 2 ∈ 푆푑 : 푤푠푛 2 (푥푛 2) is odd} � (17) and that � 1푆푑 (푛−푑−1)(푠푛 푑+2) = 2−(푛−푑−1) · � #{푥푛 푑+2 ∈ 푆푑 : 푤푠푛 푑+2(푥푛 푑+2) is even}− #{푥푛 푑+2 ∈ 푆푑 : 푤푠푛 푑+2(푥푛 푑+2) is odd} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (18) Substituting (15) and (16) in (14) and using (17) and (18), we get the first recurrence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The second recurrence relation is also proved by similar arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ We shall now explain how Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 helps compute the Fourier coefficients for a given (large) 푛, efficiently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' First, we note that a direct computation of all the Fourier coefficients of 1푆푑 at blocklength 푛, can be accomplished by the fast Walsh-Hadamard transform (FWHT) algorithm (see Exercise 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='12(b) in [21]), in time 푛 · 2푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, let us assume that we pre-compute and store the Fourier coefficients � � 1푆푑 (푚)(s) : s ∈ {0, 1}푚� , for 1 ≤ 푚 ≤ 푑 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' These Fourier coefficients help initialize the recurrences in Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, give a fixed (large) 푛, the Fourier coefficients at which blocklength we intend computing, we shall calculate, using the recurrence relations above, the Fourier coefficients at all blocklengths 푑 + 2 ≤ 푚 ≤ 푛, iteratively, beginning at length 푑 + 2, and increasing 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Assuming that the additions and multiplications in Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 take unit time, it can be seen that the time complexity of computing the Fourier coefficient at length 푛 grows as �푛 푑+2 2푖 < 2푛+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This is much less than the time that is 2푛+log2 푛, taken by the FWHT algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' However, there still remains the issue of storage cost: at “level” 푚, one needs to store all 2푚 Fourier coefficients in order to facilitate computation of the Fourier coefficients at blocklengths January 13, 2023 DRAFT JANUARY 2023 41 C 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) RM(4, 2) 83 RM(4, 3) 1292 Ham3 4 Ham4 101 TABLE VI: Table of values of 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1), for select codes C 푛 > 푚.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, assuming that the storage of a single Fourier coefficient takes up one unit of space, we see that we require at least 2푛 units of memory in order to store the Fourier coefficients at blocklength 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For 푛 ⪆ 20, for example, this storage cost becomes prohibitively expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We now use the Fourier coefficients that are numerically computed using Lemma IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5, to calculate the number of (1, ∞)-RLL constrained codewords in select codes, by applying Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' These values are listed in Table VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We denote the binary Hamming code of blocklength 2푡 −1 as Ham푡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, we assume that the coordinates of the Hamming and Reed-Muller codes follow the orderings discussed in Section III-A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we obtain upper bounds on the sizes of (푑, ∞)-RLL constrained codes with a given minimum distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Since there is no apparent symmetry group of 푆푑, we shall first directly run the Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆푑) LP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Table VII (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Table VIII) shows comparisons between the upper bounds on 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2)), obtained using our Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) LP (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) LP) with the generalized sphere packing bound of [15] and [17], when 푛 = 10, and for varying values of the minimum distance 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We also compare these upper bounds with the optimal value of Del(푛, 푑), since this is a trivial upper bound on 퐴(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), for any A ⊆ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that, from the numerical trials, for certain values of 푑, the generalized sphere-packing bound returns a value that is larger (and hence worse) than the value of Del(푛, 푑), whereas the optimal value of our Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) LP is uniformly bounded above by Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we study upper bounds on the sizes of constrained codes for a given minimum distance, for a slightly stronger variant of the (푑, ∞)-RLL constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, we require that the constraint is satisfied across the boundary when the constrained sequence is cyclically wrapped January 13, 2023 DRAFT JANUARY 2023 42 푑 Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) Del(푛, 푑) 2 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='557 144 512 3 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='762 111 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 4 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='048 111 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='667 5 12 63 12 6 6 63 6 7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 TABLE VII: Table of values of optimal values of the Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) LP, the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆1) in [15] and [17], and the Del(푛, 푑) LP, for 푛 = 10 and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푑 Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) Del(푛, 푑) 2 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='578 60 512 3 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='075 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 4 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='721 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='667 5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='856 34 12 6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='899 34 6 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='529 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 TABLE VIII: Table of values of optimal values of the Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2) LP, the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆2), and the Del(푛, 푑) LP, for 푛 = 10 and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' around.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 푆tail,푑 denote this constraint that admits only those sequences x ∈ {0, 1}푛 such that x ∈ 푆푑 and ℓ − 푠 ≤ 푛 − 푑 −1, where 푠 ∈ [푛] denotes the position of the first occurring 1 in x and ℓ ∈ [푛] denotes the position of the last occurring 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In the paragraphs that follow, we first identify the symmetry group of the more broad class of tail-biting constraints, to which 푆tail,푑 also belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let a tail-biting constraint of interest be represented by a special set A of constrained sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, A has the property that if a sequence x ∈ A, then 휋cyc,푖 · x also lies in A, where for 1 ≤ 푖 ≤ 푛, 휋cyc,푖 shifts each bit in x by 푖 bits to the left, wrapping around cyclically, if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' More formally, 푥휋cyc,푖 ( 푗) = 푥mod( 푗+푖,푛), for 1 ≤ 푗 ≤ 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Clearly, we have that the symmetry group of the constraint 퐺A contains the cyclic group 퐶푛, and it is hence possible to symmetrize Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A) using 퐶푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The orbits 휶 thus are in one-one correspondence with (fixed) 2-ary necklaces of length 푛, with turnovers prohibited (see pg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 18 in [38] and sequence A000031 of [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is known that the number of such necklaces, and hence the number of January 13, 2023 DRAFT JANUARY 2023 43 푑 Del/퐺푆tail (1,∞) (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail (1,∞)) GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail (1,∞)) Del(푛, 푑) 2 480.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='676 521 4096 3 350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='055 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 512 4 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='569 448.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='5 292.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='571 5 64 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='727 64 6 40 316.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='727 40 7 8 169 8 8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 169 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='667 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='333 TABLE IX: Table of values of optimal values of the Del/퐺푆tail,1 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail,1) LP, the generalized sphere packing bound LP GenSph(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail,1) in [17] and [15], and Del(푛, 푑), for 푛 = 13, and varying values of 푑.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' orbits, 푁cyc(푛), is such that lim푛→∞ 푁cyc(푛) 2푛/푛 = 1 (see [40]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' in other words, the sum of the number of variables and constraints in Del/퐺 A (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), is bounded above by 푐 · 2푛 푛 , for some constant 푐 > 0, and large enough 푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Thus, we obtain a slight reduction in the size of the LP as compared to Del(푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' A), which had 2푛 variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then apply this symmetrization procedure to the tail-biting (1, ∞)-RLL constraint, 푆tail,1, which admits only binary sequences x ∈ {0, 1}푛 with no consecutive ones and which are such that (푥1, 푥푛) ≠ (1, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Similar to the approach in the previous two subsections, we can set up a symmetrized LP Del/퐺푆tail,1 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail,1), using the orbits of the cyclic group 퐶푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Table IX shows numerical evaluations of Del/퐺푆tail,1 (푛, 푑;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' 푆tail,1), when 푛 = 13, for varying values of 푑, and comparisons with the generalized sphere packing bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Again, our LP provides tighter upper bounds than the generalized sphere packing bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe also that for certain values of minimum distance 푑, the optimal value of our LP coincides with the optimal value of the Delsarte LP Del(푛, 푑).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' CONCLUSION In this work, we took two approaches to the problem of estimating the sizes of binary error- correcting constrained codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' First, motivated by the application of transmission of codes over stochastic, symmetric, channel noise models—a problem for which explicit capacity-achieving linear codes have been constructed, we consider the question of computing the sizes of con- January 13, 2023 DRAFT JANUARY 2023 44 strained subcodes of linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Such constrained subcodes of capacity-achieving linear codes, for example, are resilient to symmetric errors and erasures, in that their error probabilities using the same decoding strategy as for the larger linear code, vanish as the blocklength of the code goes to infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Our approach was to view the problem through a Fourier-analytic lens, thereby transforming it into a counting problem in the space of the dual code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' As part of our method, we analyzed (analytically or numerically) the Fourier transform of the indicator function of the constraint, and provided values of the number of constrained codewords in select linear codes and algorithmic procedures for efficient counting, in the cases of certain constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For some constraints, we also obtained insights into the construction of linear codes with a large number of constrained codewords.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Next, we considered the scenario where the constrained codes were subjected to adversarial bit-flip errors or erasures, with a combinatorial bound on the number of errors or erasures that can be induced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then proposed numerical upper bounds on the sizes of constrained codes with a given resilience to such combinatorial errors and erasures (equivalently, with a prescribed minimum Hamming distance), via an extension of Delsarte’s linear program (LP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We then applied our LPs to different constraints, and observed that the optimal numerical values returned by our LP are better than those provided by the generalized sphere packing bounds of Fazeli, Vardy, and Yaakobi (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' There are many interesting directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' One line of study would be to build on the Fourier-theoretic techniques in this paper and study the asymptotics (in the limit as the blocklength goes to infinity) of the rates of constrained subcodes of specific linear codes of a given rate 푅 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Similarly, one could try to use our dual LP formulation to derive asymptotic upper bounds on the rate-distance tradeoff for constrained codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' This, for example, will help us understand if the Gilbert-Varshamov lower bounds of Kolesnik and Krachkovsky (1991) and Marcus and Roth (1992) are tight for any constrained system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Another direction of work could study the extension of results here to codes with larger alphabet sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' ACKNOWLEDGEMENTS The authors would like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Pfister for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 45 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' On the Weight Distribution of Constrained Sequences in F푛 2 Suppose that we are interested in computing the weight distribution of words in {0, 1}푛 that lie in a (constrained) set A ⊆ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Let 푎푖,A denote the number of constrained words of weight 푖 ∈ [0 : 푛] and recall the definition of the 푖th-Krawtchouk polynomial 퐾(푛) 푖 , for a given blocklength 푛, where 퐾(푛) 푖 (푧) = �푖 ℓ=0(−1)ℓ�푧 ℓ � �푛−푧 푖−ℓ �, and the notation 푊푖 = {x ∈ {0, 1}푛 : 푤(x) = 푖}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The following theorem then holds true: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The weight distribution of sequences that lie in a set A ⊆ {0, 1}푛 obeys 푎푖,A = 푛 � 푗=0 퐾(푛) 푖 ( 푗) · � s: 푤(s)=푗 � 1A(s), 푖 ∈ [0 : 푛].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The proof is again a simple application of Plancherel’s Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Observe that 푎푖,A = � x∈A 1푊푖(x) = � x∈{0,1}푛 1푊푖 (x) · 1A(x) = 2푛 · � s∈{0,1}푛 � 1푊푖 (s) · � 1A(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' (19) We now recall the well-known proof of the fact that 2푛 · � 1푊푖(s) = 퐾푖(푤(s)) (see Chapter 5 in [24] for more details on Krawtchouk polynomials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that 2푛 · � 1푊푖(s) = � x∈{0,1}푛: 푤(x)=푖 (−1)x·s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Now, the summation on the right side depends only on the weight 푤(s), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=', for any permutation of coordinates 휋 : {0, 1}푛 → {0, 1}푛, it holds that � x∈{0,1}푛: 푤(x)=푖 (−1)x·휋(s) = � x∈{0,1}푛: 푤(x)=푖 (−1)휋(x)·휋(s) = � x∈{0,1}푛: 푤(x)=푖 (−1)x·s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 46 In other words, we have that � 1푊푖 (s) = � 1푊푖 (휋(s)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Hence, for s such that 푤(s) = 푗, it suffices that we calculate 2푛 · � 1푊푖(s★), where s★ = (푠★ 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' , 푠★ 푛) is such that 푠★ 1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' = 푠★ 푗 = 1 and 푠★ 푗+1 = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' = 푠★ 푛 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By a direct computation, it can be checked that 2푛 · � 1푊푖 (s★) = �푖 ℓ=0(−1)ℓ� 푗 ℓ � �푛−푗 푖−ℓ � = 2푛 · � 1푊푖 (s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Plugging this back into (19) and simplifying, we obtain the expression in the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ As before, Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 implies that if the Fourier coefficients � 1A(s) (or, the sum of Fourier coefficients at a fixed weight � s: 푤(s)=푗 � 1A(s)) were available to us, we can easily compute the number of constrained words of a given weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Weight Distribution of Constrained Codewords in Linear Codes In this section, we briefly describe another useful computation that is facilitated by knowledge of the Fourier coefficients of the indicator function that a word belongs to a set of constrained sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Before we do so, we recall another property of Fourier transforms: Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given functions 푓 , 푔 : {0, 1}푛 → R, the Fourier transform of 푓 · 푔 is the function �푓 ⃝★ �푔, where ( �푓 ⃝★ �푔)(s) := � z∈{0,1}푛 �푓 (z)�푔(z + s) = 2푛 · ( �푓 ★ �푔)(s), s ∈ {0, 1}푛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The calculation we wish to perform puts together the theses of Theorems III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' In particular, we ask for the number of codewords of a linear code C, which lie in a fixed set A and are of a given weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We thus obtain Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' For a given v ∈ {0, 1}푛, we use the notation v + C to denote the coset of C to which v belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We let 푎푖,A(C) denote the number of constrained codewords of weight 푖 ∈ [0 : 푛], which lie in the set A and in a linear code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Given a linear code C of blocklength 푛, we have that 푎푖,A(C) = |C| 2푛 푛 � 푗=0 퐾(푛) 푖 ( 푗) � s: 푤(s)=푗 � z∈s+C⊥ � 1A(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We observe that 푎푖,A(C) = � x∈{0,1}푛 1푊푖 (x) · 1A(x) · 1C(x) January 13, 2023 DRAFT JANUARY 2023 47 =2푛 · � s∈{0,1}푛 � 1푊푖(s) · � 1A · 1C(s), where the last equality above uses Plancherel’s Theorem (see Section II-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Further, by employing Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='2 to expand the last equality, we get that 푎푖,A(C) = 2푛 · � s∈{0,1}푛 � 1푊푖 (s) · � � 1A ⃝★ � 1C � (s) (푎)= |C| 2푛 · � s∈{0,1}푛 퐾(푛) 푖 (푤(s)) � z∈{0,1}푛 � 1C⊥(z) · � 1A(s + z) = |C| 2푛 · 푛 � 푗=0 퐾(푛) 푖 ( 푗) � s: 푤(s)=푗 � z∈s+C⊥ � 1A(z), where equality (a) above uses the fact that 2푛 · � 1푊푖 (s) = 퐾(푛) 푖 (푤(s)) and that � 1C(s) = |C| 2푛 · 1C⊥(s) (see the proofs of Theorems A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 and III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='3 could prove useful in the following context: consider the transmission of code- words of a linear code C over an input-constrained binary-input memoryless symmetric (BMS) channel, which admits only binary constrained sequences that lie in the set A as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Suppose that the decoder being used is the maximum a-posteriori (MAP) (equivalently, the maximum like- lihood (ML)) decoder of the linear code C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By calculating the weight distribution of constrained codewords in a linear code C as above, it is possible to obtain an upper bound on the block error probability (via a union bounding argument), when the MAP decoder is used (see Chapter 1 of [41]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Obtaining MacWilliams’ Identities for Linear Codes Via Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 Consider the simple constraint that admits only sequences having a fixed weight 푖 ∈ [0 : 푛], where 푛 is the blocklength of the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Note that in this case, the set of constrained sequences is A = 푊푖.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By applying Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1 to this constraint, for a given linear code C, we obtain the well-known MacWilliams’ identities [22] for linear codes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' We use the notation 푎푖(C) for the number of codewords of weight 푖 ∈ [0 : 푛] in C, which equals 푁(C;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='푊푖), following the notation of Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' January 13, 2023 DRAFT JANUARY 2023 48 Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='4 (MacWilliams’ identities).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' It is true that 푎푖(C) = 1 |C⊥| 푛 � 푗=0 퐾(푛) 푖 ( 푗) · 푎 푗(C⊥).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' The proof simply uses the fact that � 1푊푖 (s) = 퐾 (푛) 푖 (푤(s)) 2푛 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' By simplifying the summation in Theorem III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content='1, and by using the fact that |C| · |C⊥| = 2푛, we obtain the required result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' □ Further, it holds that the number of constrained codewords of weight 푗 obeys 푎푖,푊푗 = �푛 푗 �, if 푗 = 푖, and 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/CdE4T4oBgHgl3EQfeQ2g/content/2301.05098v1.pdf'} +page_content=' REFERENCES [1] B.' metadata={'source': 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mode 100644 index 0000000000000000000000000000000000000000..7ebf5761c9cbc57c300565122b46f6517fdf05e9 --- /dev/null +++ b/HdFLT4oBgHgl3EQfIC9G/content/tmp_files/2301.11998v1.pdf.txt @@ -0,0 +1,1923 @@ +Augmented Reality’s Potential for Identifying and +Mitigating Home Privacy Leaks +Stefany Cruz1, Logan Danek1, Shinan Liu2, Christopher Kraemer6, Zixin Wang3 +Nick Feamster2, Danny Yuxing Huang4, Yaxing Yao5, Josiah Hester6 +1Northwestern University, 2University of Chicago, 3Zhejiang University +4New York University, 5University of Maryland, Baltimore County, 6Georgia Institute of Technology +Abstract—Users face various privacy risks in smart homes, yet +there are limited ways for them to learn about the details of such +risks, such as the data practices of smart home devices and their +data flow. In this paper, we present Privacy Plumber, a system +that enables a user to inspect and explore the privacy “leaks” in +their home using an augmented reality tool. Privacy Plumber +allows the user to learn and understand the volume of data +leaving the home and how that data may affect a user’s privacy— +in the same physical context as the devices in question, because +we visualize the privacy leaks with augmented reality. Privacy +Plumber uses ARP spoofing to gather aggregate network traffic +information and presents it through an overlay on top of the +device in an smartphone app. The increased transparency aims to +help the user make privacy decisions and mend potential privacy +leaks, such as instruct Privacy Plumber on what devices to block, +on what schedule (i.e., turn off Alexa when sleeping), etc. Our +initial user study with six participants demonstrates participants’ +increased awareness of privacy leaks in smart devices, which +further contributes to their privacy decisions (e.g., which devices +to block). +I. +INTRODUCTION +The increasing adoption of Internet-connected smart de- +vices has brought huge improvements to our lives. Yet, these +devices also raise significant privacy concerns from their users, +such as sensitive data collection [53], [51], data sharing [51], +and data misuse [22], [23], [27]. Literature has suggested +many types of privacy risks associated with smart devices. +For example, some seemingly innocent data, such as the +network traffic shapes and patterns of smart devices, may +reveal sensitive personal information, such as users’ daily +schedule, their gender, date of birth, social security number, +location, and behaviors [5], [3]. +However, many risks are not obvious to users due to the +opaque nature of the data practices of smart devices; the +average users lack an understanding of how their data is +collected, processed, and shared [51], [50], [21]. Prior research +has proposed various ways to increase users’ awareness of the +data practices in smart homes, such as data dashboards, mobile +phone apps, ambient light and sounds, and so on [44], [15], +[9], [16]. Some other mechanisms (e.g., IoT Inspector [15]) +focus on specific aspects of the data practices and present +network traffic data to users so that they can access first- +hand data of the data flow in/out of smart devices. Yet, +most mechanisms we know decouple such transparency from +the device themselves—i.e., users need to learn about the +data practices separately from the smart devices—making +the information less intuitive to consume, especially for the +average user. In addition, these mechanisms do not provide +users with the ability to take action if they notice unexpected +data practices (e.g., blocking the data from being sent out to +third parties). +In this paper, we focus on the data flow in and out +of smart devices. We build a proof-of-concept smartphone- +based augmented reality system called Privacy Plumber to +increase users’ awareness of the data flows of smart devices +and provide them with controls to block certain data flow if +needed. We focus on data flow rather than other aspects of +data practices (e.g., types of data being collected) mostly due to +practicality and feasibility reason, as we can reasonably capture +data flow and identify its source and destination using ARP +spoofing [15]. In addition, from the smart devices’ perspective, +these devices have multiple tiers of software, all of which entail +some type of tracking. Such tracking is generally embodied +in the data flow. We use augmented reality to visualize data +flows in the same physical environment as the devices in +question; this method could potentially help users establish +a connection between the devices and their data flows in the +same context. Users’ proper understanding of data flow may +help them understand the privacy implications of devices such +as smart TVs [28], voice assistants [15], children’s toys [10], +security cameras [24], [35], and smart light bulbs [8]. +The development of Privacy Plumber is inspired by the +following three gaps in the literature. First, the data flows of +smart devices are opaque and not visible to users. Second, +existing tools to monitor network traffic of smart devices (e.g., +IoT Inspector [15], open.Dash [9]) require a certain level of +technical knowledge to be able to interpret the results—not to +mention that the results are often decoupled from the physical +environment where the smart devices are situated. Oftentimes, +the results are presented on, for instance, dashboards on +computers or phones, where there is a disconnection between +the visualization of data flows and the smart devices that +create the data flow. Third, existing tools or mechanisms do +not provide users with the ability to control unnecessary or +unexpected data flows. With Privacy Plumber, we aim to bridge +the gaps and increase users’ awareness and control of the data +flow in smart devices. +arXiv:2301.11998v1 [cs.CR] 27 Jan 2023 + +Fig. 1: Privacy Plumber lets a user find and mitigate potential +privacy violations in the smart home. The figure shows a +user walking around the smart home and inspecting the traffic +and trackers coming out of a Samsung Smart Fridge using +the Augmented Reality enabled app. Furthermore, (not shown +in the picture above) users can use built-in, infrastructure- +free controls to limit traffic of devices to times of day— +without requiring any additional hardware or modifications +to the network. The graph shows the actual network traffic +as the user interacted with the Smart Fridge: A: turning on +the ice maker; B: browsing recipe; C: browsing goods; D: +interacting with the Bixby voice assistant of the fridge; E: +opening the fridge door; F: adding items to the shopping list. +During these interactions, the Smart Fridge communicated with +various advertising and tracking services, such as DoubleClick +and Tapad. +Privacy Plumber uses augmented reality (AR) techniques +and visualizes real-time network traffic flowing in and out +of smart devices through an overlay. It allows users to find +potential privacy leaks in their homes by pointing the AR- +based app at smart devices. As shown in Figure 1, the app adds +an overlay on top of the smart devices in which it displays +a real-time data flow based on the network traffic with the +necessary information for users to understand it. We chose +to use AR because, as privacy is highly contextual [32], it +can provide strong contextual connections between the actual +real-time privacy leaks, and the user actions (or inaction). +This allows the smartphone to function as a viewfinder into +the invisible world of data flow and identify potential privacy +violations. The smartphone application relies on a companion +software tool hosted on a laptop or desktop on the same home +network. This tool discovers smart devices in a user’s home, +intercepts their traffic via ARP spoofing [48], and analyzes the +data flow (e.g., what traffic is leaving the home over time)— +without requiring the user to modify their network settings +0 +1 +Nest Camera +Live streaming +0 +2 +Samsung Fridge +Door opening +Recipe browsing +0 +10 +20 +30 +40 +50 +60 +70 +80 +time [s] +0 +10 +Amazon Echo +Weather reporting +Radio playing +traffic [mbps] +Fig. 2: Outbound network traffic from various smart home IoT +devices: a Nest Camera, an Amazon Echo, and a Samsung +Smart Fridge. Traffic increases or provides a fingerprint for +many types of seemingly benign actions, creating a privacy +leak. Current systems do not provide real-time context or +ability to experiment with these devices, nor control their +leakage. +or install additional hardware. When users would like to take +action and block certain data flow, ARP-spoofing is used again +to jam specific devices’ traffic (thereby blocking the device) +at the time of day set by the user. +We build a proof-of-concept prototype and conducted a +pilot study with 6 participants in our lab to collect their +feedback on the prototype. Our initial findings have suggested +that Privacy Plumber helped participants understand the net- +work traffic, increased their awareness of potential privacy +violations, and helped them make more informed decisions +on how to handle IoT devices. +This paper makes three contributions. First, to the best of +our knowledge, Privacy Plumber is the first mechanism that +provides users with real-time information on the data flow of +their smart devices. This paper proves the possibility of using +AR-based technology as a viable option to increase users’ +awareness of the data flows of smart devices. Second, our +initial evaluation shows promising results, indicating users’ po- +tential acceptance of these technologies. Third, we summarized +lessons learned from the pilot user study to inform the design +and development of future systems that aim to improve users’ +awareness of data practices in smart homes. +II. +BACKGROUND AND RELATED WORK +In this section we discuss related work seeking to under- +stand or discover privacy leaks, and the tools that exist to +help users understand and mitigate them. Privacy Plumber is +meant to to provide a handheld and zero-cost inspection and +experimentation tool for privacy leaks of nearby smart devices +in the home, and a straightforward and low burden method for +mitigating those leaks. +A. Privacy Issues in Smart Home +Over the decades, privacy issues have been deeply dis- +closed in smart home, such as transparency of data collection, +data sharing, and accessibility [20], [50], [49], [16], [53], [30], +[50]. Some smart home devices have always-on sensors that +capture users’ offline activities in their homes and transmit +relevant information outside of the home, especially for cloud +services run by device manufacturers [6]. +2 + +02:006Privacy +PLumbe +X +LiveDeviceTraffic +720 +540 +360 +180 +10 +CurrentDeviceTraffic:984.3bytesinthelast10seconds +Sendingdatato17differentdestinations,including2 +advertisingservice(s) +Thisdataisequivalentto536.9wordsoftextor0.5 +picturesperminute +DeviceDetails&Control>DoubleClick +pedel +DoubleClick +Tapad +traffic [mbps] +6 +A +E +C +4 +B +2 +0 +10 +20 +30 +40 +50 +60 +70 +80 +time [s]In the meantime, users are concerned about leaks of +sensitive information [23], [51], [25], such as visual and +auditory information which they see as private [23], [25]. +Thus, users have a strong desire to protect themselves against +such recordings being accessed without their permission [30], +[19]. However, some information users perceived as not very +sensitive also lead to privacy leaks. For example, the home +temperature could be used to determine whether a house is +occupied or not, as a precursor to burglary [20]. +In fact, smart devices give off digital exhaust which can +be used by third parties including a user’s Internet Service +Provider, advertisers, device manufacturers, and others, to +fingerprint activities and get sensitive information. Shown in +Figure 2 is the network traffic and trackers of various smart +home devices.This network traffic forms the basis of most +leaks. +B. Tools for Enhancing Smart Home Privacy +Most related to Privacy Plumber are tools that watch or +monitor network traffic in the home and provide something +of use to the user, whether visualization and information, +education, or a mechanism for control. +Sophisticated, technically literate users can use systems that +block advertising and tracking domains (e.g., PiHole [38] and +pfSense [34]), but these methods are bespoke and often require +additional or dedicated hardware (e.g., Raspberry Pi for Pi- +Hole, and a supporting custom router for pfSense). Other tools +have provided insight into what might be exposed from web- +browsing activities, including WiFi privacy Ticker [11], but do +not consider or scale to the new problems of connected devices +with physical sensors and abilities in a space. Aretha [40] +explores this tool space and proposed (but did not deploy) +a simple firewall-based control mechanism. Aretha presents +data in aggregate instead of contextually and in real-time. +None of these techniques investigate a range of IoT devices, +usually constrained by studies with participants in their own +homes, in a time when smart home adoption is low (Aretha +had three participants, and only one had more than a phone, +tablet, and Alexa). None of these techniques develop a scalable +(no additional hardware required) way to interpret privacy +leaks and control them. Emerging smart devices are highly +contextual and location sensitive, an Alexa in the bedroom +versus the kitchen has different privacy exposure (i.e. the +former gives sleep times, the latter exposes eating habits). +Moreover, tracking these devices’ privacy exposures presents +a technical challenge because the traffic is not centralized +through a web browser or laptop. A tool is needed to visualize +privacy leaks from smart devices in real-time and in context, +educate users on the consequences of these leaks, and provide +control mechanisms for partially mitigating these leaks. +Wifi Privacy Ticker [11] demonstrated a first method for +improving the awareness of users in terms of privacy by +providing a count of the amount of sensitive data that was +being transmitted unencrypted over the network awareness. +By seeing this in real-time, users could adjust their behavior +or find encrypted means to browse the web. Of course, this +ticker was developed well before the current generation of +smart devices, however the underlying concept of surfacing the +invisible privacy leaks remains the same for Privacy Plumber, +but for smart devices. Xray-refine [46], [47] provided smart +phone users a means to visualize their exposure profile, based +on the duration of app use. This method was an educational +solution, but users had to adjust behavior to work around the +constraints of the apps they were using, in some cases, opting +out of apps to reduce privacy exposure. +Finally, recent work like Aretha [40], PriView [36], Lu- +mos [41], and IoT Inspector [15] look at making usable +visualizations and mechanisms to understand and interpret data +coming from smart devices in the home. IoT Inspector is a +simple-to-install desktop application that uses ARP spoofing +to gather network traffic on the Wifi network of the desk- +top/laptop. This information is sent and collated at a server, +and then viewed online by the user, listing different trackers +and websites that are attached to smart device usage. Because +of the ease of installation and no extra hardware requirement, +IoT Inspector was deployed by thousands of users. +In comparison, Aretha is a part research tool, part ex- +ploratory users tool for exploring a design space of privacy +tools and controls. Aretha helps users become aware of the net- +work traffic flows in their homes while also educating users to +regain their privacy in the connected home. Aretha suggests the +use of firewall mechanisms controllable by the user, but does +not implement them. Aretha, owing to a hardware requirement +(a device must be attached to the Wifi router in the home) was +only deployed in three homes, compared to the massive scale +deployment of IoT Inspector. Similarly, PriView also has a +hardware requirement; its users need to have dedicated external +thermal cameras (e.g., FLIR One [36]) attached to their phones. +For Lumos, there is no special hardware environment, although +the focus is more on identifying hidden smart devices rather +than analyzing the network traffic for privacy leaks. +Privacy Plumber is not meant as a research tool or a design +space exploration tool. It is meant as an actual, real world +system with a focus on scalability and ease of deployment in +any home, similar to IoT inspector. Unlike both IoT inspector +and Aretha, Privacy Plumber provides real-time and contextual +visualizations of privacy leaks, real-time ability to plug those +leaks (as well as automated rule setting for plugging leaks), +and enables experimentation in real-time. +Finally, other significant measurement campaigns on in- +home traffic have been conducted, focusing on the Wifi net- +work itself or devices in the home [39], [18]. These have +usually been for research purposes and need finding and are +useful for informing the design of Privacy Plumber, but are not +necessarily tools for controlling smart home device privacy. +C. Determining Home Activities from Network Traffic +Complementary to Privacy Plumber are other works which +demonstrate the ability to infer activities from network traffic: +whether on a phone, smart device, or laptop [4]. By analyzing +the patterns of network traffic in the home, occupancy, habits +such as sleeping, watching TV, listening to music, and some- +times preferences, can all be determined. HomeSnitch [33], +Peek-a-boo [1], and HoMonit [52] all utilize machine learning +with varying degrees of success to identify activities in the +home from network traffic. Other tools utilized for monitoring +Internet connected smart devices in the home, IoT Sentinel [26] +and IoT Sense [7], have shown that particular devices can be +3 + +fingerprinted by their traffic patterns. Enabling another way +for an ISP or third party to determine the activity in the +home. Each of these systems and methods are complementary +with Privacy Plumber; inferred activities from traffic would be +useful to surface in Privacy Plumber for the user to understand +privacy exposure and know when to mitigate it, and device +fingerprinting provides a way for zero-registration or setup of +Privacy Plumber in a home. +D. Challenges: Contextual, Real-time Privacy Understanding +and Control in the Home +Despite the diverse work in the smart home privacy space, +significant gaps and challenges remain, which we detail below. +C1: Users can’t model what devices are doing, especially +without context. With tools like IoT Inspector, a user might be +able to count the number of trackers and advertisers contacted +in a day from the sum of their interactions with smart devices. +But how can a user know that turning on the NPR podcast +on their smart fridge will send thousands of bytes of informa- +tion to Bloomberg News for advertising purposes? How can +they know that turning on the device sends a short burst of +traffic? Users know that data captured will often be used for +advertising, which often generates an adverse reaction [45]. +However, with smart devices, it is not always clear what +actions or contexts trigger data being transmitted [13]. Things +like Privacy labels for websites and smart devices are meant to +give a method for scoring devices privacy [42], [17]. However, +these are static representations of the privacy exposure of a +device. With tools like IoT inspector and Aretha, aggregate +views of data are seen (as opposed to real-time views), not +associated with very fine user actions: like the turn on the light, +say command to Alexa, or open the fridge door. Because of +this granularity, the mental models of what devices are doing, +and what they are sharing, are very perplexing. Privacy tools +must address this lack of action mapping to network traffic, +enabling contextual integrity [32] in real-time. +C2: Users don’t have intuitive methods to control the +privacy “valve”. Users want devices that provide helpful +features, but they do not know the cost of this ease. One option +is to just unplug the device; however, this is all or nothing. +Users need a way to valve the privacy flow to something they +are comfortable with, or to at least be able to analyze the +tradeoffs [43]. Making privacy more ”tangible” [2] is one way +this can be done; where the privacy leaks are more visceral. +Selective firewalls (such as pfSense [34]), or other more fine +grained network mechanisms may provide a means to control +the privacy valve, but this must be intuitive and understandable +to the user, and they must be able to actually ”see” the effect +of turning this valve. +C3: Smart devices are context (location, time, action) +dependent. Smart devices are necessarily scattered around +the home; and this will continue as more devices become +intelligent, and more applications are explored. Watching a +desktop or laptop traffic meter and figuring out which device +in which room is doing what at which times, is mentally trying +for the user and disassociates the device from the physical +space that defines its context and use. Just like when trying to +find leaks in pipes, physical proximity is required. Handheld +inspection tools provide mobility, and enable in-situ fixing and +experimentation. +C4: Users can’t experiment. Indeed, because of contextual +changes in how private information is leaked, experimentation +is difficult with existing tools that generally provide traffic +summaries. Interactions with smart devices can last only a few +seconds. Enabling a user to experiment with different actions +and uses of a smart device, and then see the associated network +traffic in real-time, would provide a powerful way to build a +mental model. However, providing an ability to experiment is +challenging with the current suite of tools. +C5: Technical challenge of scalability and deployment. If +a privacy tool is to be useful and translate to the general +public, it must be hardware free, or at least trivially easy to +deploy to enable scalability and broad adoption. Commercial +products like fing.com embed all functionality in a single +phone application. Large scale deployments like with IoT +Inspector are enabled through a desktop application that is +easy to install. However, these methods do not provide controls +since that is technically difficult to do without custom hardware +put between the Wifi endpoint and the user. On the other +hand, hardware requirements or custom install procedures +reduce the deployment size of tools like Aretha, or narrow +the user base by requiring technical ability, as with PiHole. +It is not clear how to implement mechanisms of control +without changing the Wifi network and infrastructure. To +create scalable, user-centered, novice friendly privacy tools, +mechanisms for enabling control of smart device traffic without +hardware intervention must be developed. +III. +SYSTEM DESIGN +We present Privacy Plumber as a proof-of-concept and +end-to-end system to address the challenges listed of scalable +and general population serving privacy tools for emerging +smart homes. Privacy Plumber is inspired by various handheld +tools for identifying and fixing faults in large and complex +systems. For example, acoustic leak finding has been used for +decades to localize leaks in gas and water pipelines. Handheld +oscilloscopes, multimeters, and RF Spectrum Analyzers have +helped engineers debug problems in large electrical systems. +These handheld devices make the invisible signals visible and +interactive. They allow real-time experimentation and debug- +ging. Inspired by these devices, Privacy Plumber is designed +to offer a general user a level of insight and control of the +invisible privacy leaks that are rampant in Internet-connected +smart devices in the home. Privacy Plumber is composed of +two pieces as shown in Figure 3: +(1) the IoT Network Analyzer, a desktop application +which collects real-time data on smart devices on the shared +Wifi network, and provides an infrastructure and hardware free +mechanism to block arbitrary devices traffic, and; +(2) the Privacy Plumber phone application, which serves +as a viewfinder or inspector for any smart devices in view, and +presents data from the desktop application, including device +network traffic and potential privacy leaks to the users, along +with educational content matched to what is known about the +device, all in real time. +4 + +ARP + Spoofer +Viewfinder +Educational +Views +Packet +Analyzer +Traffic Analyzer +Controls +IoT Network Analyzer +Router +Privacy Plumber App +Traffic and Control +Visualizer +IoT Devices +Network +Traffic +ARP +Spoofing +Leak +Information +Fig. 3: System diagram of Privacy Plumber including the IoT Network Analyzer and Privacy Plumber mobile application. IoT +Network Analyzer runs on a computer that is connected to a user’s router. IoT Network Analyzer automatically discovers and +captures IoT devices on the same network using ARP spoofing. Privacy Plumber connects with IoT Network Analyzer to present +the network analysis in AR. The user can then examine their devices’ network traffic and control when they want their devices +to be on or off. +Overview of Usage. A user would first download, install, +and run IoT Network Analyzer on their computer and the +Privacy Plumber app on their mobile phone, such that both +the computer and the phone are on the same local area +network. Let us assume that the user is interested in inspecting +a smart device like an Amazon Echo. While running the +Privacy Plumber app, the user points the phone camera to +Echo and speaks a voice command (e.g., “Alexa, what is +the weather?”) IoT Network Analyzer captures all network +traffic between Echo and the Internet, analyzes the packets, +and identifies destinations that are third-party advertising and +tracking companies. The Privacy Plumber app extracts this +information from IoT Network Analyzer and visualizes key +statistics for the user—such as real-time bandwidth usage of +the device and the number of advertising and tracking services +contacted—as an overlay in the AR view. +When the user points the phone camera at a device, the +Privacy Plumber app does not recognize devices with computer +vision algorithms. Instead, for the purpose of this prototype, +we print a QR code on each IoT device. The QR code includes +the device’s MAC address, its name, and the manufacturer. The +app uses the phone’s camera to scan for the QR code, identifies +the device based on the QR code, and displays the device with +a dial menu around it (see Figure 4a). The options in the menu +allow the user to see the outbound traffic from the device as +well as read a brief article stating what types of information +the device may be tracking. The user may also use the Device +Control menu (Figure 4c) to manually block or allow traffic +from the device. Future versions of the app will use computer +vision to recognize devices; see the discussion in Section V. +Privacy Threat Model and Assumptions. We assume that +a user’s privacy may be potentially violated if an IoT device +exhibits either or both of the following behaviors. In Threat 1, +an IoT device could contact an advertising and tracking service +on the Internet. In Threat 2, an IoT device could be sending +out network traffic to hosts on the Internet when the user does +not expect any network activities—for example, when the user +is not interacting with the device. +We design both the Privacy Plumber app and IoT Network +Analyzer with this privacy threat model in mind. IoT Network +Analyzer captures packets, analyzes the headers, identifies the +destination hosts (based on the IP addresses, domain names, +and the TLS Server Name Indication fields), and determines +if a destination host is an advertising and tracking company. +The Privacy Plumber app displays the number of advertising +and tracking services (e.g., the red text below the graph in +Figure 4b), thereby helping users toward identifying Threat 1. +Based on the byte counters from IoT Network Analyzer, the +Privacy Plumber app also shows a bandwidth graph that plots +the bytes sent per second over time (e.g., the time-series graph +in Figure 4b). This graph could help users correlate network +activities with human interactions—or the lack thereof—with +given IoT devices and thus identify possible instances of +Threat 2. Note that IoT Network Analyzer does not parse the +payload of packets to discover sensitive information within the +traffic, as the network traffic is likely encrypted. +A. Design Goals +Privacy Plumber must make the underlying behavior of the +devices in the home apparent, and enable forms of fine-grained +(informed) control of the leakage of sensitive information for +the user. Towards this end, and addressing the challenges +described in Section II-D, we are guided by the following +design goals. +(1) Handheld and Mobile. Smart devices are scattered +throughout the home. Phone adoption is nearly universal. +Using a phone as a window into the information world gives +5 + +Privacy +Jagwnnd +X +Live Device Traffic +Bytes/seo +1500 +0 +1125 +0 +7500 +3750 +-10 +Current Device Traffic: 26954.1 bytes in the last 10 +seconds +Sending data to 15 different destinations +This data is equivalent to 14702.2 words of text or 13.5 +pictures per minute +Amazon Details +Status: Allowed +BLOCK +ALLOW +Device +Device +Traffic +Traffic +Custom Time Rule +ApplyRule +Time (ex. 8:00AM) +Turn On +Turn Off: +Time (ex. 8:00PM) +RemoveRulecontext and a sense of place. The phone form factor increases +the likelihood of adoption and allows for inspection on the go; +users can trigger or interact with devices and easily watch the +movement of data, instead of having to return to the desktop. +(2) Real-time. Seeing statistics after the fact, as in most +systems, is not as impactful or helpful when developing a +model of how devices operate. Moreover, real-time analysis +enables experimentation, providing users with a mechanism for +exploring limitless scenarios and quickly associating triggers +with outcomes. +(3) Infrastructure/Hardware Free. Many other meth- +ods require custom hardware. This increases cost and raises +the barrier to entry. We hope to enable anyone, especially +those that may have limited autonomy over infrastructure (i.e. +renters, low-resourced communities) to be able to inspect the +devices put in their living space. +(4) Intuitive Controls. Complex mechanisms to control or +limit the flow of privacy are not interpretable by users, and are +possibly frustrating. Configuring a firewall is not a task most +people would enjoy. Straightforward controls, with visible +results, once those controls are put in place, are essential for +users to trust the capability of the system. +(5) Educational. The ever-changing landscape of devices +and the security/privacy arms is impossible to keep up with for +privacy tools. Assisting users in understanding what makes cer- +tain devices leakier (e.g., always-on microphone) is essential. +To realize these design goals, we build the Privacy Plumber +app—i.e., the handheld form factor—and IoT Network An- +alyzer as a two-part architecture working in tandem. Both +systems must be running on the same local area network. +IoT Network Analyzer, running on a computer, captures and +analyzes network traffic between smart devices on the network +and the Internet. IoT Network Analyzer acts as a server and +provides the above information over an HTTP REST API. The +Privacy Plumber app, acting as a client, regularly polls the +REST API and presents the analysis as an AR overlay to users. +In the following sections, we detail the pieces of the system +and how they interact to enable understanding and control of +smart devices in the home. In Section III-B we discuss the +IoT Network Analyzer and its role in capturing and curating +privacy leak information; in Section III-D we describe the +phone app design; in Section III-C we detail the mechanisms +we use for controlling devices on a schedule, and finally, in +Section III-E we describe a few ways to use Privacy Plumber. +B. Low Burden Home Network Traffic Capture +To use the Privacy Plumber app, the user must also +have IoT Network Analyzer running on a computer (macOS, +Windows, or Linux) that is on the same local area network +as the phone. For our study, we run IoT Network Analyzer +on a Raspberry Pi 3 Model B that is connected to the lab’s +network via Ethernet. We based IoT Network Analyzer’s code +on the open-source project, IoT Inspector [15], and made +modifications according to our needs. In particular, whereas +the original IoT Inspector constantly sends captured traffic’s +metadata to the researchers’ servers, IoT Network Analyzer +runs without the Internet; it processes the captured traffic +locally and exposes the traffic via a REST API. Furthermore, +whereas the original IoT Inspector runs on users’ computers +and visualizes the traffic in a browser-based dashboard, IoT +Network Analyzer uses an AR-based app, Privacy Plumber, +to visualize the network traffic; the mobile app reads the +processed traffic through the abovementioned REST API and +presents the results as an AR overlay. +Once running, IoT Network Analyzer automatically discov- +ers IoT devices on the network, captures their network traffic +via ARP spoofing, produces traffic statistics (e.g., bandwidth +usage and identifying advertising and tracking services) over +a local HTTP REST API, and blocks select devices (if desired +by the user). We explain each of these features below. +Discovering IoT devices. Upon launch, IoT Network An- +alyzer automatically broadcasts ARP packets to the local +area network and discovers active devices. To identify IoT +devices, Huang et al. [15] describe an algorithm that infers +the likely identities of IoT devices based on MAC OUI (i.e., +Organizationally Unique Identifier, basically the first three +octets of a MAC address), DNS, and UPnP messages. For the +prototype in this paper, we only use the MAC OUI. Within the +code of IoT Network Analyzer, we have already hard-coded +the mapping between OUIs and names of five IoT devices in +our lab (which we can find out beforehand). In this way, IoT +Network Analyzer can instantaneously identify the IoT device +in our lab without relying on the device identification algorithm +in Huang et al. [15]. +Capturing network traffic. Once IoT Network Analyzer iden- +tifies a known IoT device on the lab’s network, it automatically +starts intercepting network traffic between the device and the +Internet via ARP spoofing, a technique used in the original IoT +Inspector implementation and which incurs an overhead of 3.4 +Kbps, given that we have five IoT devices in the lab [15].1 +Obtaining traffic statistics. All traffic to and from IoT devices +in our lab is redirected through IoT Network Analyzer. In do- +ing so, IoT Network Analyzer is able to obtain statistics about +the network traffic for every device, including the device’s +MAC address (from which to extract the OUI and determine +the device’s identity based on our hard-coded mapping); the +number and size of packets (from which to infer the bandwidth +usage); the remote IP addresses, DNS requests and responses, +and the Server Name Indication field within TLS packets +(from which to infer the remote hostname and whether the +hostname is associated with a known advertising and tracking +company, based on the Disconnect block list [12]. IoT Network +Analyzer presents all these statistics and information via an +HTTP REST API that the Privacy Plumber app can access over +the local area network. For example, if the computer running +IoT Network Analyzer has a local IP address of Ii, then the +Privacy Plumber app (on the same local network) can access +the traffic information via http://[Ii]/get traffic. +Phone +Application: +App +Implementation. The Privacy +Plumber mobile app was implemented in Unity using C# +and is cross-platform, tested on Android and iPhone. The +1Per Huang et al. [15], our setup includes N = 5 devices. It follows that +N(N + 1) = 30 spoofed ARP packets are sent every two seconds. As each +ARP packet has 28 bytes, the overhead is 28×30/2∗8 = 3, 360 Bits/second +or 3.4 Kbps. +6 + +(a) View finder +(b) Traffic view +(c) Controls +(d) Education +Fig. 4: Illustration of mobile application design. (a) Device recognition with interactive menu. (b) Live traffic inspection. (c) +Rule-based device traffic control (i.e., blocking and unblocking). (d) Educational material on privacy details. +app works by communicating with IoT Network Analyzer via +HTTP GET requests, as described in the previous paragraph, +to obtain JSON-encoded information about the devices on the +network and their traffic. Parsing these JSON objects, the app +visualizes the information as charts and text on the AR display +(e.g., Figure 4b). The app also shows an interface where users +could block an IoT device’s traffic, e.g., Figure 4c. Once the +user confirms, the app sends the corresponding request to IoT +Network Analyzer via the HTTP REST API, and IoT Network +Analyzer would subsequently block the device by jamming the +device with corrupt ARP packets. +C. User Control of Privacy Leaks from a Phone +With Privacy Plumber we also want to help the user feel +more empowered by allowing them to take control of their +devices with the ability to block device traffic. Users can +manually block or allow device traffic indefinitely, or they can +set rules to govern when they want their device to be on or off +and for how long (Figure 4c). Users are also given the option +to physically power off their device altogether. In this way, +Privacy Plumber provides a closed-loop system where users +can analyze the information flow out of a given device, then +immediately apply direct control over that device in response +and receive immediate feedback via the traffic view. +To illustrate how a user might control an IoT device’s +traffic, let us say that a user feels uncomfortable with an IoT +device communicating with the Internet. The user can use the +Privacy Plumber app to block Internet access on the device. +As shown in Figure 4c, the user can click “Block Traffic” +on the app to indefinitely block the device, or specify when +to block and unblock the device. Moreover, the app sends +an HTTP request to IoT Network Analyzer, using the REST +API 2 (where Ii is the IP address of the running instance +of IoT Network Analyzer). During the period of blocking, +IoT Network Analyzer jams the communication of the device +by using a corrupt source MAC address in the spoofed ARP +packets (as described in Section III-B). IoT Network Analyzer +stops this process at [unblock time], upon which IoT Network +Analyzer sends out spoofed ARP packets without the corrupt +source MAC address. This gives users the ability to control +the times of day when they want their devices to be on or off. +Privacy Plumber’s software-based device blocking offers +several advantages over simply turning off or disconnecting +a device. First, users do not need physical access to the +device; for instance, many surveillance cameras are mounted +on ceilings and are difficult to power off. Second, users can +temporarily disable a device when they are feeling uncom- +fortable, e.g., blocking Amazon Echo for an hour during a +sensitive phone call or conversation, through Privacy Plumber. +Such temporary blocking is difficult to achieve through Echo’s +app (no such feature) or manually (e.g., the user has to +remind themselves to re-connect Echo again). Third, though +not currently implemented, Privacy Plumber, with the help +of IoT Network Analyzer, can block based on the context +(i.e., future work). For example, when IoT Network Analyzer +detects the presence of a user’s phone on the network (e.g., by +checking if the phone responds to periodic ARP requests), IoT +Network Analyzer automatically blocks all indoor cameras; +when the phone leaves the network (e.g., when the user is +out), IoT Network Analyzer could automatically unblock all +indoor cameras. +Technical Mechanism for Blocking Devices. A major differ- +ence with respect to IoT Inspector’s original implementation is +2http://[Ii]/block/[device id]/[block time]/[unblock time] +7 + +Privacy +Lagwnd +(A) +amagon +M +Device Details & Control >Privacy +PLumber +X +Live Device Traffic +Bytes/sec +720 +540 +360 +180 +-10 +.9 +CurrentDeviceTraffic:984.3bytes inthe last10seconds +Sending data to 17 different destinations, including 2 +advertising service(s) +This data is equivalent to 536.9 words of text or 0.5 +pictures per minute +M +amazon +Device Details & Control >Privacy +Lagwd +(A) +M +cef:co.ob: +Amazon Details +Status:Allowed +BLOCK +ALLOW +Device +Device +Traffic +Traffic +CustomTime Rule +ApplyRule +:uo ni +Time (ex. 8:00AM) +Tum Off: +Time (ex. 8:00PM) +Remove RulePrivacy +Laqwnd +X +Device Privacy Details +Echo Smart Speaker +This voice assistant is like a butler, providing voice access and +control of music and other media, enabling voice based +purchases, making phone calls, and helping to keep track of +things,among many other functions. +Potential Privacy Leaks +Location +Your physical location is leaked when interacting with this +device. Because motion can be detected, your near exact +locationwithinaroomcanbedetermined. +讠 Activity +(A) +amazo +三 +DeviceDetails&Control>that we have added the capability of blocking devices in IoT +Network Analyzer. Using the HTTP REST API 3, the Privacy +Plumber app can request IoT Network Analyzer to block a +certain device at a particular time (for instance, because the +user does not want the device to be communicating to the +Internet). Upon receiving this request, IoT Network Analyzer +jams the network communication of the device by sending it +spoofed ARP packets with corrupt MAC addresses. +To illustrate this process, let us assume that the computer +running IoT Network Analyzer has a MAC address Mi and IP +address Ii. Let us also assume that IoT Network Analyzer is +about to intercept the communication from the gateway (with +MAC address Mg and IP address Ig) to a particular device +(with MAC Md and IP Id) without blocking. To do so, every +two seconds, IoT Network Analyzer sends an ARP packet to +the device, such that the ARP packet has a source MAC of +Mi and a source IP of Ig, along with a destination MAC of +Md and a destination IP of Id. In contrast, let us assume that +IoT Network Analyzer is to block the device. It sends the +same ARP packet to the device, except that the ARP packet’s +source MAC is a series of random numbers (instead of Mi) +that represent an unreachable MAC address on the local area +network. +D. Visualizing and Understanding Traffic in Real-Time +One of the goals of Privacy Plumber is to show users +contextualized network activities of IoT devices to help them +pinpoint the potential privacy risks. In this section, we discuss +how Privacy Plumber utilizes Augmented Reality to help users +contextually visualize their devices’ network traffic informa- +tion in real-time, provide a chart of network traffic in real- +time, and provide links to other research in which the privacy +concerns of the inspected device have been studied (including +home behavior inference, sleeping behaviors, and personal +data). Lastly, users are able to send feedback and bug reports. +Use of Augmented Reality. The use of AR visualization +makes the interaction with the device the user is inspecting +more tangible and contextual. While IoT Inspector [15] and +IoT Network Analyzer are text-only data-driven analyzers that +can only be accessed using browser HTTP requests, Privacy +Plumber is a fully-fledged interactive application due to the +utilization of AR. By pointing their camera at the device +being inspected, the user can see, in their environment, the +traffic coming out of the device that they are physically +inspecting. Users can interact with their devices and receive +immediate feedback about data output and communication +with advertisers. Combined with manual device control, this is +intended to help the user feel informed and in control of the +IoT devices that physically surround them, similar to the use +of a TV remote control. +Learning About Privacy Threats. We aim to educate and +inform users on how their IoT devices expose their network +traffic information to third parties. In Figures 4d and 5, the +app shows icons surrounding the IoT device. When any of +these icons are pressed, they provide links to other research +materials—which we have manually curated in advance— +where the privacy concerns of the inspected device have been +3http://[Ii]/block/[device id]/[block time] +studied. Depending on the device, Privacy Plumber provides +the following categories of potential privacy violations repre- +sented by icons: +• +Location: Your physical location either roughly (your +address) or fine-grained (room you are in). +• +Activity: Your physical activity such as walking, talk- +ing, sleeping. +• +Screen: Your online activity, such as when you browse +videos on YouTube or surf the web. +• +Identity: Attributes that can identify you such as your +face or voice. +• +Shopping: Data on your usage of money or products. +• +Health: Can infer different health markers without +consent (heart rate, breathing, and others). +E. Privacy Plumber Example Use Cases +In this section, we illustrate two example use cases of +the Privacy Plumber app. We focus on the ability of Privacy +Plumber to enable experimentation and the usefulness of a +real-time inspector. We will describe the users’ reactions in +Section 4.3. +Example 1: Is Echo Always Listening? +A user may use the Privacy Plumber app to correlate net- +work activities on an Amazon Echo device with the user’s +interactions—or the lack thereof—with it. While pointing the +AR camera at the device, the user could invoke a voice com- +mand, such as “Alexa, what is the weather”, while observing +the device’s bandwidth usage graph on the Privacy Plumber +app. Afterward, the user may physically press the mute button +on Echo, repeat the same voice command, and observe the +bandwidth usage graph on the app. +Example 2: What is this App on My Smart Fridge? +Many smart fridges have built-in touch-screen panels. For +example, the Samsung Smart Fridge has a tablet-like touch- +screen panel to control various settings of the fridge (such as +temperature). The panel also allows users to access various +built-in apps, such as checking recipes or ordering ingredients +online. A user who is concerned with the privacy of such +apps may point the AR camera at the fridge, interact with +an app, and observe the advertising and tracking services +counter on the app. This counter shows, in real-time, the total +number of advertising and tracking services that the fridge has +communicated with, based on the Disconnect block list [12]. +IV. +PILOT USER STUDY +To test how users react to Privacy Plumber and inform its +future iteration, we conducted a pilot study with 6 participants +to experiment with, understand, and control the potential +privacy violations of IoT devices. It should be noted that the +pilot study would be best conducted in participants’ homes. +However, due to University research restrictions, the COVID- +19 pandemic has made it difficult for us to recruit real users, +distribute hardware (e.g., phones powerful enough for AR +and Raspberry Pi’s for running IoT Network Analyzer), and +conduct a free-living study. +8 + +We conducted a one-day controlled lab study in our IoT +Lab with 6 participants. Participants were invited to use +the Privacy Plumber app while interacting with several IoT +devices in the lab, including Samsung Smart Fridge, Amazon +Echo, Google Home, Samsung Smart TV, and Google’s Nest +Cam. Our goal is to assess whether using augmented reality +to display network traffic (i.e., by using Privacy Plumber) +influenced the participants’ awareness of privacy and changed +their behaviors. +In the following sections, we present the details of the pilot +study and discuss some highlights in the results as well as +lessons learned to inform the next iterative of Privacy Plumber. +A. Participants Recruitment and Demographics +We recruited 6 graduate students from our institution +through our university mailing list. We did so rather than +recruiting from a broader population sample because of the +constraints our university implemented during the COVID-19 +pandemic (i.e., external members were not permitted to enter +our buildings). Our sample consisted of four males and two +females. Three of the participants were between the ages of +18-24, two participants were between the ages of 25-34, and +one participant was between the ages of 35-44. +B. Study Procedure and Data Collection +For safety reasons and to implement social distancing +procedures, only two people were allowed in the IoT Lab +during the study. Aside from the participant, one of the co- +authors in this paper served as the research coordinator. They +were present throughout the user study to help guide the +participants or troubleshoot any technical difficulties that could +arise during the study procedure. +Before the study began, each participant filled out a back- +ground pre-survey on a computer in the IoT lab. We asked +questions about their demographics, how technically savvy +they are, their smart device experiences, their general under- +standing of privacy, and their concerns about their information +being exposed to third parties. +After completing the survey, our research coordinator +handed each participant a script and an Android mobile phone +that had Privacy Plumber installed. Following the script, each +participant opened the Privacy Plumber app, kept it running in +the foreground, and interacted with one IoT device at a time. +Regardless of the IoT device, each interaction consists of the +following steps, as prescribed in the script: +1) +Using the Privacy Plumber app, the participant +scanned the QR code that we had placed on the +IoT device. The QR code encodes the device’s MAC +address, device name, and manufacturer. Based on the +QR code, Privacy Plumber shows the corresponding +device’s AR model on the screen. +2) +The participant used the app to inspect the device’s +traffic, while not doing anything to the device. +3) +The participant interacted with the device (which we +will describe in detail). During the interactions, the +participants observed the network traffic graph on the +app. +Fig. 5: This screen on the phone application describes the +different categories of privacy leaks that different devices have, +based on a database that we manually curated in advance. +4) +Using the app, the participant clicked on any of the +icons surrounding the AR model of the device and +read the educational material. +After interacting with all the IoT devices, participants +returned the phone to the research coordinator and responded +to a post-survey that asked the same questions as in the pre- +survey, along with a usability survey. We discuss the results in +more depth in Sections IV-C and IV-D. We also include our +pre- and post-surveys in the Appendix. +Below, we describe each participant’s scripted interactions +with each device—i.e., showing Step 3 in detail. During the +interactions with the devices, users can access the educational +content which is summarized from Mozilla’s “privacy not +included” handout [29] and academic literature. Each device +is described by the categories of privacy exposure they create, +those categories are shown in Figure 5. +Samsung Smart Fridge. The fridge has a built-in touchscreen +on the door. Through the touchscreen, users have the ability +to interact with several built-in apps, such as managing the +shopping list, checking what is inside the fridge, and searching +for recipes online. Users can also interact with the touchscreen +using voice commands, using the trigger word, “Bixby.” +Per the script, the research coordinator instructed the +participant to follow the following three tasks. (i) Once the +participant scanned the QR code of the smart fridge, they +said the voice command, “Hey Bixby, do we have mangoes?” +Bixby, the fridge’s voice assistant, would say “no,”. The +participant then said, “Hey Bixby, can you add mangos to +my shopping list?” Immediately, the participant looked at the +9 + +Privacy Leak Categories +Smart devices can collect private information about you +intentionally or incidentally in the following ways. +Location +Your physical location either roughly (your address) or fine- +grained (room you are in). +Activity +Your physical activity, such as walking, talking, sleeping. +Screen +Your online activity, like when you watch videos on YouTube +or surf the web. +Identity +Collects attributes that can identify you such as your face or +voice. +Shopping +Collects data on your usage of money or products. +Health +Can infer different health markers without consent +(heartrate, breathing and others).-3 +-2 +-1 +0 +1 +2 +3 +Strongly Disagree +Strongly Agree +I think about what information I may be exposing to 3rd parties +when I interact smart devices. +I am not concerned over the +information I may be exposing to 3rd +parties when I interact with smart +devices. +Privacy Plumber has made me more aware of what information I may +be exposing to 3rd parties when I interact with smart devices. +Privacy Plumber has made me more aware of privacy concerns +regarding smart devices. +Fig. 6: Representation of participants’ average agreement ratings relating to statements about information being exposed to third +parties and privacy concerns caused by interacting with IoT devices. Participants rated the first two statements before and after +the study, while the last two statements were rated at the end of the study. The results show that after the study, participants +displayed an increase in awareness and concern about how their information is being handled when interacting with IoT devices. +Privacy Plumber app and observed the network traffic emitting +from the fridge for about 30 seconds. (ii) The participant +said, “Hey Bixby, find me a Ramen recipe.” The recipe app +popped up on the touchscreen. Using the finger, the participant +browsed through the available recipes on the screen, while +observing the network traffic on Privacy Plumber for 30 +seconds. (iii) The participant opened the fridge door and then +closed it. Once again, they inspected the fridge’s network +traffic through the Privacy Plumber app for 30 seconds. +Amazon Echo. Interactions with Echo consists of the follow- +ing 3 tasks. (i) The participant said the voice command, “Alexa, +play a thunderstorm sound.” Immediately, the participant ob- +served the network traffic on the app for 30 seconds. (ii) The +participant physically pressed the “mute” button on the Echo +and watch the device’s network traffic for 15 seconds. (iii) The +participant said the same voice command as in Task (i) and +observed the traffic in the app. +Google Home. The participant said a voice command, “Hey +Google, what was the final score in the Super Bowl last year?” +The participant immediately started observing the network +traffic on the app for 30 seconds. +Samsung Smart TV. The participant used the TV’s remote +control to navigate to the Bloomberg app on the home screen. +They then started streaming a live video on the Bloomberg app +for one minute while they observed the network traffic with +the Privacy Plumber app. +Nest Cam. Interactions with the camera consists of the follow- +ing 2 tasks. (i) The participant walked into the field of view of +the camera and stay there for five seconds, walked out of the +camera’s field of view, and observed inspect the network traffic +with the Privacy Plumber app. They repeated this task as many +times as they liked. (ii) The participant blocked the network +traffic to and from the camera using the built-in feature on the +Privacy Plumber app. The participant observed the network +traffic for 10 seconds, walked in front of the camera’s field +of view, waited for another ten seconds, and unblocked the +device using the Privacy Plumber functionality. +C. Analysis of Pre-Study and Post-Study Surveys +We asked each participant to complete two surveys: (i) a +pre-Study Survey that they filled out on a dedicated computer +at the beginning of the study, i.e., before the participants +interacted with the Privacy Plumber app or the IoT devices; +(ii) a post-Study Survey that the participants filled out on +the dedicated computer after interacting with all the five IoT +devices. We present the results below. +In Figure 6 we present the participants’ agreement rating +responses for two statements that were asked in the pre-study +survey and post-study survey. We observe that for those two +statements participants seemed less concerned by how their +information is exposed to third parties when they interact with +IoT devices before they performed the activities in the study. +After participants completed the study, they were more aware +and concerned about how their information was disclosed to +third parties. The last two statements of Figure 6 were only +given in the post-study survey, which asked participants to rate +whether Privacy Plumber was useful in helping them become +more aware of privacy concerns and how their information is +being shared with third parties. On average, participants some- +what agreed that Privacy Plumber helped raise their awareness +and privacy concerns. Participants found that Privacy Plumber +was helpful in that it helped them visualize what information +was being shared. +Additionally, we discuss the results of participants’ re- +sponses with the IoT devices before and after the study. We +show that after the study participants felt less safe with how +IoT devices handle their data. Participants were presented with +three statements and were asked to rate whether they agree or +disagree with these statements on a scale of one to five, where +a 1 meant they strongly agree and a 5 represents a strongly +disagree rating. Table I demonstrates the average change in +attitudes participants had before the study and after the study. +We note that before the study, on average participants neither +agreed nor disagreed with the statements presented in Table I. +After completing the study, the average rating agreement score +increased to “somewhat agree” on the last two statements on +all IoT devices. The exception was in the first statement, the +scores for the Amazon Echo and Google Home. This indicates +10 + +Survey Question +Smart Fridge +Amazon Echo +Google Home +Smart TV +Nest Cam +pre +post +pre +post +pre +post +pre +post +pre +post +I think this device could be (or is) useful or +valuable to my daily life and routine. +3 +3.17 +2.86 +2.5 +2.71 +2.5 +2.43 +2.33 +2.71 +3 +I am comfortable having this device in +my house and always on. +2.29 +3.5 +3.86 +4.17 +3.86 +4.17 +2.29 +3.5 +3.43 +4.17 +I am comfortable having this device in +my house if I can automatically control +when it is on, or off. +1.29 +2.17 +2.29 +2.5 +2 +2.33 +1.14 +2 +2.29 +2.83 +Strongly Disagree (5) to Strongly Agree (1) +TABLE I: Results of the survey on user awareness and comfort with smart devices, before and after using Privacy Plumber to +inspect those devices. Scores are listed for both pre- and post-study surveys for each device. The higher the scores, the more +strongly the participant disagreed with the survey question statement. +that after using Privacy Plumber in the study, participants felt +that the Amazon Echo and Google would still find it useful to +use in their households. +We also observe that the Smart Fridge, Smart TV, and +the Nest cam had the most significant change in attitude. We +gathered a few quotes from participants in which they describe +how they felt about interacting with these IoT devices and +using Privacy Plumber to inspect their network traffic: +IoT devices provide more information to third par- +ties than people thought. I think apps like Privacy +Plumber can help people to make better decisions +when using IoT devices — (P1) +Cool to see when and how much traffic each device +sends at any given moment! — (P5) +I think the app does make me more aware about +how the traffic is associated with the behavior of the +device. Having some control over the traffic is nice. +That being said, if I do have the device in my home, +I probably would like to use it, and in that case, I +have to allow traffic, which I have no control about +what could pass or could not pass. In that sense, I +can only accept certain privacy risks. — (P2) +It was interesting to see the potential privacy leaks +shown next to the device. Some leaks/ privacy im- +plications were surprising. Liked the ability to al- +low/block traffic, it was also cool to see the real- +time traffic including communication with third-party +advertisers. Liked the app interface. —(P6) +These quotes, along with results from Figure 6 and Table I, +suggest that Privacy Plumber helped participants understand +the network traffic, increased their awareness of potential +privacy violations, and helped them make more informed +decisions on how to handle IoT devices. +D. Analysis of the Usability Survey +At the end of the study, each participant completed the +usability survey. Overall, most participants indicated that they +would use Privacy Plumber in their home network, found it +easy to use and user-friendly, and agreed that most people +would learn to use Privacy Plumber quickly. We summarize +the results below: +• +When asked if they would use the Privacy Plumber +mobile app to inspect the data the IoT devices in their +homes, two participants said they strongly agreed with +the statement and four participants said they somewhat +agreed to use Privacy Plumber. +• +When participants were asked if they found Privacy +Plumber easy to use, four of them somewhat agreed, +one participant strongly agreed, and one participant +somewhat disagreed. +• +When presented the statement “I imagine that most +people would learn to use Privacy Plumber very +quickly”, the responses were across the board spec- +trum. Three participants rated that comment as +strongly agreed, one participant rated the statement +with a somewhat agree, one participant responded +that they felt neither agreed or disagreed with the +statement, and one participant somewhat disagreed. +• +When participants were asked to rate the overall user- +friendliness’s of Privacy Plumber, four participants +rated the Privacy Plumber app as good and two +participants rated Privacy Plumber as fair. +We gave participants an open-ended question if they would +improve the usability of Privacy Plumber, and if so, how. +We show their responses in Appendix B. All in all, partici- +pants seemed to respond somewhat positively towards Privacy +Plumber. It shows that Privacy Plumber may have the potential +to be distributed to the general public after further studies. +We hope to build off our current platform and implement the +suggestions our participants gave us in future work. +E. Performance: System Overhead and Battery Life Impact +Network Overhead. IoT Network Analyzer intercepts the +network traffic of select IoT devices via ARP spoofing, a +technique that could introduce network overhead especially +to the targeted IoT devices. This overhead comes from two +sources. First, the spoofed ARP packets consume extra band- +width, although the overhead is relatively small—i.e., less than +60 Kilobytes/second even if 50 IoT devices are under ARP +11 + +spoofing [15]). The second source of overhead comes from +the Raspberry Pi 3 Model B, where we run IoT Network +Analyzer in the lab. The Raspberry Pi is connected to the +lab’s network via Ethernet. For all IoT devices to which IoT +Network Analyzer sends spoofed ARP packets, all inbound +(i.e., download) and outbound (i.e., upload) traffic to and +from the IoT devices has to first go through the Raspberry +Pi before IoT Network Analyzer forwards the traffic to the +targeted device and to the Internet respectively. Effectively, +the Raspberry Pi introduces a bottleneck for the ARP-spoofed +devices. +To measure the overhead as a result of the Raspberry Pi +bottleneck, we conduct the following experiment. We install +the Ookla Speed Test app on an Android phone that is con- +nected to the the lab’s WiFi network. We have the Ookla app +run 15 back-to-back speed tests, which measure the inbound +and outbound traffic rates with respect to a server in our city, +as well as the latency of packets. Using the same setup, we +repeat the same experiment, except that we have IoT Network +Analyzer inspect the phone’s traffic via ARP spoofing. +We find significant overhead as a result of IoT Network +Analyzer. Without ARP spoofing, the app achieves, on average, +an inbound rate of 293.6 ± 15.4 Mbps, an outbound rate of +94.1±0.2 Mbps, and a latency of 5.7±0.5 milliseconds. With +ARP spoofing by IoT Network Analyzer, the app achieves, on +average, an inbound rate of 41.4±74.6 Mbps, an outbound rate +of 72.8 ± 14.1 Mbps, and a latency of 5.9 ± 0.5 milliseconds. +Compared with the case without ARP spoofing, IoT Network +Analyzer reduces the inbound rate by 85.9% and outbound rate +by 22.6%, while increasing the latency by 3.5%. +Despite the seemingly significant reduction in bandwidth, +we argue that IoT Network Analyzer is unlikely to degrade +usability, as the network analyzer is not always running (only +when inspecting, or blocking a specific device). Additionally, +the overhead can be reduced with improved hardware. Ac- +cording to Netflix, 25 Mbps of inbound rate is sufficient to +stream Ultra HD contents [31]. A user who inspects a smart +TV using IoT Network Analyzer is likely to enjoy Ultra HD +streaming given the reduced inbound rate of 41.4±74.6 Mbps +under ARP spoofing. If a user desires to reduce the network +overhead, the user could upgrade the computer that runs IoT +Network Analyzer, as Raspberry Pi 3 is anecdotally known for +its poor networking performance [37], [38]. Possible upgrade +option could include a computer—or ODroid if the user needs +the compact form factor [14]—that is shipped with a fast CPU +and a Gigabit Ethernet card. +Battery Lifetime. We used AccuBattery on android, to try to +understand the energy cost. This does not hold across phones, +so we compare the energy cost against YouTube and TikTok +for ten minutes of streaming video. With all the background +application killed, 10 minutes of Privacy Plumber impacts +3.98% (159mAh) of the battery lifetime, while YouTube costs +2.63% (105mAh) and TikTok costs 3.9% (156mAh). Privacy +Plumber is only meant for point inspection and short usage to +analyze new devices in the home, or experiment with different +setups, so it should not impact battery lifetime too much since +it is not always on. Moreover, the battery lifetime cost is +similar to that of streaming videos online, a normal function, +therefore users should not expect significant battery lifetime +loss due to usage of Privacy Plumber. +V. +DISCUSSION ON LIMITATIONS AND FUTURE WORK +Comparing users’ mental models against actual contents +of IoT network traffic. Our results show that users’ mental +model of how IoT devices communicate with the Internet may +be inconsistent with how devices appear to behave, but it is +unclear whether this mental model is consistent with the actual +contents of the communication. For example, two participants +in our study did not expect network traffic from Amazon Echo +when the device’s microphone was on mute. Presumably, the +participants expected Amazon not to send any audio data back +to Amazon during mute. In this case, Echo’s apparent behavior +was the communication with the Internet on mute; in contrast, +whether Echo actually sent out audio data was unknown. Our +system did not extract the contents of the communication, +which could be encrypted based on previous results [4]. +Despite the encrypted contents, man-in-the-middling is +possible (e.g., per Moghaddam et al. [28]). In future in-lab +studies, we plan to modify IoT Network Analyzer to intercept +and decrypt IoT traffic, assuming that devices do not validate +certificates and/or do not use certificate pinning. We hope to +extract the payload from some of the TLS connections, identify +exactly what devices are sending to the Internet, and compare +it against users’ mental models. +Automated, contextualized blocking of devices. The current +prototype allows users to set a block/unblock schedule for IoT +devices. Although this feature provides users with fine-grained +control, it requires manual effort from the user both in setting +what devices to block and when to block. +We plan to augment this feature with automated device +blocking based on contextualized information that IoT Net- +work Analyzer already collects. For example, a user could +create a rule on IoT Network Analyzer that would automat- +ically block surveillance cameras if IoT Network Analyzer +detects the presence of mobile phones (based on ARP and +pings) in the home network (which could suggest that the +residents are home); otherwise, it can unblock the cameras to +capture, say, unauthorized entry into the property. As another +example, let’s say a user has an Amazon Echo and a smart +TV in the living room. The user could create another rule that +lets IoT Network Analyzer automatically block Amazon Echo +if it detects active streaming traffic from the smart TV, as the +user may not want Echo to capture any conversations while +the family is watching TV in the living room. In short, by +leveraging the IoT traffic that IoT Network Analyzer already +collects, users could create automated, contextualized rules to +block IoT devices from collecting sensitive data. +Deployment roadmap and challenges. We plan to deploy the +Privacy Plumber app and IoT Network Analyzer to real-world +users at scale. Based on our current prototype, we plan to make +the following modifications. +Operating system support. Once deployed, our system will +have the same two-component architecture, although we will +expand the Privacy Plumber app to both iOS and Android +(current prototype), and IoT Network Analyzer to all major +non-mobile operating systems including macOS, Windows, +12 + +and Linux (current prototype). This process will likely be +straightforward, as we developed both components with cross- +OS platforms (Unity for the app and pure Python for IoT +Network Analyzer). +Network-based device identification. We will develop +network-based device identification mechanisms to help users +distinguish among their devices and identify the device(s) +of interest. The current prototype identifies devices based +on a hard-coded mapping between MAC OUIs and device +names, because we already know the inventory of IoT devices +in the lab. For real-world deployment, we will incorporate +IoT Inspector’s device identification algorithm [15], so that +our system will dynamically infer device names based on +the network signature, which includes not only OUIs, but +also DNS queries, UPnP banners, mDNS names, and DHCP +hostnames. We will also use information in the 802.11 frames +to discover and locate devices [41]. +Image-based device identification. To complement the +network-based approach, we will also develop image-based +device identification mechanisms for the AR camera. Cur- +rently, the Privacy Plumber app identifies devices based on +printed QR codes on or near select IoT devices, such that +the QR codes encode the MAC addresses and the names of +devices. For real-world deployment, we will use computer +vision to train a model of common IoT device types, such as +voice assistants, smart TVs, and surveillance cameras (where +security and privacy issues are commonly found in the litera- +ture). This model will help the AR app recognize possible IoT +devices (e.g., “likely a smart TV”). The app will then refine +the recognition with the network-based device identification +algorithm (e.g., “whether the device is indeed a smart TV based +on the network signatures”) and manual user input if necessary. +Both the network- and image-based approaches will hopefully +help the app identify IoT devices in real-world settings. +Expanded user study. The user study, as a pilot, has a small +sample size and is limited to graduate students, who may +be more inquisitive or technically-inclined than the general +population. We hope to scale out the testing to a larger +userbase, both in lab and in real homes, in future work. We will +also compare the participants’ changes in privacy awareness +against other visualization tools (e.g., IoT Inspector [15] and +Aretha [40]). Finally, we will conduct in-depth studies on +various ways to visualize privacy leaks in AR (e.g., icon +overlays and animations). +VI. +SUMMARY +This paper presented Privacy Plumber, an end-to-end sys- +tem demonstrating how a general population of end users can +potentially have insight into the network traffic of smart home +IoT devices, and how these users can control when these smart +devices could communicate with the Internet with one click of +a button. Designed after the concept of a leak detector, Privacy +Plumber is a phone app with a tethered desktop application— +IoT Network Analyzer—that provides an inspect and correct +interface supported by network traffic analysis (inspect) and +automated and timed network traffic jamming (correct). +Privacy Plumber is the first real-world inspection and +control system that can be deployed in any home without new +hardware or router modifications. Using AR, the tool aims to +help users model IoT device activities within the context of +the physical environment and of user interactions (addressing +challenges C1 and C3, per Section II-D); it gives users the +option to block IoT devices and control the privacy “valve” +(C2); it provides users with an interface to visualize IoT device +activities as users interact with devices (C4); and it requires +a modern AR-supported phone and computer, without any +dedicated or specialized hardware (C5). +We evaluated Privacy Plumber inside an instrumented smart +home space with a variety of devices not previously evaluated +for any privacy-enhancing tool, including a smart fridge, a +smart TV, voice assistants, and Internet-connected surveillance +cameras. We found that using Privacy Plumber improved users’ +awareness of potential privacy violations of devices and that +the system was generally easy to use and afforded useful +controls. In the future, we hope tools like Privacy Plumber will +give mechanisms back to the user for stymieing the flow of +private information outside the home, especially as our homes +and living spaces become smarter, often without our consent. +ACKNOWLEDGMENT +This research is based upon work supported by the National +Science Foundation under award numbers CNS-2219867, +CNS-1739809, and CNS-1915847. Any opinions, findings, and +conclusions or recommendations expressed in this material are +those of the authors and do not necessarily reflect the views of +the National Science Foundation. The research is also based +on work supported by gifts from Consumer Reports and Meta. +REFERENCES +[1] +Abbas Acar, Hossein Fereidooni, Tigist Abera, Amit Kumar Sikder, +Markus Miettinen, Hidayet Aksu, Mauro Conti, Ahmad-Reza Sadeghi, +and Selcuk Uluagac. Peek-a-boo: I see your smart home activities, even +encrypted! 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Proceedings of the ACM +on human-computer interaction, 2(CSCW):1–20, 2018. +APPENDIX +SURVEY QUESTIONS +All questions require responses in Likert scales, ranging +from “Strongly Agree” (1) to “Strongly Disagree” (5). +A. Pre-Study Survey Questions +1) +When I am in a smart home, I think about what in- +formation I may be exposing to vendors, companies, +and 3rd parties when I interact with or sit in the same +space with smart devices in the home. +2) +I am not concerned about the information I may be +exposing to 3rd parties when I interact with or sit in +the same space as smart devices in a smart home. +3) +I think this device could be (or is) useful or valuable +to my daily life and routine. +• +Smart Fridge +• +Google Home +• +Amazon Echo +• +Smart TV +• +Nest Cam +4) +I am comfortable having this device in my house and +always on. +• +Smart Fridge +• +Google Home +• +Amazon Echo +• +Smart TV +• +Nest Cam +B. Post-Study Survey Questions +1) +When I am in a smart home, I think about what in- +formation I may be exposing to vendors, companies, +and 3rd parties when I interact with or sit in the same +space with smart devices in the home. +2) +I am not concerned about the information I may be +exposing to 3rd parties when I interact with or sit in +the same space as smart devices in a smart home. +3) +Privacy Plumber has made me more aware of what +information I may be exposing to 3rd parties when I +interact with smart devices in the home. +4) +I feel Privacy Plumber has made me more aware +of privacy and security concerns surrounding IoT +devices. +5) +I think this device could be (or is) useful or valuable +to my daily life and routine. +• +Smart Fridge +• +Google Home +• +Amazon Echo +• +Smart TV +• +Nest Cam +6) +I am comfortable having this device in my house and +always on. +• +Smart Fridge +• +Google Home +• +Amazon Echo +• +Smart TV +• +Nest Cam +7) +Finally, please provide any other thoughts or obser- +vations from participating in this experiment with +Privacy Plumber (open ended). +ADDITIONAL RESPONSES FROM THE USABILITY SURVEY +We gave participants an open-ended question if they would +improve the usability if privacy plumber, if so how. We +obtained the following responses from each participant. +I would include more guidance or instructions in the app +for first-time users. (P1) +I think the app is generally easy-to-use, although I might +want more functionalities in the app. There are certain laten- +cies in the app, which can be annoying. It would be more +helpful if I can know if the device is not sending any traffic, +or it is just simply late (e.g., adding a loading icon). (P2) +Make it possible to view past trends (a la net microscope) +and scroll backwards in time, so I can get the context of how +much traffic is regularly sent. Give me a global view of the +worst offenders. Still some work to do on basic stability. It +only works on devices that people have obviously ALREADY +DECIDED TO BUY, which is a weird sample. Obviously, I +don’t have QR codes printed out on all of my household +electronics. (P3) +I had difficulties trying to access the buttons, and the +images seemed lagged a little. But the info was very useful +overall. (P4) +Fix where the traffic and ‘learn more about the device’ +buttons once you’ve scanned the QR code. It’s a bit awkward +to have to hold the phone back up to the device. Maybe add +the units (byte/kB) to the left hand side of the graph instead +of above it for the traffic visualization. (P5) +The plots are not super-intuitive but I liked the representa- +tions in terms of text/pictures which is easier to comprehend. +I would also be interested to see what advertisers the infor- +mation is being leaked to. While the AR thing is cool, I would +also like the option to just scroll through a list of devices. +That ways I do not have to be close to the device and would +also be able to monitor its activity when I am not close to +the device. In fact, I would be interested in seeing the device +communication (including interaction w/ advertisers) in that +case. (P6) +15 + diff --git a/HdFLT4oBgHgl3EQfIC9G/content/tmp_files/load_file.txt b/HdFLT4oBgHgl3EQfIC9G/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2257b9a52b74a8bca7f840902acbb405823916cd --- /dev/null +++ b/HdFLT4oBgHgl3EQfIC9G/content/tmp_files/load_file.txt @@ -0,0 +1,922 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf,len=921 +page_content='Augmented Reality’s Potential for Identifying and Mitigating Home Privacy Leaks Stefany Cruz1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Logan Danek1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Shinan Liu2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Christopher Kraemer6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Zixin Wang3 Nick Feamster2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Danny Yuxing Huang4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Yaxing Yao5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Josiah Hester6 1Northwestern University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2University of Chicago,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3Zhejiang University 4New York University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 5University of Maryland,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Baltimore County,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 6Georgia Institute of Technology Abstract—Users face various privacy risks in smart homes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' yet there are limited ways for them to learn about the details of such risks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' such as the data practices of smart home devices and their data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this paper, we present Privacy Plumber, a system that enables a user to inspect and explore the privacy “leaks” in their home using an augmented reality tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber allows the user to learn and understand the volume of data leaving the home and how that data may affect a user’s privacy— in the same physical context as the devices in question, because we visualize the privacy leaks with augmented reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber uses ARP spoofing to gather aggregate network traffic information and presents it through an overlay on top of the device in an smartphone app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The increased transparency aims to help the user make privacy decisions and mend potential privacy leaks, such as instruct Privacy Plumber on what devices to block, on what schedule (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', turn off Alexa when sleeping), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our initial user study with six participants demonstrates participants’ increased awareness of privacy leaks in smart devices, which further contributes to their privacy decisions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', which devices to block).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' INTRODUCTION The increasing adoption of Internet-connected smart de- vices has brought huge improvements to our lives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Yet, these devices also raise significant privacy concerns from their users, such as sensitive data collection [53], [51], data sharing [51], and data misuse [22], [23], [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Literature has suggested many types of privacy risks associated with smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, some seemingly innocent data, such as the network traffic shapes and patterns of smart devices, may reveal sensitive personal information, such as users’ daily schedule, their gender, date of birth, social security number, location, and behaviors [5], [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, many risks are not obvious to users due to the opaque nature of the data practices of smart devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' the average users lack an understanding of how their data is collected, processed, and shared [51], [50], [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Prior research has proposed various ways to increase users’ awareness of the data practices in smart homes, such as data dashboards, mobile phone apps, ambient light and sounds, and so on [44], [15], [9], [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Some other mechanisms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', IoT Inspector [15]) focus on specific aspects of the data practices and present network traffic data to users so that they can access first- hand data of the data flow in/out of smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Yet, most mechanisms we know decouple such transparency from the device themselves—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', users need to learn about the data practices separately from the smart devices—making the information less intuitive to consume, especially for the average user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In addition, these mechanisms do not provide users with the ability to take action if they notice unexpected data practices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', blocking the data from being sent out to third parties).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this paper, we focus on the data flow in and out of smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We build a proof-of-concept smartphone- based augmented reality system called Privacy Plumber to increase users’ awareness of the data flows of smart devices and provide them with controls to block certain data flow if needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We focus on data flow rather than other aspects of data practices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', types of data being collected) mostly due to practicality and feasibility reason, as we can reasonably capture data flow and identify its source and destination using ARP spoofing [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In addition, from the smart devices’ perspective, these devices have multiple tiers of software, all of which entail some type of tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Such tracking is generally embodied in the data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We use augmented reality to visualize data flows in the same physical environment as the devices in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' this method could potentially help users establish a connection between the devices and their data flows in the same context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users’ proper understanding of data flow may help them understand the privacy implications of devices such as smart TVs [28], voice assistants [15], children’s toys [10], security cameras [24], [35], and smart light bulbs [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The development of Privacy Plumber is inspired by the following three gaps in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' First, the data flows of smart devices are opaque and not visible to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Second, existing tools to monitor network traffic of smart devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', IoT Inspector [15], open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='Dash [9]) require a certain level of technical knowledge to be able to interpret the results—not to mention that the results are often decoupled from the physical environment where the smart devices are situated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Oftentimes, the results are presented on, for instance, dashboards on computers or phones, where there is a disconnection between the visualization of data flows and the smart devices that create the data flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Third, existing tools or mechanisms do not provide users with the ability to control unnecessary or unexpected data flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' With Privacy Plumber, we aim to bridge the gaps and increase users’ awareness and control of the data flow in smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='11998v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='CR] 27 Jan 2023 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 1: Privacy Plumber lets a user find and mitigate potential privacy violations in the smart home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The figure shows a user walking around the smart home and inspecting the traffic and trackers coming out of a Samsung Smart Fridge using the Augmented Reality enabled app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Furthermore, (not shown in the picture above) users can use built-in, infrastructure- free controls to limit traffic of devices to times of day— without requiring any additional hardware or modifications to the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The graph shows the actual network traffic as the user interacted with the Smart Fridge: A: turning on the ice maker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' B: browsing recipe;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C: browsing goods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' D: interacting with the Bixby voice assistant of the fridge;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' E: opening the fridge door;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' F: adding items to the shopping list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' During these interactions, the Smart Fridge communicated with various advertising and tracking services, such as DoubleClick and Tapad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber uses augmented reality (AR) techniques and visualizes real-time network traffic flowing in and out of smart devices through an overlay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It allows users to find potential privacy leaks in their homes by pointing the AR- based app at smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' As shown in Figure 1, the app adds an overlay on top of the smart devices in which it displays a real-time data flow based on the network traffic with the necessary information for users to understand it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We chose to use AR because, as privacy is highly contextual [32], it can provide strong contextual connections between the actual real-time privacy leaks, and the user actions (or inaction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This allows the smartphone to function as a viewfinder into the invisible world of data flow and identify potential privacy violations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The smartphone application relies on a companion software tool hosted on a laptop or desktop on the same home network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This tool discovers smart devices in a user’s home, intercepts their traffic via ARP spoofing [48], and analyzes the data flow (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', what traffic is leaving the home over time)— without requiring the user to modify their network settings 0 1 Nest Camera Live streaming 0 2 Samsung Fridge Door opening Recipe browsing 0 10 20 30 40 50 60 70 80 time [s] 0 10 Amazon Echo Weather reporting Radio playing traffic [mbps] Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2: Outbound network traffic from various smart home IoT devices: a Nest Camera, an Amazon Echo, and a Samsung Smart Fridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Traffic increases or provides a fingerprint for many types of seemingly benign actions, creating a privacy leak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Current systems do not provide real-time context or ability to experiment with these devices, nor control their leakage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' or install additional hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When users would like to take action and block certain data flow, ARP-spoofing is used again to jam specific devices’ traffic (thereby blocking the device) at the time of day set by the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We build a proof-of-concept prototype and conducted a pilot study with 6 participants in our lab to collect their feedback on the prototype.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our initial findings have suggested that Privacy Plumber helped participants understand the net- work traffic, increased their awareness of potential privacy violations, and helped them make more informed decisions on how to handle IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This paper makes three contributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' First, to the best of our knowledge, Privacy Plumber is the first mechanism that provides users with real-time information on the data flow of their smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This paper proves the possibility of using AR-based technology as a viable option to increase users’ awareness of the data flows of smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Second, our initial evaluation shows promising results, indicating users’ po- tential acceptance of these technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Third, we summarized lessons learned from the pilot user study to inform the design and development of future systems that aim to improve users’ awareness of data practices in smart homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' BACKGROUND AND RELATED WORK In this section we discuss related work seeking to under- stand or discover privacy leaks, and the tools that exist to help users understand and mitigate them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is meant to to provide a handheld and zero-cost inspection and experimentation tool for privacy leaks of nearby smart devices in the home, and a straightforward and low burden method for mitigating those leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Issues in Smart Home Over the decades, privacy issues have been deeply dis- closed in smart home, such as transparency of data collection, data sharing, and accessibility [20], [50], [49], [16], [53], [30], [50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Some smart home devices have always-on sensors that capture users’ offline activities in their homes and transmit relevant information outside of the home, especially for cloud services run by device manufacturers [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2 02:006Privacy PLumbe X LiveDeviceTraffic 720 540 360 180 10 CurrentDeviceTraffic:984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='3bytesinthelast10seconds Sendingdatato17differentdestinations,including2 advertisingservice(s) Thisdataisequivalentto536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9wordsoftextor0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 picturesperminute DeviceDetails&Control>DoubleClick pedel DoubleClick Tapad traffic [mbps] 6 A E C 4 B 2 0 10 20 30 40 50 60 70 80 time [s]In the meantime, users are concerned about leaks of sensitive information [23], [51], [25], such as visual and auditory information which they see as private [23], [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Thus, users have a strong desire to protect themselves against such recordings being accessed without their permission [30], [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, some information users perceived as not very sensitive also lead to privacy leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, the home temperature could be used to determine whether a house is occupied or not, as a precursor to burglary [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In fact, smart devices give off digital exhaust which can be used by third parties including a user’s Internet Service Provider, advertisers, device manufacturers, and others, to fingerprint activities and get sensitive information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Shown in Figure 2 is the network traffic and trackers of various smart home devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='This network traffic forms the basis of most leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Tools for Enhancing Smart Home Privacy Most related to Privacy Plumber are tools that watch or monitor network traffic in the home and provide something of use to the user, whether visualization and information, education, or a mechanism for control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Sophisticated, technically literate users can use systems that block advertising and tracking domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', PiHole [38] and pfSense [34]), but these methods are bespoke and often require additional or dedicated hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', Raspberry Pi for Pi- Hole, and a supporting custom router for pfSense).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Other tools have provided insight into what might be exposed from web- browsing activities, including WiFi privacy Ticker [11], but do not consider or scale to the new problems of connected devices with physical sensors and abilities in a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aretha [40] explores this tool space and proposed (but did not deploy) a simple firewall-based control mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aretha presents data in aggregate instead of contextually and in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' None of these techniques investigate a range of IoT devices, usually constrained by studies with participants in their own homes, in a time when smart home adoption is low (Aretha had three participants, and only one had more than a phone, tablet, and Alexa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' None of these techniques develop a scalable (no additional hardware required) way to interpret privacy leaks and control them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Emerging smart devices are highly contextual and location sensitive, an Alexa in the bedroom versus the kitchen has different privacy exposure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' the former gives sleep times, the latter exposes eating habits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Moreover, tracking these devices’ privacy exposures presents a technical challenge because the traffic is not centralized through a web browser or laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A tool is needed to visualize privacy leaks from smart devices in real-time and in context, educate users on the consequences of these leaks, and provide control mechanisms for partially mitigating these leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Wifi Privacy Ticker [11] demonstrated a first method for improving the awareness of users in terms of privacy by providing a count of the amount of sensitive data that was being transmitted unencrypted over the network awareness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' By seeing this in real-time, users could adjust their behavior or find encrypted means to browse the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Of course, this ticker was developed well before the current generation of smart devices, however the underlying concept of surfacing the invisible privacy leaks remains the same for Privacy Plumber, but for smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Xray-refine [46], [47] provided smart phone users a means to visualize their exposure profile, based on the duration of app use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This method was an educational solution, but users had to adjust behavior to work around the constraints of the apps they were using, in some cases, opting out of apps to reduce privacy exposure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Finally, recent work like Aretha [40], PriView [36], Lu- mos [41], and IoT Inspector [15] look at making usable visualizations and mechanisms to understand and interpret data coming from smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Inspector is a simple-to-install desktop application that uses ARP spoofing to gather network traffic on the Wifi network of the desk- top/laptop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This information is sent and collated at a server, and then viewed online by the user, listing different trackers and websites that are attached to smart device usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Because of the ease of installation and no extra hardware requirement, IoT Inspector was deployed by thousands of users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In comparison, Aretha is a part research tool, part ex- ploratory users tool for exploring a design space of privacy tools and controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aretha helps users become aware of the net- work traffic flows in their homes while also educating users to regain their privacy in the connected home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aretha suggests the use of firewall mechanisms controllable by the user, but does not implement them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aretha, owing to a hardware requirement (a device must be attached to the Wifi router in the home) was only deployed in three homes, compared to the massive scale deployment of IoT Inspector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Similarly, PriView also has a hardware requirement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' its users need to have dedicated external thermal cameras (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', FLIR One [36]) attached to their phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For Lumos, there is no special hardware environment, although the focus is more on identifying hidden smart devices rather than analyzing the network traffic for privacy leaks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is not meant as a research tool or a design space exploration tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It is meant as an actual, real world system with a focus on scalability and ease of deployment in any home, similar to IoT inspector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Unlike both IoT inspector and Aretha, Privacy Plumber provides real-time and contextual visualizations of privacy leaks, real-time ability to plug those leaks (as well as automated rule setting for plugging leaks), and enables experimentation in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Finally, other significant measurement campaigns on in- home traffic have been conducted, focusing on the Wifi net- work itself or devices in the home [39], [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' These have usually been for research purposes and need finding and are useful for informing the design of Privacy Plumber, but are not necessarily tools for controlling smart home device privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Determining Home Activities from Network Traffic Complementary to Privacy Plumber are other works which demonstrate the ability to infer activities from network traffic: whether on a phone, smart device, or laptop [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' By analyzing the patterns of network traffic in the home, occupancy, habits such as sleeping, watching TV, listening to music, and some- times preferences, can all be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' HomeSnitch [33], Peek-a-boo [1], and HoMonit [52] all utilize machine learning with varying degrees of success to identify activities in the home from network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Other tools utilized for monitoring Internet connected smart devices in the home, IoT Sentinel [26] and IoT Sense [7], have shown that particular devices can be 3 fingerprinted by their traffic patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Enabling another way for an ISP or third party to determine the activity in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Each of these systems and methods are complementary with Privacy Plumber;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' inferred activities from traffic would be useful to surface in Privacy Plumber for the user to understand privacy exposure and know when to mitigate it, and device fingerprinting provides a way for zero-registration or setup of Privacy Plumber in a home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Challenges: Contextual, Real-time Privacy Understanding and Control in the Home Despite the diverse work in the smart home privacy space, significant gaps and challenges remain, which we detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C1: Users can’t model what devices are doing, especially without context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' With tools like IoT Inspector, a user might be able to count the number of trackers and advertisers contacted in a day from the sum of their interactions with smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' But how can a user know that turning on the NPR podcast on their smart fridge will send thousands of bytes of informa- tion to Bloomberg News for advertising purposes?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' How can they know that turning on the device sends a short burst of traffic?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users know that data captured will often be used for advertising, which often generates an adverse reaction [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, with smart devices, it is not always clear what actions or contexts trigger data being transmitted [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Things like Privacy labels for websites and smart devices are meant to give a method for scoring devices privacy [42], [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, these are static representations of the privacy exposure of a device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' With tools like IoT inspector and Aretha, aggregate views of data are seen (as opposed to real-time views), not associated with very fine user actions: like the turn on the light, say command to Alexa, or open the fridge door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Because of this granularity, the mental models of what devices are doing, and what they are sharing, are very perplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy tools must address this lack of action mapping to network traffic, enabling contextual integrity [32] in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C2: Users don’t have intuitive methods to control the privacy “valve”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users want devices that provide helpful features, but they do not know the cost of this ease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' One option is to just unplug the device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' however, this is all or nothing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users need a way to valve the privacy flow to something they are comfortable with, or to at least be able to analyze the tradeoffs [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Making privacy more ”tangible” [2] is one way this can be done;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' where the privacy leaks are more visceral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Selective firewalls (such as pfSense [34]), or other more fine grained network mechanisms may provide a means to control the privacy valve, but this must be intuitive and understandable to the user, and they must be able to actually ”see” the effect of turning this valve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C3: Smart devices are context (location, time, action) dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart devices are necessarily scattered around the home;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' and this will continue as more devices become intelligent, and more applications are explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Watching a desktop or laptop traffic meter and figuring out which device in which room is doing what at which times, is mentally trying for the user and disassociates the device from the physical space that defines its context and use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Just like when trying to find leaks in pipes, physical proximity is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Handheld inspection tools provide mobility, and enable in-situ fixing and experimentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C4: Users can’t experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Indeed, because of contextual changes in how private information is leaked, experimentation is difficult with existing tools that generally provide traffic summaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Interactions with smart devices can last only a few seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Enabling a user to experiment with different actions and uses of a smart device, and then see the associated network traffic in real-time, would provide a powerful way to build a mental model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, providing an ability to experiment is challenging with the current suite of tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C5: Technical challenge of scalability and deployment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' If a privacy tool is to be useful and translate to the general public, it must be hardware free, or at least trivially easy to deploy to enable scalability and broad adoption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Commercial products like fing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='com embed all functionality in a single phone application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Large scale deployments like with IoT Inspector are enabled through a desktop application that is easy to install.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, these methods do not provide controls since that is technically difficult to do without custom hardware put between the Wifi endpoint and the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' On the other hand, hardware requirements or custom install procedures reduce the deployment size of tools like Aretha, or narrow the user base by requiring technical ability, as with PiHole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It is not clear how to implement mechanisms of control without changing the Wifi network and infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To create scalable, user-centered, novice friendly privacy tools, mechanisms for enabling control of smart device traffic without hardware intervention must be developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' SYSTEM DESIGN We present Privacy Plumber as a proof-of-concept and end-to-end system to address the challenges listed of scalable and general population serving privacy tools for emerging smart homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is inspired by various handheld tools for identifying and fixing faults in large and complex systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, acoustic leak finding has been used for decades to localize leaks in gas and water pipelines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Handheld oscilloscopes, multimeters, and RF Spectrum Analyzers have helped engineers debug problems in large electrical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' These handheld devices make the invisible signals visible and interactive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' They allow real-time experimentation and debug- ging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Inspired by these devices, Privacy Plumber is designed to offer a general user a level of insight and control of the invisible privacy leaks that are rampant in Internet-connected smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is composed of two pieces as shown in Figure 3: (1) the IoT Network Analyzer, a desktop application which collects real-time data on smart devices on the shared Wifi network, and provides an infrastructure and hardware free mechanism to block arbitrary devices traffic, and;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (2) the Privacy Plumber phone application, which serves as a viewfinder or inspector for any smart devices in view, and presents data from the desktop application, including device network traffic and potential privacy leaks to the users, along with educational content matched to what is known about the device, all in real time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 4 ARP Spoofer Viewfinder Educational Views Packet Analyzer Traffic Analyzer Controls IoT Network Analyzer Router Privacy Plumber App Traffic and Control Visualizer IoT Devices Network Traffic ARP Spoofing Leak Information Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3: System diagram of Privacy Plumber including the IoT Network Analyzer and Privacy Plumber mobile application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer runs on a computer that is connected to a user’s router.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer automatically discovers and captures IoT devices on the same network using ARP spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber connects with IoT Network Analyzer to present the network analysis in AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The user can then examine their devices’ network traffic and control when they want their devices to be on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Overview of Usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A user would first download, install, and run IoT Network Analyzer on their computer and the Privacy Plumber app on their mobile phone, such that both the computer and the phone are on the same local area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Let us assume that the user is interested in inspecting a smart device like an Amazon Echo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' While running the Privacy Plumber app, the user points the phone camera to Echo and speaks a voice command (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', “Alexa, what is the weather?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=') IoT Network Analyzer captures all network traffic between Echo and the Internet, analyzes the packets, and identifies destinations that are third-party advertising and tracking companies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The Privacy Plumber app extracts this information from IoT Network Analyzer and visualizes key statistics for the user—such as real-time bandwidth usage of the device and the number of advertising and tracking services contacted—as an overlay in the AR view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When the user points the phone camera at a device, the Privacy Plumber app does not recognize devices with computer vision algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Instead, for the purpose of this prototype, we print a QR code on each IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The QR code includes the device’s MAC address, its name, and the manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The app uses the phone’s camera to scan for the QR code, identifies the device based on the QR code, and displays the device with a dial menu around it (see Figure 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The options in the menu allow the user to see the outbound traffic from the device as well as read a brief article stating what types of information the device may be tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The user may also use the Device Control menu (Figure 4c) to manually block or allow traffic from the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Future versions of the app will use computer vision to recognize devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' see the discussion in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Threat Model and Assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We assume that a user’s privacy may be potentially violated if an IoT device exhibits either or both of the following behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Threat 1, an IoT device could contact an advertising and tracking service on the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Threat 2, an IoT device could be sending out network traffic to hosts on the Internet when the user does not expect any network activities—for example, when the user is not interacting with the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We design both the Privacy Plumber app and IoT Network Analyzer with this privacy threat model in mind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer captures packets, analyzes the headers, identifies the destination hosts (based on the IP addresses, domain names, and the TLS Server Name Indication fields), and determines if a destination host is an advertising and tracking company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The Privacy Plumber app displays the number of advertising and tracking services (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', the red text below the graph in Figure 4b), thereby helping users toward identifying Threat 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Based on the byte counters from IoT Network Analyzer, the Privacy Plumber app also shows a bandwidth graph that plots the bytes sent per second over time (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', the time-series graph in Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This graph could help users correlate network activities with human interactions—or the lack thereof—with given IoT devices and thus identify possible instances of Threat 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Note that IoT Network Analyzer does not parse the payload of packets to discover sensitive information within the traffic, as the network traffic is likely encrypted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Design Goals Privacy Plumber must make the underlying behavior of the devices in the home apparent, and enable forms of fine-grained (informed) control of the leakage of sensitive information for the user.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Towards this end, and addressing the challenges described in Section II-D, we are guided by the following design goals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (1) Handheld and Mobile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart devices are scattered throughout the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Phone adoption is nearly universal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Using a phone as a window into the information world gives 5 Privacy Jagwnnd X Live Device Traffic Bytes/seo 1500 0 1125 0 7500 3750 10 Current Device Traffic: 26954.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='1 bytes in the last 10 seconds Sending data to 15 different destinations This data is equivalent to 14702.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='2 words of text or 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 pictures per minute Amazon Details Status: Allowed BLOCK ALLOW Device Device Traffic Traffic Custom Time Rule ApplyRule Time (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 8:00AM) Turn On Turn Off: Time (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 8:00PM) RemoveRulecontext and a sense of place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The phone form factor increases the likelihood of adoption and allows for inspection on the go;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' users can trigger or interact with devices and easily watch the movement of data, instead of having to return to the desktop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (2) Real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Seeing statistics after the fact, as in most systems, is not as impactful or helpful when developing a model of how devices operate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Moreover, real-time analysis enables experimentation, providing users with a mechanism for exploring limitless scenarios and quickly associating triggers with outcomes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (3) Infrastructure/Hardware Free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Many other meth- ods require custom hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This increases cost and raises the barrier to entry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We hope to enable anyone, especially those that may have limited autonomy over infrastructure (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' renters, low-resourced communities) to be able to inspect the devices put in their living space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (4) Intuitive Controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Complex mechanisms to control or limit the flow of privacy are not interpretable by users, and are possibly frustrating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Configuring a firewall is not a task most people would enjoy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Straightforward controls, with visible results, once those controls are put in place, are essential for users to trust the capability of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (5) Educational.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The ever-changing landscape of devices and the security/privacy arms is impossible to keep up with for privacy tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Assisting users in understanding what makes cer- tain devices leakier (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', always-on microphone) is essential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To realize these design goals, we build the Privacy Plumber app—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', the handheld form factor—and IoT Network An- alyzer as a two-part architecture working in tandem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Both systems must be running on the same local area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer, running on a computer, captures and analyzes network traffic between smart devices on the network and the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer acts as a server and provides the above information over an HTTP REST API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The Privacy Plumber app, acting as a client, regularly polls the REST API and presents the analysis as an AR overlay to users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In the following sections, we detail the pieces of the system and how they interact to enable understanding and control of smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Section III-B we discuss the IoT Network Analyzer and its role in capturing and curating privacy leak information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' in Section III-D we describe the phone app design;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' in Section III-C we detail the mechanisms we use for controlling devices on a schedule, and finally, in Section III-E we describe a few ways to use Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Low Burden Home Network Traffic Capture To use the Privacy Plumber app, the user must also have IoT Network Analyzer running on a computer (macOS, Windows, or Linux) that is on the same local area network as the phone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For our study, we run IoT Network Analyzer on a Raspberry Pi 3 Model B that is connected to the lab’s network via Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We based IoT Network Analyzer’s code on the open-source project, IoT Inspector [15], and made modifications according to our needs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In particular, whereas the original IoT Inspector constantly sends captured traffic’s metadata to the researchers’ servers, IoT Network Analyzer runs without the Internet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' it processes the captured traffic locally and exposes the traffic via a REST API.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Furthermore, whereas the original IoT Inspector runs on users’ computers and visualizes the traffic in a browser-based dashboard, IoT Network Analyzer uses an AR-based app, Privacy Plumber, to visualize the network traffic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' the mobile app reads the processed traffic through the abovementioned REST API and presents the results as an AR overlay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Once running, IoT Network Analyzer automatically discov- ers IoT devices on the network, captures their network traffic via ARP spoofing, produces traffic statistics (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', bandwidth usage and identifying advertising and tracking services) over a local HTTP REST API, and blocks select devices (if desired by the user).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We explain each of these features below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Discovering IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Upon launch, IoT Network An- alyzer automatically broadcasts ARP packets to the local area network and discovers active devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To identify IoT devices, Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [15] describe an algorithm that infers the likely identities of IoT devices based on MAC OUI (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', Organizationally Unique Identifier, basically the first three octets of a MAC address), DNS, and UPnP messages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For the prototype in this paper, we only use the MAC OUI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Within the code of IoT Network Analyzer, we have already hard-coded the mapping between OUIs and names of five IoT devices in our lab (which we can find out beforehand).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this way, IoT Network Analyzer can instantaneously identify the IoT device in our lab without relying on the device identification algorithm in Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Capturing network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Once IoT Network Analyzer iden- tifies a known IoT device on the lab’s network, it automatically starts intercepting network traffic between the device and the Internet via ARP spoofing, a technique used in the original IoT Inspector implementation and which incurs an overhead of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='4 Kbps, given that we have five IoT devices in the lab [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='1 Obtaining traffic statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' All traffic to and from IoT devices in our lab is redirected through IoT Network Analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In do- ing so, IoT Network Analyzer is able to obtain statistics about the network traffic for every device, including the device’s MAC address (from which to extract the OUI and determine the device’s identity based on our hard-coded mapping);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' the number and size of packets (from which to infer the bandwidth usage);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' the remote IP addresses, DNS requests and responses, and the Server Name Indication field within TLS packets (from which to infer the remote hostname and whether the hostname is associated with a known advertising and tracking company, based on the Disconnect block list [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer presents all these statistics and information via an HTTP REST API that the Privacy Plumber app can access over the local area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, if the computer running IoT Network Analyzer has a local IP address of Ii, then the Privacy Plumber app (on the same local network) can access the traffic information via http://[Ii]/get traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Phone Application: App Implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The Privacy Plumber mobile app was implemented in Unity using C# and is cross-platform, tested on Android and iPhone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The 1Per Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [15], our setup includes N = 5 devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It follows that N(N + 1) = 30 spoofed ARP packets are sent every two seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' As each ARP packet has 28 bytes, the overhead is 28×30/2∗8 = 3, 360 Bits/second or 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='4 Kbps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 6 (a) View finder (b) Traffic view (c) Controls (d) Education Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 4: Illustration of mobile application design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (a) Device recognition with interactive menu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (b) Live traffic inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (c) Rule-based device traffic control (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', blocking and unblocking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (d) Educational material on privacy details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' app works by communicating with IoT Network Analyzer via HTTP GET requests, as described in the previous paragraph, to obtain JSON-encoded information about the devices on the network and their traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Parsing these JSON objects, the app visualizes the information as charts and text on the AR display (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', Figure 4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The app also shows an interface where users could block an IoT device’s traffic, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', Figure 4c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Once the user confirms, the app sends the corresponding request to IoT Network Analyzer via the HTTP REST API, and IoT Network Analyzer would subsequently block the device by jamming the device with corrupt ARP packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' User Control of Privacy Leaks from a Phone With Privacy Plumber we also want to help the user feel more empowered by allowing them to take control of their devices with the ability to block device traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users can manually block or allow device traffic indefinitely, or they can set rules to govern when they want their device to be on or off and for how long (Figure 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users are also given the option to physically power off their device altogether.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this way, Privacy Plumber provides a closed-loop system where users can analyze the information flow out of a given device, then immediately apply direct control over that device in response and receive immediate feedback via the traffic view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To illustrate how a user might control an IoT device’s traffic, let us say that a user feels uncomfortable with an IoT device communicating with the Internet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The user can use the Privacy Plumber app to block Internet access on the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' As shown in Figure 4c, the user can click “Block Traffic” on the app to indefinitely block the device, or specify when to block and unblock the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Moreover, the app sends an HTTP request to IoT Network Analyzer, using the REST API 2 (where Ii is the IP address of the running instance of IoT Network Analyzer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' During the period of blocking, IoT Network Analyzer jams the communication of the device by using a corrupt source MAC address in the spoofed ARP packets (as described in Section III-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer stops this process at [unblock time], upon which IoT Network Analyzer sends out spoofed ARP packets without the corrupt source MAC address.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This gives users the ability to control the times of day when they want their devices to be on or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber’s software-based device blocking offers several advantages over simply turning off or disconnecting a device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' First, users do not need physical access to the device;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' for instance, many surveillance cameras are mounted on ceilings and are difficult to power off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Second, users can temporarily disable a device when they are feeling uncom- fortable, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', blocking Amazon Echo for an hour during a sensitive phone call or conversation, through Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Such temporary blocking is difficult to achieve through Echo’s app (no such feature) or manually (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', the user has to remind themselves to re-connect Echo again).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Third, though not currently implemented, Privacy Plumber, with the help of IoT Network Analyzer, can block based on the context (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', future work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, when IoT Network Analyzer detects the presence of a user’s phone on the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', by checking if the phone responds to periodic ARP requests), IoT Network Analyzer automatically blocks all indoor cameras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' when the phone leaves the network (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', when the user is out), IoT Network Analyzer could automatically unblock all indoor cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Technical Mechanism for Blocking Devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A major differ- ence with respect to IoT Inspector’s original implementation is 2http://[Ii]/block/[device id]/[block time]/[unblock time] 7 Privacy Lagwnd (A) amagon M Device Details & Control >Privacy PLumber X Live Device Traffic Bytes/sec 720 540 360 180 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9 CurrentDeviceTraffic:984.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='3bytes inthe last10seconds Sending data to 17 different destinations, including 2 advertising service(s) This data is equivalent to 536.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9 words of text or 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 pictures per minute M amazon Device Details & Control >Privacy Lagwd (A) M cef:co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='ob: Amazon Details Status:Allowed BLOCK ALLOW Device Device Traffic Traffic CustomTime Rule ApplyRule :uo ni Time (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 8:00AM) Tum Off: Time (ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 8:00PM) Remove RulePrivacy Laqwnd X Device Privacy Details Echo Smart Speaker This voice assistant is like a butler, providing voice access and control of music and other media, enabling voice based purchases, making phone calls, and helping to keep track of things,among many other functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Potential Privacy Leaks Location Your physical location is leaked when interacting with this device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Because motion can be detected, your near exact locationwithinaroomcanbedetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 讠 Activity (A) amazo 三 DeviceDetails&Control>that we have added the capability of blocking devices in IoT Network Analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Using the HTTP REST API 3, the Privacy Plumber app can request IoT Network Analyzer to block a certain device at a particular time (for instance, because the user does not want the device to be communicating to the Internet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Upon receiving this request, IoT Network Analyzer jams the network communication of the device by sending it spoofed ARP packets with corrupt MAC addresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To illustrate this process, let us assume that the computer running IoT Network Analyzer has a MAC address Mi and IP address Ii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Let us also assume that IoT Network Analyzer is about to intercept the communication from the gateway (with MAC address Mg and IP address Ig) to a particular device (with MAC Md and IP Id) without blocking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To do so, every two seconds, IoT Network Analyzer sends an ARP packet to the device, such that the ARP packet has a source MAC of Mi and a source IP of Ig, along with a destination MAC of Md and a destination IP of Id.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In contrast, let us assume that IoT Network Analyzer is to block the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It sends the same ARP packet to the device, except that the ARP packet’s source MAC is a series of random numbers (instead of Mi) that represent an unreachable MAC address on the local area network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Visualizing and Understanding Traffic in Real-Time One of the goals of Privacy Plumber is to show users contextualized network activities of IoT devices to help them pinpoint the potential privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this section, we discuss how Privacy Plumber utilizes Augmented Reality to help users contextually visualize their devices’ network traffic informa- tion in real-time, provide a chart of network traffic in real- time, and provide links to other research in which the privacy concerns of the inspected device have been studied (including home behavior inference, sleeping behaviors, and personal data).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Lastly, users are able to send feedback and bug reports.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Use of Augmented Reality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The use of AR visualization makes the interaction with the device the user is inspecting more tangible and contextual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' While IoT Inspector [15] and IoT Network Analyzer are text-only data-driven analyzers that can only be accessed using browser HTTP requests, Privacy Plumber is a fully-fledged interactive application due to the utilization of AR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' By pointing their camera at the device being inspected, the user can see, in their environment, the traffic coming out of the device that they are physically inspecting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users can interact with their devices and receive immediate feedback about data output and communication with advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Combined with manual device control, this is intended to help the user feel informed and in control of the IoT devices that physically surround them, similar to the use of a TV remote control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Learning About Privacy Threats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We aim to educate and inform users on how their IoT devices expose their network traffic information to third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Figures 4d and 5, the app shows icons surrounding the IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When any of these icons are pressed, they provide links to other research materials—which we have manually curated in advance— where the privacy concerns of the inspected device have been 3http://[Ii]/block/[device id]/[block time] studied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Depending on the device, Privacy Plumber provides the following categories of potential privacy violations repre- sented by icons: Location: Your physical location either roughly (your address) or fine-grained (room you are in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Activity: Your physical activity such as walking, talk- ing, sleeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Screen: Your online activity, such as when you browse videos on YouTube or surf the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Identity: Attributes that can identify you such as your face or voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Shopping: Data on your usage of money or products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Health: Can infer different health markers without consent (heart rate, breathing, and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber Example Use Cases In this section, we illustrate two example use cases of the Privacy Plumber app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We focus on the ability of Privacy Plumber to enable experimentation and the usefulness of a real-time inspector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We will describe the users’ reactions in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Example 1: Is Echo Always Listening?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A user may use the Privacy Plumber app to correlate net- work activities on an Amazon Echo device with the user’s interactions—or the lack thereof—with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' While pointing the AR camera at the device, the user could invoke a voice com- mand, such as “Alexa, what is the weather”, while observing the device’s bandwidth usage graph on the Privacy Plumber app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Afterward, the user may physically press the mute button on Echo, repeat the same voice command, and observe the bandwidth usage graph on the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Example 2: What is this App on My Smart Fridge?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Many smart fridges have built-in touch-screen panels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, the Samsung Smart Fridge has a tablet-like touch- screen panel to control various settings of the fridge (such as temperature).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The panel also allows users to access various built-in apps, such as checking recipes or ordering ingredients online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A user who is concerned with the privacy of such apps may point the AR camera at the fridge, interact with an app, and observe the advertising and tracking services counter on the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This counter shows, in real-time, the total number of advertising and tracking services that the fridge has communicated with, based on the Disconnect block list [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' PILOT USER STUDY To test how users react to Privacy Plumber and inform its future iteration, we conducted a pilot study with 6 participants to experiment with, understand, and control the potential privacy violations of IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It should be noted that the pilot study would be best conducted in participants’ homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' However, due to University research restrictions, the COVID- 19 pandemic has made it difficult for us to recruit real users, distribute hardware (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', phones powerful enough for AR and Raspberry Pi’s for running IoT Network Analyzer), and conduct a free-living study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 8 We conducted a one-day controlled lab study in our IoT Lab with 6 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Participants were invited to use the Privacy Plumber app while interacting with several IoT devices in the lab, including Samsung Smart Fridge, Amazon Echo, Google Home, Samsung Smart TV, and Google’s Nest Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our goal is to assess whether using augmented reality to display network traffic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', by using Privacy Plumber) influenced the participants’ awareness of privacy and changed their behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In the following sections, we present the details of the pilot study and discuss some highlights in the results as well as lessons learned to inform the next iterative of Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Participants Recruitment and Demographics We recruited 6 graduate students from our institution through our university mailing list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We did so rather than recruiting from a broader population sample because of the constraints our university implemented during the COVID-19 pandemic (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', external members were not permitted to enter our buildings).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our sample consisted of four males and two females.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Three of the participants were between the ages of 18-24, two participants were between the ages of 25-34, and one participant was between the ages of 35-44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Study Procedure and Data Collection For safety reasons and to implement social distancing procedures, only two people were allowed in the IoT Lab during the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Aside from the participant, one of the co- authors in this paper served as the research coordinator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' They were present throughout the user study to help guide the participants or troubleshoot any technical difficulties that could arise during the study procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Before the study began, each participant filled out a back- ground pre-survey on a computer in the IoT lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We asked questions about their demographics, how technically savvy they are, their smart device experiences, their general under- standing of privacy, and their concerns about their information being exposed to third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' After completing the survey, our research coordinator handed each participant a script and an Android mobile phone that had Privacy Plumber installed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Following the script, each participant opened the Privacy Plumber app, kept it running in the foreground, and interacted with one IoT device at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Regardless of the IoT device, each interaction consists of the following steps, as prescribed in the script: 1) Using the Privacy Plumber app, the participant scanned the QR code that we had placed on the IoT device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The QR code encodes the device’s MAC address, device name, and manufacturer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Based on the QR code, Privacy Plumber shows the corresponding device’s AR model on the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2) The participant used the app to inspect the device’s traffic, while not doing anything to the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3) The participant interacted with the device (which we will describe in detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' During the interactions, the participants observed the network traffic graph on the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 5: This screen on the phone application describes the different categories of privacy leaks that different devices have, based on a database that we manually curated in advance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 4) Using the app, the participant clicked on any of the icons surrounding the AR model of the device and read the educational material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' After interacting with all the IoT devices, participants returned the phone to the research coordinator and responded to a post-survey that asked the same questions as in the pre- survey, along with a usability survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We discuss the results in more depth in Sections IV-C and IV-D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We also include our pre- and post-surveys in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Below, we describe each participant’s scripted interactions with each device—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', showing Step 3 in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' During the interactions with the devices, users can access the educational content which is summarized from Mozilla’s “privacy not included” handout [29] and academic literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Each device is described by the categories of privacy exposure they create, those categories are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Samsung Smart Fridge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The fridge has a built-in touchscreen on the door.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Through the touchscreen, users have the ability to interact with several built-in apps, such as managing the shopping list, checking what is inside the fridge, and searching for recipes online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Users can also interact with the touchscreen using voice commands, using the trigger word, “Bixby.” Per the script, the research coordinator instructed the participant to follow the following three tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (i) Once the participant scanned the QR code of the smart fridge, they said the voice command, “Hey Bixby, do we have mangoes?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Bixby, the fridge’s voice assistant, would say “no,”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The participant then said, “Hey Bixby, can you add mangos to my shopping list?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Immediately, the participant looked at the 9 Privacy Leak Categories Smart devices can collect private information about you intentionally or incidentally in the following ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Location Your physical location either roughly (your address) or fine- grained (room you are in).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Activity Your physical activity, such as walking, talking, sleeping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Screen Your online activity, like when you watch videos on YouTube or surf the web.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Identity Collects attributes that can identify you such as your face or voice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Shopping Collects data on your usage of money or products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Health Can infer different health markers without consent (heartrate, breathing and others).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='-3 2 1 0 1 2 3 Strongly Disagree Strongly Agree I think about what information I may be exposing to 3rd parties when I interact smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' I am not concerned over the information I may be exposing to 3rd parties when I interact with smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber has made me more aware of what information I may be exposing to 3rd parties when I interact with smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber has made me more aware of privacy concerns regarding smart devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 6: Representation of participants’ average agreement ratings relating to statements about information being exposed to third parties and privacy concerns caused by interacting with IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Participants rated the first two statements before and after the study, while the last two statements were rated at the end of the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The results show that after the study, participants displayed an increase in awareness and concern about how their information is being handled when interacting with IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber app and observed the network traffic emitting from the fridge for about 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (ii) The participant said, “Hey Bixby, find me a Ramen recipe.” The recipe app popped up on the touchscreen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Using the finger, the participant browsed through the available recipes on the screen, while observing the network traffic on Privacy Plumber for 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (iii) The participant opened the fridge door and then closed it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Once again, they inspected the fridge’s network traffic through the Privacy Plumber app for 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Amazon Echo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Interactions with Echo consists of the follow- ing 3 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (i) The participant said the voice command, “Alexa, play a thunderstorm sound.” Immediately, the participant ob- served the network traffic on the app for 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (ii) The participant physically pressed the “mute” button on the Echo and watch the device’s network traffic for 15 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (iii) The participant said the same voice command as in Task (i) and observed the traffic in the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Google Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The participant said a voice command, “Hey Google, what was the final score in the Super Bowl last year?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The participant immediately started observing the network traffic on the app for 30 seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Samsung Smart TV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The participant used the TV’s remote control to navigate to the Bloomberg app on the home screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' They then started streaming a live video on the Bloomberg app for one minute while they observed the network traffic with the Privacy Plumber app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Nest Cam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Interactions with the camera consists of the follow- ing 2 tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (i) The participant walked into the field of view of the camera and stay there for five seconds, walked out of the camera’s field of view, and observed inspect the network traffic with the Privacy Plumber app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' They repeated this task as many times as they liked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (ii) The participant blocked the network traffic to and from the camera using the built-in feature on the Privacy Plumber app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The participant observed the network traffic for 10 seconds, walked in front of the camera’s field of view, waited for another ten seconds, and unblocked the device using the Privacy Plumber functionality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Analysis of Pre-Study and Post-Study Surveys We asked each participant to complete two surveys: (i) a pre-Study Survey that they filled out on a dedicated computer at the beginning of the study, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', before the participants interacted with the Privacy Plumber app or the IoT devices;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (ii) a post-Study Survey that the participants filled out on the dedicated computer after interacting with all the five IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We present the results below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Figure 6 we present the participants’ agreement rating responses for two statements that were asked in the pre-study survey and post-study survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We observe that for those two statements participants seemed less concerned by how their information is exposed to third parties when they interact with IoT devices before they performed the activities in the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' After participants completed the study, they were more aware and concerned about how their information was disclosed to third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The last two statements of Figure 6 were only given in the post-study survey, which asked participants to rate whether Privacy Plumber was useful in helping them become more aware of privacy concerns and how their information is being shared with third parties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' On average, participants some- what agreed that Privacy Plumber helped raise their awareness and privacy concerns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Participants found that Privacy Plumber was helpful in that it helped them visualize what information was being shared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Additionally, we discuss the results of participants’ re- sponses with the IoT devices before and after the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We show that after the study participants felt less safe with how IoT devices handle their data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Participants were presented with three statements and were asked to rate whether they agree or disagree with these statements on a scale of one to five, where a 1 meant they strongly agree and a 5 represents a strongly disagree rating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Table I demonstrates the average change in attitudes participants had before the study and after the study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We note that before the study, on average participants neither agreed nor disagreed with the statements presented in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' After completing the study, the average rating agreement score increased to “somewhat agree” on the last two statements on all IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The exception was in the first statement, the scores for the Amazon Echo and Google Home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This indicates 10 Survey Question Smart Fridge Amazon Echo Google Home Smart TV Nest Cam pre post pre post pre post pre post pre post I think this device could be (or is) useful or valuable to my daily life and routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='71 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='33 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='71 3 I am comfortable having this device in my house and always on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='17 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='86 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='43 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='17 I am comfortable having this device in my house if I can automatically control when it is on, or off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='33 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='14 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='83 Strongly Disagree (5) to Strongly Agree (1) TABLE I: Results of the survey on user awareness and comfort with smart devices, before and after using Privacy Plumber to inspect those devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Scores are listed for both pre- and post-study surveys for each device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The higher the scores, the more strongly the participant disagreed with the survey question statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' that after using Privacy Plumber in the study, participants felt that the Amazon Echo and Google would still find it useful to use in their households.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We also observe that the Smart Fridge, Smart TV, and the Nest cam had the most significant change in attitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We gathered a few quotes from participants in which they describe how they felt about interacting with these IoT devices and using Privacy Plumber to inspect their network traffic: IoT devices provide more information to third par- ties than people thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' I think apps like Privacy Plumber can help people to make better decisions when using IoT devices — (P1) Cool to see when and how much traffic each device sends at any given moment!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' — (P5) I think the app does make me more aware about how the traffic is associated with the behavior of the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Having some control over the traffic is nice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' That being said, if I do have the device in my home, I probably would like to use it, and in that case, I have to allow traffic, which I have no control about what could pass or could not pass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In that sense, I can only accept certain privacy risks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' — (P2) It was interesting to see the potential privacy leaks shown next to the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Some leaks/ privacy im- plications were surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Liked the ability to al- low/block traffic, it was also cool to see the real- time traffic including communication with third-party advertisers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Liked the app interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' —(P6) These quotes, along with results from Figure 6 and Table I, suggest that Privacy Plumber helped participants understand the network traffic, increased their awareness of potential privacy violations, and helped them make more informed decisions on how to handle IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Analysis of the Usability Survey At the end of the study, each participant completed the usability survey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Overall, most participants indicated that they would use Privacy Plumber in their home network, found it easy to use and user-friendly, and agreed that most people would learn to use Privacy Plumber quickly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We summarize the results below: When asked if they would use the Privacy Plumber mobile app to inspect the data the IoT devices in their homes, two participants said they strongly agreed with the statement and four participants said they somewhat agreed to use Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When participants were asked if they found Privacy Plumber easy to use, four of them somewhat agreed, one participant strongly agreed, and one participant somewhat disagreed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When presented the statement “I imagine that most people would learn to use Privacy Plumber very quickly”, the responses were across the board spec- trum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Three participants rated that comment as strongly agreed, one participant rated the statement with a somewhat agree, one participant responded that they felt neither agreed or disagreed with the statement, and one participant somewhat disagreed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' When participants were asked to rate the overall user- friendliness’s of Privacy Plumber, four participants rated the Privacy Plumber app as good and two participants rated Privacy Plumber as fair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We gave participants an open-ended question if they would improve the usability of Privacy Plumber, and if so, how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We show their responses in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' All in all, partici- pants seemed to respond somewhat positively towards Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It shows that Privacy Plumber may have the potential to be distributed to the general public after further studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We hope to build off our current platform and implement the suggestions our participants gave us in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Performance: System Overhead and Battery Life Impact Network Overhead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' IoT Network Analyzer intercepts the network traffic of select IoT devices via ARP spoofing, a technique that could introduce network overhead especially to the targeted IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This overhead comes from two sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' First, the spoofed ARP packets consume extra band- width, although the overhead is relatively small—i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', less than 60 Kilobytes/second even if 50 IoT devices are under ARP 11 spoofing [15]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The second source of overhead comes from the Raspberry Pi 3 Model B, where we run IoT Network Analyzer in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The Raspberry Pi is connected to the lab’s network via Ethernet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For all IoT devices to which IoT Network Analyzer sends spoofed ARP packets, all inbound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', download) and outbound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', upload) traffic to and from the IoT devices has to first go through the Raspberry Pi before IoT Network Analyzer forwards the traffic to the targeted device and to the Internet respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Effectively, the Raspberry Pi introduces a bottleneck for the ARP-spoofed devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To measure the overhead as a result of the Raspberry Pi bottleneck, we conduct the following experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We install the Ookla Speed Test app on an Android phone that is con- nected to the the lab’s WiFi network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We have the Ookla app run 15 back-to-back speed tests, which measure the inbound and outbound traffic rates with respect to a server in our city, as well as the latency of packets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Using the same setup, we repeat the same experiment, except that we have IoT Network Analyzer inspect the phone’s traffic via ARP spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We find significant overhead as a result of IoT Network Analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Without ARP spoofing, the app achieves, on average, an inbound rate of 293.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='6 ± 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='4 Mbps, an outbound rate of 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='1±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='2 Mbps, and a latency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='7±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' With ARP spoofing by IoT Network Analyzer, the app achieves, on average, an inbound rate of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='4±74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='6 Mbps, an outbound rate of 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='8 ± 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='1 Mbps, and a latency of 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5 milliseconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Compared with the case without ARP spoofing, IoT Network Analyzer reduces the inbound rate by 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9% and outbound rate by 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='6%, while increasing the latency by 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Despite the seemingly significant reduction in bandwidth, we argue that IoT Network Analyzer is unlikely to degrade usability, as the network analyzer is not always running (only when inspecting, or blocking a specific device).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Additionally, the overhead can be reduced with improved hardware.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Ac- cording to Netflix, 25 Mbps of inbound rate is sufficient to stream Ultra HD contents [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A user who inspects a smart TV using IoT Network Analyzer is likely to enjoy Ultra HD streaming given the reduced inbound rate of 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='4±74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='6 Mbps under ARP spoofing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' If a user desires to reduce the network overhead, the user could upgrade the computer that runs IoT Network Analyzer, as Raspberry Pi 3 is anecdotally known for its poor networking performance [37], [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Possible upgrade option could include a computer—or ODroid if the user needs the compact form factor [14]—that is shipped with a fast CPU and a Gigabit Ethernet card.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Battery Lifetime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We used AccuBattery on android, to try to understand the energy cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This does not hold across phones, so we compare the energy cost against YouTube and TikTok for ten minutes of streaming video.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' With all the background application killed, 10 minutes of Privacy Plumber impacts 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='98% (159mAh) of the battery lifetime, while YouTube costs 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='63% (105mAh) and TikTok costs 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='9% (156mAh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is only meant for point inspection and short usage to analyze new devices in the home, or experiment with different setups, so it should not impact battery lifetime too much since it is not always on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Moreover, the battery lifetime cost is similar to that of streaming videos online, a normal function, therefore users should not expect significant battery lifetime loss due to usage of Privacy Plumber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' DISCUSSION ON LIMITATIONS AND FUTURE WORK Comparing users’ mental models against actual contents of IoT network traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our results show that users’ mental model of how IoT devices communicate with the Internet may be inconsistent with how devices appear to behave, but it is unclear whether this mental model is consistent with the actual contents of the communication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, two participants in our study did not expect network traffic from Amazon Echo when the device’s microphone was on mute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Presumably, the participants expected Amazon not to send any audio data back to Amazon during mute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In this case, Echo’s apparent behavior was the communication with the Internet on mute;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' in contrast, whether Echo actually sent out audio data was unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Our system did not extract the contents of the communication, which could be encrypted based on previous results [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Despite the encrypted contents, man-in-the-middling is possible (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', per Moghaddam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [28]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In future in-lab studies, we plan to modify IoT Network Analyzer to intercept and decrypt IoT traffic, assuming that devices do not validate certificates and/or do not use certificate pinning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We hope to extract the payload from some of the TLS connections, identify exactly what devices are sending to the Internet, and compare it against users’ mental models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Automated, contextualized blocking of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The current prototype allows users to set a block/unblock schedule for IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Although this feature provides users with fine-grained control, it requires manual effort from the user both in setting what devices to block and when to block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We plan to augment this feature with automated device blocking based on contextualized information that IoT Net- work Analyzer already collects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For example, a user could create a rule on IoT Network Analyzer that would automat- ically block surveillance cameras if IoT Network Analyzer detects the presence of mobile phones (based on ARP and pings) in the home network (which could suggest that the residents are home);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' otherwise, it can unblock the cameras to capture, say, unauthorized entry into the property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' As another example, let’s say a user has an Amazon Echo and a smart TV in the living room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The user could create another rule that lets IoT Network Analyzer automatically block Amazon Echo if it detects active streaming traffic from the smart TV, as the user may not want Echo to capture any conversations while the family is watching TV in the living room.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In short, by leveraging the IoT traffic that IoT Network Analyzer already collects, users could create automated, contextualized rules to block IoT devices from collecting sensitive data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Deployment roadmap and challenges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We plan to deploy the Privacy Plumber app and IoT Network Analyzer to real-world users at scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Based on our current prototype, we plan to make the following modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Operating system support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Once deployed, our system will have the same two-component architecture, although we will expand the Privacy Plumber app to both iOS and Android (current prototype), and IoT Network Analyzer to all major non-mobile operating systems including macOS, Windows, 12 and Linux (current prototype).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This process will likely be straightforward, as we developed both components with cross- OS platforms (Unity for the app and pure Python for IoT Network Analyzer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Network-based device identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We will develop network-based device identification mechanisms to help users distinguish among their devices and identify the device(s) of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The current prototype identifies devices based on a hard-coded mapping between MAC OUIs and device names, because we already know the inventory of IoT devices in the lab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For real-world deployment, we will incorporate IoT Inspector’s device identification algorithm [15], so that our system will dynamically infer device names based on the network signature, which includes not only OUIs, but also DNS queries, UPnP banners, mDNS names, and DHCP hostnames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We will also use information in the 802.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='11 frames to discover and locate devices [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Image-based device identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' To complement the network-based approach, we will also develop image-based device identification mechanisms for the AR camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Cur- rently, the Privacy Plumber app identifies devices based on printed QR codes on or near select IoT devices, such that the QR codes encode the MAC addresses and the names of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' For real-world deployment, we will use computer vision to train a model of common IoT device types, such as voice assistants, smart TVs, and surveillance cameras (where security and privacy issues are commonly found in the litera- ture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' This model will help the AR app recognize possible IoT devices (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', “likely a smart TV”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The app will then refine the recognition with the network-based device identification algorithm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', “whether the device is indeed a smart TV based on the network signatures”) and manual user input if necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Both the network- and image-based approaches will hopefully help the app identify IoT devices in real-world settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Expanded user study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The user study, as a pilot, has a small sample size and is limited to graduate students, who may be more inquisitive or technically-inclined than the general population.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We hope to scale out the testing to a larger userbase, both in lab and in real homes, in future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We will also compare the participants’ changes in privacy awareness against other visualization tools (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', IoT Inspector [15] and Aretha [40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Finally, we will conduct in-depth studies on various ways to visualize privacy leaks in AR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', icon overlays and animations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' SUMMARY This paper presented Privacy Plumber, an end-to-end sys- tem demonstrating how a general population of end users can potentially have insight into the network traffic of smart home IoT devices, and how these users can control when these smart devices could communicate with the Internet with one click of a button.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Designed after the concept of a leak detector, Privacy Plumber is a phone app with a tethered desktop application— IoT Network Analyzer—that provides an inspect and correct interface supported by network traffic analysis (inspect) and automated and timed network traffic jamming (correct).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Privacy Plumber is the first real-world inspection and control system that can be deployed in any home without new hardware or router modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Using AR, the tool aims to help users model IoT device activities within the context of the physical environment and of user interactions (addressing challenges C1 and C3, per Section II-D);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' it gives users the option to block IoT devices and control the privacy “valve” (C2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' it provides users with an interface to visualize IoT device activities as users interact with devices (C4);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' and it requires a modern AR-supported phone and computer, without any dedicated or specialized hardware (C5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We evaluated Privacy Plumber inside an instrumented smart home space with a variety of devices not previously evaluated for any privacy-enhancing tool, including a smart fridge, a smart TV, voice assistants, and Internet-connected surveillance cameras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We found that using Privacy Plumber improved users’ awareness of potential privacy violations of devices and that the system was generally easy to use and afforded useful controls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In the future, we hope tools like Privacy Plumber will give mechanisms back to the user for stymieing the flow of private information outside the home, especially as our homes and living spaces become smarter, often without our consent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' ACKNOWLEDGMENT This research is based upon work supported by the National Science Foundation under award numbers CNS-2219867, CNS-1739809, and CNS-1915847.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' The research is also based on work supported by gifts from Consumer Reports and Meta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' REFERENCES [1] Abbas Acar, Hossein Fereidooni, Tigist Abera, Amit Kumar Sikder, Markus Miettinen, Hidayet Aksu, Mauro Conti, Ahmad-Reza Sadeghi, and Selcuk Uluagac.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 14 [49] Peter Worthy, Ben Matthews, and Stephen Viller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Trust me: doubts and concerns living with the internet of things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Proceedings of the 2016 ACM Conference on Designing Interactive Systems, pages 427–434, 2016.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [50] Yaxing Yao, Justin Reed Basdeo, Smirity Kaushik, and Yang Wang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Defending my castle: A co-design study of privacy mechanisms for smart homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Proceedings of the 2019 CHI conference on human factors in computing systems, pages 1–12, 2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [51] Eric Zeng, Shrirang Mare, and Franziska Roesner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' End user security and privacy concerns with smart homes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Thirteenth Symposium on Usable Privacy and Security (SOUPS 2017), pages 65–80, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [52] Wei Zhang, Yan Meng, Yugeng Liu, Xiaokuan Zhang, Yinqian Zhang, and Haojin Zhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Homonit: Monitoring smart home apps from encrypted traffic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In Proceedings of the 2018 ACM SIGSAC Conference on Computer and Communications Security, pages 1074–1088, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' [53] Serena Zheng, Noah Apthorpe, Marshini Chetty, and Nick Feamster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' User perceptions of smart home iot privacy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Proceedings of the ACM on human-computer interaction, 2(CSCW):1–20, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' APPENDIX SURVEY QUESTIONS All questions require responses in Likert scales, ranging from “Strongly Agree” (1) to “Strongly Disagree” (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Pre-Study Survey Questions 1) When I am in a smart home, I think about what in- formation I may be exposing to vendors, companies, and 3rd parties when I interact with or sit in the same space with smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2) I am not concerned about the information I may be exposing to 3rd parties when I interact with or sit in the same space as smart devices in a smart home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3) I think this device could be (or is) useful or valuable to my daily life and routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart Fridge Google Home Amazon Echo Smart TV Nest Cam 4) I am comfortable having this device in my house and always on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart Fridge Google Home Amazon Echo Smart TV Nest Cam B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Post-Study Survey Questions 1) When I am in a smart home, I think about what in- formation I may be exposing to vendors, companies, and 3rd parties when I interact with or sit in the same space with smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 2) I am not concerned about the information I may be exposing to 3rd parties when I interact with or sit in the same space as smart devices in a smart home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 3) Privacy Plumber has made me more aware of what information I may be exposing to 3rd parties when I interact with smart devices in the home.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 4) I feel Privacy Plumber has made me more aware of privacy and security concerns surrounding IoT devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' 5) I think this device could be (or is) useful or valuable to my daily life and routine.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart Fridge Google Home Amazon Echo Smart TV Nest Cam 6) I am comfortable having this device in my house and always on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Smart Fridge Google Home Amazon Echo Smart TV Nest Cam 7) Finally, please provide any other thoughts or obser- vations from participating in this experiment with Privacy Plumber (open ended).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' ADDITIONAL RESPONSES FROM THE USABILITY SURVEY We gave participants an open-ended question if they would improve the usability if privacy plumber, if so how.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' We obtained the following responses from each participant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' I would include more guidance or instructions in the app for first-time users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P1) I think the app is generally easy-to-use, although I might want more functionalities in the app.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' There are certain laten- cies in the app, which can be annoying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It would be more helpful if I can know if the device is not sending any traffic, or it is just simply late (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=', adding a loading icon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P2) Make it possible to view past trends (a la net microscope) and scroll backwards in time, so I can get the context of how much traffic is regularly sent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Give me a global view of the worst offenders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Still some work to do on basic stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It only works on devices that people have obviously ALREADY DECIDED TO BUY, which is a weird sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Obviously, I don’t have QR codes printed out on all of my household electronics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P3) I had difficulties trying to access the buttons, and the images seemed lagged a little.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' But the info was very useful overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P4) Fix where the traffic and ‘learn more about the device’ buttons once you’ve scanned the QR code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' It’s a bit awkward to have to hold the phone back up to the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' Maybe add the units (byte/kB) to the left hand side of the graph instead of above it for the traffic visualization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P5) The plots are not super-intuitive but I liked the representa- tions in terms of text/pictures which is easier to comprehend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' I would also be interested to see what advertisers the infor- mation is being leaked to.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' While the AR thing is cool, I would also like the option to just scroll through a list of devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' That ways I do not have to be close to the device and would also be able to monitor its activity when I am not close to the device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' In fact, I would be interested in seeing the device communication (including interaction w/ advertisers) in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} +page_content=' (P6) 15' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/HdFLT4oBgHgl3EQfIC9G/content/2301.11998v1.pdf'} diff --git a/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/2301.00813v1.pdf.txt b/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/2301.00813v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..17e87dc8cd762b504bc3deccecc65f015d6bc5aa --- /dev/null +++ b/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/2301.00813v1.pdf.txt @@ -0,0 +1,1464 @@ +A Survey on Protein Representation Learning: Retrospect and Prospect +Lirong Wu 1,2 ∗ , Yufei Huang 1,2 ∗ , Haitao Lin 1,2 , Stan Z. Li 1† +1 AI Lab, Research Center for Industries of the Future, Westlake University +2 College of Computer Science and Technology, Zhejiang University +{wulirong,huangyufei,linhaitao,stan.zq.li}@westlake.edu.cn +Abstract +Proteins are fundamental biological entities that +play a key role in life activities. The amino acid +sequences of proteins can be folded into stable 3D +structures in the real physicochemical world, form- +ing a special kind of sequence-structure data. With +the development of Artificial Intelligence (AI) tech- +niques, Protein Representation Learning (PRL) has +recently emerged as a promising research topic +for extracting informative knowledge from mas- +sive protein sequences or structures. To pave the +way for AI researchers with little bioinformatics +background, we present a timely and comprehen- +sive review of PRL formulations and existing PRL +methods from the perspective of model architec- +tures, pretext tasks, and downstream applications. +We first briefly introduce the motivations for pro- +tein representation learning and formulate it in a +general and unified framework. Next, we divide +existing PRL methods into three main categories: +sequence-based, structure-based, and sequence- +structure co-modeling. Finally, we discuss some +technical challenges and potential directions for +improving protein representation learning. The lat- +est advances in PRL methods are summarized in +a GitHub repository https://github.com/LirongWu/ +awesome-protein-representation-learning. +1 +Introduction +Proteins perform specific biological functions that are essen- +tial for all living organisms and therefore play a key role when +investigating the most fundamental questions in the life sci- +ences. The proteins are composed of one or several chains of +amino acids that fold into a stable 3D structure to enable vari- +ous biological functionalities. Therefore, understanding, pre- +dicting, and designing proteins for biological processes are +critical for medical, pharmaceutical, and genetic research. +Previous approaches on protein modeling are mostly driven +by biological or physical priors, and they explore com- +plex sequence-structure-function relationships through en- +ergy minimization [Rohl et al., 2004; Xu and Zhang, 2011], +∗Equal contribution, † Corresponding author +dynamics simulations [Hospital et al., 2015; Karplus and +Petsko, 1990], etc. +With the development of artificial in- +telligence and low-cost sequencing technologies, data-driven +Protein Representation Learning (PRL) [Jumper et al., 2021; +Rao et al., 2019; Rives et al., 2021; Hermosilla and Ropinski, +2022; Jing et al., 2020] has made remarkable progress due to +its superior performance in modeling complex nonlinear rela- +tionships. The primary goal of protein representation learning +is to extract transferable knowledge from protein data with +well-designed model architectures and pretext tasks, and then +generalize the learned knowledge to various protein-related +downstream applications, ranging from structure prediction +to sequence design. Despite their great progress, it is still +tricky for AI researchers without bioinformatics background +to get started with protein representation learning, and one +obstacle is the vast amount of physicochemical knowledge in- +volved behind the proteins. Therefore, a survey on PRL meth- +ods that is friendly to the AI community is urgently needed. +Existing surveys related to PRL [Iuchi et al., 2021; Unsal +et al., 2020; Hu et al., 2021; Torrisi et al., 2020] are mainly +developed from the perspective of biological applications, but +do not go deeper into other important aspects, such as model +architectures and pretext tasks. Overall, our contributions can +be summarized as follows: (1) Comprehensive review. Our +survey provides a comprehensive and up-to-date review of +existing PRL methods from the perspective of the model ar- +chitectures and pretext tasks. (2) New taxonomy. We divide +existing PRL methods into three categories: sequence-based, +structure-based, and sequence-structure co-modeling. (3) De- +tailed Implementations. We summarize the paper lists and +open-source codes in a public GitHub repository, setting the +stage for the development of more future works. (4) Future +directions. We point out the technical limitations of current +research and discuss several promising directions. +2 +Notation and Problem Statement +The sequence of amino acids can be folded into a stable 3D +structure, forming a special kind of sequence-structure data, +which determines its properties and functions. Therefore, we +can model each protein as a graph G = (V, E, X, F), where V +is the ordered set of N nodes in the graph representing amino +acid residues and E ∈ V × V is the set of edges that connects +the nodes. Each node u ∈ V in graph G can be attributed with +a scalar-vector tuple xu = (su, Vu), where su ∈ RO and +arXiv:2301.00813v1 [cs.LG] 31 Dec 2022 + +Vu ∈ R3×P . Each edge e ∈ E can be attributed with a scalar- +vector tuple fe = (se, Ve), where se ∈ RT and Ve ∈ R3×D. +Given a model architecture fθ(·) and a set of K losses +of pretext tasks {L(1) +pre(θ, η1), L(2) +pre(θ, η2), · · · , L(K) +pre (θ, ηK)} +with projection heads {gηk(·)}K +k=1, Protein Representation +Learning (PRL) usually works in a two-stage manner: (1) +Pre-training the model fθ(·) with pretext tasks; and (2) Fine- +tuning the pre-trained model fθinit(·) with a projection head +gω(·) under the supervision of a specific downstream task +Ltask(θ, ω). The learning objective can be formulated as +θ∗, ω∗ = arg min +(θ,ω) Ltask(θinit, ω), +s.t. θinit, {η∗ +k}K +k=1 = arg min +θ,{ηk}K +k=1 +K +� +k=1 +λkL(k) +pre(θ, ηk) +(1) +where {λk}K +k=1 are trade-off task hyperparameters. A high- +level overview of the PRL framework is shown in Fig. 1. +In practice, if we set K = 1, ω = η1, i.e., L(1) +pre(θ, η1) = +Ltask(θ, ω), it is equivalent to learning task-specific repre- +sentations directly under downstream supervision, which in +this survey can be considered as a special case of Eq. (1). +Pretext Tasks +Prediction +Head +Prediction +Head +Encoder +Downstream Task +Encoder +Step 2 +Fine-tune +Step 1 +Pre-train +Figure 1: A general framework for protein representation learning. +In this survey, we mainly focus on the model architecture +fθ(·) and pretext tasks {L(k) +pre(θ, ηk)}K +k=1 for protein repre- +sentation learning, and defer the discussion on downstream +applications until Sec. 5. A high-level overview of this sur- +vey with some representative examples is shown in Fig. 2. +3 +Model Architectures +In this section, we summarize some commonly used model +architectures for learning protein sequences or structures. +3.1 +Sequence-based Encoder +The sequence encoder takes as input (V, X) and then aims to +capture the dependencies between amino acids. [Wang et al., +2019] treats protein sequences as a special “biological lan- +guage” and then establishes an analogy between such “bio- +logical language” and natural (textual) language. Inspired by +this, many classical model architectures developed for natural +language processing can be directly extended to handle pro- +tein sequences [Asgari et al., 2019]. Depending on whether +a single sequence or multiple sequences are to be encoded, +there are a variety of different sequence-based encoders. +Single Sequences +The commonly used sequence encoders for modeling single +sequences include Variational Auto-Encoder (VAE) [Sinai et +al., 2017; Ding et al., 2019], Recurrent Neural Networks +(RNNs) [Armenteros et al., 2020], Long Short-Term Memory +(LSTM) [Hochreiter and Schmidhuber, 1997], BERT [Devlin +et al., 2018], Transformer [Vaswani et al., 2017]. Based on +the vanilla Transformer, [Wu et al., 2022] proposes a novel +geometry-inspired transformer (Geoformer) to further distill +the structural and physical pairwise relationships between +amino acids into the learned protein representation. If we +do not consider the ordering of amino acids in the sequences, +we can also directly apply Convolutional Neural Networks +(CNNs) [LeCun et al., 1995] or ResNet [He et al., 2016] to +capture the local dependencies between adjacent amino acids. +MSA Sequences +The long-standing practices in computational biology are to +make inferences from a family of evolutionarily related se- +quences [Weigt et al., 2009; Thomas et al., 2005; Lapedes +et al., 1999]. +Therefore, there have been several multi- +ple sequences encoders proposed to capture co-evolutionary +information by taking as input a set of sequences in the +form of multiple sequence alignment (MSA). For exam- +ple, MSA Transformer [Rao et al., 2021] extends the self- +attention mechanism to the MSA setting, which interleaves +self-attention across rows and columns to capture dependen- +cies between amino acids and between sequences. As a cru- +cial component of AlphaFold2, Evoformer [Jumper et al., +2021] alternatively updates MSA and Pair representations in +each block, which encode co-evolutionary information in se- +quences and relations between residues, respectively. +3.2 +Structure-based Encoder +Despite the effectiveness of sequence-based encoders, the +power of pre-training with protein structures has been rarely +explored, even though protein structures are known to be de- +terminants of protein functions. To better utilize this critical +structural information, a large number of structure-based en- +coders have been proposed to model structural information, +which can be mainly divided into three categories: feature +map-based, message-passing GNNs, and geometric GNNs. +Feature map-based Methods +The use of deep learning to model protein 3D structures could +be traced back to a decade ago [Zhang and Zhang, 2010; +Schaap et al., 2001]. Early methods directly extracted sev- +eral hand-crafted feature maps from protein structures and +then applied 3D CNNs to model the geometric information +of proteins [Derevyanko et al., 2018; Amidi et al., 2018; +Townshend et al., 2019]. Later work extended 3D CNNs to +spherical convolution for identifying interaction patterns on +protein surfaces [Sverrisson et al., 2021; Gainza et al., 2020]. +Message-passing GNNs +To further capture the geometric relationships and biomedi- +cal interactions between amino acids, it has been proposed +to first construct a graph from the extracted feature maps by +thresholding or k Nearest Neighbors (kNN) [Preparata and +Shamos, 2012]. Then, many existing message-passing Graph +Neural Networks (GNNs) can be directly applied to model +protein structures, including Graph Convolutional Network +(GCN) [Kipf and Welling, 2016], Graph Isomorphism Net- +work (GIN) [Xu et al., 2018], and GraphSAGE [Hamilton + +PRL +Preliminaries +Notation and Problem Statement +Architectures +Sequence-based +Single Sequence +LSTM [Hochreiter and Schmidhuber, 1997], Transformer [Vaswani et al., 2017], CNNs [LeCun et al., 1995] +MSA Sequence +MSA Transformer [Rao et al., 2021], Evoformer [Jumper et al., 2021] +Structure-based +Feature map-based +3D CNNs [Derevyanko et al., 2018], Spherical CNNs [Sverrisson et al., 2021] +Message-passing GNNs +GCNs [Kipf and Welling, 2016], IEConv [Hermosilla et al., 2020], GearNet [Zhang et al., 2022] +Geometric GNNs +GVP [Jing et al., 2020], GBP [Aykent and Xia, 2022], DWP [Li et al., 2022] +Sequence-structure Co-modeling +DeepFRI [Gligorijevi´c et al., 2021], LM-GVP [Wang et al., 2021] +Pretext Tasks +Sequence-based +Supervised +PLUS [Min et al., 2021], Profile Prediction [Sturmfels et al., 2020], Progen [Madani et al., 2020] +Self-Supervised +MLM [Rao et al., 2019], PMLM [He et al., 2021], NAP [Alley et al., 2019], CPC [Lu et al., 2020] +Structure-based +Contrative +Multiview Contrast [Hermosilla and Ropinski, 2022; Zhang et al., 2022] +Predictive +Distance and Angle Prediction [Chen et al., 2022], Dihedral Prediction [Hermosilla and Ropinski, 2022] +Sequence-structure Co-modeling +Full-atomic Structure Prediction [Jumper et al., 2021; Hu et al., 2022] +Applications +Property Prediction +Stability [Rao et al., 2019], Fold Quality [Baldassarre et al., 2021], Mutation Effect [Meier et al., 2021], PPI [Wang et al., 2019] +Structure Prediction +Full-atomic or Backbone Prediction [Hiranuma et al., 2021; Wu et al., 2022], Structure Inpainting [McPartlon and Xu, 2022] +Protein Design +Template-based [Ingraham et al., 2019], De Novo [Huang et al., 2016; Koepnick et al., 2019] +Structure-Based Drug Design +Auto-regressive [Liu et al., 2022a; Peng et al., 2022], Diffusion [Lin et al., 2022; Schneuing et al., 2022] +Figure 2: A high-level overview of this survey with representative examples. +et al., 2017]. However, the edges in the protein graph may +have some key properties, such as dihedral angles and direc- +tions, which determine the biological function of proteins. +With this in mind, there have been several structure-based +encoders proposed to simultaneously leverages the node and +edge features of the protein graph. For example, [Hermosilla +et al., 2020] proposes IE convolution (IEconv) to simultane- +ously capture the primary, secondary and tertiary structures +of proteins by incorporating intrinsic and extrinsic distances +between nodes. Besides, [Hermosilla and Ropinski, 2022] +adopts a similar architecture to IEConv, but introduces seven +additional edge features to efficiently describe the relative po- +sition and orientation of neighboring nodes. +Furthermore, +GearNet [Zhang et al., 2022] proposes a simple structure en- +coder, which encodes spatial information by adding different +types of sequential or structural edges and then performs both +node-level and edge-level message passing simultaneously. +Geometric GNNs +The above message-passing GNNs incorporate the 3D geom- +etry of proteins by encoding the vector features Vu/Ve into +rotation-invariant scalars su/se. However, reducing this vec- +tor information directly to scalars may not fully capture com- +plex geometry. Therefore, geometric-aware neural networks +are proposed to bake 3D rigid transformations into network +operations, leading to SO(3)-invariant and equivariant GNNs. +For example, [Jing et al., 2020] introduces Geometric Vector +Perceptrons (GVPs), which replace standard multi-layer per- +ceptrons (MLPs) in feed-forward layers and operate directly +on both scalar and vector features under a global coordinate +system. Besides, [Aykent and Xia, 2022] proposes Geometric +Bottleneck Perceptron (GBPs) to integrate geometric features +and capture complex geometric relations in the 3D structure, +based on which a new SO(3)-equivariant message passing +neural network is proposed to support a variety of geomet- +ric representation learning tasks. To achieve more sensitive +geometric awareness in both global transformations and local +relations, [Li et al., 2022] proposes Directed Weight Percep- +trons (DWPs) by extending not only the hidden neurons but +the weights from scalars to 2D/3D vectors, naturally saturat- +ing the network with 3D structures in the Euclidean space. +3.3 +Sequence-structure Encoder +Compared to sequence- and structure-based encoders, com- +paratively less work has focused on the co-encoding of pro- +tein sequences and structures. The mainstream model archi- +tecture is to extract amino acid representations as node fea- +tures by a language model and then capture the dependencies +between amino acids using a GNN module. For example, +[Gligorijevi´c et al., 2021] introduces DeepFRI, a Graph Con- +volutional Network (GCN) for predicting protein functions +by leveraging sequence representations extracted from a pro- +tein language model (LSTM) and protein structures. Besides, +LM-GVP [Wang et al., 2021] is composed of a protein lan- +guage model (composed of Transformer blocks) and a GVP +network, where the protein LM takes protein sequences as +input to compute amino acid embeddings and the GVP net- +work is used to make predictions about protein properties on a +graph derived from the protein 3D structure. Moreover, [You + +and Shen, 2022] applies the hierarchical RNN and GAT to +encode both protein sequences and structures and proposes a +cross-interaction module to enforce a learned relationship be- +tween the encoded embeddings of the two protein modalities. +4 +Pretext Task +The pretext tasks are designed to extract meaningful repre- +sentations from massive data through optimizing some well- +designed objective functions. In this section, we summarize +some commonly used pretext tasks for learning on proteins. +4.1 +Sequence-based Pretext Task +There have been many pretext tasks proposed for pre-training +language models, including Masked Language Modeling +(MLM) and Next Sentence Prediction (NSP) [Devlin et al., +2018], which can be naturally extended to pre-train protein +sequences. We divide existing sequence-based pretext tasks +into two main categories: self-supervised and supervised. +Self-supervised Pretext Task +The self-supervised pretext tasks utilize the training data itself +as supervision signals without the need for additional annota- +tions. If we consider an amino acid in a sequence as a word +in a sentence, we can naturally extend masked language mod- +eling to protein sequences. For example, we can statically or +dynamically mask out a single or a set of contiguous amino +acids and then predict the masked amino acids from the re- +maining sequences [Rao et al., 2019; Elnaggar et al., 2020; +Rives et al., 2021; Rao et al., 2021; Nambiar et al., 2020; +Xiao et al., 2021]. Besides, [McDermott et al., 2021] com- +bines adversarial training with MLM and proposes to mask +amino acids in a learnable manner. Taking into account the +dependence between masked amino acids, Pairwise MLM +(PMLM) [He et al., 2021] proposes to model the probabil- +ity of a pair of masked amino acids instead of predicting the +probability of a single amino acid. Besides, Next Amino acid +Prediction (NAP) [Alley et al., 2019; Elnaggar et al., 2020; +Strodthoff et al., 2020] aims to predict the type of the next +amino acid based on a set of given sequence fragments. Dif- +ferent from the above methods, Contrastive Predictive Cod- +ing (CPC) [Lu et al., 2020] applies different augmentation +transformations on the input sequence to generate different +views, and then maximizes the agreement of two jointly sam- +pled pairs against that of two independently sampled pairs. +Supervised Pretext Task +The supervised pretext tasks use additional labels as auxiliary +information to guide the model to learn knowledge relevant +to downstream tasks. For example, PLUS [Min et al., 2021] +devises a protein-specific pretext task, namely Same-Family +Prediction (SFP), which trains a model to predict whether a +given protein pair belongs to the same protein family. The +protein family labels provide weak structural information and +help the model learn structurally contextualized representa- +tions. Besides, [Sturmfels et al., 2020] proposes to use HMM +profiles derived from MSA as labels and then take Profile Pre- +diction as a pretext task to help the model learn information +about protein structures. In addition, to leverage the exponen- +tially growing protein sequences that lack costly structural +annotations, Progen [Madani et al., 2020] trains a language +model with conditioning tags that encode various annotations, +such as taxonomic, functional, and locational information. +4.2 +Structure-based Pretext Task +Despite the great progress in the design of structure-based +encoders and graph-based pretext tasks [Wu et al., 2021; +Xie et al., 2022; Liu et al., 2022b], there are few efforts focus- +ing on the structure-based pre-training of proteins. Existing +structure-based pretext tasks for proteins can be mainly clas- +sified into two branches: contrastive and predictive methods. +Contrastive Pretext Task +The primary goal of contrastive methods is to maximize the +agreement of two jointly sampled positive pairs. For example, +Multiview Contrast [Hermosilla and Ropinski, 2022] pro- +poses to randomly sample two sub-structures from each pro- +tein, encoder them into two representations, and finally max- +imize the similarity between representations from the same +protein while minimizing the similarity between representa- +tions from different proteins. Besides, [Zhang et al., 2022] +adopts almost the same architecture as Multiview Contrast, +but replaces GearNet with IEConv as the structure encoder. +Predictive Pretext Task +The contrastive methods deal with the inter-data information +(data-data pairs). In contrast, the predictive methods aim to +self-generate informative labels from the data as supervision +and handle the data-label relationships. Categorized by dif- +ferent types of pseudo labels, the predictive methods have +different designs that can capture different levels of struc- +tural protein information. For example, [Chen et al., 2022] +proposes two predictive tasks, namely Distance Prediction +and Angle Prediction, which take hidden representations of +residues as input and aim to predict the relative distance be- +tween pairwise residues and the angle between two edges, +respectively, which helps to learn structure-aware protein rep- +resentations. Furthermore, [Hermosilla and Ropinski, 2022] +propose Residue Type Prediction and Dihedral Prediction +based on geometric or biochemical properties. Specifically, +Residue Type Prediction randomly masks the node features +of some residues and then lets the structure-based encoders +predict these masked residue types. Instead, Dihedral Pre- +diction constructs a learning objective by predicting the di- +hedral angle between three consecutive edges. Besides, [You +and Shen, 2022] proposes graph completion (GraphComp), +which takes as input a protein graph with partially masked +residues and then makes predictions for those masked tokens. +4.3 +Sequence-structure Pretext Task +Most of the existing methods design pretext tasks for a single +modality but ignore the dependencies between sequences and +structures. If we can design the pretext task based on both +protein sequences and structures, it should capture richer in- +formation than using single modality data. In practice, there +is no clear boundary between pretext tasks and downstream +tasks. For example, AlphaFold2 [Jumper et al., 2021] takes +full-atomic structure prediction as a downstream task. How- +ever, if we are concerned with protein property prediction, +structure prediction can also be considered as a pretext task + +Table 1: Summary of representative protein representation learning methods. +Method +Category +Architecture +Pretext Task +Year +Bio2Vec-CNN [Wang et al., 2019] +Sequence-based +CNN +- +2019 +TAPE [Rao et al., 2019] +Sequence-based +ResNet, LSTM, Transformer +Masked Language Modeling, +Next Amino Acid Prediction +2019 +UniRep [Alley et al., 2019] +Sequence-based +Multiplicative LSTM +Next Amino Acid Prediction +2019 +TripletProt [Nourani et al., 2020] +Sequence-based +Siamese Networks +Contrastive Predictive Coding +2020 +PLP-CNN [Shanehsazzadeh et al., 2020] +Sequence-based +CNN +- +2020 +CPCProt [Lu et al., 2020] +Sequence-based +GRU, LSTM +Contrastive Predictive Coding +2020 +MuPIPR [Zhou et al., 2020] +Sequence-based +GRU, LSTM +Next Amino Acid Prediction +2020 +ProtTrans [Elnaggar et al., 2020] +Sequence-based +Transformer, Bert, XLNet +Masked Language Modeling +2020 +DMPfold [Kandathil et al., 2020] +Sequence-based +GRU, ResNet +- +2020 +Profile Prediction [Sturmfels et al., 2020] +Sequence-based +Transformer +HMM Profile Prediction +2020 +PRoBERTa [Nambiar et al., 2020] +Sequence-based +Transformer +Masked Language Modeling +2020 +UDSMProt [Strodthoff et al., 2020] +Sequence-based +LSTM +Next Amino Acid Prediction +2020 +ESM-1b [Rives et al., 2021] +Sequence-based +Transformer +Masked Language Modeling +2021 +PMLM [He et al., 2021] +Sequence-based +Transformer +Pairwise Masked Language Modeling +2021 +MSA Transformer [Rao et al., 2021] +Sequence-based +MSA Transformer +Masked Language Modeling +2021 +ProteinLM [Xiao et al., 2021] +Sequence-based +BERT +Masked Language Modeling +2021 +PLUS [Min et al., 2021] +Sequence-based +Bidirectional RNN +Masked Language Modeling, +Same-Family Prediction +2021 +Adversarial MLM [McDermott et al., 2021] +Sequence-based +Transformer +Masked Language Modeling, +Adversarial Training +2021 +ProteinBERT [Brandes et al., 2022] +Sequence-based +BERT +Masked Language Modeling +2022 +CARP [Yang et al., 2022a] +Sequence-based +CNN +Masked Language Modeling +2022 +3DCNN [Derevyanko et al., 2018] +Structure-based +3DCNN +- +2018 +IEConv [Hermosilla et al., 2020] +Structure-based +IEConv +- +2020 +GVP-GNN [Jing et al., 2020] +Structure-based +GVP +- +2020 +GraphMS [Cheng et al., 2021] +Structure-based +GCN +Multiview Contrast +2021 +DL-MSFM [Gelman et al., 2021] +Structure-based +GCN +- +2021 +PG-GNN [Xia and Ku, 2021] +Structure-based +PG-GNN +- +2021 +CRL [Hermosilla and Ropinski, 2022] +Structure-based +IEConv +Multiview Contrast +2022 +DW-GNN [Li et al., 2022] +Structure-based +DWP +- +2022 +GBPNet [Aykent and Xia, 2022] +Structure-based +GBP +- +2022 +GearNet [Zhang et al., 2022] +Structure-based +GearNet +Multiview Contrast, +Distance and Dihedral Prediction, +Residue Type Prediction +2022 +ATOMRefine [Wu and Cheng, 2022] +Structure-based +SE(3) Transformer +- +2022 +STEPS [Chen et al., 2022] +Structure-based +GIN +Distance and Dihedral Prediction +2022 +GraphCPI [Quan et al., 2019] +Co-Modeling +CNN, GNN +- +2019 +MT-LSTM [Bepler and Berger, 2019] +Co-Modeling +Bidirectional LSTM +Contact prediction, +Pairwise Similarity Prediction +2019 +LM-GVP [Wang et al., 2021] +Co-Modeling +Transformer, GVP +- +2021 +AlphaFold2 [Jumper et al., 2021] +Co-Modeling +Evoformer +Masked Language Modeling, +Full-atomic Structure Prediction +2021 +DeepFRI [Gligorijevi´c et al., 2021] +Co-Modeling +LSTM, GCN +- +2021 +HJRSS [Mansoor et al., 2021] +Co-Modeling +SE(3) Transformer +Masked Language Modeling, +Graph Completion +2021 +GraSR [Xia et al., 2022] +Co-Modeling +LSTM, GCN +Momentum Contrast +2022 +CPAC [You and Shen, 2022] +Co-Modeling +Hierarchical RNN, GAT +Masked Language Modeling, +Graph Completion +2022 +MIF-ST [Yang et al., 2022b] +Co-Modeling +CNN, GNN +Masked Inverse Folding +2022 +OmegaFold [Wu et al., 2022] +Co-Modeling +Geoformer +Masked Language Modeling, +Full-atomic Structure Prediction +2022 +that enables the learned sequence representations to contain +sufficient structural information. It was found by [Hu et al., +2022] that the representations from AlphFold2’s Evoformer +could work well on various protein-related downstream tasks, +including fold classification, stability prediction, etc. More- +over, [Yang et al., 2022b] proposes a novel pre-training pre- +text task, namely Masked Inverse Folding (MIF), which trains +a model to reconstruct the original amino acids conditioned +on the corrupted sequence and the backbone structure. +5 +Downstream Tasks (Applications) +In the above, we have presented a variety of commonly used +model architectures and pretext tasks for protein representa- + +tion learning, based on which we summarized the surveyed +works in Table. 1, listing their categories, model architec- +tures, pretext tasks, and publication years. In this section, we +can divide existing downstream tasks for protein representa- +tion learning into the following four main categories: protein +property prediction, protein (complex) structure prediction, +protein design, and structure-based drug design. +It is worth noting that some downstream tasks have labels +(i.e., model outputs) that do not change with rigid body trans- +formations of the inputs (if they can, e.g., protein structures). +For example, various protein property prediction tasks take +a transformable protein structure as input and output a con- +stant prediction, usually modeled as a simple multi-label clas- +sification problem or multiple binary classification problem. +However, the labels of some downstream tasks will change +equivariantly with the inputs, and these tasks are getting more +and more attention. Typically, the learning objectives of these +tasks are structure-related, and they usually have higher re- +quirements on the model architecture, requiring the model to +be SE(3)-equivariant. We believe that from the perspective +of protein representation learning, the approaches to different +downstream tasks can also learn from each other. +5.1 +Protein Property Prediction +The protein property prediction aims to regress or classify +some important properties from protein sequences or struc- +tures that are closely related to biological functions, such +as the types of secondary structure, the strength of connec- +tions between amino acids, types of protein folding, fluo- +rescence intensity, protein stability, etc. [Rao et al., 2019]. +Besides, several protein-specific prediction tasks can also be +grouped into this category, including quality evaluation of +protein folding [Baldassarre et al., 2021], predicting the ef- +fect of mutations on protein function [Meier et al., 2021], and +predicting protein-protein interactions [Wang et al., 2019]. +5.2 +Protein (Complex) Structure Prediction +The primary goal of protein structure prediction is to pre- +dict the structural coordinates from a given set of amino +acid sequences. Some approaches aim to predict only back- +bone coordinates [Baek et al., 2021; Si et al., 2020], while +others focus on the more challenging full-atomic coordi- +nate predictions [Jumper et al., 2021; Wu et al., 2022; +Rao et al., 2021]. On the other hand, protein structure refine- +ment [Hiranuma et al., 2021; Wu and Cheng, 2022] proposes +to update a coarse protein structure to generate a more fine- +grained structure in an iterative manner. Besides, the task of +protein structure inpainting aims to reconstruct the complete +protein structure from a partially given sub-structure [McPart- +lon and Xu, 2022] or distance map [Lee and Kim, 2022]. +5.3 +Protein Design +Deep learning-based protein design has made tremendous +progress in recent years, and the major works can be di- +vided into three categories. The first one is to pre-train the +model with a large number of sequences from the same pro- +tein family, and then use it to generate new homologous se- +quences [Smith and Smith, 1990]. The structure-based meth- +ods aim to directly generate the protein sequences under the +condition of a given protein structure [Ingraham et al., 2019]. +The last and most challenging one is the de novo protein de- +sign [Huang et al., 2016; Korendovych and DeGrado, 2020; +Koepnick et al., 2019], which aims to generate both protein +sequences and structures conditioned on taxonomic and key- +word tags such as molecular function and cellular component. +5.4 +Structure-Based Drug Design +Structure-Based Drug Design (SBDD) is a promising direc- +tion for fast and cost-efficient compound discovery. Specif- +ically, SBDD designs inhibitors or activators (usually small +molecules, i.e., drugs) directly against protein targets of inter- +est, which means a high success rate and efficiency [Kuntz, +1992; Drews, 2000]. In the past two years, a line of auto- +regressive methods have been proposed for SBDD [Liu et al., +2022a; Peng et al., 2022; Masuda et al., 2020], which gener- +ate molecule atoms one by one conditioned on given structure +context of protein targets. Recently, there are some works +based on Denoising Diffusion Probabilistic Model (DDPM) +[Lin et al., 2022; Schneuing et al., 2022]. Targeting on spe- +cific protein pockets, the diffusion-based methods generate +molecule atoms as a whole from random gaussian noise. +The above methods are all dependent on a proper repre- +sentation module of protein, especially the protein structure. +The early attempt of deep generative models in this field [Luo +et al., 2021] uses 3D CNN as the protein structure context +encoder to get meaningful and roto-translation invariant fea- +tures. With the development of protein structure representa- +tion methods, particularly the geometric-aware models, sub- +sequent methods widely use geometric-(equi/in)variant net- +works, such as EGNN [Gong and Cheng, 2019], GVP [Jing +et al., 2020], and IPA [Jumper et al., 2021], as the backbones. +It is worth noting that protein representation models are not +only common in various protein structure context encoders, +but many generative decoders can also adopt its architectural +design. From this example, we can see that protein represen- +tation is a very fundamental problem and that many down- +stream tasks involving proteins can benefit from advances of +protein representation research in various aspects, including +better embeddings and more excellent model architectures. +6 +Deep Insights and Future Outlooks +6.1 +Deeper Insights +On the basis of a detailed review of the model architectures, +pretext tasks, and downstream tasks, we would like to provide +some deeper insights into protein representation learning. +Insights 1: PRL is the core of deep protein modeling +With the development of deep learning, deep protein mod- +eling is becoming a popular research topic, and one of its +core is how to learn “meaningful” representations for pro- +teins. This involves three key issues: (1) Feature Extraction: +model architectures; (2) Pre-training: pretext tasks; and (3) +Application: downstream tasks. An in-depth investigation of +the above three key issues is of great importance for the de- +velopment of more deep protein modeling methods. + +Insights 2: Task-level convertibility +Throughout this survey, one of the main points we have em- +phasized is the convertibility between downstream tasks and +pretext tasks. We believe we are the first to explain the role +of pretext tasks from this perspective, which seems to have +been rarely involved in previous work. For example, we di- +rectly categorize some well-known downstream tasks, such as +full-atomic structure prediction, as a specific kinds of pretext +tasks. The motivation behind such an understanding lies in +the fact that the definition of a task is itself a relative concept +and that different tasks can help the model extract different +aspects of information, which may be complementary to each +other. For example, full-atomic structure prediction helps the +model capture rich structural information, which is also ben- +eficial for various protein property prediction tasks, such as +folding prediction, since it is known that protein structure of- +ten determines protein function. This suggests that whether +a specific task is a downstream task or a pretext task usually +depends on what we are concerned about, and the role of a +task may keep changing from application to application. +Insights 3: Data-specific criterion for design selections +It is tricky to discuss the advantages and disadvantages of dif- +ferent methods or designs because the effectiveness of differ- +ent methods depends heavily on the size, format, and com- +plexity of the data. For example, for simple small-scale data, +Transformer is not necessarily more effective than traditional +LSTM for sequence modeling, and the situation may be com- +pletely opposite for large-scale complex data. +Therefore, +there is no “optimal” architecture or pretext task that works +for all data types and downstream tasks, and the criterion for +the selection of architecture and pretext task is data-specific. +6.2 +Future Outlooks +Despite the great progress of existing methods, challenges +still exist due to the complexity of proteins. In this section, +we suggest some promising directions for future work. +Direction 1: Broader application scenarios +The biological research topics on proteins are diverse, but +most of the existing work has delved into only a small subset +of them, due to the fact that these topics have been well for- +malized by some representative works, such as AlphaFlod2 +[Jumper et al., 2021] for protein structure prediction and +TAPE [Rao et al., 2019] for protein property prediction. As +a result, it is more worthwhile to explore the role of protein +representation learning in a wider range of biological applica- +tion scenarios than to design some overly complex modules +for subtle performance gains in a well-formalized application. +Direction 2: Unified evaluation protocols +Research in protein representation learning is now in an era of +barbarism. While a great deal of new works are emerging ev- +ery day, most of them are on unfair comparisons, such as with +different datasets, architectures, metrics, etc. For example, +some MSA-based works on structure prediction have been +blatantly compared with those single-sequence-based works +and claimed to be better. To promote the health of the field, +there is an urgent need to establish unified evaluation proto- +cols in various downstream tasks to provide fair comparisons. +Direction 3: Protein-specific designs +Previous PRL methods directly take mature architectures and +pretext tasks from the natural language processing field to +train proteins. For example, modeling protein sequences us- +ing LSTM may be a major innovation, but replacing LSTM +with Bi-LSTM for stuble performance improvements makes +little sense. Now, it is time to step out of this comfort zone +of scientific research, and we should no longer be satisfied +with simply extending techniques from other domains to the +protein domain. PRL is not only a machine learning problem +but also a biological problem, so we should consider design- +ing more protein-specific architectures and pretext tasks by +incorporating protein-related domain knowledge. In particu- +lar, most of the existing work on PRL is based on unimodal +protein sequences or structures, and it requires more work ex- +ploring sequence-structure co-modeling to fully explore the +correspondence between 1D sequences and 3D structures. +Direction 4: Margin from pre-training to fine-tuning +Currently, tremendous efforts are focusing on protein pre- +training strategies. +However, how to fine-tune these pre- +trained models to specific downstream tasks is still under- +explored. Though numerous strategies have been proposed +to address this problem in the fields of computer vision and +natural language processing [Zhuang et al., 2020], they are +difficult to be directly applied to proteins. One obstacle to +knowledge transfer is the huge variability between different +protein datasets, both in terms of sequence length and struc- +tural complexity. The second one is poor generalization of +pre-trained models especially for various tasks where collect- +ing labeled data is laborious. Therefore, it is an important +issue to design protein-specific techniques to minimize the +margin between pre-training and downstream tasks. +Direction 5: Lack of explainability +While existing protein representation learning methods have +achieved promising results on a variety of downstream tasks, +we still know little about what the model has learned from +protein data. Which of the feature patterns, sequence frag- +ments, or sequence-structure relationships has been learned? +These are important issues for understanding and interpret- +ing model behavior, especially for those privacy-secure tasks +such as drug design, but are missing in current PRL works. +Overall, the interpretability of PRL methods remains to be +explored further in many respects, which helps us understand +how the model works and provides a guide for better usage. +7 +Conclusions +A comprehensive survey of the literature on protein repre- +sentation learning is conducted in this paper. We develop a +general unified framework for PRL methods. Moreover, we +systematically divide existing PRL methods into three main +categories: sequence-based, structure-based, and sequence- +structure co-modeling from three different perspectives, in- +cluding model architectures, pretext tasks, and downstream +applications. 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Proceedings of the IEEE, 109(1):43–76, 2020. + diff --git a/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/load_file.txt b/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7b7c93e29c8d03d6a5e8c711c25fc9adfb781b36 --- /dev/null +++ b/I9AyT4oBgHgl3EQf5_o0/content/tmp_files/load_file.txt @@ -0,0 +1,820 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf,len=819 +page_content='A Survey on Protein Representation Learning: Retrospect and Prospect Lirong Wu 1,2 ∗ , Yufei Huang 1,2 ∗ , Haitao Lin 1,2 , Stan Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Li 1† 1 AI Lab, Research Center for Industries of the Future, Westlake University 2 College of Computer Science and Technology, Zhejiang University {wulirong,huangyufei,linhaitao,stan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='zq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='li}@westlake.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='cn Abstract Proteins are fundamental biological entities that play a key role in life activities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The amino acid sequences of proteins can be folded into stable 3D structures in the real physicochemical world, form- ing a special kind of sequence-structure data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' With the development of Artificial Intelligence (AI) tech- niques, Protein Representation Learning (PRL) has recently emerged as a promising research topic for extracting informative knowledge from mas- sive protein sequences or structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' To pave the way for AI researchers with little bioinformatics background, we present a timely and comprehen- sive review of PRL formulations and existing PRL methods from the perspective of model architec- tures, pretext tasks, and downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We first briefly introduce the motivations for pro- tein representation learning and formulate it in a general and unified framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Next, we divide existing PRL methods into three main categories: sequence-based, structure-based, and sequence- structure co-modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Finally, we discuss some technical challenges and potential directions for improving protein representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The lat- est advances in PRL methods are summarized in a GitHub repository https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='com/LirongWu/ awesome-protein-representation-learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 1 Introduction Proteins perform specific biological functions that are essen- tial for all living organisms and therefore play a key role when investigating the most fundamental questions in the life sci- ences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The proteins are composed of one or several chains of amino acids that fold into a stable 3D structure to enable vari- ous biological functionalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, understanding, pre- dicting, and designing proteins for biological processes are critical for medical, pharmaceutical, and genetic research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Previous approaches on protein modeling are mostly driven by biological or physical priors, and they explore com- plex sequence-structure-function relationships through en- ergy minimization [Rohl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Xu and Zhang, 2011], ∗Equal contribution, † Corresponding author dynamics simulations [Hospital et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Karplus and Petsko, 1990], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' With the development of artificial in- telligence and low-cost sequencing technologies, data-driven Protein Representation Learning (PRL) [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Hermosilla and Ropinski, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] has made remarkable progress due to its superior performance in modeling complex nonlinear rela- tionships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The primary goal of protein representation learning is to extract transferable knowledge from protein data with well-designed model architectures and pretext tasks, and then generalize the learned knowledge to various protein-related downstream applications, ranging from structure prediction to sequence design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Despite their great progress, it is still tricky for AI researchers without bioinformatics background to get started with protein representation learning, and one obstacle is the vast amount of physicochemical knowledge in- volved behind the proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, a survey on PRL meth- ods that is friendly to the AI community is urgently needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Existing surveys related to PRL [Iuchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Unsal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Torrisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] are mainly developed from the perspective of biological applications, but do not go deeper into other important aspects, such as model architectures and pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Overall, our contributions can be summarized as follows: (1) Comprehensive review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Our survey provides a comprehensive and up-to-date review of existing PRL methods from the perspective of the model ar- chitectures and pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' (2) New taxonomy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We divide existing PRL methods into three categories: sequence-based, structure-based, and sequence-structure co-modeling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' (3) De- tailed Implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We summarize the paper lists and open-source codes in a public GitHub repository, setting the stage for the development of more future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' (4) Future directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We point out the technical limitations of current research and discuss several promising directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 2 Notation and Problem Statement The sequence of amino acids can be folded into a stable 3D structure, forming a special kind of sequence-structure data, which determines its properties and functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, we can model each protein as a graph G = (V, E, X, F), where V is the ordered set of N nodes in the graph representing amino acid residues and E ∈ V × V is the set of edges that connects the nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Each node u ∈ V in graph G can be attributed with a scalar-vector tuple xu = (su, Vu), where su ∈ RO and arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='00813v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='LG] 31 Dec 2022 Vu ∈ R3×P .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Each edge e ∈ E can be attributed with a scalar- vector tuple fe = (se, Ve), where se ∈ RT and Ve ∈ R3×D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Given a model architecture fθ(·) and a set of K losses of pretext tasks {L(1) pre(θ, η1), L(2) pre(θ, η2), · · · , L(K) pre (θ, ηK)} with projection heads {gηk(·)}K k=1, Protein Representation Learning (PRL) usually works in a two-stage manner: (1) Pre-training the model fθ(·) with pretext tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' and (2) Fine- tuning the pre-trained model fθinit(·) with a projection head gω(·) under the supervision of a specific downstream task Ltask(θ, ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The learning objective can be formulated as θ∗, ω∗ = arg min (θ,ω) Ltask(θinit, ω), s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' θinit, {η∗ k}K k=1 = arg min θ,{ηk}K k=1 K � k=1 λkL(k) pre(θ, ηk) (1) where {λk}K k=1 are trade-off task hyperparameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' A high- level overview of the PRL framework is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In practice, if we set K = 1, ω = η1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', L(1) pre(θ, η1) = Ltask(θ, ω), it is equivalent to learning task-specific repre- sentations directly under downstream supervision, which in this survey can be considered as a special case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Pretext Tasks Prediction Head Prediction Head Encoder Downstream Task Encoder Step 2 Fine-tune Step 1 Pre-train Figure 1: A general framework for protein representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In this survey, we mainly focus on the model architecture fθ(·) and pretext tasks {L(k) pre(θ, ηk)}K k=1 for protein repre- sentation learning, and defer the discussion on downstream applications until Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' A high-level overview of this sur- vey with some representative examples is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 3 Model Architectures In this section, we summarize some commonly used model architectures for learning protein sequences or structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='1 Sequence-based Encoder The sequence encoder takes as input (V, X) and then aims to capture the dependencies between amino acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] treats protein sequences as a special “biological lan- guage” and then establishes an analogy between such “bio- logical language” and natural (textual) language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Inspired by this, many classical model architectures developed for natural language processing can be directly extended to handle pro- tein sequences [Asgari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Depending on whether a single sequence or multiple sequences are to be encoded, there are a variety of different sequence-based encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Single Sequences The commonly used sequence encoders for modeling single sequences include Variational Auto-Encoder (VAE) [Sinai et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Ding et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], Recurrent Neural Networks (RNNs) [Armenteros et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], Long Short-Term Memory (LSTM) [Hochreiter and Schmidhuber, 1997], BERT [Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018], Transformer [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Based on the vanilla Transformer, [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] proposes a novel geometry-inspired transformer (Geoformer) to further distill the structural and physical pairwise relationships between amino acids into the learned protein representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' If we do not consider the ordering of amino acids in the sequences, we can also directly apply Convolutional Neural Networks (CNNs) [LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 1995] or ResNet [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2016] to capture the local dependencies between adjacent amino acids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' MSA Sequences The long-standing practices in computational biology are to make inferences from a family of evolutionarily related se- quences [Weigt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Lapedes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 1999].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, there have been several multi- ple sequences encoders proposed to capture co-evolutionary information by taking as input a set of sequences in the form of multiple sequence alignment (MSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For exam- ple, MSA Transformer [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] extends the self- attention mechanism to the MSA setting, which interleaves self-attention across rows and columns to capture dependen- cies between amino acids and between sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' As a cru- cial component of AlphaFold2, Evoformer [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] alternatively updates MSA and Pair representations in each block, which encode co-evolutionary information in se- quences and relations between residues, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='2 Structure-based Encoder Despite the effectiveness of sequence-based encoders, the power of pre-training with protein structures has been rarely explored, even though protein structures are known to be de- terminants of protein functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' To better utilize this critical structural information, a large number of structure-based en- coders have been proposed to model structural information, which can be mainly divided into three categories: feature map-based, message-passing GNNs, and geometric GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Feature map-based Methods The use of deep learning to model protein 3D structures could be traced back to a decade ago [Zhang and Zhang, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Schaap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Early methods directly extracted sev- eral hand-crafted feature maps from protein structures and then applied 3D CNNs to model the geometric information of proteins [Derevyanko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Amidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Townshend et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Later work extended 3D CNNs to spherical convolution for identifying interaction patterns on protein surfaces [Sverrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Gainza et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Message-passing GNNs To further capture the geometric relationships and biomedi- cal interactions between amino acids, it has been proposed to first construct a graph from the extracted feature maps by thresholding or k Nearest Neighbors (kNN) [Preparata and Shamos, 2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Then, many existing message-passing Graph Neural Networks (GNNs) can be directly applied to model protein structures, including Graph Convolutional Network (GCN) [Kipf and Welling, 2016], Graph Isomorphism Net- work (GIN) [Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018], and GraphSAGE [Hamilton PRL Preliminaries Notation and Problem Statement Architectures Sequence-based Single Sequence LSTM [Hochreiter and Schmidhuber, 1997], Transformer [Vaswani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2017], CNNs [LeCun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 1995] MSA Sequence MSA Transformer [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], Evoformer [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Structure-based Feature map-based 3D CNNs [Derevyanko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018], Spherical CNNs [Sverrisson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Message-passing GNNs GCNs [Kipf and Welling, 2016], IEConv [Hermosilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], GearNet [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Geometric GNNs GVP [Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], GBP [Aykent and Xia, 2022], DWP [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Sequence-structure Co-modeling DeepFRI [Gligorijevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], LM-GVP [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Pretext Tasks Sequence-based Supervised PLUS [Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], Profile Prediction [Sturmfels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], Progen [Madani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Self-Supervised MLM [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], PMLM [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], NAP [Alley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], CPC [Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Structure-based Contrative Multiview Contrast [Hermosilla and Ropinski, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Predictive Distance and Angle Prediction [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022], Dihedral Prediction [Hermosilla and Ropinski, 2022] Sequence-structure Co-modeling Full-atomic Structure Prediction [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Applications Property Prediction Stability [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], Fold Quality [Baldassarre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], Mutation Effect [Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], PPI [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Structure Prediction Full-atomic or Backbone Prediction [Hiranuma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022], Structure Inpainting [McPartlon and Xu, 2022] Protein Design Template-based [Ingraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], De Novo [Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Koepnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Structure-Based Drug Design Auto-regressive [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022], Diffusion [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Schneuing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Figure 2: A high-level overview of this survey with representative examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' However, the edges in the protein graph may have some key properties, such as dihedral angles and direc- tions, which determine the biological function of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' With this in mind, there have been several structure-based encoders proposed to simultaneously leverages the node and edge features of the protein graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, [Hermosilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] proposes IE convolution (IEconv) to simultane- ously capture the primary, secondary and tertiary structures of proteins by incorporating intrinsic and extrinsic distances between nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [Hermosilla and Ropinski, 2022] adopts a similar architecture to IEConv, but introduces seven additional edge features to efficiently describe the relative po- sition and orientation of neighboring nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Furthermore, GearNet [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] proposes a simple structure en- coder, which encodes spatial information by adding different types of sequential or structural edges and then performs both node-level and edge-level message passing simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Geometric GNNs The above message-passing GNNs incorporate the 3D geom- etry of proteins by encoding the vector features Vu/Ve into rotation-invariant scalars su/se.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' However, reducing this vec- tor information directly to scalars may not fully capture com- plex geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, geometric-aware neural networks are proposed to bake 3D rigid transformations into network operations, leading to SO(3)-invariant and equivariant GNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, [Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] introduces Geometric Vector Perceptrons (GVPs), which replace standard multi-layer per- ceptrons (MLPs) in feed-forward layers and operate directly on both scalar and vector features under a global coordinate system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [Aykent and Xia, 2022] proposes Geometric Bottleneck Perceptron (GBPs) to integrate geometric features and capture complex geometric relations in the 3D structure, based on which a new SO(3)-equivariant message passing neural network is proposed to support a variety of geomet- ric representation learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' To achieve more sensitive geometric awareness in both global transformations and local relations, [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] proposes Directed Weight Percep- trons (DWPs) by extending not only the hidden neurons but the weights from scalars to 2D/3D vectors, naturally saturat- ing the network with 3D structures in the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='3 Sequence-structure Encoder Compared to sequence- and structure-based encoders, com- paratively less work has focused on the co-encoding of pro- tein sequences and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The mainstream model archi- tecture is to extract amino acid representations as node fea- tures by a language model and then capture the dependencies between amino acids using a GNN module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, [Gligorijevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] introduces DeepFRI, a Graph Con- volutional Network (GCN) for predicting protein functions by leveraging sequence representations extracted from a pro- tein language model (LSTM) and protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, LM-GVP [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] is composed of a protein lan- guage model (composed of Transformer blocks) and a GVP network, where the protein LM takes protein sequences as input to compute amino acid embeddings and the GVP net- work is used to make predictions about protein properties on a graph derived from the protein 3D structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Moreover, [You and Shen, 2022] applies the hierarchical RNN and GAT to encode both protein sequences and structures and proposes a cross-interaction module to enforce a learned relationship be- tween the encoded embeddings of the two protein modalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 4 Pretext Task The pretext tasks are designed to extract meaningful repre- sentations from massive data through optimizing some well- designed objective functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In this section, we summarize some commonly used pretext tasks for learning on proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='1 Sequence-based Pretext Task There have been many pretext tasks proposed for pre-training language models, including Masked Language Modeling (MLM) and Next Sentence Prediction (NSP) [Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018], which can be naturally extended to pre-train protein sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We divide existing sequence-based pretext tasks into two main categories: self-supervised and supervised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Self-supervised Pretext Task The self-supervised pretext tasks utilize the training data itself as supervision signals without the need for additional annota- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' If we consider an amino acid in a sequence as a word in a sentence, we can naturally extend masked language mod- eling to protein sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, we can statically or dynamically mask out a single or a set of contiguous amino acids and then predict the masked amino acids from the re- maining sequences [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Nambiar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [McDermott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] com- bines adversarial training with MLM and proposes to mask amino acids in a learnable manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Taking into account the dependence between masked amino acids, Pairwise MLM (PMLM) [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] proposes to model the probabil- ity of a pair of masked amino acids instead of predicting the probability of a single amino acid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, Next Amino acid Prediction (NAP) [Alley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Strodthoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] aims to predict the type of the next amino acid based on a set of given sequence fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Dif- ferent from the above methods, Contrastive Predictive Cod- ing (CPC) [Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] applies different augmentation transformations on the input sequence to generate different views, and then maximizes the agreement of two jointly sam- pled pairs against that of two independently sampled pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Supervised Pretext Task The supervised pretext tasks use additional labels as auxiliary information to guide the model to learn knowledge relevant to downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, PLUS [Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] devises a protein-specific pretext task, namely Same-Family Prediction (SFP), which trains a model to predict whether a given protein pair belongs to the same protein family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The protein family labels provide weak structural information and help the model learn structurally contextualized representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [Sturmfels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] proposes to use HMM profiles derived from MSA as labels and then take Profile Pre- diction as a pretext task to help the model learn information about protein structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In addition, to leverage the exponen- tially growing protein sequences that lack costly structural annotations, Progen [Madani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] trains a language model with conditioning tags that encode various annotations, such as taxonomic, functional, and locational information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='2 Structure-based Pretext Task Despite the great progress in the design of structure-based encoders and graph-based pretext tasks [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022b], there are few efforts focus- ing on the structure-based pre-training of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Existing structure-based pretext tasks for proteins can be mainly clas- sified into two branches: contrastive and predictive methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Contrastive Pretext Task The primary goal of contrastive methods is to maximize the agreement of two jointly sampled positive pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, Multiview Contrast [Hermosilla and Ropinski, 2022] pro- poses to randomly sample two sub-structures from each pro- tein, encoder them into two representations, and finally max- imize the similarity between representations from the same protein while minimizing the similarity between representa- tions from different proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] adopts almost the same architecture as Multiview Contrast, but replaces GearNet with IEConv as the structure encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Predictive Pretext Task The contrastive methods deal with the inter-data information (data-data pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In contrast, the predictive methods aim to self-generate informative labels from the data as supervision and handle the data-label relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Categorized by dif- ferent types of pseudo labels, the predictive methods have different designs that can capture different levels of struc- tural protein information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] proposes two predictive tasks, namely Distance Prediction and Angle Prediction, which take hidden representations of residues as input and aim to predict the relative distance be- tween pairwise residues and the angle between two edges, respectively, which helps to learn structure-aware protein rep- resentations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Furthermore, [Hermosilla and Ropinski, 2022] propose Residue Type Prediction and Dihedral Prediction based on geometric or biochemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Specifically, Residue Type Prediction randomly masks the node features of some residues and then lets the structure-based encoders predict these masked residue types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Instead, Dihedral Pre- diction constructs a learning objective by predicting the di- hedral angle between three consecutive edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, [You and Shen, 2022] proposes graph completion (GraphComp), which takes as input a protein graph with partially masked residues and then makes predictions for those masked tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='3 Sequence-structure Pretext Task Most of the existing methods design pretext tasks for a single modality but ignore the dependencies between sequences and structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' If we can design the pretext task based on both protein sequences and structures, it should capture richer in- formation than using single modality data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In practice, there is no clear boundary between pretext tasks and downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, AlphaFold2 [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] takes full-atomic structure prediction as a downstream task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' How- ever, if we are concerned with protein property prediction, structure prediction can also be considered as a pretext task Table 1: Summary of representative protein representation learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Method Category Architecture Pretext Task Year Bio2Vec-CNN [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Sequence-based CNN 2019 TAPE [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Sequence-based ResNet, LSTM, Transformer Masked Language Modeling, Next Amino Acid Prediction 2019 UniRep [Alley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Sequence-based Multiplicative LSTM Next Amino Acid Prediction 2019 TripletProt [Nourani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based Siamese Networks Contrastive Predictive Coding 2020 PLP-CNN [Shanehsazzadeh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based CNN 2020 CPCProt [Lu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based GRU, LSTM Contrastive Predictive Coding 2020 MuPIPR [Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based GRU, LSTM Next Amino Acid Prediction 2020 ProtTrans [Elnaggar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based Transformer, Bert, XLNet Masked Language Modeling 2020 DMPfold [Kandathil et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based GRU, ResNet 2020 Profile Prediction [Sturmfels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based Transformer HMM Profile Prediction 2020 PRoBERTa [Nambiar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based Transformer Masked Language Modeling 2020 UDSMProt [Strodthoff et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Sequence-based LSTM Next Amino Acid Prediction 2020 ESM-1b [Rives et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based Transformer Masked Language Modeling 2021 PMLM [He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based Transformer Pairwise Masked Language Modeling 2021 MSA Transformer [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based MSA Transformer Masked Language Modeling 2021 ProteinLM [Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based BERT Masked Language Modeling 2021 PLUS [Min et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based Bidirectional RNN Masked Language Modeling, Same-Family Prediction 2021 Adversarial MLM [McDermott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Sequence-based Transformer Masked Language Modeling, Adversarial Training 2021 ProteinBERT [Brandes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Sequence-based BERT Masked Language Modeling 2022 CARP [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022a] Sequence-based CNN Masked Language Modeling 2022 3DCNN [Derevyanko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2018] Structure-based 3DCNN 2018 IEConv [Hermosilla et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Structure-based IEConv 2020 GVP-GNN [Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020] Structure-based GVP 2020 GraphMS [Cheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Structure-based GCN Multiview Contrast 2021 DL-MSFM [Gelman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Structure-based GCN 2021 PG-GNN [Xia and Ku, 2021] Structure-based PG-GNN 2021 CRL [Hermosilla and Ropinski, 2022] Structure-based IEConv Multiview Contrast 2022 DW-GNN [Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Structure-based DWP 2022 GBPNet [Aykent and Xia, 2022] Structure-based GBP 2022 GearNet [Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Structure-based GearNet Multiview Contrast, Distance and Dihedral Prediction, Residue Type Prediction 2022 ATOMRefine [Wu and Cheng, 2022] Structure-based SE(3) Transformer 2022 STEPS [Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Structure-based GIN Distance and Dihedral Prediction 2022 GraphCPI [Quan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] Co-Modeling CNN, GNN 2019 MT-LSTM [Bepler and Berger, 2019] Co-Modeling Bidirectional LSTM Contact prediction, Pairwise Similarity Prediction 2019 LM-GVP [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Co-Modeling Transformer, GVP 2021 AlphaFold2 [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Co-Modeling Evoformer Masked Language Modeling, Full-atomic Structure Prediction 2021 DeepFRI [Gligorijevi´c et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Co-Modeling LSTM, GCN 2021 HJRSS [Mansoor et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] Co-Modeling SE(3) Transformer Masked Language Modeling, Graph Completion 2021 GraSR [Xia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Co-Modeling LSTM, GCN Momentum Contrast 2022 CPAC [You and Shen, 2022] Co-Modeling Hierarchical RNN, GAT Masked Language Modeling, Graph Completion 2022 MIF-ST [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022b] Co-Modeling CNN, GNN Masked Inverse Folding 2022 OmegaFold [Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] Co-Modeling Geoformer Masked Language Modeling, Full-atomic Structure Prediction 2022 that enables the learned sequence representations to contain sufficient structural information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' It was found by [Hu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022] that the representations from AlphFold2’s Evoformer could work well on various protein-related downstream tasks, including fold classification, stability prediction, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' More- over, [Yang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022b] proposes a novel pre-training pre- text task, namely Masked Inverse Folding (MIF), which trains a model to reconstruct the original amino acids conditioned on the corrupted sequence and the backbone structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5 Downstream Tasks (Applications) In the above, we have presented a variety of commonly used model architectures and pretext tasks for protein representa- tion learning, based on which we summarized the surveyed works in Table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 1, listing their categories, model architec- tures, pretext tasks, and publication years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In this section, we can divide existing downstream tasks for protein representa- tion learning into the following four main categories: protein property prediction, protein (complex) structure prediction, protein design, and structure-based drug design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' It is worth noting that some downstream tasks have labels (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', model outputs) that do not change with rigid body trans- formations of the inputs (if they can, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', protein structures).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, various protein property prediction tasks take a transformable protein structure as input and output a con- stant prediction, usually modeled as a simple multi-label clas- sification problem or multiple binary classification problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' However, the labels of some downstream tasks will change equivariantly with the inputs, and these tasks are getting more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Typically, the learning objectives of these tasks are structure-related, and they usually have higher re- quirements on the model architecture, requiring the model to be SE(3)-equivariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We believe that from the perspective of protein representation learning, the approaches to different downstream tasks can also learn from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='1 Protein Property Prediction The protein property prediction aims to regress or classify some important properties from protein sequences or struc- tures that are closely related to biological functions, such as the types of secondary structure, the strength of connec- tions between amino acids, types of protein folding, fluo- rescence intensity, protein stability, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, several protein-specific prediction tasks can also be grouped into this category, including quality evaluation of protein folding [Baldassarre et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], predicting the ef- fect of mutations on protein function [Meier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], and predicting protein-protein interactions [Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='2 Protein (Complex) Structure Prediction The primary goal of protein structure prediction is to pre- dict the structural coordinates from a given set of amino acid sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Some approaches aim to predict only back- bone coordinates [Baek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Si et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], while others focus on the more challenging full-atomic coordi- nate predictions [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' On the other hand, protein structure refine- ment [Hiranuma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Wu and Cheng, 2022] proposes to update a coarse protein structure to generate a more fine- grained structure in an iterative manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Besides, the task of protein structure inpainting aims to reconstruct the complete protein structure from a partially given sub-structure [McPart- lon and Xu, 2022] or distance map [Lee and Kim, 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='3 Protein Design Deep learning-based protein design has made tremendous progress in recent years, and the major works can be di- vided into three categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The first one is to pre-train the model with a large number of sequences from the same pro- tein family, and then use it to generate new homologous se- quences [Smith and Smith, 1990].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The structure-based meth- ods aim to directly generate the protein sequences under the condition of a given protein structure [Ingraham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The last and most challenging one is the de novo protein de- sign [Huang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Korendovych and DeGrado, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Koepnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019], which aims to generate both protein sequences and structures conditioned on taxonomic and key- word tags such as molecular function and cellular component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='4 Structure-Based Drug Design Structure-Based Drug Design (SBDD) is a promising direc- tion for fast and cost-efficient compound discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Specif- ically, SBDD designs inhibitors or activators (usually small molecules, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', drugs) directly against protein targets of inter- est, which means a high success rate and efficiency [Kuntz, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Drews, 2000].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In the past two years, a line of auto- regressive methods have been proposed for SBDD [Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Peng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Masuda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], which gener- ate molecule atoms one by one conditioned on given structure context of protein targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Recently, there are some works based on Denoising Diffusion Probabilistic Model (DDPM) [Lin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Schneuing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Targeting on spe- cific protein pockets, the diffusion-based methods generate molecule atoms as a whole from random gaussian noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The above methods are all dependent on a proper repre- sentation module of protein, especially the protein structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The early attempt of deep generative models in this field [Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] uses 3D CNN as the protein structure context encoder to get meaningful and roto-translation invariant fea- tures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' With the development of protein structure representa- tion methods, particularly the geometric-aware models, sub- sequent methods widely use geometric-(equi/in)variant net- works, such as EGNN [Gong and Cheng, 2019], GVP [Jing et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], and IPA [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021], as the backbones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' It is worth noting that protein representation models are not only common in various protein structure context encoders, but many generative decoders can also adopt its architectural design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' From this example, we can see that protein represen- tation is a very fundamental problem and that many down- stream tasks involving proteins can benefit from advances of protein representation research in various aspects, including better embeddings and more excellent model architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 6 Deep Insights and Future Outlooks 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='1 Deeper Insights On the basis of a detailed review of the model architectures, pretext tasks, and downstream tasks, we would like to provide some deeper insights into protein representation learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Insights 1: PRL is the core of deep protein modeling With the development of deep learning, deep protein mod- eling is becoming a popular research topic, and one of its core is how to learn “meaningful” representations for pro- teins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' This involves three key issues: (1) Feature Extraction: model architectures;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' (2) Pre-training: pretext tasks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' and (3) Application: downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' An in-depth investigation of the above three key issues is of great importance for the de- velopment of more deep protein modeling methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Insights 2: Task-level convertibility Throughout this survey, one of the main points we have em- phasized is the convertibility between downstream tasks and pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We believe we are the first to explain the role of pretext tasks from this perspective, which seems to have been rarely involved in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, we di- rectly categorize some well-known downstream tasks, such as full-atomic structure prediction, as a specific kinds of pretext tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The motivation behind such an understanding lies in the fact that the definition of a task is itself a relative concept and that different tasks can help the model extract different aspects of information, which may be complementary to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, full-atomic structure prediction helps the model capture rich structural information, which is also ben- eficial for various protein property prediction tasks, such as folding prediction, since it is known that protein structure of- ten determines protein function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' This suggests that whether a specific task is a downstream task or a pretext task usually depends on what we are concerned about, and the role of a task may keep changing from application to application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Insights 3: Data-specific criterion for design selections It is tricky to discuss the advantages and disadvantages of dif- ferent methods or designs because the effectiveness of differ- ent methods depends heavily on the size, format, and com- plexity of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, for simple small-scale data, Transformer is not necessarily more effective than traditional LSTM for sequence modeling, and the situation may be com- pletely opposite for large-scale complex data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, there is no “optimal” architecture or pretext task that works for all data types and downstream tasks, and the criterion for the selection of architecture and pretext task is data-specific.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content='2 Future Outlooks Despite the great progress of existing methods, challenges still exist due to the complexity of proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In this section, we suggest some promising directions for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Direction 1: Broader application scenarios The biological research topics on proteins are diverse, but most of the existing work has delved into only a small subset of them, due to the fact that these topics have been well for- malized by some representative works, such as AlphaFlod2 [Jumper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2021] for protein structure prediction and TAPE [Rao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2019] for protein property prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' As a result, it is more worthwhile to explore the role of protein representation learning in a wider range of biological applica- tion scenarios than to design some overly complex modules for subtle performance gains in a well-formalized application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Direction 2: Unified evaluation protocols Research in protein representation learning is now in an era of barbarism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' While a great deal of new works are emerging ev- ery day, most of them are on unfair comparisons, such as with different datasets, architectures, metrics, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, some MSA-based works on structure prediction have been blatantly compared with those single-sequence-based works and claimed to be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' To promote the health of the field, there is an urgent need to establish unified evaluation proto- cols in various downstream tasks to provide fair comparisons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Direction 3: Protein-specific designs Previous PRL methods directly take mature architectures and pretext tasks from the natural language processing field to train proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' For example, modeling protein sequences us- ing LSTM may be a major innovation, but replacing LSTM with Bi-LSTM for stuble performance improvements makes little sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Now, it is time to step out of this comfort zone of scientific research, and we should no longer be satisfied with simply extending techniques from other domains to the protein domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' PRL is not only a machine learning problem but also a biological problem, so we should consider design- ing more protein-specific architectures and pretext tasks by incorporating protein-related domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' In particu- lar, most of the existing work on PRL is based on unimodal protein sequences or structures, and it requires more work ex- ploring sequence-structure co-modeling to fully explore the correspondence between 1D sequences and 3D structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Direction 4: Margin from pre-training to fine-tuning Currently, tremendous efforts are focusing on protein pre- training strategies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' However, how to fine-tune these pre- trained models to specific downstream tasks is still under- explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Though numerous strategies have been proposed to address this problem in the fields of computer vision and natural language processing [Zhuang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=', 2020], they are difficult to be directly applied to proteins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' One obstacle to knowledge transfer is the huge variability between different protein datasets, both in terms of sequence length and struc- tural complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' The second one is poor generalization of pre-trained models especially for various tasks where collect- ing labeled data is laborious.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Therefore, it is an important issue to design protein-specific techniques to minimize the margin between pre-training and downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Direction 5: Lack of explainability While existing protein representation learning methods have achieved promising results on a variety of downstream tasks, we still know little about what the model has learned from protein data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Which of the feature patterns, sequence frag- ments, or sequence-structure relationships has been learned?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' These are important issues for understanding and interpret- ing model behavior, especially for those privacy-secure tasks such as drug design, but are missing in current PRL works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Overall, the interpretability of PRL methods remains to be explored further in many respects, which helps us understand how the model works and provides a guide for better usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' 7 Conclusions A comprehensive survey of the literature on protein repre- sentation learning is conducted in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We develop a general unified framework for PRL methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Moreover, we systematically divide existing PRL methods into three main categories: sequence-based, structure-based, and sequence- structure co-modeling from three different perspectives, in- cluding model architectures, pretext tasks, and downstream applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' Finally, we point out the technical limitations of the current research and provide promising directions for future work on PRL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/I9AyT4oBgHgl3EQf5_o0/content/2301.00813v1.pdf'} +page_content=' We hope this survey to pave the way for follow-up AI researchers with no bioinformatics background, setting the stage for the development of more 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b/ItAzT4oBgHgl3EQfjv0x/content/tmp_files/2301.01520v1.pdf.txt @@ -0,0 +1,680 @@ +1 +Counterfactual Explanations for Land Cover +Mapping in a Multi-class Setting +Cassio F. Dantas, Diego Marcos, Dino Ienco +Abstract—Counterfactual explanations are an emerging tool +to enhance interpretability of deep learning models. Given a +sample, these methods seek to find and display to the user similar +samples across the decision boundary. In this paper, we propose a +generative adversarial counterfactual approach for satellite image +time series in a multi-class setting for the land cover classification +task. One of the distinctive features of the proposed approach is +the lack of prior assumption on the targeted class for a given +counterfactual explanation. This inherent flexibility allows for the +discovery of interesting information on the relationship between +land cover classes. The other feature consists of encouraging the +counterfactual to differ from the original sample only in a small +and compact temporal segment. These time-contiguous perturba- +tions allow for a much sparser and, thus, interpretable solution. +Furthermore, plausibility/realism of the generated counterfactual +explanations is enforced via the proposed adversarial learning +strategy. +I. INTRODUCTION +Deep learning techniques have gained widespread popu- +larity in the remote sensing field due to impressive results +on a variety of tasks such as image super-resolution, image +restoration, biophysical variables estimation and land cover +classification from satellite image time series (SITS) data [1]. +Of particular importance, this last task provides useful knowl- +edge to support many downstream geospatial analyses [2]. +Despite the high performances achieved by recent deep learn- +ing frameworks on this task, they remain black-box models +with limited understanding on their internal behavior. Due +to this limitation, there is a growing need for improving the +interpretability of deep learning models in remote sensing with +the objective to raise up their acceptability and usefulness, as +their decision-making processes are often not transparent [3]– +[5]. Counterfactual explanation methods have recently received +increasing attention as a means to provide some level of +interpretability [6]–[8] to these black-box models. Counter- +factual explanations aim to describe the behaviour of a model +by providing minimal changes to the input data that would +result in realistic samples that result in the model predicting +a different class. +For these perturbations to be more easily interpretable it is +desirable that they are sparse and that they can be identified +with some semantic element of the input data. In the case +of time series, this would require to perturb a short and +contiguous section of the timeline [9]. +Cassio F. Dantas and Dino Ienco are with UMR-TETIS laboratory, IN- +RAE, University of Montpellier, France (email: cassio.fraga-dantas@inrae.fr; +dino.ienco@inrae.fr). +Diego Marcos is with Inria, University of Montpellier, France (email: +diego.marcos@inria.fr) +Related work: Most papers on counterfactual explana- +tions focus on image data, while much fewer concentrate on +time series [9]–[15]. To the best of our knowledge, this is the +first paper focusing more specifically on counterfactuals for +remote sensing time series data. While [9], [10] also generate +time-contiguous perturbations, counterfactual plausibility is +achieved by replacing an interval of the time series by a portion +of another sample from the dataset [9] or shapelet motifs [10] +(also used in [12]). In contrast, we use an adversarial approach +to learn a counterfactual generator. In a multivariate setting, +the approach in [11] replaces entire variables (not just a time +section) with variables from another multivariate sample in +the dataset. Related adversarial approaches are proposed in +[13], [14], but time localization is not enforced. Finally, in +many existing approaches only the binary classification case +is considered [10], [14], [15], and when applied to the multi- +class case, it usually requires explicitly picking a target class +for every counterfactual explanation [11], [13]–[15]. +Contributions: Here, we propose a counterfactual genera- +tion approach in a multi-class land cover classification setting +for satellite image time series data. The proposed approach +generates counterfactual explanations that are plausible (i.e. +belong as much as possible to the data distribution) and close +to the original data (modifying only a limited and contiguous +set of time entries by a small amount). Finally, it is not +necessary to pre-determine a target class for the generated +counterfactual. +Paper outline: In Section II we describe the considered +study case with the associated remote sensing data. After +detailing the proposed method in Section III, we present the +experimental results in Section IV. Concluding remarks and +future works are outlined in Section V. +II. STUDY AREA +The study site covers an area around the town of Koumbia, +in the Province of Tuy, Hauts-Bassins region, in the south- +west of Burkina Faso. This area has a surface of about 2338 +km2, and is situated in the sub-humid sudanian zone. The +surface is covered mainly by natural savannah (herbaceous and +shrubby) and forests, interleaved with a large portion of land +(around 35%) used for rainfed agricultural production (mostly +smallholder farming). The main crops are cereals (maize, +sorghum and millet) and cotton, followed by oleaginous and +leguminous crops. Several temporary watercourses constitute +the hydrographic network around the city of Koumbia. Fig- +ure 1 presents the study site with the reference data (ground +truth) superposed on a Sentinel-2 image. +arXiv:2301.01520v1 [cs.LG] 4 Jan 2023 + +2 +Fig. 1: Location of the Koumbia study site. The corresponding +ground truth is shown on the right. +Fig. 2: Acquisition dates of the Sentinel-2 Satellite Image Time +Series on the year 2020. +Concerning the satellite data, we collected a time series +of Sentinel-2 images spanning the year 2020 from January +to December. All images were provided by the THEIA Pole +platform1 at level-2A, which consist of atmospherically cor- +rected surface reflectances (cf. MAJA processing chain [16]) +and relative cloud/shadow masks. A standard pre-processing +was performed over each band to replace cloudy pixel values +as detected by the available cloud masks based on the method +proposed in [17]. Figure 2 depicts the acquisition dates of the +Sentinel-2 satellite image time series. Finally, from the spectral +raw bands at 10-m of spatial resolution the NDVI (Normalized +Differential Vegetation Index) was derived. +The GT (ground truth) data for the study site is a collection +of (i) digitized plots from a GPS field mission performed in +October 2020 and mostly covering classes within cropland and +(ii) additional reference plots on non-crop classes obtained by +photo-interpretation by an expert. Finally, the polygons have +been rasterized at the S2 spatial resolution (10-m), resulting +in 79961 labeled pixels. The statistics related to the GT are +reported in Table I. +Class +Label +Pixels +1 +Cereals +9 731 +2 +Cotton +6 971 +3 +Oleaginous +7 950 +4 +Grassland +12 998 +5 +Shrubland +22 546 +6 +Forest +17 435 +7 +Bare Soil/Built-up +1 125 +8 +Water +1 205 +Total +79 961 +TABLE I: Koumbia study site Ground Truth statistics. +Classi er +Real +Counterfactual +(frozen) +Noiser +Class A +Class B +Discriminator +Fig. 3: Schematic representation of the proposed approach. +III. PROPOSED METHOD +A. Architecture overview +For the counterfactual generation, we propose a GAN +(generative adversarial network) inspired architecture which +is summarized in Fig. 3. +A counterfactual xCF is obtained for each input sample x +by adding a perturbation δ to the original signal: +xCF = x + δ +(1) +The perturbation δ is generated by a Noiser module which is +learned with the goal to swap the prediction of the Classifier. +Finally, a Discriminator module is leveraged to ensure the +generation of realistic counterfactual examples. +B. Networks implementation and training +Regarding the different components on which our frame- +work is built on, we get inspiration by state of the art +literature in the field of satellite image time series land cover +mapping. For the Classifier network we leverage the Temporal +Convolutional Neural Network (TempCNN) model proposed +in [18]. This architecture has an encoder based on several +one-dimensional convolutional layers to explicitly cope with +the temporal dimension of the time series data followed by +two fully connected layers and a final output layer to provide +the multi-class decision. +For the Discriminator network we adopt the same archi- +tecture as the Classifier network and we replace the output +layer with a single neuron with sigmoid activation function +as commonly done for discriminator networks in adversarial +learning [19]. +Concerning the Noiser module, it is implemented as a multi- +layer perceptron network with two hidden layers (each with +128 neurons) and an output layer with the same dimensionality +of the time series data. For each of the hidden layers, batch +normalization, tangent activation function and a drop-out reg- +ularization are employed in this order while for the output +layer only the tangent activation function is used. The tangent +activation function allows us to restrict the output domain +between -1 and +1 thus, facilitating the learning process of +the different networks. +The Classifier model is pre-trained on the training set and, +successively, frozen during the adversarial learning stage since +this stage is devoted to learn the model weights associated to +the Noiser and the Discriminator (see section III-D). +1http://theia.cnes.fr + +Legend: +000000 +Cereals +Cotton +Oleag./Legum +Grassland +Shrubland +Forest +B. Soil/Built-up +WaterDD +DD +DDD +B +2020-01 +2020-03 +2020-05 +2020-07 +2020-09 +2020-11 +2021-013 +The Noiser module is updated with respect to a composite +loss made of three parts detailed in sections III-C to III-E. +Lnoiser = Lcl + λgenLgen + λw-ℓ1Lw-ℓ1 +(2) +C. Class-swapping loss +To generate counterfactuals that effectively change the pre- +dicted class for a given input we use the following loss: +Lcl = − 1 +n +n +� +i=1 +y(i) log(1 − p(y(i))) +(3) +It enforces the reduction of the classifier’s softmax output for +the original label y(i), here denoted p(y(i)), eventually leading +to a change on the predicted class. +Note that, conversely to standard literature [13], [15] in +which a target class for the counterfactual example is chosen +a priori, here we purposely do not enforce the prediction of +a predefined target class. Instead, we let the Noiser free to +generate a perturbation δ that will change the classifier output +to any other class different from yi. +D. GAN-based regularization for plausibility +Counterfactual plausibility is enforced via a GAN-inspired +architecture, where a discriminator is trained to identify unreal- +istic counterfactuals while, simultaneously, the Noiser module +acts as a generator with the goal to fool the discriminator in +a two player game. +The Discriminator is updated with respect to a standard +GAN loss classifying real versus fake (counterfactual) sam- +ples: +Ldsc = − 1 +n +n +� +i=1 +� +log D(x(i)) + log +� +1 − D(x(i) +CF) +�� +(4) +where D(x(i)) denotes the discriminator’s output for a real +input x(i) (with expected output 1) and D(x(i) +CF) its output for +a fake input x(i) +CF (with expected output 0). +The following non-saturating generator loss is used in the +Noiser update: +Lgen = − 1 +n +n +� +i=1 +log +� +D(x(i) +CF) +� +(5) +Lgen is minimized when the discriminator wrongly identifies +the counterfactuals as real inputs. +E. Unimodal regularization for time-contiguity +To generate perturbations concentrated around a contiguous +time frame we employ a weighted L1-norm penalization, +with weights growing quadratically around a central time ˜t(i) +chosen independently for each sample i ∈ {1, . . . , n}: +Lw-ℓ1 = 1 +n +n +� +i=1 +T +� +t=1 +d(t, ˜t(i))2|δ(i) +t | +(6) +where, for the i-th sample, ˜t(i) is chosen as the time step with +the highest absolute value perturbation ˜t(i) = argmaxt |δ(i) +t |. +To avoid biasing ˜t towards the center, we use the modulo +distance d(t, ˜t) = min +� +(t − ˜t)%T, (˜t − t)%T +� +which treats +the time samples as a circular list. +This regularization also brings a degree of sparsity to the +generated perturbation δ, since its entries will tend to vanish +when getting far away from ˜t. Finally, penalizing the entries +of δ enforces the proximity (similarity) between xCF and x. +IV. RESULTS +In this section we inspect the behaviour of the proposed +method considering the study case introduced in Section II. +More precisely, we first provide a general analysis of the class +transitions induced by the counterfactual generation process. +Secondly, we discuss per-class average perturbations generated +by our framework as well as specific counterfactual examples. +Then, we assess the plausibility of the generated counterfactual +examples via anomaly detection strategies as suggested in [15]. +Finally, we perform an ablation analysis to assess the role of +the different loss functions involved in the learning process of +our framework. +A. Experimental setup +The Koumbia study case described in Section II was split +into training, validation and test sets containing respectively +50-17-33% of the 79961 samples. Each data sample cor- +responds to a (univariate) NDVI time series with 24 time +samples (cf. Fig. 2). +First, the Classifier was trained over 1000 epochs with batch +size 32 and Adam optimizer with learning rate 10−4 and +weight decay of same value. The model weights corresponding +to the best obtained F1-score on the validation set were kept. +Then, with the classifier weights frozen, the Noiser and +Discriminator modules are simultaneously trained over 100 +epochs with batch size 128 and Adam optimizer. +Regularization parameters: we set λgen = 5 · 10−1 and +λw-ℓ1 = 5 · 10−2 on the reported results. In practice, in- +creasing these weights implies in further constraining the +set of admissible perturbations which, in turn, leads to a +smaller rate of successful counterfactual samples –i.e., those +that actually change the classifier’s prediction (see details in +section IV-E). The chosen values lead to a success rate of +about 50%. Naturally, by further relaxing these constraints +(reducing λgen and λw-ℓ1) would lead to higher success rates, +but the generated counterfactual samples would be of lesser +quality in terms of plausibility (due to λgen) as well as time +localization and proximity (due to λw-ℓ1). +B. Visualizing class relationships +The class transitions induced by the counterfactual samples +are summarized in Fig. +4. The left (resp. right) graph was +generated by feeding the obtained network with each of the +training (resp. test) data samples. They present very similar +behavior, which attests the fact that the proposed method +generalizes well to previously unseen data. We recall that the +class transitions are to no extent pre-defined on our approach; +on the contrary, our method allows input samples from the + +4 +CEREALS +COTTON +OLEAGINOUS +GRASSLAND +SHRUBLAND +FOREST +B. +W. +CEREALS +COTTON +OLEAGINOUS +GRASSLAND +SHRUBLAND +FOREST +B. +W. +Fig. 4: Summary of class transitions induced by the counter- +factuals. Training data (left) and test data (right), where B. +stands for Bare Soil and W. for Water classes. +Fig. 5: Examples of average counterfactual perturbations be- +tween classes Cereals and Grassland on both ways. Shaded +area corresponds to the standard deviation. +same class to freely split-up into multiple target classes. +Transitions obtained in such a way thus bring up valuable +insights on the relation between classes. +The obtained transitions are very much in line with the +intuitive relation between the different classes. For instance, +the three crop-related classes (Cereals, Cotton and Oleaginous) +form a very coherent cluster, with almost all transitions staying +within the sub-group. The vegetation classes Shrubland and +Forest are most often sent to one another, while Grassland +remains much closer to the crop classes (especially Oleagi- +nous). The Bare Soil class is also most often transformed into +Oleaginous. Finally, the Water class is very rarely modified +by the counterfactual learning process, which is somewhat ex- +pected due to its very distinct characteristic (NDVI signature) +compared to the other classes. +The ratio of successful class-swapping counterfactual sam- +ples –i.e., those that actually change the classifier’s prediction– +was 52.7% (17947 over 34066) for the training data and 43.8% +(8765 over 20006) for the test data, considering only the +samples that were correctly classified before counterfactuals. +C. Counterfactual examples +Examples of average perturbation profiles for two different +class transitions are depicted in Fig 5. +It is interesting to notice how the perturbations correspond +roughly to the opposite of each other, which is quite suitable +since they correspond to opposite transitions between the same +two classes. +Fig. 6: Examples of original time series with corresponding +counterfactual from classes Shrubland (4) and Forest (5) on +both ways. +Two illustrative examples of counterfactual explanations are +shown in Fig. 6. It is interesting to observe the similarity +between the generated counterfactual and a real data example +from the same class (on the neighboring plot). +To transform a Shrubland sample into a Forest one, NDVI is +added between the months of July and October. The opposite +is done to obtain the reverse transition, which matches the +general knowledge of such land cover classes on the consid- +ered study area. Also note that the NDVI peak is slight shifted +from one class to another. +From the provided examples, one can verify that the ob- +tained counterfactual do look realistic (this aspect is further +evaluated in section IV-D) besides differing from the real +signal only on a contiguous time window. These two properties +have been explicitly enforced via the losses in eqs. (5) and (6). +D. Plausibility analysis +In this section, we quantify to what extent the proposed +counterfactual explanations fit the original data distribution. +To do so, we run an anomaly detection method, Isolation +Forest [20], on both the original data and corresponding +counterfactuals. To attest the importance of the proposed +adversarial training for the generation of realistic/plausible +counterfactuals, we perform an ablation study confronting +the proposed model trained with and without the generator +loss in Eq. (5). Fig. 7 shows contingency matrices relating +the isolation forest outputs on the original data (rows) and +on the corresponding counterfactual explanations (columns). +Two counterfactual generation approaches are investigated: the +proposed method (left matrix) and its non-adversarial variant +(right matrix). In the figures, diagonal entries correspond +to matching isolation forest outputs –i.e., same prediction +(inlier/outlier) for both real and counterfactual data. Later, +in Table II we compute some metrics on such contingency +matrices to further quantify and summarize the behaviour of +the compared methods. The proposed counterfactual model +achieves impressive results, even leading to more samples +identified as inliers than the real data itself (23806 against +23755), since proposed approach converts less inliers into +outliers (164) than the other way around (215). +The non-adversarial variant, on the other hand, obtains +considerably more degraded results, as it converts as many +as 4338 real inlier samples into outliers (about 20 times +more). Such a gap becomes evident when looking at the + +Cereals -→Grassland(876 CFs) +0.3 +0.2 +0.1 +0.0 +0.1 +-0.20.9 +0.8 +0.7 - +NDVI +0.6 +0.5 +0.4 +Real (Shrubland) +CF (Forest) +0.3 +20- +-03 +2020- +2020- +-09 +2020- +-01 +2020- +2020- +2021Grassland-→Cereals(1394CFs) +0.1 +0.0 +0.1- +0.2 +心0.9 +0.8 - +0.7 - +0.6 +0.5 +0.4 - +Real (Forest) +0.3 +CF (Shrubland) +2020- +2020- +-09 +-03 +2020- +2020- +2020- +2021-5 +Inlier +Outlier +Counterfactual +Inlier +Outlier +Real +99.3% +(23591) +0.7% +(164) +7.1% +(215) +92.9% +(2820) +Proposed model +Inlier +Outlier +Counterfactual +Inlier +Outlier +Real +81.7% +(19417) +18.3% +(4338) +1.2% +(35) +98.8% +(3000) +Non-adversarial +Fig. 7: Isolation forest results on real (rows) and counterfactual +data (columns). Proposed model with (left) and without (right) +adversarial loss during training. Row-normalized percentages. +corresponding accuracy and normalized mutual information +(NMI) computed w.r.t. the isolation forest results on the +original data (cf. Table II). Such scores measure to what degree +the inlier/outlier partitioning obtained on the counterfactual +samples (for each of the two compared variants) matches the +one obtained on the original data. The higher they are the +better the two partitions match. The obtained results clearly +show that counterfactual plausibility is achieved thanks to the +adversarial training process. +Method +Accuracy +NMI +Inliers ratio +Proposed +98.6% +0.808 +88.9% +Non-adversarial +83.7% +0.337 +72.6% +TABLE II: Plausibility analysis using different performance +metrics. Isolation Forest results on the real data were used as +ground truth for the accuracy and NMI scores. +E. Other ablation studies +In Table III we compare the number of successful class- +swapping counterfactual samples as well as the average ℓ2 +and ℓ1 norms of the perturbations δ generated by the proposed +model and two variants ignoring the generator loss (Lgen) and +the weighted-ℓ1 loss (Lw-ℓ1), respectively. +One can see that the removal of the auxiliary losses signif- +icantly bumps the class-swapping rate, but it happens at the +expense of either: 1) counterfactual plausibility, as shown in +the Section IV-D for the removal of Lgen; 2) counterfactual +proximity/similarity, as demonstrated by the dramatic increase +on the norm of the generated perturbations (or, equivalently, +the distance between x and xCF) upon removal of Lw-ℓ1. +Method +Class-swap CF +Average ∥δ∥2 +Average ∥δ∥1 +Proposed +43.8% +0.24 ± 0.18 +0.76 ± 0.54 +Without Lgen +83.7% +0.97 ± 0.47 +1.69 ± 0.99 +Without Lw-ℓ1 +99.6% +4.79 ± 0.07 +23.3 ± 0.53 +TABLE III: Ablation study on test data. +V. CONCLUSION +In this letter we have presented a new framework to generate +counterfactual SITS samples of vegetation indices (i.e. NDVI) +for the land cover classification task. The proposed method +overcomes the restriction to apriori define the source and the +target classes for the counterfactual generation process while +it exploits adversarial learning to ensure realistic counterfac- +tual samples. As possible future work, we would extend the +framework to the case of multivariate time series satellite data +as well as leverage the feedback provided by the generated +counterfactual samples to improve the robustness of the land +cover classifier regarding the most frequent class confusions. +REFERENCES +[1] Q. Yuan, H. Shen, T. Li, Z. Li, S. Li, Y. Jiang, H. Xu, W. Tan, Q. Yang, +J. Wang, J. Gao, and L. Zhang, “Deep learning in environmental remote +sensing: Achievements and challenges,” Remote Sensing of Environment, +vol. 241, p. 111716, 2020. +[2] J. Inglada, A. Vincent, M. 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Rudin, “Deep learning for case- +based reasoning through prototypes: A neural network that explains its +predictions,” AAAI Conference on Artificial Intelligence, vol. 32, no. 1, +Apr. 2018. + diff --git a/ItAzT4oBgHgl3EQfjv0x/content/tmp_files/load_file.txt b/ItAzT4oBgHgl3EQfjv0x/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f9e0d7efa3b8dbbecc280dd18a0b82f4299189e7 --- /dev/null +++ b/ItAzT4oBgHgl3EQfjv0x/content/tmp_files/load_file.txt @@ -0,0 +1,435 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf,len=434 +page_content='1 Counterfactual Explanations for Land Cover Mapping in a Multi-class Setting Cassio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Dantas, Diego Marcos, Dino Ienco Abstract—Counterfactual explanations are an emerging tool to enhance interpretability of deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Given a sample, these methods seek to find and display to the user similar samples across the decision boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In this paper, we propose a generative adversarial counterfactual approach for satellite image time series in a multi-class setting for the land cover classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' One of the distinctive features of the proposed approach is the lack of prior assumption on the targeted class for a given counterfactual explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' This inherent flexibility allows for the discovery of interesting information on the relationship between land cover classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The other feature consists of encouraging the counterfactual to differ from the original sample only in a small and compact temporal segment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' These time-contiguous perturba- tions allow for a much sparser and, thus, interpretable solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Furthermore, plausibility/realism of the generated counterfactual explanations is enforced via the proposed adversarial learning strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' INTRODUCTION Deep learning techniques have gained widespread popu- larity in the remote sensing field due to impressive results on a variety of tasks such as image super-resolution, image restoration, biophysical variables estimation and land cover classification from satellite image time series (SITS) data [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Of particular importance, this last task provides useful knowl- edge to support many downstream geospatial analyses [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Despite the high performances achieved by recent deep learn- ing frameworks on this task, they remain black-box models with limited understanding on their internal behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Due to this limitation, there is a growing need for improving the interpretability of deep learning models in remote sensing with the objective to raise up their acceptability and usefulness, as their decision-making processes are often not transparent [3]– [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Counterfactual explanation methods have recently received increasing attention as a means to provide some level of interpretability [6]–[8] to these black-box models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Counter- factual explanations aim to describe the behaviour of a model by providing minimal changes to the input data that would result in realistic samples that result in the model predicting a different class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' For these perturbations to be more easily interpretable it is desirable that they are sparse and that they can be identified with some semantic element of the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In the case of time series, this would require to perturb a short and contiguous section of the timeline [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Cassio F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Dantas and Dino Ienco are with UMR-TETIS laboratory, IN- RAE, University of Montpellier, France (email: cassio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='fraga-dantas@inrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='fr;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' dino.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='ienco@inrae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='fr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Diego Marcos is with Inria, University of Montpellier, France (email: diego.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='marcos@inria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='fr) Related work: Most papers on counterfactual explana- tions focus on image data, while much fewer concentrate on time series [9]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' To the best of our knowledge, this is the first paper focusing more specifically on counterfactuals for remote sensing time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' While [9], [10] also generate time-contiguous perturbations, counterfactual plausibility is achieved by replacing an interval of the time series by a portion of another sample from the dataset [9] or shapelet motifs [10] (also used in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In contrast, we use an adversarial approach to learn a counterfactual generator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In a multivariate setting, the approach in [11] replaces entire variables (not just a time section) with variables from another multivariate sample in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Related adversarial approaches are proposed in [13], [14], but time localization is not enforced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, in many existing approaches only the binary classification case is considered [10], [14], [15], and when applied to the multi- class case, it usually requires explicitly picking a target class for every counterfactual explanation [11], [13]–[15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Contributions: Here, we propose a counterfactual genera- tion approach in a multi-class land cover classification setting for satellite image time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The proposed approach generates counterfactual explanations that are plausible (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' belong as much as possible to the data distribution) and close to the original data (modifying only a limited and contiguous set of time entries by a small amount).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, it is not necessary to pre-determine a target class for the generated counterfactual.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Paper outline: In Section II we describe the considered study case with the associated remote sensing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' After detailing the proposed method in Section III, we present the experimental results in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Concluding remarks and future works are outlined in Section V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' STUDY AREA The study site covers an area around the town of Koumbia, in the Province of Tuy, Hauts-Bassins region, in the south- west of Burkina Faso.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' This area has a surface of about 2338 km2, and is situated in the sub-humid sudanian zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The surface is covered mainly by natural savannah (herbaceous and shrubby) and forests, interleaved with a large portion of land (around 35%) used for rainfed agricultural production (mostly smallholder farming).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The main crops are cereals (maize, sorghum and millet) and cotton, followed by oleaginous and leguminous crops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Several temporary watercourses constitute the hydrographic network around the city of Koumbia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig- ure 1 presents the study site with the reference data (ground truth) superposed on a Sentinel-2 image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='01520v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='LG] 4 Jan 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 1: Location of the Koumbia study site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The corresponding ground truth is shown on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 2: Acquisition dates of the Sentinel-2 Satellite Image Time Series on the year 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Concerning the satellite data, we collected a time series of Sentinel-2 images spanning the year 2020 from January to December.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' All images were provided by the THEIA Pole platform1 at level-2A, which consist of atmospherically cor- rected surface reflectances (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' MAJA processing chain [16]) and relative cloud/shadow masks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' A standard pre-processing was performed over each band to replace cloudy pixel values as detected by the available cloud masks based on the method proposed in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Figure 2 depicts the acquisition dates of the Sentinel-2 satellite image time series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, from the spectral raw bands at 10-m of spatial resolution the NDVI (Normalized Differential Vegetation Index) was derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The GT (ground truth) data for the study site is a collection of (i) digitized plots from a GPS field mission performed in October 2020 and mostly covering classes within cropland and (ii) additional reference plots on non-crop classes obtained by photo-interpretation by an expert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, the polygons have been rasterized at the S2 spatial resolution (10-m), resulting in 79961 labeled pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The statistics related to the GT are reported in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Class Label Pixels 1 Cereals 9 731 2 Cotton 6 971 3 Oleaginous 7 950 4 Grassland 12 998 5 Shrubland 22 546 6 Forest 17 435 7 Bare Soil/Built-up 1 125 8 Water 1 205 Total 79 961 TABLE I: Koumbia study site Ground Truth statistics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Classi er Real Counterfactual (frozen) Noiser Class A Class B Discriminator Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 3: Schematic representation of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' PROPOSED METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Architecture overview For the counterfactual generation, we propose a GAN (generative adversarial network) inspired architecture which is summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' A counterfactual xCF is obtained for each input sample x by adding a perturbation δ to the original signal: xCF = x + δ (1) The perturbation δ is generated by a Noiser module which is learned with the goal to swap the prediction of the Classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, a Discriminator module is leveraged to ensure the generation of realistic counterfactual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Networks implementation and training Regarding the different components on which our frame- work is built on, we get inspiration by state of the art literature in the field of satellite image time series land cover mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' For the Classifier network we leverage the Temporal Convolutional Neural Network (TempCNN) model proposed in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' This architecture has an encoder based on several one-dimensional convolutional layers to explicitly cope with the temporal dimension of the time series data followed by two fully connected layers and a final output layer to provide the multi-class decision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' For the Discriminator network we adopt the same archi- tecture as the Classifier network and we replace the output layer with a single neuron with sigmoid activation function as commonly done for discriminator networks in adversarial learning [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Concerning the Noiser module, it is implemented as a multi- layer perceptron network with two hidden layers (each with 128 neurons) and an output layer with the same dimensionality of the time series data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' For each of the hidden layers, batch normalization, tangent activation function and a drop-out reg- ularization are employed in this order while for the output layer only the tangent activation function is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The tangent activation function allows us to restrict the output domain between -1 and +1 thus, facilitating the learning process of the different networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The Classifier model is pre-trained on the training set and, successively, frozen during the adversarial learning stage since this stage is devoted to learn the model weights associated to the Noiser and the Discriminator (see section III-D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 1http://theia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='cnes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='fr Legend: 000000 Cereals Cotton Oleag.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='/Legum Grassland Shrubland Forest B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Soil/Built-up WaterDD DD DDD B 2020-01 2020-03 2020-05 2020-07 2020-09 2020-11 2021-013 The Noiser module is updated with respect to a composite loss made of three parts detailed in sections III-C to III-E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Lnoiser = Lcl + λgenLgen + λw-ℓ1Lw-ℓ1 (2) C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Class-swapping loss To generate counterfactuals that effectively change the pre- dicted class for a given input we use the following loss: Lcl = − 1 n n � i=1 y(i) log(1 − p(y(i))) (3) It enforces the reduction of the classifier’s softmax output for the original label y(i), here denoted p(y(i)), eventually leading to a change on the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Note that, conversely to standard literature [13], [15] in which a target class for the counterfactual example is chosen a priori, here we purposely do not enforce the prediction of a predefined target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Instead, we let the Noiser free to generate a perturbation δ that will change the classifier output to any other class different from yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' GAN-based regularization for plausibility Counterfactual plausibility is enforced via a GAN-inspired architecture, where a discriminator is trained to identify unreal- istic counterfactuals while, simultaneously, the Noiser module acts as a generator with the goal to fool the discriminator in a two player game.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The Discriminator is updated with respect to a standard GAN loss classifying real versus fake (counterfactual) sam- ples: Ldsc = − 1 n n � i=1 � log D(x(i)) + log � 1 − D(x(i) CF) �� (4) where D(x(i)) denotes the discriminator’s output for a real input x(i) (with expected output 1) and D(x(i) CF) its output for a fake input x(i) CF (with expected output 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The following non-saturating generator loss is used in the Noiser update: Lgen = − 1 n n � i=1 log � D(x(i) CF) � (5) Lgen is minimized when the discriminator wrongly identifies the counterfactuals as real inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Unimodal regularization for time-contiguity To generate perturbations concentrated around a contiguous time frame we employ a weighted L1-norm penalization, with weights growing quadratically around a central time ˜t(i) chosen independently for each sample i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' , n}: Lw-ℓ1 = 1 n n � i=1 T � t=1 d(t, ˜t(i))2|δ(i) t | (6) where, for the i-th sample, ˜t(i) is chosen as the time step with the highest absolute value perturbation ˜t(i) = argmaxt |δ(i) t |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' To avoid biasing ˜t towards the center, we use the modulo distance d(t, ˜t) = min � (t − ˜t)%T, (˜t − t)%T � which treats the time samples as a circular list.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' This regularization also brings a degree of sparsity to the generated perturbation δ, since its entries will tend to vanish when getting far away from ˜t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, penalizing the entries of δ enforces the proximity (similarity) between xCF and x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' RESULTS In this section we inspect the behaviour of the proposed method considering the study case introduced in Section II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' More precisely, we first provide a general analysis of the class transitions induced by the counterfactual generation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Secondly, we discuss per-class average perturbations generated by our framework as well as specific counterfactual examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Then, we assess the plausibility of the generated counterfactual examples via anomaly detection strategies as suggested in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, we perform an ablation analysis to assess the role of the different loss functions involved in the learning process of our framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Experimental setup The Koumbia study case described in Section II was split into training, validation and test sets containing respectively 50-17-33% of the 79961 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Each data sample cor- responds to a (univariate) NDVI time series with 24 time samples (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' First, the Classifier was trained over 1000 epochs with batch size 32 and Adam optimizer with learning rate 10−4 and weight decay of same value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The model weights corresponding to the best obtained F1-score on the validation set were kept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Then, with the classifier weights frozen, the Noiser and Discriminator modules are simultaneously trained over 100 epochs with batch size 128 and Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Regularization parameters: we set λgen = 5 · 10−1 and λw-ℓ1 = 5 · 10−2 on the reported results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In practice, in- creasing these weights implies in further constraining the set of admissible perturbations which, in turn, leads to a smaller rate of successful counterfactual samples –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=', those that actually change the classifier’s prediction (see details in section IV-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The chosen values lead to a success rate of about 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Naturally, by further relaxing these constraints (reducing λgen and λw-ℓ1) would lead to higher success rates, but the generated counterfactual samples would be of lesser quality in terms of plausibility (due to λgen) as well as time localization and proximity (due to λw-ℓ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Visualizing class relationships The class transitions induced by the counterfactual samples are summarized in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The left (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' right) graph was generated by feeding the obtained network with each of the training (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' test) data samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' They present very similar behavior, which attests the fact that the proposed method generalizes well to previously unseen data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' We recall that the class transitions are to no extent pre-defined on our approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' on the contrary, our method allows input samples from the 4 CEREALS COTTON OLEAGINOUS GRASSLAND SHRUBLAND FOREST B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' CEREALS COTTON OLEAGINOUS GRASSLAND SHRUBLAND FOREST B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 4: Summary of class transitions induced by the counter- factuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Training data (left) and test data (right), where B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' stands for Bare Soil and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' for Water classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 5: Examples of average counterfactual perturbations be- tween classes Cereals and Grassland on both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Shaded area corresponds to the standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' same class to freely split-up into multiple target classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Transitions obtained in such a way thus bring up valuable insights on the relation between classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The obtained transitions are very much in line with the intuitive relation between the different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' For instance, the three crop-related classes (Cereals, Cotton and Oleaginous) form a very coherent cluster, with almost all transitions staying within the sub-group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The vegetation classes Shrubland and Forest are most often sent to one another, while Grassland remains much closer to the crop classes (especially Oleagi- nous).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The Bare Soil class is also most often transformed into Oleaginous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Finally, the Water class is very rarely modified by the counterfactual learning process, which is somewhat ex- pected due to its very distinct characteristic (NDVI signature) compared to the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The ratio of successful class-swapping counterfactual sam- ples –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=', those that actually change the classifier’s prediction– was 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7% (17947 over 34066) for the training data and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='8% (8765 over 20006) for the test data, considering only the samples that were correctly classified before counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Counterfactual examples Examples of average perturbation profiles for two different class transitions are depicted in Fig 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' It is interesting to notice how the perturbations correspond roughly to the opposite of each other, which is quite suitable since they correspond to opposite transitions between the same two classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 6: Examples of original time series with corresponding counterfactual from classes Shrubland (4) and Forest (5) on both ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Two illustrative examples of counterfactual explanations are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' It is interesting to observe the similarity between the generated counterfactual and a real data example from the same class (on the neighboring plot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' To transform a Shrubland sample into a Forest one, NDVI is added between the months of July and October.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The opposite is done to obtain the reverse transition, which matches the general knowledge of such land cover classes on the consid- ered study area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Also note that the NDVI peak is slight shifted from one class to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' From the provided examples, one can verify that the ob- tained counterfactual do look realistic (this aspect is further evaluated in section IV-D) besides differing from the real signal only on a contiguous time window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' These two properties have been explicitly enforced via the losses in eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' (5) and (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Plausibility analysis In this section, we quantify to what extent the proposed counterfactual explanations fit the original data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' To do so, we run an anomaly detection method, Isolation Forest [20], on both the original data and corresponding counterfactuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' To attest the importance of the proposed adversarial training for the generation of realistic/plausible counterfactuals, we perform an ablation study confronting the proposed model trained with and without the generator loss in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 7 shows contingency matrices relating the isolation forest outputs on the original data (rows) and on the corresponding counterfactual explanations (columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Two counterfactual generation approaches are investigated: the proposed method (left matrix) and its non-adversarial variant (right matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' In the figures, diagonal entries correspond to matching isolation forest outputs –i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=', same prediction (inlier/outlier) for both real and counterfactual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Later, in Table II we compute some metrics on such contingency matrices to further quantify and summarize the behaviour of the compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The proposed counterfactual model achieves impressive results, even leading to more samples identified as inliers than the real data itself (23806 against 23755), since proposed approach converts less inliers into outliers (164) than the other way around (215).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The non-adversarial variant, on the other hand, obtains considerably more degraded results, as it converts as many as 4338 real inlier samples into outliers (about 20 times more).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Such a gap becomes evident when looking at the Cereals -→Grassland(876 CFs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7 - NDVI 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='4 Real (Shrubland) CF (Forest) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3 20- 03 2020- 2020- 09 2020- 01 2020- 2020- 2021Grassland-→Cereals(1394CFs) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='1- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='2 心0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='4 - Real (Forest) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3 CF (Shrubland) 2020- 2020- 09 03 2020- 2020- 2020- 2021-5 Inlier Outlier Counterfactual Inlier Outlier Real 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3% (23591) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7% (164) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='1% (215) 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='9% (2820) Proposed model Inlier Outlier Counterfactual Inlier Outlier Real 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7% (19417) 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3% (4338) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='2% (35) 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='8% (3000) Non-adversarial Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 7: Isolation forest results on real (rows) and counterfactual data (columns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Proposed model with (left) and without (right) adversarial loss during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Row-normalized percentages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' corresponding accuracy and normalized mutual information (NMI) computed w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' the isolation forest results on the original data (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Table II).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Such scores measure to what degree the inlier/outlier partitioning obtained on the counterfactual samples (for each of the two compared variants) matches the one obtained on the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The higher they are the better the two partitions match.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The obtained results clearly show that counterfactual plausibility is achieved thanks to the adversarial training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Method Accuracy NMI Inliers ratio Proposed 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='6% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='808 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='9% Non-adversarial 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='337 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='6% TABLE II: Plausibility analysis using different performance metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Isolation Forest results on the real data were used as ground truth for the accuracy and NMI scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Other ablation studies In Table III we compare the number of successful class- swapping counterfactual samples as well as the average ℓ2 and ℓ1 norms of the perturbations δ generated by the proposed model and two variants ignoring the generator loss (Lgen) and the weighted-ℓ1 loss (Lw-ℓ1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' One can see that the removal of the auxiliary losses signif- icantly bumps the class-swapping rate, but it happens at the expense of either: 1) counterfactual plausibility, as shown in the Section IV-D for the removal of Lgen;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' 2) counterfactual proximity/similarity, as demonstrated by the dramatic increase on the norm of the generated perturbations (or, equivalently, the distance between x and xCF) upon removal of Lw-ℓ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' Method Class-swap CF Average ∥δ∥2 Average ∥δ∥1 Proposed 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='8% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='54 Without Lgen 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='7% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='47 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='69 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='99 Without Lw-ℓ1 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='6% 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='79 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='07 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='53 TABLE III: Ablation study on test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' CONCLUSION In this letter we have presented a new framework to generate counterfactual SITS samples of vegetation indices (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' NDVI) for the land cover classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' The proposed method overcomes the restriction to apriori define the source and the target classes for the counterfactual generation process while it exploits adversarial learning to ensure realistic counterfac- tual samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' As possible future work, we would extend the framework to the case of multivariate time series satellite data as well as leverage the feedback provided by the generated counterfactual samples to improve the robustness of the land cover classifier regarding the most frequent class confusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ItAzT4oBgHgl3EQfjv0x/content/2301.01520v1.pdf'} +page_content=' REFERENCES [1] Q.' metadata={'source': 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b/KNE1T4oBgHgl3EQfYgRo/content/tmp_files/2301.03139v1.pdf.txt @@ -0,0 +1,2425 @@ +arXiv:2301.03139v1 [math.OC] 9 Jan 2023 +A Newton-CG based augmented Lagrangian method for finding a +second-order stationary point of nonconvex equality constrained +optimization with complexity guarantees +Chuan He∗ +Zhaosong Lu∗ +Ting Kei Pong† +April 10, 2022 (Revised: September 22, 2022; December 31, 2022) +Abstract +In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality con- +strained optimization when a nearly feasible point is known. In particular, we first propose a new Newton- +CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a +substantially better complexity than the Newton-CG method [56]. We then propose a Newton-CG based +augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained +optimization, in which the proposed Newton-CG method is used as a subproblem solver. We show that +under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total +inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding +an (ǫ, √ǫ)-SOSP of nonconvex equality constrained optimization with high probability, which are signif- +icantly better than the ones achieved by the proximal AL method [60]. Besides, we show that it has +a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) +when the GLICQ does not hold. To the best of our knowledge, all the complexity results obtained in +this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization +with high probability. Preliminary numerical results also demonstrate the superiority of our proposed +methods over the ones in [56, 60]. +Keywords: Nonconvex equality constrained optimization, second-order stationary point, augmented Lagrangian +method, Newton-conjugate gradient method, iteration complexity, operation complexity +Mathematics Subject Classification: 49M15, 68Q25, 90C06, 90C26, 90C30, 90C60 +1 +Introduction +In this paper we consider nonconvex equality constrained optimization problem +min +x∈Rn f(x) +s. t. c(x) = 0, +(1) +where f : Rn → R and c : Rn → Rm are twice continuously differentiable, and we assume that problem (1) +has at least one optimal solution. Since (1) is a nonconvex optimization problem, it may have many local but +non-global minimizers and finding its global minimizer is generally NP-hard. A first-order stationary point +(FOSP) of it is usually found in practice instead. Nevertheless, a mere FOSP may sometimes not suit our +needs and a second-order stationary point (SOSP) needs to be sought. For example, in the context of linear +semidefinite programming (SDP), a powerful approach to solving it is by solving an equivalent nonconvex +∗Department +of +Industrial +and +Systems +Engineering, +University +of +Minnesota, +USA +(email: +he000233@umn.edu, +zhaosong@umn.edu). The work of the second author was partially supported by NSF Award IIS-2211491. +†Department of Applied Mathematics, the Hong Kong Polytechnic University, Hong Kong, People’s Republic of China +(email: tk.pong@polyu.edu.hk). The work of this author was partially supported by a Research Scheme of the Research Grants +Council of Hong Kong SAR, China (Project No. T22-504/21R). +1 + +equality constrained optimization problem [17, 18]. It was shown in [18, 15] that under some mild conditions +an SOSP of the latter problem can yield an optimal solution of the linear SDP, while a mere FOSP generally +cannot. It is therefore important to find an SOSP of problem (1). +In recent years, numerous methods with complexity guarantees have been developed for finding an ap- +proximate SOSP of several types of nonconvex optimization. For example, cubic regularized Newton methods +[52, 25, 1, 22], accelerated gradient methods [23, 24], trust-region methods [34, 35, 50], quadratic regulariza- +tion method [12], second-order line-search method [57], and Newton-conjugate gradient (Newton-CG) method +[56] were developed for nonconvex unconstrained optimization. In addition, interior-point method [8] and +log-barrier method [54] were proposed for nonconvex optimization with sign constraints. The interior-point +method [8] was also generalized in [38] to solve nonconvex optimization with sign constraints and additional +linear equality constraints. Furthermore, a projected gradient descent method with random perturbations +was proposed in [47] for nonconvex optimization with linear inequality constraints. Iteration complexity was +established for these methods for finding an approximate SOSP. Besides, operation complexity measured +by the amount of fundamental operations such as gradient evaluations and matrix-vector products was also +studied in [1, 23, 34, 41, 24, 57, 22, 56]. +Several methods including trust-region methods [21, 33], sequential quadratic programming method [14], +two-phase method [9, 30, 32] and augmented Lagrangian (AL) type methods [4, 10, 58, 60] were proposed +for finding an SOSP of problem (1). However, only a few of them have complexity guarantees for finding +an approximate SOSP of (1). In particular, the inexact AL method [58] has a worst-case complexity in +terms of the number of calls to a second-order oracle. Yet its operation complexity, measured by the amount +of fundamental operations such as gradient evaluations and Hessian-vector products, is unknown. To the +best of our knowledge, the proximal AL method in [60] appears to be the only existing method that enjoys +a worst-case complexity for finding an approximate SOSP of (1) in terms of fundamental operations. In +this method, given an iterate xk and a multiplier estimate λk at the kth iteration, the next iterate xk+1 is +obtained by finding an approximate stochastic SOSP of the proximal AL subproblem: +min +x∈Rn L(x, λk; ρ) + β∥x − xk∥2/2 +for some suitable positive ρ and β using a Newton-CG method proposed in [56], where L is the AL function +of (1) defined as +L(x, λ; ρ) := f(x) + λT c(x) + ρ∥c(x)∥2/2. +Then the multiplier estimate is updated using the classical scheme, i.e., λk+1 = λk + ρc(xk+1) (e.g., see +[39, 55]). The authors of [60] studied the worst-case complexity of their proximal AL method including: (i) +total inner iteration complexity, which measures the total number of iterations of the Newton-CG method [56] +performed in their method; (ii) operation complexity, which measures the total number of gradient evaluations +and matrix-vector products involving the Hessian of the AL function that are evaluated in their method. +Under some suitable assumptions, including that a generalized linear independence constraint qualification +(GLICQ) holds at all iterates, it was established in [60] that their proximal AL method enjoys a total inner +iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−3/4}) for finding an +(ǫ, √ǫ)-SOSP of problem (1) with high probability.1 Yet, there is a big gap between these complexities and the +iteration complexity of �O(ǫ−3/2) and the operation complexity of �O(ǫ−3/2 min{n, ǫ−1/4}) that are achieved +by the methods in [1, 24, 57, 56] for finding an (ǫ, √ǫ)-SOSP of nonconvex unconstrained optimization with +high probability, which is a special case of (1) with c ≡ 0. Also, there is a lack of complexity guarantees for +this proximal AL method when the GLICQ does not hold. It shall be mentioned that Newton-CG based AL +methods were also developed for efficiently solving various convex optimization problems (e.g., see [61, 62]), +though their complexities remain unknown. +In this paper we propose a Newton-CG based AL method for finding an approximate SOSP of problem (1) +with high probability, and study its worst-case complexity with and without the assumption of a GLICQ. In +1In fact, a total inner iteration complexity of �O(ǫ−7) and an operation complexity of �O(ǫ−7 min{n, ǫ−1}) were established +in [60] for finding an (ǫ, ǫ)-SOSP of problem (1) with high probability; see [60, Theorem 4(ii), Corollary 3(ii), Theorem 5]. +Nonetheless, they can be modified to obtain the aforementioned complexity for finding an (ǫ, √ǫ)-SOSP of (1) with high +probability. +2 + +particular, we show that this method enjoys a total inner iteration complexity of �O(ǫ−7/2) and an operation +complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding a stochastic (ǫ, √ǫ)-SOSP of (1) under the GLICQ, which +are significantly better than the aforementioned ones achieved by the proximal AL method in [60]. Besides, +when the GLICQ does not hold, we show that it has a total inner iteration complexity of �O(ǫ−11/2) and +an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding a stochastic (ǫ, √ǫ)-SOSP of (1), which fills +the research gap in this topic. Specifically, our AL method (Algorithm 2) proceeds in the following manner. +Instead of directly solving problem (1), it solves a perturbed problem of (1) with c replaced by its perturbed +counterpart ˜c constructed by using a nearly feasible point of (1) (see (25) for details). At the kth iteration, +an approximate stochastic SOSP xk+1 of the AL subproblem of this perturbed problem is found by our +newly proposed Newton-CG method (Algorithm 1) for a penalty parameter ρk and a truncated Lagrangian +multiplier λk, which results from projecting onto a Euclidean ball the standard multiplier estimate ˜λk +obtained by the classical scheme ˜λk = λk−1 + ρk˜c(xk).2 The penalty parameter ρk+1 is then updated by the +following practical scheme (e.g., see [7, Section 4.2]): +ρk+1 = +� rρk +if ∥˜c(xk+1)∥ > α∥˜c(xk)∥, +ρk +otherwise +for some r > 1 and α ∈ (0, 1). It shall be mentioned that in contrast with the classical AL method, our +method has two distinct features: (i) the values of the AL function along the iterates are bounded from above; +(ii) the multiplier estimates associated with the AL subproblems are bounded. In addition, to solve the AL +subproblems with better complexity guarantees, we propose a variant of the Newton-CG method in [56] for +finding an approximate stochastic SOSP of unconstrained optimization, whose complexity has significantly +less dependence on the Lipschitz constant of the Hessian of the objective than that of the Newton-CG method +in [56], while improving or retaining the same order of dependence on tolerance parameter. Given that such +a Lipschitz constant is typically large for the AL subproblems, our Newton-CG method (Algorithm 1) is a +much more favorable subproblem solver than the Newton-CG method in [56] that is used in the proximal +AL method in [60] from theoretical complexity perspective. +The main contributions of this paper are summarized below. +• We propose a new Newton-CG method for finding an approximate SOSP of unconstrained optimization +and show that it enjoys an iteration and operation complexity with a quadratic dependence on the +Lipschitz constant of the Hessian of the objective that improves the cubic dependence achieved by the +Newton-CG method in [56], while improving or retaining the same order of dependence on tolerance +parameter. In addition, our complexity results are established under the assumption that the Hessian +of the objective is Lipschitz continuous in a convex neighborhood of a level set of the objective. This +assumption is weaker than the one commonly imposed for the Newton-CG method in [56] and some +other methods (e.g., [12, 35]) that the Hessian of the objective is Lipschitz continuous in a convex set +containing this neighborhood and also all the trial points arising in the line search or trust region steps +of the methods (see Section 3 for more detailed discussion). +• We propose a Newton-CG based AL method for finding an approximate SOSP of nonconvex equality +constrained optimization (1) with high probability, and study its worst-case complexity with and +without the assumption of a GLICQ. Prior to our work, there was no complexity study on finding +an approximate SOSP of problem (1) without imposing a GLICQ. Besides, under the GLICQ and +some other suitable assumptions, we show that our method enjoys a total inner iteration complexity +of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding an (ǫ, √ǫ)-SOSP of (1) +with high probability, which are significantly better than the respective complexity of �O(ǫ−11/2) and +�O(ǫ−11/2 min{n, ǫ−3/4}) achieved by the proximal AL method in [60]. To the best of our knowledge, all +the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex +equality constrained optimization with high probability. +2The λk obtained by projecting ˜λk onto a compact set is also called a safeguarded Lagrangian multiplier in the relevant +literature [11, 42, 13], which has been shown to enjoy many practical and theoretical advantages (see [11] for discussions). +3 + +For ease of comparison, we summarize in Table 1 the total inner iteration and operation complexity of +our AL method and the proximal AL method in [60] for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) +with or without assuming GLICQ. +Table 1: Total inner iteration and operation complexity of finding a stochastic (ǫ, √ǫ)-SOSP of (1). +Method +GLICQ +Total inner iteration complexity +Operation complexity +Proximal AL method [60] +✓ +�O(ǫ−11/2) +�O(ǫ−11/2 min{n, ǫ−3/4}) +Proximal AL method [60] +✗ +unknown +unknown +Our AL method +✓ +�O(ǫ−7/2) +�O(ǫ−7/2 min{n, ǫ−3/4}) +Our AL method +✗ +�O(ǫ−11/2) +�O(ǫ−11/2 min{n, ǫ−5/4}) +It shall be mentioned that there are many works other than [60] studying complexity of AL methods for +nonconvex constrained optimization. However, they aim to find an approximate FOSP rather than SOSP +of the problem (e.g., see [40, 37, 13, 51, 45]). +Since our main focus is on the complexity of finding an +approximate SOSP by AL methods, we do not include them in the above table for comparison. +The rest of this paper is organized as follows. In Section 2, we introduce some notation and optimality +conditions. In Section 3, we propose a Newton-CG method for unconstrained optimization and study its +worst-case complexity. +In Section 4, we propose a Newton-CG based AL method for (1) and study its +worst-case complexity. We present numerical results and the proof of the main results in Sections 5 and 6, +respectively. In Section 7, we discuss some future research directions. +2 +Notation and preliminaries +Throughout this paper, we let Rn denote the n-dimensional Euclidean space. We use ∥ · ∥ to denote the +Euclidean norm of a vector or the spectral norm of a matrix. For a real symmetric matrix H, we use λmin(H) +to denote its minimum eigenvalue. The Euclidean ball centered at the origin with radius R ≥ 0 is denoted +by BR := {x : ∥x∥ ≤ R}, and we use ΠBR(v) to denote the Euclidean projection of a vector v onto BR. For +a given finite set A, we let | A | denote its cardinality. For any s ∈ R, we let sgn(s) be 1 if s ≥ 0 and let it +be −1 otherwise. In addition, �O(·) represents O(·) with logarithmic terms omitted. +Suppose that x∗ is a local minimizer of problem (1) and the linear independence constraint qualification +holds at x∗, i.e., ∇c(x∗) := [∇c1(x∗) ∇c2(x∗) · · · ∇cm(x∗)] has full column rank. Then there exists a +Lagrangian multiplier λ∗ ∈ Rm such that +∇f(x∗) + ∇c(x∗)λ∗ = 0, +(2) +dT +� +∇2f(x∗) + +m +� +i=1 +λ∗ +i ∇2ci(x∗) +� +d ≥ 0, +∀d ∈ C(x∗), +(3) +where C(·) is defined as +C(x) := {d ∈ Rn : ∇c(x)T d = 0}. +(4) +The relations (2) and (3) are respectively known as the first- and second-order optimality conditions for (1) +in the literature (e.g., see [53]). Note that it is in general impossible to find a point that exactly satisfies (2) +and (3). Thus, we are instead interested in finding a point that satisfies their approximate counterparts. In +particular, we introduce the following definitions of an approximate first-order stationary point (FOSP) and +second-order stationary point (SOSP), which are similar to those considered in [4, 10, 60]. The rationality +of them can be justified by the study of the sequential optimality conditions for constrained optimization +[3, 4]. +Definition 2.1 (ǫ1-first-order stationary point). Let ǫ1 > 0. We say that x ∈ Rn is an ǫ1-first-order +stationary point (ǫ1-FOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies +∥∇f(x) + ∇c(x)λ∥ ≤ ǫ1, +∥c(x)∥ ≤ ǫ1. +(5) +4 + +Definition 2.2 ((ǫ1, ǫ2)-second-order stationary point). Let ǫ1, ǫ2 > 0. We say that x ∈ Rn is an (ǫ1, ǫ2)- +second-order stationary point ((ǫ1, ǫ2)-SOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies (5) +and additionally +dT +� +∇2f(x) + +m +� +i=1 +λi∇2ci(x) +� +d ≥ −ǫ2∥d∥2, +∀d ∈ C(x), +(6) +where C(·) is defined as in (4). +3 +A Newton-CG method for unconstrained optimization +In this section we propose a variant of Newton-CG method [56, Algorithm 3] for finding an approximate +SOSP of a class of unconstrained optimization problems, which will be used as a subproblem solver for the +AL method proposed in the next section. In particular, we consider an unconstrained optimization problem +min +x∈Rn F(x), +(7) +where the function F satisfies the following assumptions. +Assumption 3.1. (a) The level set LF (u0) := {x : F(x) ≤ F(u0)} is compact for some u0 ∈ Rn. +(b) The function F is twice Lipschitz continuously differentiable in a convex open neighborhood, denoted by +Ω, of LF (u0), that is, there exists LF +H > 0 such that +∥∇2F(x) − ∇2F(y)∥ ≤ LF +H∥x − y∥, +∀x, y ∈ Ω. +(8) +By Assumption 3.1, there exist Flow ∈ R, U F +g > 0 and U F +H > 0 such that +F(x) ≥ Flow, +∥∇F(x)∥ ≤ U F +g , +∥∇2F(x)∥ ≤ U F +H, +∀x ∈ LF(u0). +(9) +Recently, a Newton-CG method [56, Algorithm 3] was developed to find an approximate stochastic SOSP +of problem (7), which is not only easy to implement but also enjoys a nice feature that the main computation +consists only of gradient evaluations and Hessian-vector products associated with the function F. Under +the assumption that ∇2F is Lipschitz continuous in a convex open set containing LF (u0) and also all the +trial points arising in the line search steps of this method (see [56, Assumption 2]), it was established in [56, +Theorem 4, Corollary 2] that the iteration and operation complexity of this method for finding a stochastic +(ǫg, ǫH)-SOSP of (7) (namely, a point x satisfying ∥∇F(x)∥ ≤ ǫg deterministically and λmin(∇2F(x)) ≥ −ǫH +with high probability) are +O((LF +H)3 max{ǫ−3 +g ǫ3 +H, ǫ−3 +H }) +and +�O((LF +H)3 max{ǫ−3 +g ǫ3 +H, ǫ−3 +H } min{n, (U F +H/ǫH)1/2}), +(10) +respectively, where ǫg, ǫH ∈ (0, 1) are prescribed tolerances. +Yet, this assumption can be hard to check +because these trial points are unknown before the method terminates and moreover the distance between +the origin and them depends on the tolerance ǫH in O(ǫ−1 +H ) (see [56, Lemma 3]). +In addition, as seen +from (10), iteration and operation complexity of the Newton-CG method in [56] depend cubically on LF +H. +Notice that LF +H can sometimes be very large. For example, the AL subproblems arising in Algorithm 2 have +LF +H = O(ǫ−2 +1 ) or O(ǫ−1 +1 ), where ǫ1 ∈ (0, 1) is a prescribed tolerance for problem (1) (see Section 4). The +cubic dependence on LF +H makes such a Newton-CG method not appealing as an AL subproblem solver from +theoretical complexity perspective. +In the rest of this section, we propose a variant of the Newton-CG method [56, Algorithm 3] and show +that under Assumption 3.1, it enjoys an iteration and operation complexity of +O((LF +H)2 max{ǫ−2 +g ǫH, ǫ−3 +H }) +and +�O((LF +H)2 max{ǫ−2 +g ǫH, ǫ−3 +H } min{n, (U F +H/ǫH)1/2}), +(11) +for finding a stochastic (ǫg, ǫH)-SOSP of problem (7), respectively. These complexities are substantially +superior to those in (10) achieved by the Newton-CG method in [56]. +Indeed, the complexities in (11) +depend quadratically on LF +H, while those in (10) depend cubically on LF +H. In addition, it can be verified that +they improve or retain the order of dependence on ǫg and ǫH given in (10). +5 + +3.1 +Main components of a Newton-CG method +In this subsection we briefly discuss two main components of the Newton-CG method in [56], which will be +used to propose a variant of this method for finding an approximate stochastic SOSP of problem (7) in the +next subsection. +The first main component of the Newton-CG method in [56] is a capped CG method [56, Algorithm 1], +which is a modified CG method, for solving a possibly indefinite linear system +(H + 2εI)d = −g, +(12) +where 0 ̸= g ∈ Rn, ε > 0, and H ∈ Rn×n is a symmetric matrix. This capped CG method terminates within a +finite number of iterations. It outputs either an approximate solution d to (12) such that ∥(H +2εI)d+g∥ ≤ +�ζ∥g∥ and dT Hd ≥ −ε∥d∥2 for some �ζ ∈ (0, 1) or a sufficiently negative curvature direction d of H with +dT Hd < −ε∥d∥2. The second main component of the Newton-CG method in [56] is a minimum eigenvalue +oracle that either produces a sufficiently negative curvature direction v of H with ∥v∥ = 1 and vT Hv ≤ −ε/2 +or certifies that λmin(H) ≥ −ε holds with high probability. For ease of reference, we present these two +components in Algorithms 3 and 4 in Appendices A and B, respectively. +Algorithm 1 A Newton-CG method for problem (7) +Input: Tolerances ǫg, ǫH ∈ (0, 1), backtracking ratio θ ∈ (0, 1), starting point u0, CG-accuracy parameter ζ ∈ (0, 1), line- +search parameter η ∈ (0, 1), probability parameter δ ∈ (0, 1). +Set x0 = u0; +for t = 0, 1, 2, . . . do +if ∥∇F (xt)∥ > ǫg then +Call Algorithm 3 with H = ∇2F (xt), ε = ǫH, g = ∇F (xt), accuracy parameter ζ, and U = 0 to obtain outputs d, +d type; +if d type=NC then +dt ← − sgn(dT ∇F (xt))|dT ∇2F (xt)d| +∥d∥3 +d; +(13) +else {d type=SOL} +dt ← d; +(14) +end if +Go to Line Search; +else +Call Algorithm 4 with H = ∇2F (xt), ε = ǫH, and probability parameter δ; +if Algorithm 4 certifies that λmin(∇2F (xt)) ≥ −ǫH then +Output xt and terminate; +else {Sufficiently negative curvature direction v returned by Algorithm 4} +Set d type=NC and +dt ← − sgn(vT ∇F (xt))|vT ∇2F (xt)v|v; +(15) +Go to Line Search; +end if +end if +Line Search: +if d type=SOL then +Find αt = θjt, where jt is the smallest nonnegative integer j such that +F (xt + θjdt) < F (xt) − ηǫHθ2j∥dt∥2; +(16) +else {d type=NC} +Find αt = θjt, where jt is the smallest nonnegative integer j such that +F (xt + θjdt) < F (xt) − ηθ2j∥dt∥3/2; +(17) +end if +xt+1 = xt + αtdt; +end for +3.2 +A Newton-CG method for problem (7) +In this subsection we propose a Newton-CG method in Algorithm 1, which is a variant of the Newton-CG +method [56, Algorithm 3], for finding an approximate stochastic SOSP of problem (7). +6 + +Our Newton-CG method (Algorithm 1) follows the same framework as [56, Algorithm 3]. In particular, +at each iteration, if the gradient of F at the current iterate is not desirably small, then the capped CG +method (Algorithm 3) is called to solve a damped Newton system for obtaining a descent direction and a +subsequent line search along this direction results in a sufficient reduction on F. Otherwise, the current +iterate is already an approximate first-order stationary point of (7), and the minimum eigenvalue oracle +(Algorithm 4) is then called, which either produces a sufficiently negative curvature direction for F and a +subsequent line search along this direction results in a sufficient reduction on F, or certifies that the current +iterate is an approximate SOSP of (7) with high probability and terminates the algorithm. More details +about this framework can be found in [56]. +Despite sharing the same framework, our Newton-CG method and [56, Algorithm 3] use different line +search criteria. Indeed, our Newton-CG method uses a hybrid line search criterion adopted from [59], which +is a combination of the quadratic descent criterion (16) and the cubic descent criterion (17). Specifically, it +uses the quadratic descent criterion (16) when the search direction is of type ‘SOL’. On the other hand, it +uses the cubic descent criterion (17) when the search direction is of type ‘NC’.3 In contrast, the Newton-CG +method in [56] always uses a cubic descent criterion regardless of the type of search directions. As observed +from Theorem 3.1 below, our Newton-CG method achieves an iteration and operation complexity given in +(11), which are superior to those in (10) achieved by [56, Algorithm 3] in terms of the order dependence +on LF +H, while improving or retaining the order of dependence on ǫg and ǫH as given in (10). Consequently, +our Newton-CG method is more appealing than [56, Algorithm 3] as an AL subproblem solver for the AL +method proposed in Section 4 from theoretical complexity perspective. +The following theorem states the iteration and operation complexity of Algorithm 1, whose proof is +deferred to Section 6.1. +Theorem 3.1. Suppose that Assumption 3.1 holds. Let +T1 := +� Fhi − Flow +min{csol, cnc} max{ǫ−2 +g ǫH, ǫ−3 +H } +� ++ +�Fhi − Flow +cnc +ǫ−3 +H +� ++ 1, T2 := +�Fhi − Flow +cnc +ǫ−3 +H +� ++ 1, +(18) +where Fhi = F(u0), Flow is given in (9), and +csol := η min + + + + + + + +4 +4 + ζ + +� +(4 + ζ)2 + 8LF +H + + +2 +, +�min{6(1 − η), 2}θ +LF +H +�2 + + + + + +, +(19) +cnc := η +16 min +� +1, +�min{3(1 − η), 1}θ +LF +H +�2� +. +(20) +Then the following statements hold. +(i) The total number of calls of Algorithm 4 in Algorithm 1 is at most T2. +(ii) The total number of calls of Algorithm 3 in Algorithm 1 is at most T1. +(iii) (iteration complexity) Algorithm 1 terminates in at most T1 + T2 iterations with +T1 + T2 = O((Fhi − Flow)(LF +H)2 max{ǫ−2 +g ǫH, ǫ−3 +H }). +(21) +Also, its output xt satisfies ∥∇F(xt)∥ ≤ ǫg deterministically and λmin(∇2F(xt)) ≥ −ǫH with probability +at least 1 − δ for some 0 ≤ t ≤ T1 + T2. +(iv) (operation complexity) Algorithm 1 requires at most +�O((Fhi − Flow)(LF +H)2 max{ǫ−2 +g ǫH, ǫ−3 +H } min{n, (U F +H/ǫH)1/2}) +matrix-vector products, where U F +H is given in (9). +3SOL and NC stand for “approximate solution” and “negative curvature”, respectively. +7 + +4 +A Newton-CG based AL method for problem (1) +In this section we propose a Newton-CG based AL method for finding a stochastic (ǫ1, ǫ2)-SOSP of problem +(1) for any prescribed tolerances ǫ1, ǫ2 ∈ (0, 1). Before proceeding, we make some additional assumptions on +problem (1). +Assumption 4.1. (a) An ǫ1/2-approximately feasible point zǫ1 of problem (1), namely satisfying ∥c(zǫ1)∥ ≤ +ǫ1/2, is known. +(b) There exist constants fhi, flow and γ > 0, independent of ǫ1 and ǫ2, such that +f(zǫ1) ≤ fhi, +(22) +f(x) + γ∥c(x)∥2/2 ≥ flow, +∀x ∈ Rn, +(23) +where zǫ1 is given in (a). +(c) There exist some δf, δc > 0 such that the set +S(δf, δc) := {x : f(x) ≤ fhi + δf, ∥c(x)∥ ≤ 1 + δc} +(24) +is compact with fhi given above. Also, ∇2f and ∇2ci, i = 1, 2, . . ., m, are Lipschitz continuous in a +convex open neighborhood, denoted by Ω(δf, δc), of S(δf, δc). +We now make some remarks on Assumption 4.1. +Remark 4.1. +(i) A very similar assumption as Assumption 4.1(a) was considered in [31, 37, 49, 60]. By +imposing Assumption 4.1(a), we restrict our study on problem (1) for which an ǫ1/2-approximately fea- +sible point zǫ1 can be found by an inexpensive procedure. One example of such problem instances arises +when there exists v0 such that {x : ∥c(x)∥ ≤ ∥c(v0)∥} is compact, ∇2ci, 1 ≤ i ≤ m, is Lipschitz contin- +uous on a convex neighborhood of this set, and the LICQ holds on this set. Indeed, for this instance, a +point zǫ1 satisfying ∥c(zǫ1)∥ ≤ ǫ1/2 can be computed by applying our Newton-CG method (Algorithm 1) +to the problem minx∈Rn ∥c(x)∥2. As seen from Theorem 3.1, the resulting iteration and operation com- +plexity of Algorithm 1 for finding such zǫ1 are respectively O(ǫ−3/2 +1 +) and �O(ǫ−3/2 +1 +min{n, ǫ−1/4 +1 +}), which +are negligible compared with those of our AL method (see Theorems 4.4 and 4.5 below). As another +example, when the standard error bound condition ∥c(x)∥2 = O(∥∇(∥c(x)∥2)∥ν) holds on a level set +of ∥c(x)∥ for some ν > 0, one can find the above zǫ1 by applying a gradient method to the problem +minx∈Rn ∥c(x)∥2 (e.g., see [46, 58]). In addition, the Newton-CG based AL method (Algorithm 2) pro- +posed below is a second-order method with the aim to find a second-order stationary point. It is more +expensive than a first-order method in general. To make best use of such an AL method in practice, +it is natural to run a first-order method in advance to obtain an ǫ1/2-first-order stationary point zǫ1 +and then run the AL method using zǫ1 as an ǫ1/2-approximately feasible point. Therefore, Assump- +tion 4.1(a) is met in practice, provided that an ǫ1/2-first-order stationary point of (1) can be found by +a first-order method. +(ii) Assumption 4.1(b) is mild. In particular, the assumption in (22) holds if f(x) ≤ fhi holds for all x with +∥c(x)∥ ≤ 1, which is imposed in [60, Assumption 3]. It also holds if problem (1) has a known feasible +point, which is often imposed for designing AL methods for nonconvex constrained optimization (e.g., +see [49, 31, 48, 37]). Besides, the assumption in (23) implies that the quadratic penalty function is +bounded below when the associated penalty parameter is sufficiently large, which is typically used in the +study of quadratic penalty and AL methods for solving problem (1) (e.g., see [40, 37, 60, 43]). Clearly, +when infx∈Rn f(x) > −∞, one can see that (23) holds for any γ > 0. In general, one possible approach +to identifying γ is to apply the techniques on infeasibility detection developed in the literature (e.g., +[20, 19, 6]) to check the infeasibility of the level set {x : f(x)+γ∥c(x)∥2/2 ≤ ˜flow} for some sufficiently +small ˜flow. Note that this level set being infeasible for some ˜flow implies that (23) holds for the given +γ and flow = ˜flow. +8 + +(iii) Assumption 4.1(c) is not too restrictive. Indeed, the set S(δf, δc) is compact if f or f(·)+γ∥c(·)∥2/2 is +level-bounded. The latter level-boundedness assumption is commonly imposed for studying AL methods +(e.g., see [37, 60]), which is stronger than our assumption. +We next propose a Newton-CG based AL method in Algorithm 2 for finding a stochastic (ǫ1, ǫ2)-SOSP +of problem (1) under Assumption 4.1. Instead of solving (1) directly, this method solves the perturbed +problem: +min +x∈Rn f(x) +s. t. ˜c(x) := c(x) − c(zǫ1) = 0, +(25) +where zǫ1 is given in Assumption 4.1(a). Specifically, at the kth iteration, this method applies the Newton- +CG method (Algorithm 1) to find an approximate stochastic SOSP xk+1 of the AL subproblem associated +with (25): +min +x∈Rn +��L(x, λk, ρk) := f(x) + (λk)T ˜c(x) + ρk∥˜c(x)∥2/2 +� +(26) +such that �L(xk+1, λk; ρk) is below a threshold (see (27) and (28)), where λk is a truncated Lagrangian +multiplier, i.e., the one that results from projecting the standard multiplier estimate ˜λk onto an Euclidean +ball (see step 6 of Algorithm 2). The standard multiplier estimate ˜λk+1 is then updated by the classical +scheme described in step 4 of Algorithm 2. Finally, the penalty parameter ρk+1 is adaptively updated based +on the improvement on constraint violation (see step 7 of Algorithm 2). Such a practical update scheme is +often adopted in the literature (e.g., see [7, 2, 31]). +We would like to point out that the truncated Lagrangian multiplier sequence {λk} is used in the AL +subproblems of Algorithm 2 and is bounded, while the standard Lagrangian multiplier sequence {˜λk} is used +in those of the classical AL methods and can be unbounded. Therefore, Algorithm 2 can be viewed as a +safeguarded AL method. Truncated Lagrangian multipliers have been used in the literature for designing +some AL methods [2, 11, 42, 13], and will play a crucial role in the subsequent complexity analysis of +Algorithm 2. +Algorithm 2 A Newton-CG based AL method for problem (1) +Let γ be given in Assumption 4.1. +Input: ǫ1, ǫ2 ∈ (0, 1), Λ > 0, x0 ∈ Rn, λ0 ∈ BΛ, ρ0 > 2γ, α ∈ (0, 1), r > 1, δ ∈ (0, 1), and zǫ1 given in Assumption 4.1. +1: Set k = 0. +2: Set τ g +k = max{ǫ1, rk log ǫ1/ log 2} and τ H +k = max{ǫ2, rk log ǫ2/ log 2}. +3: Call Algorithm 1 with ǫg += τ g +k, ǫH += τ H +k +and u0 += xk +init to find an approximate solution xk+1 to +minx∈Rn �L(x, λk; ρk) such that +�L(xk+1, λk; ρk) ≤ f(zǫ1), ∥∇x�L(xk+1, λk; ρk)∥ ≤ τ g +k , +(27) +λmin(∇2 +xx�L(xk+1, λk; ρk)) ≥ −τ H +k with probability at least 1 − δ, +(28) +where +xk +init = +� +zǫ1 +if �L(xk, λk; ρk) > f(zǫ1), +xk +otherwise, +for k ≥ 0. +(29) +4: Set ˜λk+1 = λk + ρk˜c(xk+1). +5: If τ g +k ≤ ǫ1, τ H +k ≤ ǫ2 and ∥c(xk+1)∥ ≤ ǫ1, then output (xk+1, ˜λk+1) and terminate. +6: Set λk+1 = ΠBΛ(˜λk+1). +7: If k = 0 or ∥˜c(xk+1)∥ > α∥˜c(xk)∥, set ρk+1 = rρk. Otherwise, set ρk+1 = ρk. +8: Set k ← k + 1, and go to step 2. +Remark 4.2. +(i) Notice that the starting point x0 +init of Algorithm 2 can be different from zǫ1 and it may be +rather infeasible, though zǫ1 is a nearly feasible point of (1). Besides, zǫ1 is used to ensure convergence +of Algorithm 2. Specifically, if the algorithm runs into a “poorly infeasible point” xk, namely satisfying +�L(xk, λk; ρk) > f(zǫ1), it will be superseded by zǫ1 (see (29)), which prevents the iterates {xk} from +converging to an infeasible point. Yet, xk may be rather infeasible when k is not large. Thus, Algorithm +2 substantially differs from a funneling or two-phase type algorithm, in which a nearly feasible point +9 + +is found in Phase 1, and then approximate stationarity is sought while near feasibility is maintained +throughout Phase 2 (e.g., see [9, 16, 26, 27, 28, 29, 30, 36]). +(ii) The choice of ρ0 in Algorithm 2 is mainly for the simplicity of complexity analysis. Yet, it may be overly +large and lead to highly ill-conditioned AL subproblems in practice. To make Algorithm 2 practically +more efficient, one can possibly modify it by choosing a relatively small initial penalty parameter, then +solving the subsequent AL subproblems by a first-order method until an ǫ1-first-order stationary point +ˆx of (1) along with a Lagrangian multiplier ˆλ is found, and finally performing the steps described in +Algorithm 2 but with x0 = ˆx and λ0 = ΠBΛ(ˆλ). +Before analyzing the complexity of Algorithm 2, we first argue that it is well-defined if ρ0 is suitably +chosen. Specifically, we will show that when ρ0 is sufficiently large, one can apply the Newton-CG method +(Algorithm 1) to the AL subproblem minx∈Rn �L(x, λk; ρk) with xk +init as the initial point to find an xk+1 +satisfying (27) and (28). To this end, we start by noting from (22), (25), (26) and (29) that +�L(xk +init, λk; ρk) ≤ max{�L(zǫ1, λk; ρk), f(zǫ1)} = f(zǫ1) ≤ fhi. +(30) +Based on the above observation, we show in the next lemma that when ρ0 is sufficiently large, �L(·, λk; ρk) is +bounded below and its certain level set is bounded, whose proof is deferred to Section 6.2. +Lemma 4.1. Suppose that Assumption 4.1 holds. Let (λk, ρk) be generated at the kth iteration of Algorithm 2 +for some k ≥ 0, and S(δf, δc) and xk +init be defined in (24) and (29), respectively, and let fhi, flow, δf and δc +be given in Assumption 4.1. Suppose that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where +δf,1 := Λ2/(2ρ0) +and +δc,1 := +� +2(fhi − flow + γ) +ρ0 − 2γ ++ +Λ2 +(ρ0 − 2γ)2 + +Λ +ρ0 − 2γ . +(31) +Then the following statements hold. +(i) {x : �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk)} ⊆ S(δf, δc). +(ii) infx∈Rn �L(x, λk; ρk) ≥ flow − γ − Λδc. +Using Lemma 4.1, we can verify that the Newton-CG method (Algorithm 1), starting with u0 = xk +init, +is capable of finding an approximate solution xk+1 of the AL subproblem minx∈Rn �L(x, λk; ρk) satisfying +(27) and (28). +Indeed, let F(·) = �L(·, λk; ρk) and u0 = xk +init. +By these and Lemma 4.1, one can see +that {x : F(x) ≤ F(u0)} ⊆ S(δf, δc). +It then follows from this and Assumption 4.1(c) that the level +set {x : F(x) ≤ F(u0)} is compact and ∇2F is Lipschitz continuous on a convex open neighborhood of +{x : F(x) ≤ F(u0)}. Thus, such F and u0 satisfy Assumption 3.1. Based on this and the discussion in +Section 3, one can conclude that Algorithm 1, starting with u0 = xk +init, is applicable to the AL subproblem +minx∈Rn �L(x, λk; ρk). Moreover, it follows from Theorem 3.1 that this algorithm with (ǫg, ǫH) = (τ g +k , τ H +k ) can +produce a point xk+1 satisfying (28) and also the second relation in (27). In addition, since this algorithm is +descent and its starting point is xk +init, its output xk+1 must satisfy �L(xk+1, λk; ρk) ≤ �L(xk +init, λk; ρk), which +along with (30) implies that �L(xk+1, λk; ρk) ≤ f(zǫ1) and thus xk+1 also satisfies the first relation in (27). +The above discussion leads to the following conclusion concerning the well-definedness of Algorithm 2. +Theorem 4.1. Under the same settings as in Lemma 4.1, the Newton-CG method (Algorithm 1) applied to +the AL subproblem minx∈Rn �L(x, λk; ρk) with u0 = xk +init finds a point xk+1 satisfying (27) and (28). +The following theorem characterizes the output of Algorithm 2. Its proof is deferred to Section 6.2. +Theorem 4.2. Suppose that Assumption 4.1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and +δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). If Algorithm 2 terminates at some iteration k, then xk+1 +is a deterministic ǫ1-FOSP of problem (1), and moreover, it is an (ǫ1, ǫ2)-SOSP of (1) with probability at +least 1 − δ. +10 + +Remark 4.3. As seen from this theorem, the output of Algorithm 2 is a stochastic (ǫ1, ǫ2)-SOSP of prob- +lem (1). Nevertheless, one can easily modify Algorithm 2 to seek some other approximate solutions. For +example, if one is only interested in finding an ǫ1-FOSP of (1), one can remove the condition (28) from +Algorithm 2. In addition, if one aims to find a deterministic (ǫ1, ǫ2)-SOSP of (1), one can replace the condi- +tion (28) and Algorithm 1 by λmin(∇2 +xx�L(xk+1, λk; ρk)) ≥ −τ H +k and a deterministic counterpart, respectively. +The purpose of imposing high probability in the condition (28) is to enable us to derive operation complexity +of Algorithm 2 measured by the number of matrix-vector products. +In the rest of this section, we study the worst-case complexity of Algorithm 2. Since our method has +two nested loops, particularly, outer loops executed by the AL method and inner loops executed by the +Newton-CG method for solving the AL subproblems, we consider the following measures of complexity for +Algorithm 2. +• Outer iteration complexity, which measures the number of outer iterations of Algorithm 2; +• Total inner iteration complexity, which measures the total number of iterations of the Newton-CG +method that are performed in Algorithm 2; +• Operation complexity, which measures the total number of matrix-vector products involving the Hessian +of the augmented Lagrangian function that are evaluated in Algorithm 2. +4.1 +Outer iteration complexity of Algorithm 2 +In this subsection we establish outer iteration complexity of Algorithm 2. For notational convenience, we +rewrite (τ g +k , τ H +k ) arising in Algorithm 2 as +(τ g +k , τ H +k ) = (max{ǫ1, ωk +1}, max{ǫ2, ωk +2}) with (ω1, ω2) := (rlog ǫ1/ log 2, rlog ǫ2/ log 2), +(32) +where ǫ1, ǫ2 and r are the input parameters of Algorithm 2. Since r > 1 and ǫ1, ǫ2 ∈ (0, 1), it is not hard to +verify that ω1, ω2 ∈ (0, 1). Also, we introduce the following quantity that will be used frequently later: +Kǫ1 := +� +min{k ≥ 0 : ωk +1 ≤ ǫ1} +� += ⌈log ǫ1/ log ω1⌉ . +(33) +In view of (32), (33) and the fact that +log ǫ1/ log ω1 = log ǫ2/ log ω2 = log 2/ log r, +(34) +we see that (τ g +k , τ H +k ) = (ǫ1, ǫ2) for all k ≥ Kǫ1. This along with the termination criterion of Algorithm 2 +implies that it runs for at least Kǫ1 iterations and terminates once ∥c(xk+1)∥ ≤ ǫ1 for some k ≥ Kǫ1. As +a result, to establish outer iteration complexity of Algorithm 2, it suffices to bound such k. The resulting +outer iteration complexity of Algorithm 2 is presented below, whose proof is deferred to Section 6.2. +Theorem 4.3. Suppose that Assumption 4.1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and +δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). Let +ρǫ1 := max +� +8(fhi − flow + γ)ǫ−2 +1 ++ 4Λǫ−1 +1 ++ 2γ, 2ρ0 +� +, +(35) +Kǫ1 := inf{k ≥ Kǫ1 : ∥c(xk+1)∥ ≤ ǫ1}, +(36) +where Kǫ1 is defined in (33), and γ, fhi and flow are given in Assumption 4.1. Then Kǫ1 is finite, and +Algorithm 2 terminates at iteration Kǫ1 with +Kǫ1 ≤ +�log(ρǫ1ρ−1 +0 ) +log r ++ 1 +� ����� +log(ǫ1(2δc,1)−1) +log α +���� + 2 +� ++ 1. +(37) +Moreover, ρk ≤ rρǫ1 holds for 0 ≤ k ≤ Kǫ1 +Remark 4.4 (Upper bounds for Kǫ1 and {ρk}). As observed from Theorem 4.3, the number of outer +iterations of Algorithm 2 for finding a stochastic (ǫ1, ǫ2)-SOSP of problem (1) is Kǫ1 + 1, which is at most +of O(| log ǫ1|2). In addition, the penalty parameters {ρk} generated in this algorithm are at most of O(ǫ−2 +1 ). +11 + +4.2 +Total inner iteration and operation complexity of Algorithm 2 +We present the total inner iteration and operation complexity of Algorithm 2 for finding a stochastic (ǫ1, ǫ2)- +SOSP of (1), whose proof is deferred to Section 6.2. +Theorem 4.4. Suppose that Assumption 4.1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and +δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). Then the following statements hold. +(i) The total number of iterations of Algorithm 1 performed in Algorithm 2 is at most �O(ǫ−4 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 }). +If c is further assumed to be affine, then it is at most �O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 }). +(ii) The total number of matrix-vector products performed by Algorithm 1 in Algorithm 2 is at most +�O(ǫ−4 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1 +1 ǫ−1/2 +2 +}). +If c is further assumed to be affine, then it is at most +�O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1 +1 ǫ−1/2 +2 +}). +Remark 4.5. +(i) Note that the above complexity results of Algorithm 2 are established without assuming +any constraint qualification (CQ). In contrast, similar complexity results are obtained in [60] for a +proximal AL method under a generalized LICQ condition. To the best of our knowledge, our work +provides the first study on complexity for finding a stochastic SOSP of (1) without CQ. +(ii) Letting (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), we see that Algorithm 2 achieves a total inner iteration +complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding a stochastic +(ǫ, √ǫ)-SOSP of problem (1) without constraint qualification. +4.3 +Enhanced complexity of Algorithm 2 under constraint qualification +In this subsection we study complexity of Algorithm 2 under one additional assumption that a generalized +linear independence constraint qualification (GLICQ) holds for problem (1), which is introduced below. In +particular, under GLICQ we will obtain an enhanced total inner iteration and operation complexity for +Algorithm 2, which are significantly better than the ones in Theorem 4.4 when problem (1) has nonlinear +constraints. Moreover, when (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), our enhanced complexity bounds are also +better than those obtained in [60] for a proximal AL method. We now introduce the GLICQ assumption for +problem (1). +Assumption 4.2 (GLICQ). ∇c(x) has full column rank for all x ∈ S(δf, δc), where S(δf, δc) is as in (24). +Remark 4.6. A related yet different GLICQ is imposed in [60, Assumption 2(ii)] for problem (1), which +assumes that ∇c(x) has full column rank for all x in a level set of f(·) + γ∥c(·)∥2/2. It is not hard to verify +that this assumption is generally stronger than the above GLICQ assumption. +The following theorem shows that under Assumption 4.2, the total inner iteration and operation com- +plexity results presented in Theorem 4.4 can be significantly improved, whose proof is deferred to Section +6.2. +Theorem 4.5. Suppose that Assumptions 4.1 and 4.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf +and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). Then the following statements hold. +(i) The total number of iterations of Algorithm 1 performed in Algorithm 2 is at most �O(ǫ−2 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 }). +If c is further assumed to be affine, then it is at most �O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 }). +(ii) The total number of matrix-vector products performed by Algorithm 1 in Algorithm 2 is at most +�O(ǫ−2 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1/2 +1 +ǫ−1/2 +2 +}). If c is further assumed to be affine, then it is at most +�O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1/2 +1 +ǫ−1/2 +2 +}). +Remark 4.7. +(i) As seen from Theorem 4.5, when problem (1) has nonlinear constraints, under GLICQ +and some other suitable assumptions, Algorithm 2 achieves significantly better complexity bounds than +the ones in Theorem 4.4 without constraint qualification. +12 + +(ii) Letting (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), we see that when problem (1) has nonlinear constraints, +under GLICQ and some other suitable assumptions, Algorithm 2 achieves a total inner iteration com- +plexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}). They are vastly better than +the total inner iteration complexity of �O(ǫ−11/2) and the operation complexity of �O(ǫ−11/2 min{n, ǫ−3/4}) +that are achieved by a proximal AL method in [60] for finding a stochastic (ǫ, √ǫ)-SOSP of (1) yet under +a generally stronger GLICQ. +5 +Numerical results +We conduct some preliminary experiments to test the performance of our proposed methods (Algorithms 1 +and 2), and compare them with the Newton-CG method in [56] and the proximal AL method in [60], +respectively. All the algorithms are coded in Matlab and all the computations are performed on a desktop +with a 3.79 GHz AMD 3900XT 12-Core processor and 32 GB of RAM. +5.1 +Regularized robust regression +In this subsection we consider the regularized robust regression problem +min +x∈Rn +m +� +i=1 +φ(aT +i x − bi) + µ∥x∥4 +4, +(38) +where φ(t) = t2/(1 + t2), ∥x∥p = (�n +i=1 |xi|p)1/p for any p ≥ 1, and µ > 0. +For each triple (n, m, µ), we randomly generate 10 instances of problem (38). In particular, we first +randomly generate ai, 1 ≤ i ≤ m, with all the entries independently chosen from the standard normal +distribution. We then randomly generate ¯bi according to the standard normal distribution and set bi = 2m¯bi +for i = 1, . . . , m. +Our aim is to find a (10−5, 10−5/2)-SOSP of (38) for the above instances by Algorithm 1 and the Newton- +CG method in [56] and compare their performance. For a fair comparison, we use a minimum eigenvalue +oracle that returns a deterministic output for them so that they both certainly output an approximate +second-order stationary point. Specifically, we use the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) as +the minimum eigenvalue oracle to find the minimum eigenvalue λ and its associated unit eigenvector v of a +real symmetric matrix H. Also, for both methods, we choose the all-ones vector as the initial point, and set +θ = 0.8, ζ = 0.5, and η = 0.2. +The computational results of Algorithm 1 and the Newton-CG method in [56] for the instances randomly +generated above are presented in Table 2. In detail, the value of n, m, and µ is listed in the first three +columns, respectively. For each triple (n, m, µ), the average CPU time (in seconds), the average number +of iterations, and the average final objective value over 10 random instances are given in the rest of the +columns. One can observe that both methods output an approximate solution with a similar objective value, +while our Algorithm 1 substantially outperforms the Newton-CG method in [56] in terms of CPU time. This +is consistent with our theoretical finding that Algorithm 1 achieves a better iteration complexity than the +Newton-CG method in [56] in terms of dependence on the Lipschitz constant of the Hessian for finding an +approximate SOSP. +5.2 +Spherically constrained regularized robust regression +In subsection we consider the spherically constrained regularized robust regression problem +min +x∈Rn +m +� +i=1 +φ(aT +i x − bi) + µ∥x∥4 +4 +s. t. +∥x∥2 +2 = 1, +(39) +where φ(t) = t2/(1 + t2), ∥x∥p = (�n +i=1 |xi|p)1/p for any p ≥ 1, and µ > 0 is a tuning parameter. For +each triple (n, m, µ), we randomly generate 10 instances of problem (39) in the same manner as described +in Subsection 5.1. +13 + +Objective value +Iterations +CPU time (seconds) +n +m +µ +Algorithm 1 +Newton-CG +Algorithm 1 +Newton-CG +Algorithm 1 +Newton-CG +100 +10 +1 +5.9 +5.9 +85.7 +116.3 +1.4 +1.6 +100 +50 +1 +45.9 +45.9 +82.6 +158.2 +1.0 +2.7 +100 +90 +1 +84.8 +84.8 +102.2 +224.7 +2.0 +4.2 +500 +50 +5 +42.2 +42.5 +173.1 +344.7 +44.2 +72.2 +500 +250 +5 +243.0 +242.9 +145.5 +362.4 +41.9 +95.0 +500 +450 +5 +442.2 +442.2 +163.7 +425.2 +47.6 +138.3 +1000 +100 +10 +90.1 +90.4 +162.5 +361.0 +110.8 +259.0 +1000 +500 +10 +491.1 +491.2 +158.3 +475.4 +129.1 +558.4 +1000 +900 +10 +891.1 +891.1 +193.5 +300.7 +187.0 +298.5 +Table 2: Numerical results for problem (38) +Objective value +Feasibility violation (×10−4) +Total inner iterations +CPU time (seconds) +n +m +µ +Algorithm 2 +Prox-AL +Algorithm 2 +Prox-AL +Algorithm 2 +Prox-AL +Algorithm 2 +Prox-AL +100 +10 +1 +7.1 +7.1 +0.18 +0.27 +40.9 +97.3 +0.73 +2.2 +100 +50 +1 +46.6 +46.6 +0.21 +0.30 +37.0 +86.3 +0.78 +1.7 +100 +90 +1 +87.0 +87.0 +0.12 +0.40 +39.5 +68.6 +1.1 +1.9 +500 +50 +5 +44.4 +44.4 +0.40 +0.68 +59.0 +343.4 +11.4 +134.9 +500 +250 +5 +244.3 +244.3 +0.37 +0.47 +59.0 +543.3 +11.7 +178.2 +500 +450 +5 +444.0 +444.0 +0.27 +0.53 +66.7 +634.1 +17.1 +158.2 +1000 +100 +10 +92.8 +92.8 +0.28 +0.42 +95.0 +2054.6 +46.3 +1516.8 +1000 +500 +10 +491.9 +491.9 +0.22 +0.72 +68.3 +756.2 +39.5 +558.6 +1000 +900 +10 +893.4 +893.4 +0.19 +0.37 +81.8 +1281.4 +57.7 +1099.6 +Table 3: Numerical results for problem (39) +Our aim is to find a (10−4, 10−2)-SOSP of (39) for the above instances by Algorithm 2 and the proximal +AL method [60, Algorithm 3] and compare their performance. For a fair comparison, we use a minimum eigen- +value oracle that returns a deterministic output for them so that they both certainly output an approximate +second-order stationary point. Specifically, we use the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) as the +minimum eigenvalue oracle to find the minimum eigenvalue λ and its associated unit eigenvector v of a real +symmetric matrix H. In addition, for both methods, we choose the initial point as z0 = (1/√n, . . . , 1/√n)T , +the initial Lagrangian multiplier as λ0 = 0, and the other parameters as +• Λ = 100, ρ0 = 10, α = 0.25, and r = 10 for Algorithm 2; +• η = 1, q = 10 and T0 = 2 for the proximal AL method ([60]). +The computational results of Algorithm 2 and the proximal AL method in [60] (abbreviated as Prox-AL) +for solving problem (39) for the instances randomly generated above are presented in Table 3. In detail, +the value of n, m, and µ is listed in the first three columns, respectively. For each triple (n, m, µ), the +average CPU time (in seconds), the average total number of inner iterations, the average final objective +value, and the average final feasibility violation over 10 random instances are given in the rest columns. +One can observe that both methods output an approximate solution of similar quality in terms of objective +value and feasibility violation, while our Algorithm 2 vastly outperforms the proximal AL method in [60] +in terms of CPU time. This corroborates our theoretical finding that Algorithm 2 achieves a significantly +better operation complexity than the proximal AL method in [60] for finding an approximate SOSP. +6 +Proof of the main results +We provide proofs of our main results in Sections 3 and 4, including Theorem 3.1, Lemma 4.1, and Theorems +4.2, 4.3, 4.4 and 4.5. +6.1 +Proof of the main results in Section 3 +In this subsection we first establish several technical lemmas and then use them to prove Theorem 3.1. +14 + +One can observe from Assumption 3.1(b) that for all x and y ∈ Ω, +∥∇F(y) − ∇F(x) − ∇2F(x)(y − x)∥ ≤ LF +H∥y − x∥2/2, +(40) +F(y) ≤ F(x) + ∇F(x)T (y − x) + (y − x)T ∇2F(x)(y − x)/2 + LF +H∥y − x∥3/6. +(41) +The next lemma provides useful properties of the output of Algorithm 3, whose proof is similar to the +ones in [56, Lemma 3] and [54, Lemma 7] and thus omitted here. +Lemma 6.1. Suppose that Assumption 3.1 holds and the direction dt results from the output d of Algorithm +3 with a type specified in d type at some iteration t of Algorithm 1. Then the following statements hold. +(i) If d type=SOL, then dt satisfies +ǫH∥dt∥2 ≤ (dt)T � +∇2F(xt) + 2ǫHI +� +dt, +(42) +∥dt∥ ≤ 1.1ǫ−1 +H ∥∇F(xt)∥, +(43) +(dt)T ∇F(xt) = −(dt)T � +∇2F(xt) + 2ǫHI +� +dt, +(44) +∥(∇2F(xt) + 2ǫHI)dt + ∇F(xt)∥ ≤ ǫHζ∥dt∥/2. +(45) +(ii) If d type=NC, then dt satisfies (dt)T ∇F(xt) ≤ 0 and +(dt)T ∇2F(xt)dt/∥dt∥2 = −∥dt∥ ≤ −ǫH. +(46) +The next lemma shows that when the search direction dt in Algorithm 1 is of type ‘SOL’, the line search +step results in a sufficient reduction on F. +Lemma 6.2. Suppose that Assumption 3.1 holds and the direction dt results from the output d of Algorithm 3 +with d type=SOL at some iteration t of Algorithm 1. Let U F +g and csol be given in (9) and (19), respectively. +Then the following statements hold. +(i) The step length αt is well-defined, and moreover, +αt ≥ min +� +1, +� +min{6(1 − η), 2} +1.1LF +HU F +g +θǫH +� +. +(47) +(ii) The next iterate xt+1 = xt + αtdt satisfies +F(xt) − F(xt+1) ≥ csol min{∥∇F(xt+1)∥2ǫ−1 +H , ǫ3 +H}. +(48) +Proof. One can observe that F is descent along the iterates (whenever well-defined) generated by Algorithm 1, +which together with x0 = u0 implies that F(xt) ≤ F(u0) and hence ∥∇F(xt)∥ ≤ U F +g due to (9). In addition, +since dt results from the output d of Algorithm 3 with d type=SOL, one can see that ∥∇F(xt)∥ > ǫg and +(42)-(45) hold for dt. Moreover, by ∥∇F(xt)∥ > ǫg and (45), one can conclude that dt ̸= 0. +We first prove statement (i). If (16) holds for j = 0, then αt = 1, which clearly implies that (47) holds. +We now suppose that (16) fails for j = 0. Claim that for all j ≥ 0 that violate (16), it holds that +θ2j ≥ min{6(1 − η), 2}ǫH(LF +H)−1∥dt∥−1. +(49) +Indeed, suppose that (16) is violated by some j ≥ 0. We now show that (49) holds for such j by considering +two separate cases below. +Case 1) F(xt + θjdt) > F(xt). Let φ(α) = F(xt + αdt). Then φ(θj) > φ(0). Also, since dt ̸= 0, by (42) +and (44), one has +φ′(0) = ∇F(xt)T dt = −(dt)T (∇2F(xt) + 2ǫHI)dt ≤ −ǫH∥dt∥2 < 0. +15 + +Using these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of φ such that φ′(α∗) = +∇F(xt + α∗dt)T dt = 0 and φ(α∗) < φ(0), which implies that F(xt + α∗dt) < F(xt) ≤ F(u0). Hence, (40) +holds for x = xt and y = xt + α∗dt. Using this, 0 < α∗ < θj ≤ 1 and ∇F(xt + α∗dt)T dt = 0, we obtain +(α∗)2LF +H +2 +∥dt∥3 (40) +≥ ∥dt∥∥∇F(xt + α∗dt) − ∇F(xt) − α∗∇2F(xt)dt∥ +≥ (dt)T (∇F(xt + α∗dt) − ∇F(xt) − α∗∇2F(xt)dt) = −(dt)T ∇F(xt) − α∗(dt)T ∇2F(xt)dt +(44) += (1 − α∗)(dt)T (∇2F(xt) + 2ǫHI)dt + 2α∗ǫH∥dt∥2 +(42) +≥ (1 + α∗)ǫH∥dt∥2 ≥ ǫH∥dt∥2, +which along with dt ̸= 0 implies that (α∗)2 ≥ 2ǫH(LF +H)−1∥dt∥−1. Using this and θj > α∗, we conclude that +(49) holds in this case. +Case 2) F(xt + θjdt) ≤ F(xt). This together with F(xt) ≤ F(u0) implies that (41) holds for x = xt and +y = xt + θjdt. Then, because j violates (16), we obtain +−ηǫHθ2j∥dt∥2 ≤ F(xt + θjdt) − F(xt) +(41) +≤ θj∇F(xt)T dt + θ2j +2 (dt)T ∇2F(xt)dt + LF +H +6 θ3j∥dt∥3 +(44) += −θj(dt)T (∇2F(xt) + 2ǫHI)dt + θ2j +2 (dt)T ∇2F(xt)dt + LF +H +6 θ3j∥dt∥3 += −θj +� +1 − θj +2 +� +(dt)T (∇2F(xt) + 2ǫHI)dt − θ2jǫH∥dt∥2 + LF +H +6 θ3j∥dt∥3 +(42) +≤ −θj +� +1 − θj +2 +� +ǫH∥dt∥2 − θ2jǫH∥dt∥2 + LF +H +6 θ3j∥dt∥3 ≤ −θjǫH∥dt∥2 + LF +H +6 θ3j∥dt∥3. +(50) +Recall that dt ̸= 0. Dividing both sides of (50) by LF +Hθj∥dt∥3/6 and using η, θ ∈ (0, 1), we obtain that +θ2j ≥ 6(1 − θjη)ǫH(LF +H)−1∥dt∥−1 ≥ 6(1 − η)ǫH(LF +H)−1∥dt∥−1. +Hence, (49) also holds in this case. +Combining the above two cases, we conclude that (49) holds for any j ≥ 0 that violates (16). By this +and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (16) must be bounded above. It then follows that the +step length αt associated with (16) is well-defined. We next prove (47). Observe from the definition of jt in +Algorithm 1 that j = jt − 1 violates (16) and hence (49) holds for j = jt − 1. Then, by (49) with j = jt − 1 +and αt = θjt, one has +αt = θjt ≥ +� +min{6(1 − η), 2}ǫH(LF +H)−1 θ∥dt∥−1/2, +(51) +which, along with (43) and ∥∇F(xt)∥ ≤ U F +g , implies (47). This proves statement (i). +We next prove statement (ii) by considering two separate cases below. +Case 1) αt = 1. By this, one knows that (16) holds for j = 0. It then follows that F(xt + dt) ≤ F(xt) ≤ +F(u0), which implies that (40) holds for x = xt and y = xt + dt. By this and (45), one has +∥∇F(xt+1)∥ = ∥∇F(xt + dt)∥ +≤ +∥∇F(xt + dt) − ∇F(xt) − ∇2F(xt)dt∥ ++∥(∇2F(xt) + 2ǫHI)dt + ∇F(xt)∥ + 2ǫH∥dt∥ +≤ +LF +H +2 ∥dt∥2 + 4+ζ +2 ǫH∥dt∥, +where the last inequality follows from (40) and (45). Solving the above inequality for ∥dt∥ and using the fact +that ∥dt∥ > 0, we obtain that +∥dt∥ +≥ +−(4+ζ)ǫH+√ +(4+ζ)2ǫ2 +H+8LF +H∥∇F (xt+1)∥ +2LF +H +≥ +−(4+ζ)ǫH+√ +(4+ζ)2ǫ2 +H+8LF +Hǫ2 +H +2LF +H +min{∥∇F(xt+1)∥/ǫ2 +H, 1} += +4 +4+ζ+√ +(4+ζ)2+8LF +H +min{∥∇F(xt+1)∥/ǫH, ǫH}, +where the second inequality follows from the inequality −a + +√ +a2 + bs ≥ (−a + +√ +a2 + b) min{s, 1} for all +a, b, s ≥ 0, which can be verified by performing a rationalization to the terms −a+ +√ +a2 + b and −a+ +√ +a2 + bs, +respectively. By this, αt = 1, (16) and (19), one can see that (48) holds. +16 + +Case 2) αt < 1. It then follows that j = 0 violates (16) and hence (49) holds for j = 0. Now, letting +j = 0 in (49), we obtain that ∥dt∥ ≥ min{6(1 − η), 2}ǫH/LF +H, which together with (16) and (51) implies that +F(xt) − F(xt+1) ≥ ηǫHθ2jt∥dt∥2 ≥ η min{6(1 − η), 2}ǫ2 +H +LF +H +θ2∥dt∥ ≥ η +�min{6(1 − η), 2}θ +LF +H +�2 +ǫ3 +H. +By this and (19), one can see that (48) also holds in this case. +The following lemma shows that when the search direction dt in Algorithm 1 is of type ‘NC’, the line +search step results in a sufficient reduction on F as well. +Lemma 6.3. Suppose that Assumption 3.1 holds and the direction dt results from either the output d of +Algorithm 3 with d type=NC or the output v of Algorithm 4 at some iteration t of Algorithm 1. Let cnc be +defined in (20). Then the following statements hold. +(i) The step length αt is well-defined, and αt ≥ min{1, θ/LF +H, 3(1 − η)θ/LF +H}. +(ii) The next iterate xt+1 = xt + αtdt satisfies F(xt) − F(xt+1) ≥ cncǫ3 +H. +Proof. Observe that F is descent along the iterates (whenever well-defined) generated by Algorithm 1. Using +this and x0 = u0, we have F(xt) ≤ F(u0). By the assumption on dt, one can see from Algorithm 1 that dt is +a negative curvature direction given in (13) or (15). Also, notice that the vector v returned from Algorithm 4 +satisfies ∥v∥ = 1. By these, Lemma 6.1(ii), (13) and (15), one can observe that +∇F(xt)T dt ≤ 0, +(dt)T ∇2F(xt)dt = −∥dt∥3 < 0. +(52) +We first prove statement (i). If (17) holds for j = 0, then αt = 1, which clearly implies that αt ≥ +min{1, θ/LF +H, 3(1 − η)θ/LF +H}. We now suppose that (17) fails for j = 0. Claim that for all j ≥ 0 that violate +(17), it holds that +θj ≥ min{1/LF +H, 3(1 − η)/LF +H}. +(53) +Indeed, suppose that (17) is violated by some j ≥ 0. We now show that (53) holds for such j by considering +two separate cases below. +Case 1) F(xt + θjdt) > F(xt). Let φ(α) = F(xt + αdt). Then φ(θj) > φ(0). Also, by (52), one has +φ′(0) = ∇F(xt)T dt ≤ 0, +φ′′(0) = (dt)T ∇2F(xt)dt < 0. +Using these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of φ such that φ(α∗) < φ(0), +namely, F(xt + α∗dt) < F(xt). By the second-order optimality condition of φ at α∗, one has φ′′(α∗) = +(dt)T ∇2F(xt + α∗dt)dt ≥ 0. Since F(xt + α∗dt) < F(xt) ≤ F(u0), it follows that (8) holds for x = xt and +y = xt + α∗dt. Using this, the second relation in (52) and (dt)T ∇2F(xt + α∗dt)dt ≥ 0, we obtain that in (52) +and (dt)T ∇2F(xt + α∗dt)dt ≥ 0, we obtain that +LF +Hα∗∥dt∥3 +(8) +≥ +∥dt∥2∥∇2F(xt + α∗dt) − ∇2F(xt)∥ ≥ (dt)T (∇2F(xt + α∗dt) − ∇2F(xt))dt +≥ +−(dt)T ∇2F(xt)dt = ∥dt∥3. +(54) +Recall from (52) that dt ̸= 0. It then follows from (54) that α∗ ≥ 1/LF +H, which along with θj > α∗ implies +that θj > 1/LF +H. Hence, (53) holds in this case. +Case 2) F(xt + θjdt) ≤ F(xt). It follows from this and F(xt) ≤ F(u0) that (41) holds for x = xt and +y = xt + θjdt. By this and the fact that j violates (17), one has +− η +2θ2j∥dt∥3 +≤ +F(xt + θjdt) − F(xt) +(41) +≤ θj∇F(xt)T dt + θ2j +2 (dt)T ∇2F(xt)dt + LF +H +6 θ3j∥dt∥3 +(52) +≤ +− θ2j +2 ∥dt∥3 + LF +H +6 θ3j∥dt∥3, +which together with dt ̸= 0 implies that θj ≥ 3(1 − η)/LF +H. Hence, (53) also holds in this case. +17 + +Combining the above two cases, we conclude that (53) holds for any j ≥ 0 that violates (17). By this +and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (17) must be bounded above. It then follows that the +step length αt associated with (17) is well-defined. We next derive a lower bound for αt. Notice from the +definition of jt in Algorithm 1 that j = jt − 1 violates (17) and hence (53) holds for j = jt − 1. Then, by +(53) with j = jt − 1 and αt = θjt, one has αt = θjt ≥ min{θ/LF +H, 3(1 − η)θ/LF +H}, which immediately yields +αt ≥ min{1, θ/LF +H, 3(1 − η)θ/LF +H} as desired. +We next prove statement (ii) by considering two separate cases below. +Case 1) dt results from the output d of Algorithm 3 with d type=NC. It then follows from (46) that +∥dt∥ ≥ ǫH. This together with (17) and statement (i) implies that statement (ii) holds. +Case 2) dt results from the output v of Algorithm 4. +Notice from Algorithm 4 that ∥v∥ = 1 and +vT ∇2F(xt)v ≤ −ǫH/2, which along with (15) yields ∥dt∥ ≥ ǫH/2. By this, (17) and statement (i), one can +see that statement (ii) again holds. +Proof of Theorem 3.1. For notational convenience, we let {xt}t∈T denote all the iterates generated by Algo- +rithm 1, where T is a set of consecutive nonnegative integers starting from 0. Notice that F is descent along +the iterates generated by Algorithm 1, which together with x0 = u0 implies that xt ∈ {x : F(x) ≤ F(u0)}. +It then follows from (9) that ∥∇2F(xt)∥ ≤ U F +H holds for all t ∈ T. +(i) Suppose for contradiction that the total number of calls of Algorithm 4 in Algorithm 1 is more than T2. +Notice from Algorithm 1 and Lemma 6.3(ii) that each of these calls, except the last one, returns a sufficiently +negative curvature direction, and each of them results in a reduction on F of at least cncǫ3 +H. Hence, +T2cncǫ3 +H ≤ +� +t∈T +[F(xt) − F(xt+1)] ≤ F(x0) − Flow = Fhi − Flow, +which contradicts the definition of T2 given in (18). Hence, statement (i) of Theorem 3.1 holds. +(ii) Suppose for contradiction that the total number of calls of Algorithm 3 in Algorithm 1 is more +than T1. +Observe that if Algorithm 3 is called at some iteration t and generates the next iterate xt+1 +satisfying ∥∇F(xt+1)∥ ≤ ǫg, then Algorithm 4 must be called at the next iteration t + 1. In view of this +and statement (i) of Theorem 3.1, we see that the total number of such iterations t is at most T2. Hence, +the total number of iterations t of Algorithm 1 at which Algorithm 3 is called and generates the next iterate +xt+1 satisfying ∥∇F(xt+1)∥ > ǫg is at least T1 − T2 + 1. Moreover, for each of such iterations t, we observe +from Lemmas 6.2(ii) and 6.3(ii) that F(xt) − F(xt+1) ≥ min{csol, cnc} min{ǫ2 +gǫ−1 +H , ǫ3 +H}. It then follows that +(T1 − T2 + 1) min{csol, cnc} min{ǫ2 +gǫ−1 +H , ǫ3 +H} ≤ +� +t∈T +[F(xt) − F(xt+1)] ≤ Fhi − Flow, +which contradicts the definition of T1 and T2 given in (18). Hence, statement (ii) of Theorem 3.1 holds. +(iii) Notice that either Algorithm 3 or 4 is called at each iteration of Algorithm 1. It follows from this +and statements (i) and (ii) of Theorem 3.1 that the total number of iterations of Algorithm 1 is at most +T1 +T2. In addition, the relation (21) follows from (19), (20) and (18). One can also observe that the output +xt of Algorithm 1 satisfies ∥∇F(xt)∥ ≤ ǫg deterministically and λmin(∇2F(xt)) ≥ −ǫH with probability at +least 1 − δ for some 0 ≤ t ≤ T1 + T2, where the latter part is due to Algorithm 4. This completes the proof +of statement (ii) of Theorem 3.1. +(iv) By Theorem A.1 with (H, ε) = (∇2F(xt), ǫH) and the fact that ∥∇2F(xt)∥ ≤ U F +H, one can observe +that the number of Hessian-vector products required by each call of Algorithm 3 with input U = 0 is at +most �O(min{n, (U F +H/ǫH)1/2}). In addition, by Theorem B.1 with (H, ε) = (∇2F(xt), ǫH), ∥∇2F(xt)∥ ≤ U F +H, +and the fact that each iteration of the Lanczos method requires only one matrix-vector product, one can +observe that the number of Hessian-vector products required by each call of Algorithm 4 is also at most +�O(min{n, (U F +H/ǫH)1/2}). +Based on these observations and statement (iii) of Theorem 3.1, we see that +statement (iv) of this theorem holds. +18 + +6.2 +Proof of the main results in Section 4 +Recall from Assumption 4.1(a) that ∥c(zǫ1)∥ ≤ ǫ1/2 < 1. By virtue of this, (23) and the definition of ˜c in +(25), we obtain that +f(x) + γ∥˜c(x)∥2 ≥ f(x) + γ∥c(x)∥2/2 − γ∥c(zǫ1)∥2 ≥ flow − γ, +∀x ∈ Rn . +(55) +We now prove the following auxiliary lemma that will be used frequently later. +Lemma 6.4. Suppose that Assumption 4.1 holds. Let γ, fhi and flow be given in Assumption 4.1. Assume +that ρ > 2γ, λ ∈ Rm, and x ∈ Rn satisfy +�L(x, λ; ρ) ≤ fhi, +(56) +where �L is defined in (26). Then the following statements hold. +(i) f(x) ≤ fhi + ∥λ∥2/(2ρ). +(ii) ∥˜c(x)∥ ≤ +� +2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ). +(iii) If ρ ≥ ∥λ∥2/(2˜δf) for some ˜δf > 0, then f(x) ≤ fhi + ˜δf. +(iv) If +ρ ≥ 2(fhi − flow + γ)˜δ−2 +c ++ 2∥λ∥˜δ−1 +c ++ 2γ +(57) +for some ˜δc > 0, then ∥˜c(x)∥ ≤ ˜δc. +Proof. (i) It follows from (56) and the definition of �L in (26) that +fhi ≥ f(x) + λT ˜c(x) + ρ +2∥˜c(x)∥2 = f(x) + ρ +2 +���˜c(x) + λ +ρ +��� +2 +− ∥λ∥2 +2ρ +≥ f(x) − ∥λ∥2 +2ρ . +Hence, statement (i) holds. +(ii) In view of (55) and (56), one has +fhi +(56) +≥ f(x) + λT ˜c(x) + ρ +2∥˜c(x)∥2 = f(x) + γ∥˜c(x)∥2 + ρ−2γ +2 +���˜c(x) + +λ +ρ−2γ +��� +2 +− +∥λ∥2 +2(ρ−2γ) +(55) +≥ flow − γ + ρ−2γ +2 +���˜c(x) + +λ +ρ−2γ +��� +2 +− +∥λ∥2 +2(ρ−2γ). +It then follows that +���˜c(x) + +λ +ρ−2γ +��� ≤ +� +2(fhi−flow+γ) +ρ−2γ ++ +∥λ∥2 +(ρ−2γ)2 , which implies that statement (ii) holds. +(iii) Statement (iii) immediately follows from statement (i) and ρ ≥ ∥λ∥2/(2˜δf). +(iv) Suppose that (57) holds. Multiplying both sides of (57) by ˜δ2 +c and rearranging the terms, we have +(ρ − 2γ)˜δ2 +c − 2∥λ∥˜δc − 2(fhi − flow + γ) ≥ 0. +Recall that ρ > 2γ and ˜δc > 0. Solving this inequality for ˜δc yields +˜δc ≥ +� +2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ), +which along with statement (ii) implies that ∥˜c(x)∥ ≤ ˜δc. Hence, statement (iv) holds. +Proof of Lemma 4.1. (i) Let x be any point such that �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk). It then follows from +(30) that �L(x, λk; ρk) ≤ fhi. By this, ∥λk∥ ≤ Λ, ρk ≥ ρ0 > 2γ, δf,1 ≤ δf, δc,1 ≤ δc, and Lemma 6.4 with +(λ, ρ) = (λk, ρk), one has +f(x) ≤ fhi + ∥λk∥2/(2ρk) ≤ fhi + Λ2/(2ρ0) = fhi + δf,1 ≤ fhi + δf, +∥˜c(x)∥ ≤ +� +2(fhi−flow+γ) +ρk−2γ ++ +∥λk∥2 +(ρk−2γ)2 + +∥λk∥ +ρk−2γ ≤ +� +2(fhi−flow+γ) +ρ0−2γ ++ +Λ2 +(ρ0−2γ)2 + +Λ +ρ0−2γ = δc,1 ≤ δc. +(58) +Also, recall from the definition of ˜c in (25) and ∥c(zǫ1)∥ ≤ 1 that ∥c(x)∥ ≤ 1 + ∥˜c(x)∥. This together with +the above inequalities and (24) implies x ∈ S(δf, δc). Hence, statement (i) of Lemma 4.1 holds. +19 + +(ii) Note that inf +x∈Rn �L(x, λk; ρk) = inf +x∈Rn{�L(x, λk; ρk) : �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk)}. Consequently, to +prove statement (ii) of Lemma 4.1, it suffices to show that +inf +x∈Rn{�L(x, λk; ρk) : �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk)} ≥ flow − γ − Λδc. +(59) +To this end, let x be any point satisfying �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk). +We then know from (58) that +∥˜c(x)∥ ≤ δc. By this, ∥λk∥ ≤ Λ, ρk > 2γ, and (55), one has +�L(x, λk; ρk) = f(x) + γ∥˜c(x)∥2 + (λk)T ˜c(x) + ρk−2γ +2 +∥˜c(x)∥2 +≥ f(x) + γ∥˜c(x)∥2 − Λ∥˜c(x)∥ ≥ flow − γ − Λδc, +and hence (59) holds as desired. +Proof of Theorem 4.2. Suppose that Algorithm 2 terminates at some iteration k, that is, τg +k ≤ ǫ1, τ H +k ≤ ǫ2, +and ∥c(xk+1)∥ ≤ ǫ1 hold. Then, by τg +k ≤ ǫ1, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c and the second relation in +(27), one has +∥∇f(xk+1) + ∇c(xk+1)˜λk+1∥ = ∥∇f(xk+1) + ∇˜c(xk+1)(λk + ρk˜c(xk+1))∥ += ∥∇x�L(xk+1, λk; ρk)∥ ≤ τ g +k ≤ ǫ1. +Hence, (xk+1, ˜λk+1) satisfies the first relation in (5). In addition, by (28) and τ H +k ≤ ǫ2, one can show that +λmin(∇2 +xx�L(xk+1, λk; ρk)) ≥ −ǫ2 with probability at least 1 − δ, which leads to dT ∇2 +xx�L(xk+1, λk; ρk)d ≥ +−ǫ2∥d∥2 for all d ∈ Rn with probability at least 1 − δ. Using this, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c, and +∇2˜ci = ∇2ci for 1 ≤ i ≤ m, we see that with probability at least 1 − δ, it holds that +dT +� +∇2f(xk+1) + +m +� +i=1 +˜λk+1 +i +∇2ci(xk+1) + ρk∇c(xk+1)∇c(xk+1)T +� +d ≥ −ǫ2∥d∥2 ∀d ∈ Rn, +which implies dT (∇2f(xk+1) + �m +i=1 ˜λk+1 +i +∇2ci(xk+1))d ≥ −ǫ2∥d∥2 for all d ∈ C(xk+1), where C(·) is defined +in (4). Hence, (xk+1, ˜λk+1) satisfies (6) with probability at least 1−δ. Combining these with ∥c(xk+1)∥ ≤ ǫ1, +we conclude that xk+1 is a deterministic ǫ1-FOSP of (1) and an (ǫ1, ǫ2)-SOSP of (1) with probability at least +1 − δ. Hence, Theorem 4.2 holds. +Proof of Theorem 4.3. It follows from (35) that ρǫ1 ≥ 2ρ0. By this, one has +Kǫ1 +(33) += +⌈log ǫ1/ log ω1⌉ +(32) += +⌈log 2/ log r⌉ ≤ log(ρǫ1ρ−1 +0 )/ log r + 1. +(60) +Notice that {ρk} is either unchanged or increased by a ratio r as k increases. By this fact and (60), we see +that +max +0≤k≤Kǫ1 +ρk ≤ rKǫ1 ρ0 +(60) +≤ r +log(ρǫ1 ρ−1 +0 +) +log r ++1ρ0 = rρǫ1. +(61) +In addition, notice that ρk > 2γ and ∥λk∥ ≤ Λ. Using these, (22), the first relation in (27), and Lemma +6.4(ii) with (x, λ, ρ) = (xk+1, λk, ρk), we obtain that +∥˜c(xk+1)∥ ≤ +� +2(fhi−flow+γ) +ρk−2γ ++ +∥λk∥2 +(ρk−2γ)2 + +∥λk∥ +ρk−2γ ≤ +� +2(fhi−flow+γ) +ρk−2γ ++ +Λ2 +(ρk−2γ)2 + +Λ +ρk−2γ . +(62) +Also, we observe from ∥c(zǫ1)∥ ≤ ǫ1/2 and the definition of ˜c in (25) that +∥c(xk+1)∥ ≤ ∥˜c(xk+1)∥ + ∥c(zǫ1)∥ ≤ ∥˜c(xk+1)∥ + ǫ1/2. +(63) +We now prove that Kǫ1 is finite. Suppose for contradiction that Kǫ1 is infinite. It then follows from this +and (36) that ∥c(xk+1)∥ > ǫ1 for all k ≥ Kǫ1, which along with (63) implies that ∥˜c(xk+1)∥ > ǫ1/2 for all +k ≥ Kǫ1. It then follows that ∥˜c(xk+1)∥ > α∥˜c(xk)∥ must hold for infinitely many k’s. Using this and the +update scheme on {ρk}, we deduce that ρk+1 = rρk holds for infinitely many k’s, which together with the +20 + +monotonicity of {ρk} implies that ρk → ∞ as k → ∞. By this and (62), one can see that ∥˜c(xk+1)∥ → 0 +as k → ∞, which contradicts the fact that ∥˜c(xk+1)∥ > ǫ1/2 holds for all k ≥ Kǫ1. Hence, Kǫ1 is finite. +In addition, notice from (32), (33) and (34) that (τ g +k , τ H +k ) = (ǫ1, ǫ2) for all k ≥ Kǫ1. This along with the +termination criterion of Algorithm 2 and the definition of Kǫ1 implies that Algorithm 2 must terminate at +iteration Kǫ1. +We next show that (37) and ρk ≤ rρǫ1 hold for 0 ≤ k ≤ Kǫ1 by considering two separate cases below. +Case 1) ∥c(xKǫ1 +1)∥ ≤ ǫ1. By this and (36), one can see that Kǫ1 = Kǫ1, which together with (60) and +(61) implies that (37) and ρk ≤ rρǫ1 hold for 0 ≤ k ≤ Kǫ1. +Case 2) ∥c(xKǫ1 +1)∥ > ǫ1. By this and (36), one can observe that Kǫ1 > Kǫ1 and also ∥c(xk+1)∥ > ǫ1 +for all Kǫ1 ≤ k ≤ Kǫ1 − 1, which together with (63) implies +∥˜c(xk+1)∥ > ǫ1/2, +∀Kǫ1 ≤ k ≤ Kǫ1 − 1. +(64) +It then follows from ∥λk∥ ≤ Λ, (22), the first relation in (27), and Lemma 6.4(iv) with (x, λ, ρ, ˜δc) = +(xk+1, λk, ρk, ǫ1/2) that +ρk < 8(fhi − flow + γ)ǫ−2 +1 ++ 4∥λk∥ǫ−1 +1 ++ 2γ +≤ 8(fhi − flow + γ)ǫ−2 +1 ++ 4Λǫ−1 +1 ++ 2γ +(35) +≤ ρǫ1, +∀Kǫ1 ≤ k ≤ Kǫ1 − 1. +(65) +Combining this relation, (61), and the fact ρKǫ1 ≤ rρKǫ1 −1, we conclude that ρk ≤ rρǫ1 holds for 0 ≤ k ≤ Kǫ1. +It remains to show that (37) holds. To this end, let +K = {k : ρk+1 = rρk, Kǫ1 ≤ k ≤ Kǫ1 − 2}. +It follows from (65) and the update scheme of ρk that +r| K |ρKǫ1 = +max +Kǫ1 ≤k≤Kǫ1 −1 +{ρk} ≤ ρǫ1, +which together with ρKǫ1 ≥ ρ0 implies that +| K | ≤ log(ρǫ1ρ−1 +Kǫ1 )/ log r ≤ log(ρǫ1ρ−1 +0 )/ log r. +(66) +Let {k1, k2, . . . , k| K |} denote all the elements of K arranged in ascending order, and let k0 = Kǫ1 and +k| K |+1 = Kǫ1 − 1. We next derive an upper bound for kj+1 − kj for j = 0, 1, . . ., | K |. By the definition of +K, one can observe that ρk = ρk′ for kj < k, k′ ≤ kj+1. Using this and the update scheme of ρk, we deduce +that +∥˜c(xk+1)∥ ≤ α∥˜c(xk)∥, +∀kj < k < kj+1. +(67) +On the other hand, by (31), (62) and ρk ≥ ρ0, one has ∥˜c(xk+1)∥ ≤ δc,1 for 0 ≤ k ≤ Kǫ1. By this and (64), +one can see that +ǫ1/2 < ∥˜c(xk+1)∥ ≤ δc,1, +∀Kǫ1 ≤ k ≤ Kǫ1 − 1. +(68) +Now, note that either kj+1 − kj = 1 or kj+1 − kj > 1. In the latter case, we can apply (67) with k = +kj+1 − 1, . . . , kj + 1 together with (68) to deduce that +ǫ1/2 < ∥˜c(xkj+1)∥ ≤ α∥˜c(xkj+1−1)∥ ≤ · · · ≤ αkj+1−kj−1∥˜c(xkj+1)∥ ≤ αkj+1−kj−1δc,1, +∀j = 0, 1, . . . , | K |. +Combining these two cases, we have +kj+1 − kj ≤ | log(ǫ1(2δc,1)−1))/ log α| + 1, +∀j = 0, 1, . . ., | K |. +(69) +Summing up these inequalities, and using (60), (66), k0 = Kǫ1 and k| K |+1 = Kǫ1 − 1, we have +Kǫ1 = 1 + k| K |+1 = 1 + k0 + �| K | +j=0(kj+1 − kj) +(69) +≤ 1 + Kǫ1 + (| K | + 1) +���� log(ǫ1(2δc,1)−1) +log α +��� + 1 +� +≤ 2 + log(ρǫ1 ρ−1 +0 +) +log r ++ +� log(ρǫ1 ρ−1 +0 +) +log r ++ 1 +� ���� log(ǫ1(2δc,1)−1) +log α +��� + 1 +� += 1 + +� log(ρǫ1 ρ−1 +0 +) +log r ++ 1 +� ���� log(ǫ1(2δc,1)−1) +log α +��� + 2 +� +, +where the second inequality is due to (60) and (66). Hence, (37) also holds in this case. +21 + +We next prove Theorem 4.4. Before proceeding, we introduce some notation that will be used shortly. +Let Lk,H denote the Lipschitz constant of ∇2 +xx�L(x, λk; ρk) on the convex open neighborhood Ω(δf, δc) of +S(δf, δc), where S(δf, δc) is defined in (24), and let Uk,H = supx∈S(δf ,δc) ∥∇2 +xx�L(x, λk; ρk)∥. Notice from (25) +and (26) that +∇2 +xx�L(x, λk; ρk) = ∇2f(x) + +m +� +i=1 +λk +i ∇2ci(x) + ρk +� +∇c(x)∇c(x)T + +m +� +i=1 +˜ci(x)∇2ci(x) +� +. +(70) +By this, ∥λk∥ ≤ Λ, the definition of ˜c, and the Lipschitz continuity of ∇2f and ∇2ci’s (see Assumption 4.1(c)), +one can observe that there exist some constants L1, L2, U1 and U2, depending only on f, c, Λ, δf and δc, +such that +Lk,H ≤ L1 + ρkL2, +Uk,H ≤ U1 + ρkU2. +(71) +Proof of Theorem 4.4. Let Tk and Nk denote the number of iterations and matrix-vector products performed +by Algorithm 1 at the outer iteration k of Algorithm 2, respectively. It then follows from Theorem 4.3 that +the total number of iterations and matrix-vector products performed by Algorithm 1 in Algorithm 2 are +�Kǫ1 +k=0 Tk and �Kǫ1 +k=0 Nk, respectively. In addition, notice from (35) and Theorem 4.3 that ρǫ1 = O(ǫ−2 +1 ) and +ρk ≤ rρǫ1, which yield ρk = O(ǫ−2 +1 ). +We first claim that (τ g +k )2/τ H +k +≥ min{ǫ2 +1/ǫ2, ǫ3 +2} holds for any k ≥ 0. Indeed, let ¯t = log ǫ1/ log ω1 and +ψ(t) = max{ǫ1, ωt +1}2/ max{ǫ2, ωt +2} for all t ∈ R. It then follows from (34) that ω¯t +1 = ǫ1 and ω¯t +2 = ǫ2. By this +and ω1, ω2 ∈ (0, 1), one can observe that ψ(t) = (ω2 +1/ω2)t if t ≤ ¯t and ψ(t) = ǫ2 +1/ǫ2 otherwise. This along +with ǫ2 ∈ (0, 1) implies that +min +t∈[0,∞) ψ(t) = min{ψ(0), ψ(¯t)} = min{1, ǫ2 +1/ǫ2} ≥ min{ǫ2 +1/ǫ2, ǫ3 +2}, +which together with (32) yields (τ g +k )2/τ H +k = ψ(k) ≥ min{ǫ2 +1/ǫ2, ǫ3 +2} for all k ≥ 0. +(i) From Lemma 4.1(i) and the definitions of Ω(δf, δc) and Lk,H, we see that Lk,H is a Lipschitz constant +of ∇2 +xx�L(x, λk; ρk) on a convex open neighborhood of {x : �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk)}. Also, recall from +Lemma 4.1(ii) that infx∈Rn �L(x, λk; ρk) ≥ flow − γ − Λδc. By these, �L(xk +init, λk; ρk) ≤ fhi (see (30)) and +Theorem 3.1(iii) with (Fhi, Flow, LF +H, ǫg, ǫH) = (�L(xk +init, λk; ρk), flow − γ − Λδc, Lk,H, τ g +k , τ H +k ), one has +Tk += +O((fhi − flow + γ + Λδc)L2 +k,H max{(τ g +k )−2τ H +k , (τ H +k )−3}) +(71) += +O(ρ2 +k max{(τ g +k )−2τ H +k , (τ H +k )−3}) = O(ǫ−4 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 }), +(72) +where the last equality is from (τ g +k )2/τ H +k ≥ min{ǫ2 +1/ǫ2, ǫ3 +2}, τ H +k ≥ ǫ2, and ρk = O(ǫ−2 +1 ). +Next, if c(x) = Ax − b for some A ∈ Rm×n and b ∈ Rm, then ∇c(x) = AT and ∇2ci(x) = 0 for +1 ≤ i ≤ m. By these and (70), one has Lk,H = O(1). Using this and similar arguments as for (72), we obtain +that Tk = O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 }). By this, (72) and Kǫ1 = O(| log ǫ1|2) (see Remark 4.4), we conclude that +statement (i) of Theorem 4.4 holds. +(ii) In view of Lemma 4.1(i) and the definition of Uk,H, one can see that +Uk,H ≥ sup +x∈Rn{∥∇2 +xx�L(x, λk; ρk)∥ : �L(x, λk; ρk) ≤ �L(xk +init, λk; ρk)}. +Using this, �L(xk +init, λk; ρk) ≤ fhi and Theorem 3.1(iv) with (Fhi, Flow, LF +H, U F +H, ǫg, ǫH) = (�L(xk +init, λk; ρk), flow− +γ − Λδc, Lk,H, Uk,H, τ g +k , τ H +k ), we obtain that +Nk += +�O((fhi − flow + γ + Λδc)L2 +k,H max{(τ g +k )−2τ H +k , (τ H +k )−3} min{n, (Uk,H/τ H +k )1/2}) +(71) += +�O(ρ2 +k max{(τ g +k )−2τ H +k , (τ H +k )−3} min{n, (ρk/τ H +k )1/2}) += +�O(ǫ−4 +1 +max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1 +1 ǫ−1/2 +2 +}), +(73) +where the last equality is from (τ g +k )2/τ H +k ≥ min{ǫ2 +1/ǫ2, ǫ3 +2}, τ H +k ≥ ǫ2, and ρk = O(ǫ−2 +1 ). +On the other hand, if c is assumed to be affine, it follows from the above discussion that Lk,H = O(1). Us- +ing this, Uk,H ≤ U1+ρkU2, and similar arguments as for (73), we obtain that Nk = �O(max{ǫ−2 +1 ǫ2, ǫ−3 +2 } min{n, ǫ−1 +1 ǫ−1/2 +2 +}). +By this, (73) and Kǫ1 = O(| log ǫ1|2) (see Remark 4.4), we conclude that statement (ii) of Theorem 4.4 +holds. +22 + +Next, we provide a proof of Theorem 4.5. To proceed, we first observe from Assumptions 4.1(c) and 4.2 +that there exist U f +g > 0, U c +g > 0 and σ > 0 such that +∥∇f(x)∥ ≤ U f +g , +∥∇c(x)∥ ≤ U c +g, +λmin(∇c(x)T ∇c(x)) ≥ σ2, +∀x ∈ S(δf, δc). +(74) +We next establish several technical lemmas that will be used shortly. +Lemma 6.5. Suppose that Assumptions 4.1 and 4.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf +and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). Let {(xk, λk, ρk)} be generated by Algorithm 2. Suppose +that +ρk ≥ max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 +c ++ 2Λδ−1 +c ++ 2γ, 2(U f +g + U c +gΛ + 1)(σǫ1)−1} +(75) +for some k ≥ 0, where γ, fhi, flow, δf and δc are given in Assumption 4.1, and U f +g , U c +g and σ are given in +(74). Then it holds that ∥c(xk+1)∥ ≤ ǫ1. +Proof. By (75) and ∥λk∥ ≤ Λ (see step 6 of Algorithm 2), one can see that +ρk ≥ max{∥λk∥2(2δf)−1, 2(fhi − flow + γ)δ−2 +c ++ 2∥λk∥δ−1 +c ++ 2γ}. +Using this, (22), the first relation in (27), and Lemma 6.4(iii) and (iv) with (x, λ, ρ, ˜δf, ˜δc) = (xk+1, λk, ρk, δf, δc), +we obtain that f(xk+1) ≤ fhi + δf and ∥˜c(xk+1)∥ ≤ δc. In addition, recall from ∥c(zǫ1)∥ ≤ 1 and the defini- +tion of ˜c in (25) that ∥c(xk+1)∥ ≤ 1 + ∥˜c(xk+1)∥. These together with (24) show that xk+1 ∈ S(δf, δc). It +then follows from (74) that ∥∇f(xk+1)∥ ≤ U f +g , ∥∇c(xk+1)∥ ≤ U c +g, and λmin(∇c(xk+1)T ∇c(xk+1)) ≥ σ2. By +∥∇f(xk+1)∥ ≤ U f +g , ∥∇c(xk+1)∥ ≤ U c +g, τ g +k ≤ 1, ∥λk∥ ≤ Λ, (25) and (27), one has +ρk∥∇c(xk+1)˜c(xk+1)∥ ≤ ∥∇f(xk+1) + ∇c(xk+1)λk∥ + ∥∇x�L(xk+1, λk; ρk)∥ +(27) +≤ ∥∇f(xk+1)∥ + ∥∇c(xk+1)∥∥λk∥ + τ g +k ≤ U f +g + U c +gΛ + 1. +(76) +In addition, note that λmin(∇c(xk+1)T ∇c(xk+1)) ≥ σ2 implies that ∇c(xk+1)T ∇c(xk+1) is invertible. Using +this fact and (76), we obtain +∥˜c(xk+1)∥ ≤ ∥(∇c(xk+1)T ∇c(xk+1))−1∇c(xk+1)T ∥∥∇c(xk+1)˜c(xk+1)∥ += λmin(∇c(xk+1)T ∇c(xk+1))− 1 +2 ∥∇c(xk+1)˜c(xk+1)∥ +(76) +≤ (U f +g + U c +gΛ + 1)/(σρk). +(77) +We also observe from (75) that ρk ≥ 2(U f +g +U c +gΛ+1)(σǫ1)−1, which along with (77) proves ∥˜c(xk+1)∥ ≤ ǫ1/2. +Combining this with the definition of ˜c in (25) and ∥c(zǫ1)∥ ≤ ǫ1/2, we conclude that ∥c(xk+1)∥ ≤ ǫ1 holds +as desired. +The next lemma provides a stronger upper bound for {ρk} than the one in Theorem 4.3. +Lemma 6.6. Suppose that Assumptions 4.1 and 4.2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf +and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31). Let {ρk} be generated by Algorithm 2 and +˜ρǫ1 := max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 +c ++ 2Λδ−1 +c ++ 2γ, 2(U f +g + U c +gΛ + 1)(σǫ1)−1, 2ρ0}, +(78) +where γ, fhi, flow, δf and δc are given in Assumption 4.1, and U f +g , U c +g and σ are given in (74). Then +ρk ≤ r˜ρǫ1 holds for 0 ≤ k ≤ Kǫ1, where Kǫ1 is defined in (36). +Proof. It follows from (78) that ˜ρǫ1 ≥ 2ρ0. +By this and similar arguments as for (60), one has Kǫ1 ≤ +log(˜ρǫ1ρ−1 +0 )/ log r + 1, where Kǫ1 is defined in (33). Using this, the update scheme for {ρk}, and similar +arguments as for (61), we obtain +max +0≤k≤Kǫ1 +ρk ≤ r˜ρǫ1. +(79) +If ∥c(xKǫ1 +1)∥ ≤ ǫ1, it follows from (36) that Kǫ1 = Kǫ1, which together with (79) implies that ρk ≤ r˜ρǫ1 +holds for 0 ≤ k ≤ Kǫ1. On the other hand, if ∥c(xKǫ1 +1)∥ > ǫ1, it follows from (36) that ∥c(xk+1)∥ > ǫ1 for +Kǫ1 ≤ k ≤ Kǫ1 − 1. This together with Lemma 6.5 and (78) implies that for all Kǫ1 ≤ k ≤ Kǫ1 − 1, +ρk < max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 +c ++ 2Λδ−1 +c ++ 2γ, 2(U f +g + U c +gΛ + 1)(σǫ1)−1} +(78) +≤ ˜ρǫ1. +By this, (79), and ρKǫ1 ≤ rρKǫ1 −1, we also see that ρk ≤ r˜ρǫ1 holds for 0 ≤ k ≤ Kǫ1. +23 + +Proof of Theorem 4.5. Notice from (78) and Lemma 6.6 that ˜ρǫ1 = O(ǫ−1 +1 ) and ρk ≤ r˜ρǫ1, which yield +ρk = O(ǫ−1 +1 ). The conclusion of Theorem 4.5 then follows from this and the same arguments as for the proof +of Theorem 4.4 with ρk = O(ǫ−2 +1 ) replaced by ρk = O(ǫ−1 +1 ). +7 +Future work +There are several possible future studies on this work. First, it would be interesting to extend our AL method +to seek an approximate SOSP of nonconvex optimization with inequality or more general constraints. Indeed, +for nonconvex optimization with inequality constraints, one can reformulate it as an equality constrained +problem using squared slack variables (e.g., see [7]). It can be shown that an SOSP of the latter problem +induces a weak SOSP of the original problem and also linear independence constraint qualification holds for +the latter problem if it holds for the original problem. As a result, it is promising to find an approximate +weak SOSP of an inequality constrained problem by applying our AL method to the equivalent equality +constrained problem. Second, it is worth studying whether the enhanced complexity results in Section 4.3 +can be derived under weaker constraint qualification (e.g., see [5]). Third, the development of our AL method +is based on a strong assumption that a nearly feasible solution of the problem is known. 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Toh, A Newton-CG augmented Lagrangian method for semidefinite +programming, SIAM J. Optim., 20 (2010), pp. 1737–1765. +Appendix +A +A capped conjugate gradient method +In this part we present the capped CG method proposed in [56, Algorithm 1] for finding either an approximate +solution to the linear system (12) or a sufficiently negative curvature direction of the associated matrix H, +which has been briefly discussed in Section 3.1. Its details can be found in [56, Section 3.1]. +The following theorem presents the iteration complexity of Algorithm 3. +Theorem A.1 (iteration complexity of Algorithm 3). Consider applying Algorithm 3 with input U = 0 +to the linear system (12) with g ̸= 0, ε > 0, and H being an n × n symmetric matrix. Then the number of +iterations of Algorithm 3 is �O(min{n, +� +∥H∥/ε}). +27 + +Algorithm 3 A capped conjugate gradient method +Inputs: symmetric matrix H ∈ Rn×n, vector g ̸= 0, damping parameter ε ∈ (0, 1), desired relative accuracy ζ ∈ (0, 1). +Optional input: scalar U ≥ 0 (set to 0 if not provided). +Outputs: d type, d. +Secondary outputs: final values of U, κ, �ζ, τ, and T. +Set +¯ +H := H + 2εI, +κ := U+2ε +ε +, +�ζ := +ζ +3κ, +τ := +√κ +√κ+1, +T := +4κ4 +(1−√τ)2 , +y0 ← 0, r0 ← g, p0 ← −g, j ← 0. +if (p0)T ¯ +Hp0 < ε∥p0∥2 then +Set d ← p0 and terminate with d type = NC; +else if +∥Hp0∥ > U∥p0∥ then +Set U ← ∥Hp0∥/∥p0∥ and update κ, �ζ, τ, T accordingly; +end if +while TRUE do +αj ← (rj)T rj/(pj)T ¯ +Hpj; {Begin Standard CG Operations} +yj+1 ← yj + αjpj; +rj+1 ← rj + αj ¯ +Hpj; +βj+1 ← ∥rj+1∥2/∥rj∥2; +pj+1 ← −rj+1 + βj+1pj; {End Standard CG Operations} +j ← j + 1; +if ∥Hpj∥ > U∥pj∥ then +Set U ← ∥Hpj∥/∥pj∥ and update κ, �ζ, τ, T accordingly; +end if +if +∥Hyj∥ > U∥yj∥ then +Set U ← ∥Hyj∥/∥yj∥ and update κ, �ζ, τ, T accordingly; +end if +if +∥Hrj∥ > U∥rj∥ then +Set U ← ∥Hrj∥/∥rj∥ and update κ, �ζ, τ, T accordingly; +end if +if (yj)T ¯Hyj < ε∥yj∥2 then +Set d ← yj and terminate with d type = NC; +else if +∥rj∥ ≤ �ζ∥r0∥ then +Set d ← yj and terminate with d type = SOL; +else if +(pj)T ¯ +Hpj < ε∥pj∥2 then +Set d ← pj and terminate with d type = NC; +else if +∥rj∥ > +√ +Tτ j/2∥r0∥ then +Compute αj, yj+1 as in the main loop above; +Find i ∈ {0, . . . , j − 1} such that +(yj+1 − yi)T ¯ +H(yj+1 − yi) < ε∥yj+1 − yi∥2; +Set d ← yj+1 − yi and terminate with d type = NC; +end if +end while +Proof. From [56, Lemma 1], we know that the number of iterations of Algorithm 3 is bounded by min{n, J(U, ε, ζ)}, +where J(U, ε, ζ) is the smallest integer J such that +√ +Tτ J/2 ≤ �ζ, with U, �ζ, T and τ being the values returned +by Algorithm 3. In addition, it was shown in [56, Section 3.1] that J(U, ε, ζ) ≤ +��√κ + 1 +2 +� +ln +� +144(√κ+1)2κ6 +ζ2 +�� +, +where κ = O(U/ε) is an output by Algorithm 3. Then one can see that J(U, ε, ζ) = �O( +� +U/ε). Notice from +Algorithm 3 that the output U ≤ ∥H∥. Combining these, we obtain the conclusion as desired. +B +A randomized Lanczos based minimum eigenvalue oracle +In this part we present the randomized Lanczos method proposed in [56, Section 3.2], which can be used as +a minimum eigenvalue oracle for Algorithm 1. As briefly discussed in Section 3.1, this oracle outputs either +a sufficiently negative curvature direction of H or a certificate that H is nearly positive semidefinite with +high probability. More detailed motivation and explanation of it can be found in [56, Section 3.2]. +The following theorem justifies that Algorithm 4 is a suitable minimum eigenvalue oracle for Algorithm +1. Its proof is identical to that of [56, Lemma 2] and thus omitted. +28 + +Algorithm 4 A randomized Lanczos based minimum eigenvalue oracle +Input: symmetric matrix H ∈ Rn×n, tolerance ε > 0, and probability parameter δ ∈ (0, 1). +Output: a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1; or a certificate +that λmin(H) ≥ −ε with probability at least 1 − δ. +Apply the Lanczos method [44] to estimate λmin(H) starting with a random vector uniformly generated on +the unit sphere, and run it for at most +N(ε, δ) := min +� +n, 1 + +� +ln(2.75n/δ2) +2 +� +∥H∥ +ε +�� +(80) +iterations. If a unit vector v with vT Hv ≤ −ε/2 is found at some iteration, terminate immediately and +return v. +Theorem B.1 (iteration complexity of Algorithm 4). Consider Algorithm 4 with tolerance ε > 0, +probability parameter δ ∈ (0, 1), and symmetric matrix H ∈ Rn×n as its input. +Then it either finds a +sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1 or certifies that λmin(H) ≥ +−ε holds with probability at least 1 − δ in at most N(ε, δ) iterations, where N(ε, δ) is defined in (80). +Notice that ∥H∥ is required in Algorithm 4. In general, computing ∥H∥ may not be cheap when n is +large. Nevertheless, ∥H∥ can be efficiently estimated via a randomization scheme with high confidence (e.g., +see the discussion in [56, Appendix B3]). +29 + diff --git a/KNE1T4oBgHgl3EQfYgRo/content/tmp_files/load_file.txt b/KNE1T4oBgHgl3EQfYgRo/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..89da173ac4b3087d224d5331c412ea4114b38fc6 --- /dev/null +++ b/KNE1T4oBgHgl3EQfYgRo/content/tmp_files/load_file.txt @@ -0,0 +1,1593 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf,len=1592 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='03139v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='OC] 9 Jan 2023 A Newton-CG based augmented Lagrangian method for finding a second-order stationary point of nonconvex equality constrained optimization with complexity guarantees Chuan He∗ Zhaosong Lu∗ Ting Kei Pong† April 10, 2022 (Revised: September 22, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' December 31, 2022) Abstract In this paper we consider finding a second-order stationary point (SOSP) of nonconvex equality con- strained optimization when a nearly feasible point is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, we first propose a new Newton- CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys a substantially better complexity than the Newton-CG method [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We then propose a Newton-CG based augmented Lagrangian (AL) method for finding an approximate SOSP of nonconvex equality constrained optimization, in which the proposed Newton-CG method is used as a subproblem solver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We show that under a generalized linear independence constraint qualification (GLICQ), our AL method enjoys a total inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding an (ǫ, √ǫ)-SOSP of nonconvex equality constrained optimization with high probability, which are signif- icantly better than the ones achieved by the proximal AL method [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, we show that it has a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) when the GLICQ does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Preliminary numerical results also demonstrate the superiority of our proposed methods over the ones in [56, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Keywords: Nonconvex equality constrained optimization, second-order stationary point, augmented Lagrangian method, Newton-conjugate gradient method, iteration complexity, operation complexity Mathematics Subject Classification: 49M15, 68Q25, 90C06, 90C26, 90C30, 90C60 1 Introduction In this paper we consider nonconvex equality constrained optimization problem min x∈Rn f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' c(x) = 0, (1) where f : Rn → R and c : Rn → Rm are twice continuously differentiable, and we assume that problem (1) has at least one optimal solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Since (1) is a nonconvex optimization problem, it may have many local but non-global minimizers and finding its global minimizer is generally NP-hard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' A first-order stationary point (FOSP) of it is usually found in practice instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Nevertheless, a mere FOSP may sometimes not suit our needs and a second-order stationary point (SOSP) needs to be sought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For example, in the context of linear semidefinite programming (SDP), a powerful approach to solving it is by solving an equivalent nonconvex ∗Department of Industrial and Systems Engineering, University of Minnesota, USA (email: he000233@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='edu, zhaosong@umn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='edu).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The work of the second author was partially supported by NSF Award IIS-2211491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' †Department of Applied Mathematics, the Hong Kong Polytechnic University, Hong Kong, People’s Republic of China (email: tk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='pong@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='hk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The work of this author was partially supported by a Research Scheme of the Research Grants Council of Hong Kong SAR, China (Project No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' T22-504/21R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1 equality constrained optimization problem [17, 18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It was shown in [18, 15] that under some mild conditions an SOSP of the latter problem can yield an optimal solution of the linear SDP, while a mere FOSP generally cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It is therefore important to find an SOSP of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In recent years, numerous methods with complexity guarantees have been developed for finding an ap- proximate SOSP of several types of nonconvex optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For example, cubic regularized Newton methods [52, 25, 1, 22], accelerated gradient methods [23, 24], trust-region methods [34, 35, 50], quadratic regulariza- tion method [12], second-order line-search method [57], and Newton-conjugate gradient (Newton-CG) method [56] were developed for nonconvex unconstrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, interior-point method [8] and log-barrier method [54] were proposed for nonconvex optimization with sign constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The interior-point method [8] was also generalized in [38] to solve nonconvex optimization with sign constraints and additional linear equality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Furthermore, a projected gradient descent method with random perturbations was proposed in [47] for nonconvex optimization with linear inequality constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Iteration complexity was established for these methods for finding an approximate SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, operation complexity measured by the amount of fundamental operations such as gradient evaluations and matrix-vector products was also studied in [1, 23, 34, 41, 24, 57, 22, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Several methods including trust-region methods [21, 33], sequential quadratic programming method [14], two-phase method [9, 30, 32] and augmented Lagrangian (AL) type methods [4, 10, 58, 60] were proposed for finding an SOSP of problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' However, only a few of them have complexity guarantees for finding an approximate SOSP of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, the inexact AL method [58] has a worst-case complexity in terms of the number of calls to a second-order oracle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Yet its operation complexity, measured by the amount of fundamental operations such as gradient evaluations and Hessian-vector products, is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To the best of our knowledge, the proximal AL method in [60] appears to be the only existing method that enjoys a worst-case complexity for finding an approximate SOSP of (1) in terms of fundamental operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In this method, given an iterate xk and a multiplier estimate λk at the kth iteration, the next iterate xk+1 is obtained by finding an approximate stochastic SOSP of the proximal AL subproblem: min x∈Rn L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρ) + β∥x − xk∥2/2 for some suitable positive ρ and β using a Newton-CG method proposed in [56], where L is the AL function of (1) defined as L(x, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρ) := f(x) + λT c(x) + ρ∥c(x)∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the multiplier estimate is updated using the classical scheme, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', λk+1 = λk + ρc(xk+1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [39, 55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The authors of [60] studied the worst-case complexity of their proximal AL method including: (i) total inner iteration complexity, which measures the total number of iterations of the Newton-CG method [56] performed in their method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) operation complexity, which measures the total number of gradient evaluations and matrix-vector products involving the Hessian of the AL function that are evaluated in their method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Under some suitable assumptions, including that a generalized linear independence constraint qualification (GLICQ) holds at all iterates, it was established in [60] that their proximal AL method enjoys a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−3/4}) for finding an (ǫ, √ǫ)-SOSP of problem (1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 Yet, there is a big gap between these complexities and the iteration complexity of �O(ǫ−3/2) and the operation complexity of �O(ǫ−3/2 min{n, ǫ−1/4}) that are achieved by the methods in [1, 24, 57, 56] for finding an (ǫ, √ǫ)-SOSP of nonconvex unconstrained optimization with high probability, which is a special case of (1) with c ≡ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, there is a lack of complexity guarantees for this proximal AL method when the GLICQ does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It shall be mentioned that Newton-CG based AL methods were also developed for efficiently solving various convex optimization problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [61, 62]), though their complexities remain unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In this paper we propose a Newton-CG based AL method for finding an approximate SOSP of problem (1) with high probability, and study its worst-case complexity with and without the assumption of a GLICQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In 1In fact, a total inner iteration complexity of �O(ǫ−7) and an operation complexity of �O(ǫ−7 min{n, ǫ−1}) were established in [60] for finding an (ǫ, ǫ)-SOSP of problem (1) with high probability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' see [60, Theorem 4(ii), Corollary 3(ii), Theorem 5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Nonetheless, they can be modified to obtain the aforementioned complexity for finding an (ǫ, √ǫ)-SOSP of (1) with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 2 particular, we show that this method enjoys a total inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding a stochastic (ǫ, √ǫ)-SOSP of (1) under the GLICQ, which are significantly better than the aforementioned ones achieved by the proximal AL method in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, when the GLICQ does not hold, we show that it has a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding a stochastic (ǫ, √ǫ)-SOSP of (1), which fills the research gap in this topic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, our AL method (Algorithm 2) proceeds in the following manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Instead of directly solving problem (1), it solves a perturbed problem of (1) with c replaced by its perturbed counterpart ˜c constructed by using a nearly feasible point of (1) (see (25) for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' At the kth iteration, an approximate stochastic SOSP xk+1 of the AL subproblem of this perturbed problem is found by our newly proposed Newton-CG method (Algorithm 1) for a penalty parameter ρk and a truncated Lagrangian multiplier λk, which results from projecting onto a Euclidean ball the standard multiplier estimate ˜λk obtained by the classical scheme ˜λk = λk−1 + ρk˜c(xk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 The penalty parameter ρk+1 is then updated by the following practical scheme (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [7, Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2]): ρk+1 = � rρk if ∥˜c(xk+1)∥ > α∥˜c(xk)∥, ρk otherwise for some r > 1 and α ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It shall be mentioned that in contrast with the classical AL method, our method has two distinct features: (i) the values of the AL function along the iterates are bounded from above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) the multiplier estimates associated with the AL subproblems are bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, to solve the AL subproblems with better complexity guarantees, we propose a variant of the Newton-CG method in [56] for finding an approximate stochastic SOSP of unconstrained optimization, whose complexity has significantly less dependence on the Lipschitz constant of the Hessian of the objective than that of the Newton-CG method in [56], while improving or retaining the same order of dependence on tolerance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Given that such a Lipschitz constant is typically large for the AL subproblems, our Newton-CG method (Algorithm 1) is a much more favorable subproblem solver than the Newton-CG method in [56] that is used in the proximal AL method in [60] from theoretical complexity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The main contributions of this paper are summarized below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We propose a new Newton-CG method for finding an approximate SOSP of unconstrained optimization and show that it enjoys an iteration and operation complexity with a quadratic dependence on the Lipschitz constant of the Hessian of the objective that improves the cubic dependence achieved by the Newton-CG method in [56], while improving or retaining the same order of dependence on tolerance parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, our complexity results are established under the assumption that the Hessian of the objective is Lipschitz continuous in a convex neighborhood of a level set of the objective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This assumption is weaker than the one commonly imposed for the Newton-CG method in [56] and some other methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', [12, 35]) that the Hessian of the objective is Lipschitz continuous in a convex set containing this neighborhood and also all the trial points arising in the line search or trust region steps of the methods (see Section 3 for more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We propose a Newton-CG based AL method for finding an approximate SOSP of nonconvex equality constrained optimization (1) with high probability, and study its worst-case complexity with and without the assumption of a GLICQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Prior to our work, there was no complexity study on finding an approximate SOSP of problem (1) without imposing a GLICQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, under the GLICQ and some other suitable assumptions, we show that our method enjoys a total inner iteration complexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}) for finding an (ǫ, √ǫ)-SOSP of (1) with high probability, which are significantly better than the respective complexity of �O(ǫ−11/2) and �O(ǫ−11/2 min{n, ǫ−3/4}) achieved by the proximal AL method in [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To the best of our knowledge, all the complexity results obtained in this paper are new for finding an approximate SOSP of nonconvex equality constrained optimization with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 2The λk obtained by projecting ˜λk onto a compact set is also called a safeguarded Lagrangian multiplier in the relevant literature [11, 42, 13], which has been shown to enjoy many practical and theoretical advantages (see [11] for discussions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 3 For ease of comparison, we summarize in Table 1 the total inner iteration and operation complexity of our AL method and the proximal AL method in [60] for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) with or without assuming GLICQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Table 1: Total inner iteration and operation complexity of finding a stochastic (ǫ, √ǫ)-SOSP of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Method GLICQ Total inner iteration complexity Operation complexity Proximal AL method [60] ✓ �O(ǫ−11/2) �O(ǫ−11/2 min{n, ǫ−3/4}) Proximal AL method [60] ✗ unknown unknown Our AL method ✓ �O(ǫ−7/2) �O(ǫ−7/2 min{n, ǫ−3/4}) Our AL method ✗ �O(ǫ−11/2) �O(ǫ−11/2 min{n, ǫ−5/4}) It shall be mentioned that there are many works other than [60] studying complexity of AL methods for nonconvex constrained optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' However, they aim to find an approximate FOSP rather than SOSP of the problem (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [40, 37, 13, 51, 45]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Since our main focus is on the complexity of finding an approximate SOSP by AL methods, we do not include them in the above table for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In Section 2, we introduce some notation and optimality conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In Section 3, we propose a Newton-CG method for unconstrained optimization and study its worst-case complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In Section 4, we propose a Newton-CG based AL method for (1) and study its worst-case complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We present numerical results and the proof of the main results in Sections 5 and 6, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In Section 7, we discuss some future research directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 2 Notation and preliminaries Throughout this paper, we let Rn denote the n-dimensional Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We use ∥ · ∥ to denote the Euclidean norm of a vector or the spectral norm of a matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For a real symmetric matrix H, we use λmin(H) to denote its minimum eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The Euclidean ball centered at the origin with radius R ≥ 0 is denoted by BR := {x : ∥x∥ ≤ R}, and we use ΠBR(v) to denote the Euclidean projection of a vector v onto BR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For a given finite set A, we let | A | denote its cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For any s ∈ R, we let sgn(s) be 1 if s ≥ 0 and let it be −1 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, �O(·) represents O(·) with logarithmic terms omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that x∗ is a local minimizer of problem (1) and the linear independence constraint qualification holds at x∗, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', ∇c(x∗) := [∇c1(x∗) ∇c2(x∗) · · · ∇cm(x∗)] has full column rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then there exists a Lagrangian multiplier λ∗ ∈ Rm such that ∇f(x∗) + ∇c(x∗)λ∗ = 0, (2) dT � ∇2f(x∗) + m � i=1 λ∗ i ∇2ci(x∗) � d ≥ 0, ∀d ∈ C(x∗), (3) where C(·) is defined as C(x) := {d ∈ Rn : ∇c(x)T d = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (4) The relations (2) and (3) are respectively known as the first- and second-order optimality conditions for (1) in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [53]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Note that it is in general impossible to find a point that exactly satisfies (2) and (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Thus, we are instead interested in finding a point that satisfies their approximate counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, we introduce the following definitions of an approximate first-order stationary point (FOSP) and second-order stationary point (SOSP), which are similar to those considered in [4, 10, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The rationality of them can be justified by the study of the sequential optimality conditions for constrained optimization [3, 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 (ǫ1-first-order stationary point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let ǫ1 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We say that x ∈ Rn is an ǫ1-first-order stationary point (ǫ1-FOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies ∥∇f(x) + ∇c(x)λ∥ ≤ ǫ1, ∥c(x)∥ ≤ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (5) 4 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 ((ǫ1, ǫ2)-second-order stationary point).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let ǫ1, ǫ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We say that x ∈ Rn is an (ǫ1, ǫ2)- second-order stationary point ((ǫ1, ǫ2)-SOSP) of problem (1) if it, together with some λ ∈ Rm, satisfies (5) and additionally dT � ∇2f(x) + m � i=1 λi∇2ci(x) � d ≥ −ǫ2∥d∥2, ∀d ∈ C(x), (6) where C(·) is defined as in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 3 A Newton-CG method for unconstrained optimization In this section we propose a variant of Newton-CG method [56, Algorithm 3] for finding an approximate SOSP of a class of unconstrained optimization problems, which will be used as a subproblem solver for the AL method proposed in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, we consider an unconstrained optimization problem min x∈Rn F(x), (7) where the function F satisfies the following assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (a) The level set LF (u0) := {x : F(x) ≤ F(u0)} is compact for some u0 ∈ Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (b) The function F is twice Lipschitz continuously differentiable in a convex open neighborhood, denoted by Ω, of LF (u0), that is, there exists LF H > 0 such that ∥∇2F(x) − ∇2F(y)∥ ≤ LF H∥x − y∥, ∀x, y ∈ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (8) By Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, there exist Flow ∈ R, U F g > 0 and U F H > 0 such that F(x) ≥ Flow, ∥∇F(x)∥ ≤ U F g , ∥∇2F(x)∥ ≤ U F H, ∀x ∈ LF(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (9) Recently, a Newton-CG method [56, Algorithm 3] was developed to find an approximate stochastic SOSP of problem (7), which is not only easy to implement but also enjoys a nice feature that the main computation consists only of gradient evaluations and Hessian-vector products associated with the function F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Under the assumption that ∇2F is Lipschitz continuous in a convex open set containing LF (u0) and also all the trial points arising in the line search steps of this method (see [56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Assumption 2]),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' it was established in [56,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Corollary 2] that the iteration and operation complexity of this method for finding a stochastic (ǫg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ǫH)-SOSP of (7) (namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' a point x satisfying ∥∇F(x)∥ ≤ ǫg deterministically and λmin(∇2F(x)) ≥ −ǫH with high probability) are O((LF H)3 max{ǫ−3 g ǫ3 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ǫ−3 H }) and �O((LF H)3 max{ǫ−3 g ǫ3 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ǫ−3 H } min{n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (U F H/ǫH)1/2}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (10) respectively,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' where ǫg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ǫH ∈ (0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1) are prescribed tolerances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Yet, this assumption can be hard to check because these trial points are unknown before the method terminates and moreover the distance between the origin and them depends on the tolerance ǫH in O(ǫ−1 H ) (see [56, Lemma 3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, as seen from (10), iteration and operation complexity of the Newton-CG method in [56] depend cubically on LF H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice that LF H can sometimes be very large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For example, the AL subproblems arising in Algorithm 2 have LF H = O(ǫ−2 1 ) or O(ǫ−1 1 ), where ǫ1 ∈ (0, 1) is a prescribed tolerance for problem (1) (see Section 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The cubic dependence on LF H makes such a Newton-CG method not appealing as an AL subproblem solver from theoretical complexity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In the rest of this section, we propose a variant of the Newton-CG method [56, Algorithm 3] and show that under Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, it enjoys an iteration and operation complexity of O((LF H)2 max{ǫ−2 g ǫH, ǫ−3 H }) and �O((LF H)2 max{ǫ−2 g ǫH, ǫ−3 H } min{n, (U F H/ǫH)1/2}), (11) for finding a stochastic (ǫg, ǫH)-SOSP of problem (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' These complexities are substantially superior to those in (10) achieved by the Newton-CG method in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, the complexities in (11) depend quadratically on LF H, while those in (10) depend cubically on LF H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, it can be verified that they improve or retain the order of dependence on ǫg and ǫH given in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 Main components of a Newton-CG method In this subsection we briefly discuss two main components of the Newton-CG method in [56], which will be used to propose a variant of this method for finding an approximate stochastic SOSP of problem (7) in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The first main component of the Newton-CG method in [56] is a capped CG method [56, Algorithm 1], which is a modified CG method, for solving a possibly indefinite linear system (H + 2εI)d = −g, (12) where 0 ̸= g ∈ Rn, ε > 0, and H ∈ Rn×n is a symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This capped CG method terminates within a finite number of iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It outputs either an approximate solution d to (12) such that ∥(H +2εI)d+g∥ ≤ �ζ∥g∥ and dT Hd ≥ −ε∥d∥2 for some �ζ ∈ (0, 1) or a sufficiently negative curvature direction d of H with dT Hd < −ε∥d∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The second main component of the Newton-CG method in [56] is a minimum eigenvalue oracle that either produces a sufficiently negative curvature direction v of H with ∥v∥ = 1 and vT Hv ≤ −ε/2 or certifies that λmin(H) ≥ −ε holds with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For ease of reference, we present these two components in Algorithms 3 and 4 in Appendices A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Algorithm 1 A Newton-CG method for problem (7) Input: Tolerances ǫg, ǫH ∈ (0, 1), backtracking ratio θ ∈ (0, 1), starting point u0, CG-accuracy parameter ζ ∈ (0, 1), line- search parameter η ∈ (0, 1), probability parameter δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Set x0 = u0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' for t = 0, 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' do if ∥∇F (xt)∥ > ǫg then Call Algorithm 3 with H = ∇2F (xt), ε = ǫH, g = ∇F (xt), accuracy parameter ζ, and U = 0 to obtain outputs d, d type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' if d type=NC then dt ← − sgn(dT ∇F (xt))|dT ∇2F (xt)d| ∥d∥3 d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (13) else {d type=SOL} dt ← d;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (14) end if Go to Line Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else Call Algorithm 4 with H = ∇2F (xt), ε = ǫH, and probability parameter δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' if Algorithm 4 certifies that λmin(∇2F (xt)) ≥ −ǫH then Output xt and terminate;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else {Sufficiently negative curvature direction v returned by Algorithm 4} Set d type=NC and dt ← − sgn(vT ∇F (xt))|vT ∇2F (xt)v|v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (15) Go to Line Search;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if end if Line Search: if d type=SOL then Find αt = θjt, where jt is the smallest nonnegative integer j such that F (xt + θjdt) < F (xt) − ηǫHθ2j∥dt∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (16) else {d type=NC} Find αt = θjt, where jt is the smallest nonnegative integer j such that F (xt + θjdt) < F (xt) − ηθ2j∥dt∥3/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (17) end if xt+1 = xt + αtdt;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end for 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 A Newton-CG method for problem (7) In this subsection we propose a Newton-CG method in Algorithm 1, which is a variant of the Newton-CG method [56, Algorithm 3], for finding an approximate stochastic SOSP of problem (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 6 Our Newton-CG method (Algorithm 1) follows the same framework as [56, Algorithm 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, at each iteration, if the gradient of F at the current iterate is not desirably small, then the capped CG method (Algorithm 3) is called to solve a damped Newton system for obtaining a descent direction and a subsequent line search along this direction results in a sufficient reduction on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Otherwise, the current iterate is already an approximate first-order stationary point of (7), and the minimum eigenvalue oracle (Algorithm 4) is then called, which either produces a sufficiently negative curvature direction for F and a subsequent line search along this direction results in a sufficient reduction on F, or certifies that the current iterate is an approximate SOSP of (7) with high probability and terminates the algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' More details about this framework can be found in [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Despite sharing the same framework, our Newton-CG method and [56, Algorithm 3] use different line search criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, our Newton-CG method uses a hybrid line search criterion adopted from [59], which is a combination of the quadratic descent criterion (16) and the cubic descent criterion (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, it uses the quadratic descent criterion (16) when the search direction is of type ‘SOL’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' On the other hand, it uses the cubic descent criterion (17) when the search direction is of type ‘NC’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 In contrast, the Newton-CG method in [56] always uses a cubic descent criterion regardless of the type of search directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As observed from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 below, our Newton-CG method achieves an iteration and operation complexity given in (11), which are superior to those in (10) achieved by [56, Algorithm 3] in terms of the order dependence on LF H, while improving or retaining the order of dependence on ǫg and ǫH as given in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Consequently, our Newton-CG method is more appealing than [56, Algorithm 3] as an AL subproblem solver for the AL method proposed in Section 4 from theoretical complexity perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following theorem states the iteration and operation complexity of Algorithm 1, whose proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let T1 := � Fhi − Flow min{csol, cnc} max{ǫ−2 g ǫH, ǫ−3 H } � + �Fhi − Flow cnc ǫ−3 H � + 1, T2 := �Fhi − Flow cnc ǫ−3 H � + 1, (18) where Fhi = F(u0), Flow is given in (9), and csol := η min \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 \uf8ee \uf8f0 4 4 + ζ + � (4 + ζ)2 + 8LF H \uf8f9 \uf8fb 2 , �min{6(1 − η), 2}θ LF H �2 \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe , (19) cnc := η 16 min � 1, �min{3(1 − η), 1}θ LF H �2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (20) Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) The total number of calls of Algorithm 4 in Algorithm 1 is at most T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) The total number of calls of Algorithm 3 in Algorithm 1 is at most T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iii) (iteration complexity) Algorithm 1 terminates in at most T1 + T2 iterations with T1 + T2 = O((Fhi − Flow)(LF H)2 max{ǫ−2 g ǫH, ǫ−3 H }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (21) Also, its output xt satisfies ∥∇F(xt)∥ ≤ ǫg deterministically and λmin(∇2F(xt)) ≥ −ǫH with probability at least 1 − δ for some 0 ≤ t ≤ T1 + T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iv) (operation complexity) Algorithm 1 requires at most �O((Fhi − Flow)(LF H)2 max{ǫ−2 g ǫH, ǫ−3 H } min{n, (U F H/ǫH)1/2}) matrix-vector products, where U F H is given in (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 3SOL and NC stand for “approximate solution” and “negative curvature”, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 7 4 A Newton-CG based AL method for problem (1) In this section we propose a Newton-CG based AL method for finding a stochastic (ǫ1, ǫ2)-SOSP of problem (1) for any prescribed tolerances ǫ1, ǫ2 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Before proceeding, we make some additional assumptions on problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (a) An ǫ1/2-approximately feasible point zǫ1 of problem (1), namely satisfying ∥c(zǫ1)∥ ≤ ǫ1/2, is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (b) There exist constants fhi, flow and γ > 0, independent of ǫ1 and ǫ2, such that f(zǫ1) ≤ fhi, (22) f(x) + γ∥c(x)∥2/2 ≥ flow, ∀x ∈ Rn, (23) where zǫ1 is given in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (c) There exist some δf, δc > 0 such that the set S(δf, δc) := {x : f(x) ≤ fhi + δf, ∥c(x)∥ ≤ 1 + δc} (24) is compact with fhi given above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, ∇2f and ∇2ci, i = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', m, are Lipschitz continuous in a convex open neighborhood, denoted by Ω(δf, δc), of S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now make some remarks on Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) A very similar assumption as Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(a) was considered in [31, 37, 49, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By imposing Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(a), we restrict our study on problem (1) for which an ǫ1/2-approximately fea- sible point zǫ1 can be found by an inexpensive procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' One example of such problem instances arises when there exists v0 such that {x : ∥c(x)∥ ≤ ∥c(v0)∥} is compact, ∇2ci, 1 ≤ i ≤ m, is Lipschitz contin- uous on a convex neighborhood of this set, and the LICQ holds on this set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, for this instance, a point zǫ1 satisfying ∥c(zǫ1)∥ ≤ ǫ1/2 can be computed by applying our Newton-CG method (Algorithm 1) to the problem minx∈Rn ∥c(x)∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As seen from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, the resulting iteration and operation com- plexity of Algorithm 1 for finding such zǫ1 are respectively O(ǫ−3/2 1 ) and �O(ǫ−3/2 1 min{n, ǫ−1/4 1 }), which are negligible compared with those of our AL method (see Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As another example, when the standard error bound condition ∥c(x)∥2 = O(∥∇(∥c(x)∥2)∥ν) holds on a level set of ∥c(x)∥ for some ν > 0, one can find the above zǫ1 by applying a gradient method to the problem minx∈Rn ∥c(x)∥2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [46, 58]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, the Newton-CG based AL method (Algorithm 2) pro- posed below is a second-order method with the aim to find a second-order stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It is more expensive than a first-order method in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To make best use of such an AL method in practice, it is natural to run a first-order method in advance to obtain an ǫ1/2-first-order stationary point zǫ1 and then run the AL method using zǫ1 as an ǫ1/2-approximately feasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Therefore, Assump- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(a) is met in practice, provided that an ǫ1/2-first-order stationary point of (1) can be found by a first-order method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(b) is mild.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, the assumption in (22) holds if f(x) ≤ fhi holds for all x with ∥c(x)∥ ≤ 1, which is imposed in [60, Assumption 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It also holds if problem (1) has a known feasible point, which is often imposed for designing AL methods for nonconvex constrained optimization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [49, 31, 48, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, the assumption in (23) implies that the quadratic penalty function is bounded below when the associated penalty parameter is sufficiently large, which is typically used in the study of quadratic penalty and AL methods for solving problem (1) (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [40, 37, 60, 43]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Clearly, when infx∈Rn f(x) > −∞, one can see that (23) holds for any γ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In general, one possible approach to identifying γ is to apply the techniques on infeasibility detection developed in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', [20, 19, 6]) to check the infeasibility of the level set {x : f(x)+γ∥c(x)∥2/2 ≤ ˜flow} for some sufficiently small ˜flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Note that this level set being infeasible for some ˜flow implies that (23) holds for the given γ and flow = ˜flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 8 (iii) Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(c) is not too restrictive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, the set S(δf, δc) is compact if f or f(·)+γ∥c(·)∥2/2 is level-bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The latter level-boundedness assumption is commonly imposed for studying AL methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [37, 60]), which is stronger than our assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next propose a Newton-CG based AL method in Algorithm 2 for finding a stochastic (ǫ1, ǫ2)-SOSP of problem (1) under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Instead of solving (1) directly, this method solves the perturbed problem: min x∈Rn f(x) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ˜c(x) := c(x) − c(zǫ1) = 0, (25) where zǫ1 is given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, at the kth iteration, this method applies the Newton- CG method (Algorithm 1) to find an approximate stochastic SOSP xk+1 of the AL subproblem associated with (25): min x∈Rn ��L(x, λk, ρk) := f(x) + (λk)T ˜c(x) + ρk∥˜c(x)∥2/2 � (26) such that �L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) is below a threshold (see (27) and (28)), where λk is a truncated Lagrangian multiplier, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', the one that results from projecting the standard multiplier estimate ˜λk onto an Euclidean ball (see step 6 of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The standard multiplier estimate ˜λk+1 is then updated by the classical scheme described in step 4 of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Finally, the penalty parameter ρk+1 is adaptively updated based on the improvement on constraint violation (see step 7 of Algorithm 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Such a practical update scheme is often adopted in the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [7, 2, 31]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We would like to point out that the truncated Lagrangian multiplier sequence {λk} is used in the AL subproblems of Algorithm 2 and is bounded, while the standard Lagrangian multiplier sequence {˜λk} is used in those of the classical AL methods and can be unbounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Therefore, Algorithm 2 can be viewed as a safeguarded AL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Truncated Lagrangian multipliers have been used in the literature for designing some AL methods [2, 11, 42, 13], and will play a crucial role in the subsequent complexity analysis of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Algorithm 2 A Newton-CG based AL method for problem (1) Let γ be given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Input: ǫ1, ǫ2 ∈ (0, 1), Λ > 0, x0 ∈ Rn, λ0 ∈ BΛ, ρ0 > 2γ, α ∈ (0, 1), r > 1, δ ∈ (0, 1), and zǫ1 given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1: Set k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 2: Set τ g k = max{ǫ1, rk log ǫ1/ log 2} and τ H k = max{ǫ2, rk log ǫ2/ log 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 3: Call Algorithm 1 with ǫg = τ g k, ǫH = τ H k and u0 = xk init to find an approximate solution xk+1 to minx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) such that �L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ f(zǫ1), ∥∇x�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)∥ ≤ τ g k , (27) λmin(∇2 xx�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)) ≥ −τ H k with probability at least 1 − δ, (28) where xk init = � zǫ1 if �L(xk, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) > f(zǫ1), xk otherwise, for k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (29) 4: Set ˜λk+1 = λk + ρk˜c(xk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 5: If τ g k ≤ ǫ1, τ H k ≤ ǫ2 and ∥c(xk+1)∥ ≤ ǫ1, then output (xk+1, ˜λk+1) and terminate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 6: Set λk+1 = ΠBΛ(˜λk+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 7: If k = 0 or ∥˜c(xk+1)∥ > α∥˜c(xk)∥, set ρk+1 = rρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Otherwise, set ρk+1 = ρk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 8: Set k ← k + 1, and go to step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) Notice that the starting point x0 init of Algorithm 2 can be different from zǫ1 and it may be rather infeasible, though zǫ1 is a nearly feasible point of (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Besides, zǫ1 is used to ensure convergence of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, if the algorithm runs into a “poorly infeasible point” xk, namely satisfying �L(xk, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) > f(zǫ1), it will be superseded by zǫ1 (see (29)), which prevents the iterates {xk} from converging to an infeasible point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Yet, xk may be rather infeasible when k is not large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Thus, Algorithm 2 substantially differs from a funneling or two-phase type algorithm, in which a nearly feasible point 9 is found in Phase 1, and then approximate stationarity is sought while near feasibility is maintained throughout Phase 2 (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [9, 16, 26, 27, 28, 29, 30, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) The choice of ρ0 in Algorithm 2 is mainly for the simplicity of complexity analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Yet, it may be overly large and lead to highly ill-conditioned AL subproblems in practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To make Algorithm 2 practically more efficient, one can possibly modify it by choosing a relatively small initial penalty parameter, then solving the subsequent AL subproblems by a first-order method until an ǫ1-first-order stationary point ˆx of (1) along with a Lagrangian multiplier ˆλ is found, and finally performing the steps described in Algorithm 2 but with x0 = ˆx and λ0 = ΠBΛ(ˆλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Before analyzing the complexity of Algorithm 2, we first argue that it is well-defined if ρ0 is suitably chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, we will show that when ρ0 is sufficiently large, one can apply the Newton-CG method (Algorithm 1) to the AL subproblem minx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) with xk init as the initial point to find an xk+1 satisfying (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To this end, we start by noting from (22), (25), (26) and (29) that �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ max{�L(zǫ1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk), f(zǫ1)} = f(zǫ1) ≤ fhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (30) Based on the above observation, we show in the next lemma that when ρ0 is sufficiently large, �L(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) is bounded below and its certain level set is bounded, whose proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let (λk, ρk) be generated at the kth iteration of Algorithm 2 for some k ≥ 0, and S(δf, δc) and xk init be defined in (24) and (29), respectively, and let fhi, flow, δf and δc be given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 := Λ2/(2ρ0) and δc,1 := � 2(fhi − flow + γ) ρ0 − 2γ + Λ2 (ρ0 − 2γ)2 + Λ ρ0 − 2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (31) Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) {x : �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)} ⊆ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) infx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, we can verify that the Newton-CG method (Algorithm 1), starting with u0 = xk init, is capable of finding an approximate solution xk+1 of the AL subproblem minx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) satisfying (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, let F(·) = �L(·, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) and u0 = xk init.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By these and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, one can see that {x : F(x) ≤ F(u0)} ⊆ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from this and Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(c) that the level set {x : F(x) ≤ F(u0)} is compact and ∇2F is Lipschitz continuous on a convex open neighborhood of {x : F(x) ≤ F(u0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Thus, such F and u0 satisfy Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Based on this and the discussion in Section 3, one can conclude that Algorithm 1, starting with u0 = xk init, is applicable to the AL subproblem minx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Moreover, it follows from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 that this algorithm with (ǫg, ǫH) = (τ g k , τ H k ) can produce a point xk+1 satisfying (28) and also the second relation in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, since this algorithm is descent and its starting point is xk init, its output xk+1 must satisfy �L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk), which along with (30) implies that �L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ f(zǫ1) and thus xk+1 also satisfies the first relation in (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The above discussion leads to the following conclusion concerning the well-definedness of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Under the same settings as in Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, the Newton-CG method (Algorithm 1) applied to the AL subproblem minx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) with u0 = xk init finds a point xk+1 satisfying (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following theorem characterizes the output of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Its proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If Algorithm 2 terminates at some iteration k, then xk+1 is a deterministic ǫ1-FOSP of problem (1), and moreover, it is an (ǫ1, ǫ2)-SOSP of (1) with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 10 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As seen from this theorem, the output of Algorithm 2 is a stochastic (ǫ1, ǫ2)-SOSP of prob- lem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Nevertheless, one can easily modify Algorithm 2 to seek some other approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For example, if one is only interested in finding an ǫ1-FOSP of (1), one can remove the condition (28) from Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, if one aims to find a deterministic (ǫ1, ǫ2)-SOSP of (1), one can replace the condi- tion (28) and Algorithm 1 by λmin(∇2 xx�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)) ≥ −τ H k and a deterministic counterpart, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The purpose of imposing high probability in the condition (28) is to enable us to derive operation complexity of Algorithm 2 measured by the number of matrix-vector products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In the rest of this section, we study the worst-case complexity of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Since our method has two nested loops, particularly, outer loops executed by the AL method and inner loops executed by the Newton-CG method for solving the AL subproblems, we consider the following measures of complexity for Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Outer iteration complexity, which measures the number of outer iterations of Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Total inner iteration complexity, which measures the total number of iterations of the Newton-CG method that are performed in Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Operation complexity, which measures the total number of matrix-vector products involving the Hessian of the augmented Lagrangian function that are evaluated in Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 Outer iteration complexity of Algorithm 2 In this subsection we establish outer iteration complexity of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For notational convenience, we rewrite (τ g k , τ H k ) arising in Algorithm 2 as (τ g k , τ H k ) = (max{ǫ1, ωk 1}, max{ǫ2, ωk 2}) with (ω1, ω2) := (rlog ǫ1/ log 2, rlog ǫ2/ log 2), (32) where ǫ1, ǫ2 and r are the input parameters of Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Since r > 1 and ǫ1, ǫ2 ∈ (0, 1), it is not hard to verify that ω1, ω2 ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, we introduce the following quantity that will be used frequently later: Kǫ1 := � min{k ≥ 0 : ωk 1 ≤ ǫ1} � = ⌈log ǫ1/ log ω1⌉ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (33) In view of (32), (33) and the fact that log ǫ1/ log ω1 = log ǫ2/ log ω2 = log 2/ log r, (34) we see that (τ g k , τ H k ) = (ǫ1, ǫ2) for all k ≥ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This along with the termination criterion of Algorithm 2 implies that it runs for at least Kǫ1 iterations and terminates once ∥c(xk+1)∥ ≤ ǫ1 for some k ≥ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As a result, to establish outer iteration complexity of Algorithm 2, it suffices to bound such k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The resulting outer iteration complexity of Algorithm 2 is presented below, whose proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let ρǫ1 := max � 8(fhi − flow + γ)ǫ−2 1 + 4Λǫ−1 1 + 2γ, 2ρ0 � , (35) Kǫ1 := inf{k ≥ Kǫ1 : ∥c(xk+1)∥ ≤ ǫ1}, (36) where Kǫ1 is defined in (33), and γ, fhi and flow are given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then Kǫ1 is finite, and Algorithm 2 terminates at iteration Kǫ1 with Kǫ1 ≤ �log(ρǫ1ρ−1 0 ) log r + 1 � ����� log(ǫ1(2δc,1)−1) log α ���� + 2 � + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (37) Moreover, ρk ≤ rρǫ1 holds for 0 ≤ k ≤ Kǫ1 Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 (Upper bounds for Kǫ1 and {ρk}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As observed from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3, the number of outer iterations of Algorithm 2 for finding a stochastic (ǫ1, ǫ2)-SOSP of problem (1) is Kǫ1 + 1, which is at most of O(| log ǫ1|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, the penalty parameters {ρk} generated in this algorithm are at most of O(ǫ−2 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 Total inner iteration and operation complexity of Algorithm 2 We present the total inner iteration and operation complexity of Algorithm 2 for finding a stochastic (ǫ1, ǫ2)- SOSP of (1), whose proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) The total number of iterations of Algorithm 1 performed in Algorithm 2 is at most �O(ǫ−4 1 max{ǫ−2 1 ǫ2, ǫ−3 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If c is further assumed to be affine, then it is at most �O(max{ǫ−2 1 ǫ2, ǫ−3 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) The total number of matrix-vector products performed by Algorithm 1 in Algorithm 2 is at most �O(ǫ−4 1 max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1 1 ǫ−1/2 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If c is further assumed to be affine, then it is at most �O(max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1 1 ǫ−1/2 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) Note that the above complexity results of Algorithm 2 are established without assuming any constraint qualification (CQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In contrast, similar complexity results are obtained in [60] for a proximal AL method under a generalized LICQ condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To the best of our knowledge, our work provides the first study on complexity for finding a stochastic SOSP of (1) without CQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) Letting (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), we see that Algorithm 2 achieves a total inner iteration complexity of �O(ǫ−11/2) and an operation complexity of �O(ǫ−11/2 min{n, ǫ−5/4}) for finding a stochastic (ǫ, √ǫ)-SOSP of problem (1) without constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 Enhanced complexity of Algorithm 2 under constraint qualification In this subsection we study complexity of Algorithm 2 under one additional assumption that a generalized linear independence constraint qualification (GLICQ) holds for problem (1), which is introduced below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, under GLICQ we will obtain an enhanced total inner iteration and operation complexity for Algorithm 2, which are significantly better than the ones in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 when problem (1) has nonlinear constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Moreover, when (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), our enhanced complexity bounds are also better than those obtained in [60] for a proximal AL method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now introduce the GLICQ assumption for problem (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 (GLICQ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ∇c(x) has full column rank for all x ∈ S(δf, δc), where S(δf, δc) is as in (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' A related yet different GLICQ is imposed in [60, Assumption 2(ii)] for problem (1), which assumes that ∇c(x) has full column rank for all x in a level set of f(·) + γ∥c(·)∥2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It is not hard to verify that this assumption is generally stronger than the above GLICQ assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following theorem shows that under Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2, the total inner iteration and operation com- plexity results presented in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 can be significantly improved, whose proof is deferred to Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) The total number of iterations of Algorithm 1 performed in Algorithm 2 is at most �O(ǫ−2 1 max{ǫ−2 1 ǫ2, ǫ−3 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If c is further assumed to be affine, then it is at most �O(max{ǫ−2 1 ǫ2, ǫ−3 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) The total number of matrix-vector products performed by Algorithm 1 in Algorithm 2 is at most �O(ǫ−2 1 max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1/2 1 ǫ−1/2 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If c is further assumed to be affine, then it is at most �O(max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1/2 1 ǫ−1/2 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) As seen from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5, when problem (1) has nonlinear constraints, under GLICQ and some other suitable assumptions, Algorithm 2 achieves significantly better complexity bounds than the ones in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 without constraint qualification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 12 (ii) Letting (ǫ1, ǫ2) = (ǫ, √ǫ) for some ǫ ∈ (0, 1), we see that when problem (1) has nonlinear constraints, under GLICQ and some other suitable assumptions, Algorithm 2 achieves a total inner iteration com- plexity of �O(ǫ−7/2) and an operation complexity of �O(ǫ−7/2 min{n, ǫ−3/4}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' They are vastly better than the total inner iteration complexity of �O(ǫ−11/2) and the operation complexity of �O(ǫ−11/2 min{n, ǫ−3/4}) that are achieved by a proximal AL method in [60] for finding a stochastic (ǫ, √ǫ)-SOSP of (1) yet under a generally stronger GLICQ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 5 Numerical results We conduct some preliminary experiments to test the performance of our proposed methods (Algorithms 1 and 2), and compare them with the Newton-CG method in [56] and the proximal AL method in [60], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' All the algorithms are coded in Matlab and all the computations are performed on a desktop with a 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='79 GHz AMD 3900XT 12-Core processor and 32 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 Regularized robust regression In this subsection we consider the regularized robust regression problem min x∈Rn m � i=1 φ(aT i x − bi) + µ∥x∥4 4, (38) where φ(t) = t2/(1 + t2), ∥x∥p = (�n i=1 |xi|p)1/p for any p ≥ 1, and µ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For each triple (n, m, µ), we randomly generate 10 instances of problem (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In particular, we first randomly generate ai, 1 ≤ i ≤ m, with all the entries independently chosen from the standard normal distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We then randomly generate ¯bi according to the standard normal distribution and set bi = 2m¯bi for i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Our aim is to find a (10−5, 10−5/2)-SOSP of (38) for the above instances by Algorithm 1 and the Newton- CG method in [56] and compare their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For a fair comparison, we use a minimum eigenvalue oracle that returns a deterministic output for them so that they both certainly output an approximate second-order stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, we use the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) as the minimum eigenvalue oracle to find the minimum eigenvalue λ and its associated unit eigenvector v of a real symmetric matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, for both methods, we choose the all-ones vector as the initial point, and set θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='8, ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5, and η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The computational results of Algorithm 1 and the Newton-CG method in [56] for the instances randomly generated above are presented in Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In detail, the value of n, m, and µ is listed in the first three columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For each triple (n, m, µ), the average CPU time (in seconds), the average number of iterations, and the average final objective value over 10 random instances are given in the rest of the columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' One can observe that both methods output an approximate solution with a similar objective value, while our Algorithm 1 substantially outperforms the Newton-CG method in [56] in terms of CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This is consistent with our theoretical finding that Algorithm 1 achieves a better iteration complexity than the Newton-CG method in [56] in terms of dependence on the Lipschitz constant of the Hessian for finding an approximate SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 Spherically constrained regularized robust regression In subsection we consider the spherically constrained regularized robust regression problem min x∈Rn m � i=1 φ(aT i x − bi) + µ∥x∥4 4 s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ∥x∥2 2 = 1, (39) where φ(t) = t2/(1 + t2), ∥x∥p = (�n i=1 |xi|p)1/p for any p ≥ 1, and µ > 0 is a tuning parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For each triple (n, m, µ), we randomly generate 10 instances of problem (39) in the same manner as described in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 13 Objective value Iterations CPU time (seconds) n m µ Algorithm 1 Newton-CG Algorithm 1 Newton-CG Algorithm 1 Newton-CG 100 10 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='9 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='9 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6 100 50 1 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='9 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='9 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7 100 90 1 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='8 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='8 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 500 50 5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 42.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='8 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='0 1000 500 10 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 491.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 475.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 1000 900 10 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 891.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 300.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='0 298.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 Table 2: Numerical results for problem (38) Objective value Feasibility violation (×10−4) Total inner iterations CPU time (seconds) n m µ Algorithm 2 Prox-AL Algorithm 2 Prox-AL Algorithm 2 Prox-AL Algorithm 2 Prox-AL 100 10 1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='27 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='9 97.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 558.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6 1000 900 10 893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 893.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='19 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='37 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='8 1281.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='7 1099.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6 Table 3: Numerical results for problem (39) Our aim is to find a (10−4, 10−2)-SOSP of (39) for the above instances by Algorithm 2 and the proximal AL method [60, Algorithm 3] and compare their performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For a fair comparison, we use a minimum eigen- value oracle that returns a deterministic output for them so that they both certainly output an approximate second-order stationary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Specifically, we use the Matlab subroutine [v,λ] = eigs(H,1,’smallestreal’) as the minimum eigenvalue oracle to find the minimum eigenvalue λ and its associated unit eigenvector v of a real symmetric matrix H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, for both methods, we choose the initial point as z0 = (1/√n, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , 1/√n)T , the initial Lagrangian multiplier as λ0 = 0, and the other parameters as Λ = 100, ρ0 = 10, α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='25, and r = 10 for Algorithm 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' η = 1, q = 10 and T0 = 2 for the proximal AL method ([60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The computational results of Algorithm 2 and the proximal AL method in [60] (abbreviated as Prox-AL) for solving problem (39) for the instances randomly generated above are presented in Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In detail, the value of n, m, and µ is listed in the first three columns, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For each triple (n, m, µ), the average CPU time (in seconds), the average total number of inner iterations, the average final objective value, and the average final feasibility violation over 10 random instances are given in the rest columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' One can observe that both methods output an approximate solution of similar quality in terms of objective value and feasibility violation, while our Algorithm 2 vastly outperforms the proximal AL method in [60] in terms of CPU time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This corroborates our theoretical finding that Algorithm 2 achieves a significantly better operation complexity than the proximal AL method in [60] for finding an approximate SOSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 6 Proof of the main results We provide proofs of our main results in Sections 3 and 4, including Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, and Theorems 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 Proof of the main results in Section 3 In this subsection we first establish several technical lemmas and then use them to prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 14 One can observe from Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(b) that for all x and y ∈ Ω, ∥∇F(y) − ∇F(x) − ∇2F(x)(y − x)∥ ≤ LF H∥y − x∥2/2, (40) F(y) ≤ F(x) + ∇F(x)T (y − x) + (y − x)T ∇2F(x)(y − x)/2 + LF H∥y − x∥3/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (41) The next lemma provides useful properties of the output of Algorithm 3, whose proof is similar to the ones in [56, Lemma 3] and [54, Lemma 7] and thus omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and the direction dt results from the output d of Algorithm 3 with a type specified in d type at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) If d type=SOL, then dt satisfies ǫH∥dt∥2 ≤ (dt)T � ∇2F(xt) + 2ǫHI � dt, (42) ∥dt∥ ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1ǫ−1 H ∥∇F(xt)∥, (43) (dt)T ∇F(xt) = −(dt)T � ∇2F(xt) + 2ǫHI � dt, (44) ∥(∇2F(xt) + 2ǫHI)dt + ∇F(xt)∥ ≤ ǫHζ∥dt∥/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (45) (ii) If d type=NC, then dt satisfies (dt)T ∇F(xt) ≤ 0 and (dt)T ∇2F(xt)dt/∥dt∥2 = −∥dt∥ ≤ −ǫH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (46) The next lemma shows that when the search direction dt in Algorithm 1 is of type ‘SOL’, the line search step results in a sufficient reduction on F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and the direction dt results from the output d of Algorithm 3 with d type=SOL at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let U F g and csol be given in (9) and (19), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) The step length αt is well-defined, and moreover, αt ≥ min � 1, � min{6(1 − η), 2} 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1LF HU F g θǫH � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (47) (ii) The next iterate xt+1 = xt + αtdt satisfies F(xt) − F(xt+1) ≥ csol min{∥∇F(xt+1)∥2ǫ−1 H , ǫ3 H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (48) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' One can observe that F is descent along the iterates (whenever well-defined) generated by Algorithm 1, which together with x0 = u0 implies that F(xt) ≤ F(u0) and hence ∥∇F(xt)∥ ≤ U F g due to (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, since dt results from the output d of Algorithm 3 with d type=SOL, one can see that ∥∇F(xt)∥ > ǫg and (42)-(45) hold for dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Moreover, by ∥∇F(xt)∥ > ǫg and (45), one can conclude that dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We first prove statement (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If (16) holds for j = 0, then αt = 1, which clearly implies that (47) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now suppose that (16) fails for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Claim that for all j ≥ 0 that violate (16), it holds that θ2j ≥ min{6(1 − η), 2}ǫH(LF H)−1∥dt∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (49) Indeed, suppose that (16) is violated by some j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now show that (49) holds for such j by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 1) F(xt + θjdt) > F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let φ(α) = F(xt + αdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then φ(θj) > φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, since dt ̸= 0, by (42) and (44), one has φ′(0) = ∇F(xt)T dt = −(dt)T (∇2F(xt) + 2ǫHI)dt ≤ −ǫH∥dt∥2 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 15 Using these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of φ such that φ′(α∗) = ∇F(xt + α∗dt)T dt = 0 and φ(α∗) < φ(0), which implies that F(xt + α∗dt) < F(xt) ≤ F(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (40) holds for x = xt and y = xt + α∗dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, 0 < α∗ < θj ≤ 1 and ∇F(xt + α∗dt)T dt = 0, we obtain (α∗)2LF H 2 ∥dt∥3 (40) ≥ ∥dt∥∥∇F(xt + α∗dt) − ∇F(xt) − α∗∇2F(xt)dt∥ ≥ (dt)T (∇F(xt + α∗dt) − ∇F(xt) − α∗∇2F(xt)dt) = −(dt)T ∇F(xt) − α∗(dt)T ∇2F(xt)dt (44) = (1 − α∗)(dt)T (∇2F(xt) + 2ǫHI)dt + 2α∗ǫH∥dt∥2 (42) ≥ (1 + α∗)ǫH∥dt∥2 ≥ ǫH∥dt∥2, which along with dt ̸= 0 implies that (α∗)2 ≥ 2ǫH(LF H)−1∥dt∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this and θj > α∗, we conclude that (49) holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 2) F(xt + θjdt) ≤ F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This together with F(xt) ≤ F(u0) implies that (41) holds for x = xt and y = xt + θjdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then, because j violates (16), we obtain −ηǫHθ2j∥dt∥2 ≤ F(xt + θjdt) − F(xt) (41) ≤ θj∇F(xt)T dt + θ2j 2 (dt)T ∇2F(xt)dt + LF H 6 θ3j∥dt∥3 (44) = −θj(dt)T (∇2F(xt) + 2ǫHI)dt + θ2j 2 (dt)T ∇2F(xt)dt + LF H 6 θ3j∥dt∥3 = −θj � 1 − θj 2 � (dt)T (∇2F(xt) + 2ǫHI)dt − θ2jǫH∥dt∥2 + LF H 6 θ3j∥dt∥3 (42) ≤ −θj � 1 − θj 2 � ǫH∥dt∥2 − θ2jǫH∥dt∥2 + LF H 6 θ3j∥dt∥3 ≤ −θjǫH∥dt∥2 + LF H 6 θ3j∥dt∥3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (50) Recall that dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Dividing both sides of (50) by LF Hθj∥dt∥3/6 and using η, θ ∈ (0, 1), we obtain that θ2j ≥ 6(1 − θjη)ǫH(LF H)−1∥dt∥−1 ≥ 6(1 − η)ǫH(LF H)−1∥dt∥−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (49) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Combining the above two cases, we conclude that (49) holds for any j ≥ 0 that violates (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (16) must be bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that the step length αt associated with (16) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next prove (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Observe from the definition of jt in Algorithm 1 that j = jt − 1 violates (16) and hence (49) holds for j = jt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then, by (49) with j = jt − 1 and αt = θjt, one has αt = θjt ≥ � min{6(1 − η), 2}ǫH(LF H)−1 θ∥dt∥−1/2, (51) which, along with (43) and ∥∇F(xt)∥ ≤ U F g , implies (47).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This proves statement (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next prove statement (ii) by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 1) αt = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, one knows that (16) holds for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that F(xt + dt) ≤ F(xt) ≤ F(u0), which implies that (40) holds for x = xt and y = xt + dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (45), one has ∥∇F(xt+1)∥ = ∥∇F(xt + dt)∥ ≤ ∥∇F(xt + dt) − ∇F(xt) − ∇2F(xt)dt∥ +∥(∇2F(xt) + 2ǫHI)dt + ∇F(xt)∥ + 2ǫH∥dt∥ ≤ LF H 2 ∥dt∥2 + 4+ζ 2 ǫH∥dt∥, where the last inequality follows from (40) and (45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Solving the above inequality for ∥dt∥ and using the fact that ∥dt∥ > 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' we obtain that ∥dt∥ ≥ −(4+ζ)ǫH+√ (4+ζ)2ǫ2 H+8LF H∥∇F (xt+1)∥ 2LF H ≥ −(4+ζ)ǫH+√ (4+ζ)2ǫ2 H+8LF Hǫ2 H 2LF H min{∥∇F(xt+1)∥/ǫ2 H,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1} = 4 4+ζ+√ (4+ζ)2+8LF H min{∥∇F(xt+1)∥/ǫH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ǫH},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' where the second inequality follows from the inequality −a + √ a2 + bs ≥ (−a + √ a2 + b) min{s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1} for all a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' s ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' which can be verified by performing a rationalization to the terms −a+ √ a2 + b and −a+ √ a2 + bs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, αt = 1, (16) and (19), one can see that (48) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 16 Case 2) αt < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that j = 0 violates (16) and hence (49) holds for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Now, letting j = 0 in (49), we obtain that ∥dt∥ ≥ min{6(1 − η), 2}ǫH/LF H, which together with (16) and (51) implies that F(xt) − F(xt+1) ≥ ηǫHθ2jt∥dt∥2 ≥ η min{6(1 − η), 2}ǫ2 H LF H θ2∥dt∥ ≥ η �min{6(1 − η), 2}θ LF H �2 ǫ3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (19), one can see that (48) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following lemma shows that when the search direction dt in Algorithm 1 is of type ‘NC’, the line search step results in a sufficient reduction on F as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds and the direction dt results from either the output d of Algorithm 3 with d type=NC or the output v of Algorithm 4 at some iteration t of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let cnc be defined in (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) The step length αt is well-defined, and αt ≥ min{1, θ/LF H, 3(1 − η)θ/LF H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) The next iterate xt+1 = xt + αtdt satisfies F(xt) − F(xt+1) ≥ cncǫ3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Observe that F is descent along the iterates (whenever well-defined) generated by Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this and x0 = u0, we have F(xt) ≤ F(u0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By the assumption on dt, one can see from Algorithm 1 that dt is a negative curvature direction given in (13) or (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, notice that the vector v returned from Algorithm 4 satisfies ∥v∥ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By these, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(ii), (13) and (15), one can observe that ∇F(xt)T dt ≤ 0, (dt)T ∇2F(xt)dt = −∥dt∥3 < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (52) We first prove statement (i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If (17) holds for j = 0, then αt = 1, which clearly implies that αt ≥ min{1, θ/LF H, 3(1 − η)θ/LF H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now suppose that (17) fails for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Claim that for all j ≥ 0 that violate (17), it holds that θj ≥ min{1/LF H, 3(1 − η)/LF H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (53) Indeed, suppose that (17) is violated by some j ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We now show that (53) holds for such j by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 1) F(xt + θjdt) > F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let φ(α) = F(xt + αdt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then φ(θj) > φ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, by (52), one has φ′(0) = ∇F(xt)T dt ≤ 0, φ′′(0) = (dt)T ∇2F(xt)dt < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using these, we can observe that there exists a local minimizer α∗ ∈ (0, θj) of φ such that φ(α∗) < φ(0), namely, F(xt + α∗dt) < F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By the second-order optimality condition of φ at α∗, one has φ′′(α∗) = (dt)T ∇2F(xt + α∗dt)dt ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Since F(xt + α∗dt) < F(xt) ≤ F(u0), it follows that (8) holds for x = xt and y = xt + α∗dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, the second relation in (52) and (dt)T ∇2F(xt + α∗dt)dt ≥ 0, we obtain that in (52) and (dt)T ∇2F(xt + α∗dt)dt ≥ 0, we obtain that LF Hα∗∥dt∥3 (8) ≥ ∥dt∥2∥∇2F(xt + α∗dt) − ∇2F(xt)∥ ≥ (dt)T (∇2F(xt + α∗dt) − ∇2F(xt))dt ≥ −(dt)T ∇2F(xt)dt = ∥dt∥3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (54) Recall from (52) that dt ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (54) that α∗ ≥ 1/LF H, which along with θj > α∗ implies that θj > 1/LF H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (53) holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 2) F(xt + θjdt) ≤ F(xt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It follows from this and F(xt) ≤ F(u0) that (41) holds for x = xt and y = xt + θjdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and the fact that j violates (17), one has − η 2θ2j∥dt∥3 ≤ F(xt + θjdt) − F(xt) (41) ≤ θj∇F(xt)T dt + θ2j 2 (dt)T ∇2F(xt)dt + LF H 6 θ3j∥dt∥3 (52) ≤ − θ2j 2 ∥dt∥3 + LF H 6 θ3j∥dt∥3, which together with dt ̸= 0 implies that θj ≥ 3(1 − η)/LF H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (53) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 17 Combining the above two cases, we conclude that (53) holds for any j ≥ 0 that violates (17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and θ ∈ (0, 1), one can see that all j ≥ 0 that violate (17) must be bounded above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that the step length αt associated with (17) is well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next derive a lower bound for αt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from the definition of jt in Algorithm 1 that j = jt − 1 violates (17) and hence (53) holds for j = jt − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then, by (53) with j = jt − 1 and αt = θjt, one has αt = θjt ≥ min{θ/LF H, 3(1 − η)θ/LF H}, which immediately yields αt ≥ min{1, θ/LF H, 3(1 − η)θ/LF H} as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next prove statement (ii) by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 1) dt results from the output d of Algorithm 3 with d type=NC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (46) that ∥dt∥ ≥ ǫH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This together with (17) and statement (i) implies that statement (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 2) dt results from the output v of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from Algorithm 4 that ∥v∥ = 1 and vT ∇2F(xt)v ≤ −ǫH/2, which along with (15) yields ∥dt∥ ≥ ǫH/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, (17) and statement (i), one can see that statement (ii) again holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' For notational convenience, we let {xt}t∈T denote all the iterates generated by Algo- rithm 1, where T is a set of consecutive nonnegative integers starting from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice that F is descent along the iterates generated by Algorithm 1, which together with x0 = u0 implies that xt ∈ {x : F(x) ≤ F(u0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (9) that ∥∇2F(xt)∥ ≤ U F H holds for all t ∈ T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) Suppose for contradiction that the total number of calls of Algorithm 4 in Algorithm 1 is more than T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from Algorithm 1 and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3(ii) that each of these calls, except the last one, returns a sufficiently negative curvature direction, and each of them results in a reduction on F of at least cncǫ3 H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, T2cncǫ3 H ≤ � t∈T [F(xt) − F(xt+1)] ≤ F(x0) − Flow = Fhi − Flow, which contradicts the definition of T2 given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, statement (i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) Suppose for contradiction that the total number of calls of Algorithm 3 in Algorithm 1 is more than T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Observe that if Algorithm 3 is called at some iteration t and generates the next iterate xt+1 satisfying ∥∇F(xt+1)∥ ≤ ǫg, then Algorithm 4 must be called at the next iteration t + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In view of this and statement (i) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, we see that the total number of such iterations t is at most T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, the total number of iterations t of Algorithm 1 at which Algorithm 3 is called and generates the next iterate xt+1 satisfying ∥∇F(xt+1)∥ > ǫg is at least T1 − T2 + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Moreover, for each of such iterations t, we observe from Lemmas 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2(ii) and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3(ii) that F(xt) − F(xt+1) ≥ min{csol, cnc} min{ǫ2 gǫ−1 H , ǫ3 H}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that (T1 − T2 + 1) min{csol, cnc} min{ǫ2 gǫ−1 H , ǫ3 H} ≤ � t∈T [F(xt) − F(xt+1)] ≤ Fhi − Flow, which contradicts the definition of T1 and T2 given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, statement (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iii) Notice that either Algorithm 3 or 4 is called at each iteration of Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It follows from this and statements (i) and (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 that the total number of iterations of Algorithm 1 is at most T1 +T2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, the relation (21) follows from (19), (20) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' One can also observe that the output xt of Algorithm 1 satisfies ∥∇F(xt)∥ ≤ ǫg deterministically and λmin(∇2F(xt)) ≥ −ǫH with probability at least 1 − δ for some 0 ≤ t ≤ T1 + T2, where the latter part is due to Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This completes the proof of statement (ii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iv) By Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 with (H, ε) = (∇2F(xt), ǫH) and the fact that ∥∇2F(xt)∥ ≤ U F H, one can observe that the number of Hessian-vector products required by each call of Algorithm 3 with input U = 0 is at most �O(min{n, (U F H/ǫH)1/2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, by Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 with (H, ε) = (∇2F(xt), ǫH), ∥∇2F(xt)∥ ≤ U F H, and the fact that each iteration of the Lanczos method requires only one matrix-vector product, one can observe that the number of Hessian-vector products required by each call of Algorithm 4 is also at most �O(min{n, (U F H/ǫH)1/2}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Based on these observations and statement (iii) of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, we see that statement (iv) of this theorem holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 18 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 Proof of the main results in Section 4 Recall from Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(a) that ∥c(zǫ1)∥ ≤ ǫ1/2 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By virtue of this, (23) and the definition of ˜c in (25), we obtain that f(x) + γ∥˜c(x)∥2 ≥ f(x) + γ∥c(x)∥2/2 − γ∥c(zǫ1)∥2 ≥ flow − γ, ∀x ∈ Rn .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (55) We now prove the following auxiliary lemma that will be used frequently later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let γ, fhi and flow be given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Assume that ρ > 2γ, λ ∈ Rm, and x ∈ Rn satisfy �L(x, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρ) ≤ fhi, (56) where �L is defined in (26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the following statements hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) f(x) ≤ fhi + ∥λ∥2/(2ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) ∥˜c(x)∥ ≤ � 2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iii) If ρ ≥ ∥λ∥2/(2˜δf) for some ˜δf > 0, then f(x) ≤ fhi + ˜δf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iv) If ρ ≥ 2(fhi − flow + γ)˜δ−2 c + 2∥λ∥˜δ−1 c + 2γ (57) for some ˜δc > 0, then ∥˜c(x)∥ ≤ ˜δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) It follows from (56) and the definition of �L in (26) that fhi ≥ f(x) + λT ˜c(x) + ρ 2∥˜c(x)∥2 = f(x) + ρ 2 ���˜c(x) + λ ρ ��� 2 − ∥λ∥2 2ρ ≥ f(x) − ∥λ∥2 2ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, statement (i) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) In view of (55) and (56), one has fhi (56) ≥ f(x) + λT ˜c(x) + ρ 2∥˜c(x)∥2 = f(x) + γ∥˜c(x)∥2 + ρ−2γ 2 ���˜c(x) + λ ρ−2γ ��� 2 − ∥λ∥2 2(ρ−2γ) (55) ≥ flow − γ + ρ−2γ 2 ���˜c(x) + λ ρ−2γ ��� 2 − ∥λ∥2 2(ρ−2γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that ���˜c(x) + λ ρ−2γ ��� ≤ � 2(fhi−flow+γ) ρ−2γ + ∥λ∥2 (ρ−2γ)2 , which implies that statement (ii) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iii) Statement (iii) immediately follows from statement (i) and ρ ≥ ∥λ∥2/(2˜δf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (iv) Suppose that (57) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Multiplying both sides of (57) by ˜δ2 c and rearranging the terms, we have (ρ − 2γ)˜δ2 c − 2∥λ∥˜δc − 2(fhi − flow + γ) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Recall that ρ > 2γ and ˜δc > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Solving this inequality for ˜δc yields ˜δc ≥ � 2(fhi − flow + γ)/(ρ − 2γ) + ∥λ∥2/(ρ − 2γ)2 + ∥λ∥/(ρ − 2γ), which along with statement (ii) implies that ∥˜c(x)∥ ≤ ˜δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, statement (iv) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) Let x be any point such that �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (30) that �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ fhi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, ∥λk∥ ≤ Λ, ρk ≥ ρ0 > 2γ, δf,1 ≤ δf, δc,1 ≤ δc, and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 with (λ, ρ) = (λk, ρk), one has f(x) ≤ fhi + ∥λk∥2/(2ρk) ≤ fhi + Λ2/(2ρ0) = fhi + δf,1 ≤ fhi + δf, ∥˜c(x)∥ ≤ � 2(fhi−flow+γ) ρk−2γ + ∥λk∥2 (ρk−2γ)2 + ∥λk∥ ρk−2γ ≤ � 2(fhi−flow+γ) ρ0−2γ + Λ2 (ρ0−2γ)2 + Λ ρ0−2γ = δc,1 ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (58) Also, recall from the definition of ˜c in (25) and ∥c(zǫ1)∥ ≤ 1 that ∥c(x)∥ ≤ 1 + ∥˜c(x)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This together with the above inequalities and (24) implies x ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, statement (i) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 19 (ii) Note that inf x∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) = inf x∈Rn{�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) : �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Consequently, to prove statement (ii) of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, it suffices to show that inf x∈Rn{�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) : �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)} ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (59) To this end, let x be any point satisfying �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We then know from (58) that ∥˜c(x)∥ ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, ∥λk∥ ≤ Λ, ρk > 2γ, and (55), one has �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) = f(x) + γ∥˜c(x)∥2 + (λk)T ˜c(x) + ρk−2γ 2 ∥˜c(x)∥2 ≥ f(x) + γ∥˜c(x)∥2 − Λ∥˜c(x)∥ ≥ flow − γ − Λδc, and hence (59) holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Algorithm 2 terminates at some iteration k, that is, τg k ≤ ǫ1, τ H k ≤ ǫ2, and ∥c(xk+1)∥ ≤ ǫ1 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then, by τg k ≤ ǫ1, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c and the second relation in (27), one has ∥∇f(xk+1) + ∇c(xk+1)˜λk+1∥ = ∥∇f(xk+1) + ∇˜c(xk+1)(λk + ρk˜c(xk+1))∥ = ∥∇x�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)∥ ≤ τ g k ≤ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (xk+1, ˜λk+1) satisfies the first relation in (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, by (28) and τ H k ≤ ǫ2, one can show that λmin(∇2 xx�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)) ≥ −ǫ2 with probability at least 1 − δ, which leads to dT ∇2 xx�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)d ≥ −ǫ2∥d∥2 for all d ∈ Rn with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, ˜λk+1 = λk + ρk˜c(xk+1), ∇˜c = ∇c, and ∇2˜ci = ∇2ci for 1 ≤ i ≤ m, we see that with probability at least 1 − δ, it holds that dT � ∇2f(xk+1) + m � i=1 ˜λk+1 i ∇2ci(xk+1) + ρk∇c(xk+1)∇c(xk+1)T � d ≥ −ǫ2∥d∥2 ∀d ∈ Rn, which implies dT (∇2f(xk+1) + �m i=1 ˜λk+1 i ∇2ci(xk+1))d ≥ −ǫ2∥d∥2 for all d ∈ C(xk+1), where C(·) is defined in (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (xk+1, ˜λk+1) satisfies (6) with probability at least 1−δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Combining these with ∥c(xk+1)∥ ≤ ǫ1, we conclude that xk+1 is a deterministic ǫ1-FOSP of (1) and an (ǫ1, ǫ2)-SOSP of (1) with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It follows from (35) that ρǫ1 ≥ 2ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, one has Kǫ1 (33) = ⌈log ǫ1/ log ω1⌉ (32) = ⌈log 2/ log r⌉ ≤ log(ρǫ1ρ−1 0 )/ log r + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (60) Notice that {ρk} is either unchanged or increased by a ratio r as k increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this fact and (60), we see that max 0≤k≤Kǫ1 ρk ≤ rKǫ1 ρ0 (60) ≤ r log(ρǫ1 ρ−1 0 ) log r +1ρ0 = rρǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (61) In addition, notice that ρk > 2γ and ∥λk∥ ≤ Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using these, (22), the first relation in (27), and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4(ii) with (x, λ, ρ) = (xk+1, λk, ρk), we obtain that ∥˜c(xk+1)∥ ≤ � 2(fhi−flow+γ) ρk−2γ + ∥λk∥2 (ρk−2γ)2 + ∥λk∥ ρk−2γ ≤ � 2(fhi−flow+γ) ρk−2γ + Λ2 (ρk−2γ)2 + Λ ρk−2γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (62) Also, we observe from ∥c(zǫ1)∥ ≤ ǫ1/2 and the definition of ˜c in (25) that ∥c(xk+1)∥ ≤ ∥˜c(xk+1)∥ + ∥c(zǫ1)∥ ≤ ∥˜c(xk+1)∥ + ǫ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (63) We now prove that Kǫ1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose for contradiction that Kǫ1 is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from this and (36) that ∥c(xk+1)∥ > ǫ1 for all k ≥ Kǫ1, which along with (63) implies that ∥˜c(xk+1)∥ > ǫ1/2 for all k ≥ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows that ∥˜c(xk+1)∥ > α∥˜c(xk)∥ must hold for infinitely many k’s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this and the update scheme on {ρk}, we deduce that ρk+1 = rρk holds for infinitely many k’s, which together with the 20 monotonicity of {ρk} implies that ρk → ∞ as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (62), one can see that ∥˜c(xk+1)∥ → 0 as k → ∞, which contradicts the fact that ∥˜c(xk+1)∥ > ǫ1/2 holds for all k ≥ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, Kǫ1 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, notice from (32), (33) and (34) that (τ g k , τ H k ) = (ǫ1, ǫ2) for all k ≥ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This along with the termination criterion of Algorithm 2 and the definition of Kǫ1 implies that Algorithm 2 must terminate at iteration Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next show that (37) and ρk ≤ rρǫ1 hold for 0 ≤ k ≤ Kǫ1 by considering two separate cases below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 1) ∥c(xKǫ1 +1)∥ ≤ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (36), one can see that Kǫ1 = Kǫ1, which together with (60) and (61) implies that (37) and ρk ≤ rρǫ1 hold for 0 ≤ k ≤ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Case 2) ∥c(xKǫ1 +1)∥ > ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (36), one can observe that Kǫ1 > Kǫ1 and also ∥c(xk+1)∥ > ǫ1 for all Kǫ1 ≤ k ≤ Kǫ1 − 1, which together with (63) implies ∥˜c(xk+1)∥ > ǫ1/2, ∀Kǫ1 ≤ k ≤ Kǫ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (64) It then follows from ∥λk∥ ≤ Λ, (22), the first relation in (27), and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4(iv) with (x, λ, ρ, ˜δc) = (xk+1, λk, ρk, ǫ1/2) that ρk < 8(fhi − flow + γ)ǫ−2 1 + 4∥λk∥ǫ−1 1 + 2γ ≤ 8(fhi − flow + γ)ǫ−2 1 + 4Λǫ−1 1 + 2γ (35) ≤ ρǫ1, ∀Kǫ1 ≤ k ≤ Kǫ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (65) Combining this relation, (61), and the fact ρKǫ1 ≤ rρKǫ1 −1, we conclude that ρk ≤ rρǫ1 holds for 0 ≤ k ≤ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It remains to show that (37) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To this end, let K = {k : ρk+1 = rρk, Kǫ1 ≤ k ≤ Kǫ1 − 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It follows from (65) and the update scheme of ρk that r| K |ρKǫ1 = max Kǫ1 ≤k≤Kǫ1 −1 {ρk} ≤ ρǫ1, which together with ρKǫ1 ≥ ρ0 implies that | K | ≤ log(ρǫ1ρ−1 Kǫ1 )/ log r ≤ log(ρǫ1ρ−1 0 )/ log r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (66) Let {k1, k2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , k| K |} denote all the elements of K arranged in ascending order, and let k0 = Kǫ1 and k| K |+1 = Kǫ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We next derive an upper bound for kj+1 − kj for j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By the definition of K, one can observe that ρk = ρk′ for kj < k, k′ ≤ kj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this and the update scheme of ρk, we deduce that ∥˜c(xk+1)∥ ≤ α∥˜c(xk)∥, ∀kj < k < kj+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (67) On the other hand, by (31), (62) and ρk ≥ ρ0, one has ∥˜c(xk+1)∥ ≤ δc,1 for 0 ≤ k ≤ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and (64), one can see that ǫ1/2 < ∥˜c(xk+1)∥ ≤ δc,1, ∀Kǫ1 ≤ k ≤ Kǫ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (68) Now, note that either kj+1 − kj = 1 or kj+1 − kj > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In the latter case, we can apply (67) with k = kj+1 − 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , kj + 1 together with (68) to deduce that ǫ1/2 < ∥˜c(xkj+1)∥ ≤ α∥˜c(xkj+1−1)∥ ≤ · · · ≤ αkj+1−kj−1∥˜c(xkj+1)∥ ≤ αkj+1−kj−1δc,1, ∀j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Combining these two cases, we have kj+1 − kj ≤ | log(ǫ1(2δc,1)−1))/ log α| + 1, ∀j = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', | K |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (69) Summing up these inequalities, and using (60), (66), k0 = Kǫ1 and k| K |+1 = Kǫ1 − 1, we have Kǫ1 = 1 + k| K |+1 = 1 + k0 + �| K | j=0(kj+1 − kj) (69) ≤ 1 + Kǫ1 + (| K | + 1) ���� log(ǫ1(2δc,1)−1) log α ��� + 1 � ≤ 2 + log(ρǫ1 ρ−1 0 ) log r + � log(ρǫ1 ρ−1 0 ) log r + 1 � ���� log(ǫ1(2δc,1)−1) log α ��� + 1 � = 1 + � log(ρǫ1 ρ−1 0 ) log r + 1 � ���� log(ǫ1(2δc,1)−1) log α ��� + 2 � , where the second inequality is due to (60) and (66).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hence, (37) also holds in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 21 We next prove Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Before proceeding, we introduce some notation that will be used shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let Lk,H denote the Lipschitz constant of ∇2 xx�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) on the convex open neighborhood Ω(δf, δc) of S(δf, δc), where S(δf, δc) is defined in (24), and let Uk,H = supx∈S(δf ,δc) ∥∇2 xx�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from (25) and (26) that ∇2 xx�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) = ∇2f(x) + m � i=1 λk i ∇2ci(x) + ρk � ∇c(x)∇c(x)T + m � i=1 ˜ci(x)∇2ci(x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (70) By this, ∥λk∥ ≤ Λ, the definition of ˜c, and the Lipschitz continuity of ∇2f and ∇2ci’s (see Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(c)), one can observe that there exist some constants L1, L2, U1 and U2, depending only on f, c, Λ, δf and δc, such that Lk,H ≤ L1 + ρkL2, Uk,H ≤ U1 + ρkU2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (71) Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let Tk and Nk denote the number of iterations and matrix-vector products performed by Algorithm 1 at the outer iteration k of Algorithm 2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 that the total number of iterations and matrix-vector products performed by Algorithm 1 in Algorithm 2 are �Kǫ1 k=0 Tk and �Kǫ1 k=0 Nk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, notice from (35) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 that ρǫ1 = O(ǫ−2 1 ) and ρk ≤ rρǫ1, which yield ρk = O(ǫ−2 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' We first claim that (τ g k )2/τ H k ≥ min{ǫ2 1/ǫ2, ǫ3 2} holds for any k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, let ¯t = log ǫ1/ log ω1 and ψ(t) = max{ǫ1, ωt 1}2/ max{ǫ2, ωt 2} for all t ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (34) that ω¯t 1 = ǫ1 and ω¯t 2 = ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and ω1, ω2 ∈ (0, 1), one can observe that ψ(t) = (ω2 1/ω2)t if t ≤ ¯t and ψ(t) = ǫ2 1/ǫ2 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This along with ǫ2 ∈ (0, 1) implies that min t∈[0,∞) ψ(t) = min{ψ(0), ψ(¯t)} = min{1, ǫ2 1/ǫ2} ≥ min{ǫ2 1/ǫ2, ǫ3 2}, which together with (32) yields (τ g k )2/τ H k = ψ(k) ≥ min{ǫ2 1/ǫ2, ǫ3 2} for all k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (i) From Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(i) and the definitions of Ω(δf, δc) and Lk,H, we see that Lk,H is a Lipschitz constant of ∇2 xx�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) on a convex open neighborhood of {x : �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Also, recall from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(ii) that infx∈Rn �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≥ flow − γ − Λδc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By these, �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ fhi (see (30)) and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(iii) with (Fhi, Flow, LF H, ǫg, ǫH) = (�L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk), flow − γ − Λδc, Lk,H, τ g k , τ H k ), one has Tk = O((fhi − flow + γ + Λδc)L2 k,H max{(τ g k )−2τ H k , (τ H k )−3}) (71) = O(ρ2 k max{(τ g k )−2τ H k , (τ H k )−3}) = O(ǫ−4 1 max{ǫ−2 1 ǫ2, ǫ−3 2 }), (72) where the last equality is from (τ g k )2/τ H k ≥ min{ǫ2 1/ǫ2, ǫ3 2}, τ H k ≥ ǫ2, and ρk = O(ǫ−2 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Next, if c(x) = Ax − b for some A ∈ Rm×n and b ∈ Rm, then ∇c(x) = AT and ∇2ci(x) = 0 for 1 ≤ i ≤ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By these and (70), one has Lk,H = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this and similar arguments as for (72), we obtain that Tk = O(max{ǫ−2 1 ǫ2, ǫ−3 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, (72) and Kǫ1 = O(| log ǫ1|2) (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4), we conclude that statement (i) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (ii) In view of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(i) and the definition of Uk,H, one can see that Uk,H ≥ sup x∈Rn{∥∇2 xx�L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)∥ : �L(x, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, �L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk) ≤ fhi and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(iv) with (Fhi, Flow, LF H, U F H, ǫg, ǫH) = (�L(xk init, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk), flow− γ − Λδc, Lk,H, Uk,H, τ g k , τ H k ), we obtain that Nk = �O((fhi − flow + γ + Λδc)L2 k,H max{(τ g k )−2τ H k , (τ H k )−3} min{n, (Uk,H/τ H k )1/2}) (71) = �O(ρ2 k max{(τ g k )−2τ H k , (τ H k )−3} min{n, (ρk/τ H k )1/2}) = �O(ǫ−4 1 max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1 1 ǫ−1/2 2 }), (73) where the last equality is from (τ g k )2/τ H k ≥ min{ǫ2 1/ǫ2, ǫ3 2}, τ H k ≥ ǫ2, and ρk = O(ǫ−2 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' On the other hand, if c is assumed to be affine, it follows from the above discussion that Lk,H = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Us- ing this, Uk,H ≤ U1+ρkU2, and similar arguments as for (73), we obtain that Nk = �O(max{ǫ−2 1 ǫ2, ǫ−3 2 } min{n, ǫ−1 1 ǫ−1/2 2 }).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, (73) and Kǫ1 = O(| log ǫ1|2) (see Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4), we conclude that statement (ii) of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 22 Next, we provide a proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' To proceed, we first observe from Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1(c) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 that there exist U f g > 0, U c g > 0 and σ > 0 such that ∥∇f(x)∥ ≤ U f g , ∥∇c(x)∥ ≤ U c g, λmin(∇c(x)T ∇c(x)) ≥ σ2, ∀x ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (74) We next establish several technical lemmas that will be used shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let {(xk, λk, ρk)} be generated by Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that ρk ≥ max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2Λδ−1 c + 2γ, 2(U f g + U c gΛ + 1)(σǫ1)−1} (75) for some k ≥ 0, where γ, fhi, flow, δf and δc are given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, and U f g , U c g and σ are given in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then it holds that ∥c(xk+1)∥ ≤ ǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By (75) and ∥λk∥ ≤ Λ (see step 6 of Algorithm 2), one can see that ρk ≥ max{∥λk∥2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2∥λk∥δ−1 c + 2γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, (22), the first relation in (27), and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4(iii) and (iv) with (x, λ, ρ, ˜δf, ˜δc) = (xk+1, λk, ρk, δf, δc), we obtain that f(xk+1) ≤ fhi + δf and ∥˜c(xk+1)∥ ≤ δc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, recall from ∥c(zǫ1)∥ ≤ 1 and the defini- tion of ˜c in (25) that ∥c(xk+1)∥ ≤ 1 + ∥˜c(xk+1)∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' These together with (24) show that xk+1 ∈ S(δf, δc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It then follows from (74) that ∥∇f(xk+1)∥ ≤ U f g , ∥∇c(xk+1)∥ ≤ U c g, and λmin(∇c(xk+1)T ∇c(xk+1)) ≥ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By ∥∇f(xk+1)∥ ≤ U f g , ∥∇c(xk+1)∥ ≤ U c g, τ g k ≤ 1, ∥λk∥ ≤ Λ, (25) and (27), one has ρk∥∇c(xk+1)˜c(xk+1)∥ ≤ ∥∇f(xk+1) + ∇c(xk+1)λk∥ + ∥∇x�L(xk+1, λk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' ρk)∥ (27) ≤ ∥∇f(xk+1)∥ + ∥∇c(xk+1)∥∥λk∥ + τ g k ≤ U f g + U c gΛ + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (76) In addition, note that λmin(∇c(xk+1)T ∇c(xk+1)) ≥ σ2 implies that ∇c(xk+1)T ∇c(xk+1) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this fact and (76), we obtain ∥˜c(xk+1)∥ ≤ ∥(∇c(xk+1)T ∇c(xk+1))−1∇c(xk+1)T ∥∥∇c(xk+1)˜c(xk+1)∥ = λmin(∇c(xk+1)T ∇c(xk+1))− 1 2 ∥∇c(xk+1)˜c(xk+1)∥ (76) ≤ (U f g + U c gΛ + 1)/(σρk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (77) We also observe from (75) that ρk ≥ 2(U f g +U c gΛ+1)(σǫ1)−1, which along with (77) proves ∥˜c(xk+1)∥ ≤ ǫ1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Combining this with the definition of ˜c in (25) and ∥c(zǫ1)∥ ≤ ǫ1/2, we conclude that ∥c(xk+1)∥ ≤ ǫ1 holds as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The next lemma provides a stronger upper bound for {ρk} than the one in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Suppose that Assumptions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2 hold and that ρ0 is sufficiently large such that δf,1 ≤ δf and δc,1 ≤ δc, where δf,1 and δc,1 are defined in (31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Let {ρk} be generated by Algorithm 2 and ˜ρǫ1 := max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2Λδ−1 c + 2γ, 2(U f g + U c gΛ + 1)(σǫ1)−1, 2ρ0}, (78) where γ, fhi, flow, δf and δc are given in Assumption 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, and U f g , U c g and σ are given in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then ρk ≤ r˜ρǫ1 holds for 0 ≤ k ≤ Kǫ1, where Kǫ1 is defined in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It follows from (78) that ˜ρǫ1 ≥ 2ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this and similar arguments as for (60), one has Kǫ1 ≤ log(˜ρǫ1ρ−1 0 )/ log r + 1, where Kǫ1 is defined in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Using this, the update scheme for {ρk}, and similar arguments as for (61), we obtain max 0≤k≤Kǫ1 ρk ≤ r˜ρǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' (79) If ∥c(xKǫ1 +1)∥ ≤ ǫ1, it follows from (36) that Kǫ1 = Kǫ1, which together with (79) implies that ρk ≤ r˜ρǫ1 holds for 0 ≤ k ≤ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' On the other hand, if ∥c(xKǫ1 +1)∥ > ǫ1, it follows from (36) that ∥c(xk+1)∥ > ǫ1 for Kǫ1 ≤ k ≤ Kǫ1 − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' This together with Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 and (78) implies that for all Kǫ1 ≤ k ≤ Kǫ1 − 1, ρk < max{Λ2(2δf)−1, 2(fhi − flow + γ)δ−2 c + 2Λδ−1 c + 2γ, 2(U f g + U c gΛ + 1)(σǫ1)−1} (78) ≤ ˜ρǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' By this, (79), and ρKǫ1 ≤ rρKǫ1 −1, we also see that ρk ≤ r˜ρǫ1 holds for 0 ≤ k ≤ Kǫ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 23 Proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from (78) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='6 that ˜ρǫ1 = O(ǫ−1 1 ) and ρk ≤ r˜ρǫ1, which yield ρk = O(ǫ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The conclusion of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='5 then follows from this and the same arguments as for the proof of Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='4 with ρk = O(ǫ−2 1 ) replaced by ρk = O(ǫ−1 1 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 7 Future work There are several possible future studies on this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' First, it would be interesting to extend our AL method to seek an approximate SOSP of nonconvex optimization with inequality or more general constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Indeed, for nonconvex optimization with inequality constraints, one can reformulate it as an equality constrained problem using squared slack variables (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It can be shown that an SOSP of the latter problem induces a weak SOSP of the original problem and also linear independence constraint qualification holds for the latter problem if it holds for the original problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As a result, it is promising to find an approximate weak SOSP of an inequality constrained problem by applying our AL method to the equivalent equality constrained problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Second, it is worth studying whether the enhanced complexity results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='3 can be derived under weaker constraint qualification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [5]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Third, the development of our AL method is based on a strong assumption that a nearly feasible solution of the problem is known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' It would make the method applicable to a broader class of problems if such an assumption could be removed by modifying the method possibly through the use of infeasibility detection techniques (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see [19]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Lastly, more numerical studies would be helpful to further improve our AL method from a practical perspective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Agarwal, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Allen-Zhu, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Bullins, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Hazan, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Ma, Finding approximate 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', 7 (2015), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 331–366.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' [62] X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Zhao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Sun, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Toh, A Newton-CG augmented Lagrangian method for semidefinite programming, SIAM J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Optim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', 20 (2010), pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 1737–1765.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Appendix A A capped conjugate gradient method In this part we present the capped CG method proposed in [56, Algorithm 1] for finding either an approximate solution to the linear system (12) or a sufficiently negative curvature direction of the associated matrix H, which has been briefly discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Its details can be found in [56, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following theorem presents the iteration complexity of Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 (iteration complexity of Algorithm 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Consider applying Algorithm 3 with input U = 0 to the linear system (12) with g ̸= 0, ε > 0, and H being an n × n symmetric matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then the number of iterations of Algorithm 3 is �O(min{n, � ∥H∥/ε}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 27 Algorithm 3 A capped conjugate gradient method Inputs: symmetric matrix H ∈ Rn×n, vector g ̸= 0, damping parameter ε ∈ (0, 1), desired relative accuracy ζ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Optional input: scalar U ≥ 0 (set to 0 if not provided).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Outputs: d type, d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Secondary outputs: final values of U, κ, �ζ, τ, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Set ¯ H := H + 2εI, κ := U+2ε ε , �ζ := ζ 3κ, τ := √κ √κ+1, T := 4κ4 (1−√τ)2 , y0 ← 0, r0 ← g, p0 ← −g, j ← 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' if (p0)T ¯ Hp0 < ε∥p0∥2 then Set d ← p0 and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else if ∥Hp0∥ > U∥p0∥ then Set U ← ∥Hp0∥/∥p0∥ and update κ, �ζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if while TRUE do αj ← (rj)T rj/(pj)T ¯ Hpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' {Begin Standard CG Operations} yj+1 ← yj + αjpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' rj+1 ← rj + αj ¯ Hpj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' βj+1 ← ∥rj+1∥2/∥rj∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' pj+1 ← −rj+1 + βj+1pj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' {End Standard CG Operations} j ← j + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' if ∥Hpj∥ > U∥pj∥ then Set U ← ∥Hpj∥/∥pj∥ and update κ, �ζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if if ∥Hyj∥ > U∥yj∥ then Set U ← ∥Hyj∥/∥yj∥ and update κ, �ζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if if ∥Hrj∥ > U∥rj∥ then Set U ← ∥Hrj∥/∥rj∥ and update κ, �ζ, τ, T accordingly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if if (yj)T ¯Hyj < ε∥yj∥2 then Set d ← yj and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else if ∥rj∥ ≤ �ζ∥r0∥ then Set d ← yj and terminate with d type = SOL;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else if (pj)T ¯ Hpj < ε∥pj∥2 then Set d ← pj and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' else if ∥rj∥ > √ Tτ j/2∥r0∥ then Compute αj, yj+1 as in the main loop above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Find i ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' , j − 1} such that (yj+1 − yi)T ¯ H(yj+1 − yi) < ε∥yj+1 − yi∥2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Set d ← yj+1 − yi and terminate with d type = NC;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' end if end while Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' From [56, Lemma 1], we know that the number of iterations of Algorithm 3 is bounded by min{n, J(U, ε, ζ)}, where J(U, ε, ζ) is the smallest integer J such that √ Tτ J/2 ≤ �ζ, with U, �ζ, T and τ being the values returned by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In addition, it was shown in [56, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1] that J(U, ε, ζ) ≤ ��√κ + 1 2 � ln � 144(√κ+1)2κ6 ζ2 �� , where κ = O(U/ε) is an output by Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then one can see that J(U, ε, ζ) = �O( � U/ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice from Algorithm 3 that the output U ≤ ∥H∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Combining these, we obtain the conclusion as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' B A randomized Lanczos based minimum eigenvalue oracle In this part we present the randomized Lanczos method proposed in [56, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2], which can be used as a minimum eigenvalue oracle for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' As briefly discussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1, this oracle outputs either a sufficiently negative curvature direction of H or a certificate that H is nearly positive semidefinite with high probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' More detailed motivation and explanation of it can be found in [56, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' The following theorem justifies that Algorithm 4 is a suitable minimum eigenvalue oracle for Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Its proof is identical to that of [56, Lemma 2] and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 28 Algorithm 4 A randomized Lanczos based minimum eigenvalue oracle Input: symmetric matrix H ∈ Rn×n, tolerance ε > 0, and probability parameter δ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Output: a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' or a certificate that λmin(H) ≥ −ε with probability at least 1 − δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Apply the Lanczos method [44] to estimate λmin(H) starting with a random vector uniformly generated on the unit sphere, and run it for at most N(ε, δ) := min � n, 1 + � ln(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='75n/δ2) 2 � ∥H∥ ε �� (80) iterations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' If a unit vector v with vT Hv ≤ −ε/2 is found at some iteration, terminate immediately and return v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Theorem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='1 (iteration complexity of Algorithm 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Consider Algorithm 4 with tolerance ε > 0, probability parameter δ ∈ (0, 1), and symmetric matrix H ∈ Rn×n as its input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Then it either finds a sufficiently negative curvature direction v satisfying vT Hv ≤ −ε/2 and ∥v∥ = 1 or certifies that λmin(H) ≥ −ε holds with probability at least 1 − δ in at most N(ε, δ) iterations, where N(ε, δ) is defined in (80).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Notice that ∥H∥ is required in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' In general, computing ∥H∥ may not be cheap when n is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' Nevertheless, ∥H∥ can be efficiently estimated via a randomization scheme with high confidence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=', see the discussion in [56, Appendix B3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} +page_content=' 29' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/KNE1T4oBgHgl3EQfYgRo/content/2301.03139v1.pdf'} diff --git a/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/2301.01762v1.pdf.txt b/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/2301.01762v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..4ce2d8ef31dc4ee9d3d26b935fc53855b73cc97f --- /dev/null +++ b/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/2301.01762v1.pdf.txt @@ -0,0 +1,2466 @@ +1 +Modeling Sequential Recommendation as +Missing Information Imputation +Yujie Lin, Zhumin Chen, Zhaochun Ren, Chenyang Wang, Qiang Yan, Maarten de Rijke, Xiuzhen Cheng, +Fellow, IEEE, and Pengjie Ren +Abstract—Side information is being used extensively to improve the effectiveness of sequential recommendation models. It is said to +help capture the transition patterns among items. Most previous work on sequential recommendation that uses side information models +item IDs and side information separately. This can only model part of relations between items and their side information. Moreover, in +real-world systems, not all values of item feature fields are available. This hurts the performance of models that rely on side +information. Existing methods tend to neglect the context of missing item feature fields, and fill them with generic or special values, +e.g., unknown, which might lead to sub-optimal performance. To address the limitation of sequential recommenders with side +information, we define a way to fuse side information and alleviate the problem of missing side information by proposing a unified task, +namely the missing information imputation (MII), which randomly masks some feature fields in a given sequence of items, including +item IDs, and then forces a predictive model to recover them. By considering the next item as a missing feature field, sequential +recommendation can be formulated as a special case of MII. We propose a sequential recommendation model, called missing +information imputation recommender (MIIR), that builds on the idea of MII and simultaneously imputes missing item feature values and +predicts the next item. We devise a dense fusion self-attention (DFSA) for MIIR to capture all pairwise relations between items and +their side information. Empirical studies on three benchmark datasets demonstrate that MIIR, supervised by MII, achieves a +significantly better sequential recommendation performance than state-of-the-art baselines. +Index Terms—Sequential recommendation, side information fusion, missing information imputation +! +1 +INTRODUCTION +S +EQUENTIAL recommendation models transition patterns +among items and generates a recommendation for the +next item [1]. Traditional sequential recommendation so- +lutions use the item ID as the only item feature field [2, +3, 4, 5, 6, 7, 8]. In real-world cases, however, there is +rich side information in the form of multiple types of +structural feature fields, such as categories and brands, +and unstructured feature fields, e.g., titles and descriptions, +that can help to better model transitions between items. +In recent years, several publications have exploited side +information to improve sequential recommendation perfor- +mance [9, 10, 11, 12, 13, 14, 15]. Most focus on designing +different mechanisms to fuse side information into rec- +ommendation models. For example, Hidasi et al. [9] use +parallel recurrent neural networks (RNNs) [16] to encode +• +Yujie Lin, School of Computer Science and Technology, Shandong Univer- +sity, Qingdao, China, E-mail: yu.jie.lin@outlook.com +• +Zhumin Chen, School of Computer Science and Technology, Shandong +University, Qingdao, China, E-mail: chenzhumin@sdu.edu.cn +• +Zhaochun Ren, School of Computer Science and Technology, Shandong +University, Qingdao, China, E-mail: zhaochun.ren@sdu.edu.cn +• +Chenyang Wang, School of Computer Science and Technology, Shandong +University, Qingdao, China, E-mail: 201900122032@mail.sdu.edu.cn +• +Qiang +Yan, +WeChat, +Tencent, +Guangzhou, +China, +E-mail: +rolanyan@tencent.com +• +Maarten de Rijke, Informatics Institute, University of Amsterdam, Ams- +terdam, The Netherlands, E-mail: m.derijke@uva.nl +• +Xiuzhen Cheng, School of Computer Science and Technology, Shandong +University, Qingdao, China, E-mail: xzcheng@sdu.edu.cn +• +Pengjie Ren, School of Computer Science and Technology, Shandong +University, Qingdao, China, E-mail: renpengjie@sdu.edu.cn +the information in item IDs and attributes, respectively, and +then combine the outputs of RNNs for item recommenda- +tion. Zhang et al. [10] employ two groups of self-attention +blocks [17] for modeling items and features, and fuse them +in the final stage. +Importantly, previous work for sequential recommenda- +tion with side information usually regards side information +as an auxiliary representation of the item, so models item +IDs and side information separately. As a result, such meth- +ods only encode partial relations in item sequences, e.g., the +relation between an item and its side information, while the +relation between an item and the side information of other +items in the sequence is not well captured. +Even more importantly, previous studies often assume +that all side information is available, which is rarely the +case in real-world scenarios. As illustrated in Fig. 1(a), +i.e., the second and third items lack category and title +information, respectively. Previous work has proposed to +fill such gaps with special values, such as a general category +and a padding text, to make models trainable and produce +outputs. However, for different items and item sequences, +these special values are the same: they do not provide useful +and specific information for recommendations and might +introduce biases into the model learning instead [18]. As a +result, as illustrated in Fig. 1(b), a model might recommend +the wrong item. Instead, we propose to impute the missing +side information, so that the recommendation model can use +information from missing feature fields based on contexts, +as illustrated in Fig. 1(c). +Some recent studies address the probem of missing +side information in recommendation data. Wang et al. [19] +arXiv:2301.01762v1 [cs.IR] 4 Jan 2023 + +2 +(a) Original sequence. +(b) Existing work without imputation. +(c) Our work with imputation. +Fig. 1. Sequential recommendation of items with side information. Gray +blocks represent missing information. “[PAD]” (in (b)) indicates padding +with generic or special values as often done in existing work. “[Impute]” +(in (c)) indicates imputation with actual values for missing feature fields. +employ an auto-encoder (AE) with a modality dropout +to recover the missing rating and side information. Shi +et al. [18] propose an adaptive feature sampling strategy +to introduce more missing feature fields into the training +process, which increases the robustness of the recommen- +dation model against missing side information. Wu et al. +[20] define item recommendation and attribute inference in +a user-item bipartite graph with attributes, and propose a +graph convolutional network (GCN) [21] based model to +join these two tasks. However, the work just listed mainly +targets non-sequential recommendation. Moreover, it treats +item recommendation and side information imputation as +different tasks. +In this work, we seek to design a sequential recommen- +dation model that can handle missing feature fields of items +in items sequences. The main challenge is how to adap- +tively impute missing information, including missing side +information and the next item, according to the information +available in the item sequence. First, we propose a task, +the missing information imputation (MII) task that randomly +masks some non-missing feature fields, including item IDs, +in the input sequence, and then asks the model to recover +them in the output. Since the next item to be recommended +can also be seen as a missing feature field in the sequence, +MII unifies the missing side information imputation task +with the next item prediction task. MII can be considered +as the extension of the masked item prediction task [22] +that only considers and masks item IDs. Based on the +MII task, we propose a sequential recommendation model, +called missing information imputation recommender (MIIR), +that jointly imputes missing side information and predicts +the next item for the given item sequence. MIIR employs +a dense fusion self-attention (DFSA) mechanism to fuse the +information in IDs and other feature fields for predicting +both missing side information and the next item. DFSA +captures the relation between any pair of feature fields in +the input sequence, allowing it to fully fuse various types +of (side) information to impute missing feature values and +address the main recommendation challenge. +We conduct extensive experiments on three public +datasets and show that MIIR significantly outperforms +state-of-the-art sequential recommendation baselines. We +also confirm that (i) imputing missing side information +and (ii) DFSA both help to improve the performance of +sequential recommendation. +The main contributions of this work are as follows: +• We propose to unify the missing side information imputa- +tion task and the sequential recommendation task through +missing information imputation (MII). To the best of our +knowledge, this is the first work of its kind in sequential +recommendation. +• We present a novel sequential recommendation model, +missing information imputation recommender (MIIR), +that employs MII to provide the signal for simultaneously +imputing the missing item side information and predict- +ing the next item and dense fusion self-attention (DFSA) +to fuse various information. +• We conduct extensive experiments to verify the effective- +ness of MII, MIIR, and DFSA in sequential recommenda- +tion. +2 +RELATED WORK +2.1 +Sequential recommendation with side information +Side information fusion has been widely used in sequential +recommendation because it can help to capture transition +patterns among items. We classify existing work into work +that uses self-attention and work that does not. +As to work that does not use self-attention, Hidasi et al. +[9] employ parallel RNNs to extract the information from +ID sequences of item IDs and sequences of features; they +then examine different ways of combining the outputs of +the RNNs. Zhou et al. [23] propose self-supervised tasks +to maximize the mutual information between an item and +its attributes or between a sequence of item IDs and the +sequence of their attributes. Yuan et al. [24] construct a het- +erogeneous graph to aggregate different types of categorical +attributes, then aggregate the representations of attribute +types to get item representations. +Inspired by the success of self-attention mechanisms +[25, 26, 27], some work uses self-attention to fuse items and +side information. Zhang et al. [10] first use a vanilla atten- +tion mechanism to fuse different types of side information +on each item, and then use two branches of self-attention +blocks to model transition patterns between IDs and side +information; they then concatenate the hidden states of the +two blocks for item recommendation. Liu et al. [28] pro- +pose a non-invasive self-attention mechanism that uses pure +item ID representations as values and representations that +integrate side information as queries and keys to calculate +the attention. Xie et al. [15] decouple the non-invasive self- +attention of different types of side information to get fused +attention matrices for items. +Although many methods have been proposed for se- +quential recommendation with side information, they (i) ne- +glect the missing information problem, and use fixed special +values to fill missing feature fields, which might harm the +performance, and (ii) hardly explore the relation between +an item and the side information of other items in the same +sequence. These are aspects that we contribute on top of +prior work. + +Title +Title +Truth +Title +Item +Item +Item +Item +Category +Category +CategoryTitle +Title +[PAD] +Title +Predict +Item +Item +Item +Item +Category +[PAD] +Category +CategoryTitle +Title +[Impute] +Title +Predict +Item +Item +Item +Item +Category +Category +[Impute] +Category3 +2.2 +Missing side information in recommendation +In real-world applications, the side information of users and +items may be incomplete or missing, which may hurt the +performance of recommendation models that rely on side +information. +The traditional way to solve the problem of missing side +information is to fill the missing feature fields with heuristic +values [29, 30, 18], such as the most frequent feature values, +average values, randomized values, the value unknown, +or padding. As some studies have reported, these special +values are independent of the context, and using them +may lead to biased parameter estimation and prediction +[31, 32]. Another way to deal with missing feature fields +is to impute their missing values. Early approaches use +KNN-based methods [33] or auto-encoders (AEs) [34, 35] +to predict the missing data. Wang et al. [19] propose an AE- +based model with modality dropout, which randomly drops +representations of user or item information of different +modalities in hidden states and reconstructs them by an AE. +Cao et al. [36] present a translation-based recommendation +model that models preferences as translations from users to +items, and jointly trains it with a knowledge graph (KG) +completion model that predicts the missing relations in the +KG for incorporating knowledge into the recommendation +model. Instead of imputing the missing side information, +Shi et al. [18] propose an adaptive feature sampling strategy, +which employs layer-wise relevance propagation [37] to +calculate the importance of different features and samples +features to make the model more robust against unknown +features. Wu et al. [20] propose a GCN-based model to +jointly predict users’ preferences to items and predict the +missing attribute values of users or items. +What we add on top of prior work on missing infor- +mation in recommendation is that we focus on missing +information in the context of sequential recommendation. +3 +METHOD +3.1 +Overview +Before going into details of the proposed MII task and +MIIR model, we introduce notation used in this paper. +We denote the item set as I = {i1, . . . , iNi}, where Ni +is the number of items and each item ID ik ∈ RNi is +represented as a one-hot vector. In addition to IDs, items +have other feature fields corresponding to their side infor- +mation. In this work, we consider categorical feature fields, +including category and brand, and textual feature fields, +including title and description. We denote the category set +as C = {c1, . . . , cNc}, where Nc is the number of categories +and each category ck ∈ RNc is a one-hot vector. Similarly, +we denote the brand set as B = {b1, . . . , bNb}, where Nb +is the number of brands and each brand bk ∈ RNb. For +titles and descriptions of items, we employ BERT [38] to +encode them into fixed-length vectors of size 768. We denote +all titles and all descriptions as T = {t1, . . . , tNi} and +D = {d1, . . . , dNi}, respectively, where tk and dk ∈ R768. +We use S = [s1, . . . , sn] to denote a sequence with n items, +where sk = [si +k, sc +k, sb +k, st +k, sd +k] is the sequence of features +fields of the k-th item, si +k ∈ I, sc +k ⊆ C, sb +k ∈ B, st +k ∈ T, +and sd +k ∈ D. As an item may have multiple categories, +(a) Sequential recommendation task. +(b) Missing information imputation task. +Fig. 2. Comparing the sequential recommendation task and the missing +information imputation task. (Same visual conventions as in Fig. 1.) +we let sc +k be a subset of C, which can be represented as +a multi-hot vector sc +k ∈ RNc. For missing item IDs, cate- +gories and brands, we have special one-hot vectors denoted +as imiss ∈ I, cmiss ∈ C and bmiss ∈ B, respectively. +For missing titles and descriptions, we use the vector of +“[CLS][SEP]” encoded by BERT to represent them, which +are denoted as tmiss ∈ T and dmiss ∈ D, respectively. These +missing representations will be used in both MIIR and the +baselines. It is worth noting that other feature fields can be +formalized and modeled in a similar way. +The +missing +information +imputation +task +is +to +im- +pute +the +values +of +the +missing +feature +fields +in S. +The sequential recommendation task is to predict the next +item sn+1 for S. By appending a new item sn+1 += +[imiss, cmiss, bmiss, tmiss, dmiss] to the end of S and im- +puting the imiss of sn+1, we can formulate the next item +prediction task as a special case of missing information +imputation task. In Fig. 2, we compare the sequential rec- +ommendation task and the missing information imputation +task. In the sequential recommendation task, the next item is +not considered as a missing data. In the missing information +imputation task, the next item is simply a missing feature +field. A model for the missing information imputation task +that follows a unified way to impute both the next item +and the other missing side information can be used for +sequential recommendation. +To unify the missing side information imputation and +next item recommendation tasks, we propose a sequential +recommendation model called missing information imputa- +tion recommender (MIIR). As we illustrate in Fig. 3, MIIR +consists of three main components: (i) an embedding layer, +(ii) a dense fusion self-attention (DFSA) mechanims, and + +Fusion +T +ID +Side information808 +Missing information imputation +0808 +8.084 +Fig. 3. Architecture of the missing information imputation recommender +(MIIR). MIIR takes a sequence of randomly masked feature fields as +input. It transforms the input sequence into embeddings using the em- +bedding layer. Then it employs a dense fusion self-attention mechanism +to fuse information in the sequence. Finally, MIIR uses an output layer +to reconstruct the input sequence and calculate the MII loss on masked +feature fields. (Same visual conventions as in Fig. 1.) +(iii) an output layer. First, the embedding layer translates +the input sequence into a series of embeddings. Then, the +DFSA mechanism employs several transformer [17] layers +to model the relation between any pair of feature fields in +the sequence and fuse side information into the model for +both imputation and recommendation. Finally, the output +layer imputes the missing feature values including item IDs +in the sequence based on the output of DFSA. Next, we will +introduce the details of these main components. +3.2 +Embedding layer +The embedding layer projects all item feature fields in the +input sequence into low-dimensional dense vectors with a +unified length. +For the k-th item sk = [si +k, sc +k, sb +k, st +k, sd +k] in the given +sequence S, the embedding layer uses different ways to +translate different feature fields. For the high-dimensional +sparse vectors of si +k, sc +k and sb +k, we follow Eq. 1 to get the +item embedding ei +k ∈ Re, the category embedding ec +k ∈ Re, +and the brand embedding eb +k ∈ Re: +ei +k = Eisi +k, +ec +k = Ecsc +k, +eb +k = Ebsb +k, +(1) +where Ei ∈ Re×Ni is the item embedding matrix, Ec ∈ +Re×Nc is the category embedding matrix, Eb ∈ Re×Nb is +the brand embedding matrix, and e is the embedding size. +For the high-dimensional dense vectors of st +k and sd +k, we +project them into low-dimensional embeddings, i.e., the title +embedding et +k ∈ Re and the description embedding ed +k ∈ +Re, respectively, using Eq. 2: +et +k = Etst +k, +ed +k = Edsd +k, +(2) +where Et ∈ Re×768 and Ed ∈ Re×768 are the projection +matrices. +In order to distinguish different types of feature fields +in the same item, we learn a field embedding for each +type of feature fields. We denote the field embeddings of +ID, category, brand, title and description as f i, f c, f b, f t +and f d ∈ Re, respectively. To distinguish different items +in different positions in the same sequence, we also inject +the position information into the model by learning position +embeddings, where the k-th position embedding is denoted +as pk ∈ Re. Finally, we add each field embedding to the +corresponding item or feature embedding of sk, and add pk +to all embeddings of sk, as shown in Eq. 3: +Hk = +� +����� +hi +k +hc +k +hb +k +ht +k +hd +k +� +����� += +� +����� +ei +k + f i + pk +ec +k + f c + pk +eb +k + f b + pk +et +k + f t + pk +ed +k + f d + pk +� +����� +, +(3) +where hi +k, hc +k, hb +k, ht +k, hd +k ∈ Re, and Hk ∈ R5×e is the hidden +state of sk that is the stack of all embeddings of its feature +fields in order. +3.3 +Dense fusion self-attention +The dense fusion self-attention (DFSA) mechanism follows a +unified way to impute missing feature fields, both item IDs +and side information. To exploit the information in a given +context for imputation, we need to model the relations be- +tween different feature fields and fuse the representations of +various feature fields. DFSA calculates the attention values +between any pair of feature fields and fuses the information +of other feature fields based on the attention value. By +calculating the attention value, DFSA captures all possible +(hence dense) pairwise relations between feature fields to +facilitate missing information imputation. +Specifically, we first stack the hidden states of all items +in S in order by Eq. 4: +H = +� +���� +H1 +H2 +... +Hn +� +���� , +(4) +where H ∈ R5n×e is the hidden state matrix of S. Then, +DFSA employs a transformer with L layers to update H. +Each transformer layer Trm(·) is composed of two sub- +layers: (i) multi-head self-attention MH(·) and (ii) position– +wise feed-forward PFFN(·), as defined in Eq. 5: +Hl+1 = Trm(Hl) = LN( �Hl + Dropout(PFFN( �Hl))) +�Hl = LN(Hl + Dropout(MH(Hl))) +MH(Hl) = [head1; . . . ; headh]WH +headi = Attn(HlWQ +i , HlWK +i , HlWV +i ) +Attn(Q, K, V) = softmax(QK⊤/√e + M)V +PFFN( �Hl) = GELU( �HlWF +1 + bF +1 )WF +2 + bF +2 , +(5) + +808 +Output sequence +Output layer +王 +Dense fusion self-attention +Position embedding ++ +Field embedding ++ +Item/feature embedding +Embedding layer +0808 +08 +Randomly masked +Mask +Input sequence5 +where LN is layer normalization [39], Dropout is dropout +[40], Attn is attention, GELU is a Gaussian error linear unit +activation [41], [. . . ; . . .] is the concatenation operation, h +is the number of heads, WH ∈ Re×e, WQ +i , WK +i , WV +i +∈ +Re×e/h, WF +1 ∈ Re×4e, WF +2 ∈ R4e×e, bF +1 ∈ R4e and bF +2 ∈ +Re are trainable parameters, Hl and Hl+1 ∈ R5n×e are the +output hidden state matrices in the l-th layer and the (l+1)- +th layer, and H0 = H. +The matrix M ∈ R5n×5n in Eq. 5 is the attention mask +which is defined as: +Mj,y +i,x = +� 0, +allow to attend, +−∞, +prevent from attending, +(6) +where i and j ∈ {1, . . . , n}, x and y ∈ {i, c, b, t, d}, Mj,y +i,x ∈ +M is the mask to control whether the feature field sy +j can +attend to the feature field sx +i . We set all Mj,y +i,x = 0,1 which +means we allow to attend between any pair of feature fields +in the sequence. Therefore, the DFSA can model relations +and fuse information between all possible pairs of feature +fields to facilitate both imputation and recommendation. +3.4 +Output layer +The output layer reconstructs the input feature fields based +on the output hidden states of DFSA. First, we split the final +output hidden state matrix HL of DFSA by Eq. 7: +HL = �E = +� +����� +�E1 +�E2 +... +�En +� +����� +, +where �Ek = +� +����� +ˆei +k +ˆec +k +ˆeb +k +ˆet +k +ˆed +k +� +����� +, +(7) +and ˆei +k, ˆec +k, ˆeb +k, ˆet +k, ˆed +k ∈ Re. Similar to the embedding layer, +the output layer takes different ways to reconstruct different +types of feature fields. Specifically, for the categorical feature +fields, we calculate the probability distributions pi +k ∈ RNi, +pc +k ∈ RNc and pb +k ∈ RNb of the item ID, category and brand +of the k-th item sk as follows: +pi +k = softmax(Ei⊤ˆei +k) +pc +k = sigmoid(Ec⊤ˆec +k) +pb +k = softmax(Eb⊤ˆeb +k), +(8) +where Ei ∈ Re×Ni, Ec ∈ Re×Nc, Eb ∈ Re×Nb are the re- +used item embedding matrix, category embedding matrix +and brand embedding matrix in the embedding layer, re- +spectively. Note that we see the category prediction as a +series of binary classifications, because an item may contain +multiple categories. Then we get the reconstructed item ID +ˆsi +k ∈ RNi, category ˆsc +k ∈ RNc and brand ˆsb +k ∈ RNb based on +the probability distributions, as shown in Eq. 9: +ˆsi +k = argmax(pi +k) +ˆsc +k = 1(pc +k > 0.5) +ˆsb +k = argmax(pb +k), +(9) +where 1(α) is the indicator function that equals 1 if α is true +and 0 otherwise. Meanwhile, for the textual feature fields, +1Here we neglect the padding items. +we follow Eq. 10 to get the reconstructed title ˆst +k ∈ R768 and +description ˆsd +k ∈ R768 directly: +ˆst +k = Otˆet +k, +ˆsd +k = Odˆed +k, +(10) +where Ot ∈ R768×e and Od ∈ R768×e are the projection +matrices. +3.5 +Missing information imputation loss +We train MIIR with MII. MII first randomly masks feature +fields in the sequence with probability p, i.e., replacing a +non-missing feature value with the corresponding missing +feature value imiss, cmiss, bmiss, tmiss or dmiss. For the +k-th item sk in the sequence S, we use mi +k, mc +k, mb +k, mt +k +and md +k ∈ {true, false} to denote whether its ID, category, +brand, title and description are masked. Then, MIIR learns +to recover the masked feature fields by MII and impute the +missing feature values based on the context. +Specifically, there are differences in the calculation of +the missing information imputation loss for different types +of feature fields. For the categorical feature fields (i.e., ID, +category and brand), our goal is to minimize the cross- +entropy loss: +Li +k = −1(mi +k)si +k +⊤ log(pi +k) +Lc +k = −1(mc +k)(sc +k +⊤ log(pc +k) + (1 − sc +k +⊤) log(1 − pc +k))/Nc +Lb +k = −1(mb +k)sb +k +⊤ log(pb +k), +(11) +where Li +k, Lc +k and Lb +k are the imputation loss for the item +ID, category and brand of sk, respectively. For the textual +feature fields (i.e., title and description), our goal is to +minimize the mean square error loss: +Lt +k = 1(mt +k)∥st +k − ˆst +k∥2 +Ld +k = 1(md +k)∥sd +k − ˆsd +k∥2, +(12) +where Lt +k and Ld +k are the imputation loss for the title and +description of sk. The missing information imputation objective +of the entire model on S is shown in Eq. 13: +Lmii +S += 1/n +n +� +k=1 +Lmii +k +Lmii +k += Li +k + Lc +k + Lb +k + Lt +k + Ld +k. +(13) +Note that since the item ID is one of the feature fields and +the next item prediction is a MII task, MIIR trained by MII +can directly be applied to sequential recommendation. +In experiments, we also consider to further fine-tune +MIIR or directly train MIIR with the masked item prediction +loss to make the model only focus on the item prediction +task. Specifically, we randomly mask some items with their +all feature fields in the given sequence, and then let MIIR +predict the masked item IDs only. The recommendation loss +(i.e., the masked item prediction loss) on S is defined as: +Lrec +S += 1/n +n +� +k=1 +Lrec +k +Lrec +k += Li +k = −1(mi +k)si +k +⊤log(pi +k), +(14) +where Lrec +k +is the recommendation loss for sk. + +6 +TABLE 1 +Summary of the datasets. The missing rate is the percentage of +missing feature fields in all feature fields. Especially, “Missing rate D” is +the missing rate on the dataset after discarding side information. +Dataset +Beauty Sports and Outdoors Toys and Games +#items +121,291 +194,715 +164,978 +#sequences +52,374 +84,368 +58,314 +Average length +8.97 +8.50 +8.99 +#categories +656 +3,035 +957 +#brands +13,188 +14,163 +14,135 +Missing rate +12.54% +20.11% +11.20% +Missing rate D +56.32% +60.12% +55.51% +4 +EXPERIMENTAL SETUP +4.1 +Research questions +In this paper, we seek to answer the following research +questions: +(RQ1) How does MIIR perform on the sequential recom- +mendation task compared to state-of-the-art meth- +ods? +(RQ2) What are the benefits of training MIIR with MII? +(RQ3) Does modeling the relation between any pair of +feature fields in item sequences help sequential rec- +ommendation? +(RQ4) How about the performance of MIIR on imputing +the missing side information? +(RQ5) What can we find about MIIR by the case study? +4.2 +Datasets +There are many public datasets for experimenting with +sequential recommendation; see [1]. However, we need +sequential recommendation datasets that come with side +information. We conduct experiments on three public +datasets: “Beauty”, “Sports and Outdoors” and “Toys and +Games” [42], as they have rich item side information, in- +cluding category, brand, title and description. +We follow common practices [10, 28] to process the +datasets. We sort each user’s records in chronological order +to construct an item sequence. We filter out item sequences +whose length is less than 5 to avoid noise from the cold- +start problem. For each item sequence, we use the last +item for test, the second last item for validation, and the +rest items for training. For each test or validation item, +we randomly sample 99 negative items for ranking. We +randomly discard side information of items with probability +0.5. We use “Beauty D”, “Sports and Outdoors D” and +“Toys and Games D” to denote the datasets after discarding +side information. The statistics of the datasets after pre- +processing are summarized in Table 1. +4.3 +Baselines +We compare MIIR with the following recommendation +baselines, which can be grouped into (i) methods without +side information fusion, (ii) methods with side information +fusion and (iii) methods with missing feature values. +• Methods without side information fusion: +– GRU4Rec employs RNNs to capture sequential pat- +terns between items for sequential recommendation [2]. +– SASRec uses the self-attention mechanism to model +item sequences for next item recommendations [6]. +– BERT4Rec uses a bidirectional self-attention network +train-ed by a masked item prediction task for sequential +recommendation [8]. +• Methods with side information fusion: +– PRNN employs parallel RNNs to process items and +their side information respectively, then combines the +hidden states of the RNNs for next item prediction [9]. +– FDSA leverages two separate self-attention networks +to model the ID transition patterns and the feature +transition patterns respectively, then concatenates the +outputs of two networks for next item prediction [10]. +– NOVA adopts a non-invasive self-attention mecha- +nism to leverage side information under the BERT4Rec +framework for sequential recommendation [28]. +• Methods with missing feature values: +– RFS randomly samples feature fields to introduce more +missing information when training [18]. RFS is to make +the model more robust with missing feature values +instead of imputing missing feature fields. We combine +RFS with FDSA and NOVA, and denote the variants as +FDSA+RFS and NOVA+RFS. +– LRMM designs an auto-encoder with the modality +dropout to impute both user ratings and missing side +information for each item [19]. Note that LRMM is not +a sequential model. Furthermore, we use the imputed +missing side information by LRMM to train FDSA +and NOVA, and denote them as FDSA+LRMM and +NOVA+LRMM. +Other methods with side information fusion, such as [23, +24], can only model categorical item side information; for +a fair comparison, we do not consider them as baselines. +In addition to the baselines listed above, we compare MIIR +against four variants, namely MIIR-F, MIIR-R, MIIR-M, and +Sparse-MIIR, to be defined in Section 5.1, 5.2 and 5.3. +We unify the sequential recommendation loss in all base- +lines, MIIR, and its variants to the cross-entropy loss, rather +than the pairwise loss [43], to avoid noise due to negative +sampling in the pairwise loss. +4.4 +Metrics and implementation +To evaluate the performance of sequential recommendation +methods, we employ two widely used evaluation metrics: +HR@k (hit ratio) and MRR (mean reciprocal rank) [1], where +k ∈ {5, 10}. +• HR measures the proportion of the sequences whose +ground-truth items are amongst the top ranked items in +all test sequences. +• MRR is the average of reciprocal ranks of the ground- +truth items. +For all baselines and our proposed model, we initialize +the trainable parameters randomly with the Xavier method +[44]. We train all methods with the Adam optimizer [45] +for 100 epochs, with a batch size of 128 and a learning rate +of 0.0001. We also apply gradient clipping [46] with range +[−5, 5] during training. According to the average length in +Table 1, we set the maximum sequence length to 20 for both +datasets for all methods. + +7 +TABLE 2 +Performance comparison of MIIR, variants, and the baselines on the +“Beauty” dataset. MIIR-F is a variant of MIIR that is fine-tuned using the +recommendation loss (see Section 5.1) and MIIR-R is a variant trained +using the recommendation loss only (see Section 5.2). The highest +overall performance is denoted in bold face. The highest performance +among the baselines is underlined. Impr. (%) is the performance gain +of MIIR against the best baseline method. ∗ indicates that an +improvement is statistically significant based on a two-sided paired +t-test with p < 0.05. +Beauty +Beauty D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +GRU4Rec +31.58 +42.50 +21.47 +31.58 +42.50 +21.47 +SASRec +32.83 +43.61 +23.16 +32.83 +43.61 +23.16 +BERT4Rec +33.22 +43.77 +23.58 +33.22 +43.77 +23.58 +PRNN +32.27 +42.70 +23.08 +31.80 +42.55 +22.23 +FDSA +35.22 +44.83 +25.39 +35.02 +44.68 +25.33 +NOVA +34.99 +45.07 +25.02 +34.21 +44.38 +24.80 +FDSA+RFS +35.45 +45.40 +25.68 +34.73 +44.56 +25.17 +NOVA+RFS +35.57 +45.61 +25.74 +34.26 +44.24 +24.97 +LRMM +22.74 +32.95 +17.09 +18.04 +26.94 +13.96 +FDSA+LRMM +35.35 +45.15 +25.62 +35.10 +44.73 +25.52 +NOVA+LRMM +35.35 +45.31 +25.50 +34.31 +44.53 +25.01 +MIIR +38.92 +48.61 +29.46 +37.30 +46.85 +27.90 +MIIR-F +38.73 +48.01 +29.28 +37.12 +46.48 +27.95 +MIIR-R +35.59 +45.60 +25.85 +34.92 +44.96 +25.41 +Impr. (%) ++3.35∗ ++3.00∗ +3.72∗ +2.20∗ ++2.12∗ +2.38∗ +TABLE 3 +Performance comparison of MIIR, variants, and the baselines on the +“Sports and Outdoors” dataset. +Sports and Outdoors +Sports and Outdoors D +Method +HR@5 HR@10 MRR HR@5 HR@10 +MRR +GRU4Rec +33.54 +44.57 +23.70 +33.54 +44.57 +23.70 +SASRec +34.46 +44.69 +25.41 +34.46 +44.69 +25.41 +BERT4Rec +35.12 +45.24 +26.11 +35.12 +45.24 +26.11 +PRNN +37.41 +47.25 +27.23 +36.01 +46.18 +26.12 +FDSA +39.16 +48.08 +29.27 +37.30 +46.74 +27.20 +NOVA +37.95 +47.54 +28.08 +36.15 +45.96 +26.90 +FDSA+RFS +38.18 +47.18 +28.31 +37.17 +46.65 +27.01 +NOVA+RFS +37.63 +47.41 +27.33 +35.86 +45.52 +26.84 +LRMM +28.65 +41.36 +20.50 +19.79 +30.34 +15.13 +FDSA+LRMM +39.48 +48.52 +29.41 +38.46 +47.67 +28.24 +NOVA+LRMM +38.18 +47.76 +28.30 +37.28 +46.78 +27.32 +MIIR +43.66 +52.63 +32.66 +40.55 +49.80 +30.04 +MIIR-F +42.66 +51.49 +32.01 +39.98 +48.98 +29.86 +MIIR-R +40.01 +49.70 +29.40 +38.07 +47.82 +27.77 +Impr. (%) ++4.18∗ ++4.11∗ +3.25∗ +2.09∗ ++2.13∗ ++1.80∗ +All hyper-parameters of the baselines are set following +the suggestions from the original papers. For the hyper- +parameters of MIIR, we set the embedding size e to 64, the +number of heads h to 4, and the number of layers L to 3. We +set the dropout rate in DFSA and the mask probability p in +MII to 0.5. +5 +EXPERIMENTAL RESULTS +5.1 +Overall performance +To answer RQ1, we compare MIIR against the recommenda- +tion models listed in Section 4.3 on the three datasets from +TABLE 4 +Performance comparison of MIIR, variants, and the baselines on the +“Toys and Games” dataset. +Toys and Games +Toys and Games D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +GRU4Rec +31.19 +42.15 +21.90 +31.19 +42.15 +21.90 +SASRec +31.74 +41.22 +24.51 +31.74 +41.22 +24.51 +BERT4Rec +31.45 +41.22 +23.25 +31.45 +41.22 +23.25 +PRNN +34.00 +44.25 +24.32 +32.71 +42.98 +23.23 +FDSA +34.44 +43.89 +26.03 +32.70 +42.33 +24.69 +NOVA +34.50 +44.34 +25.86 +34.00 +43.74 +25.06 +FDSA+RFS +34.81 +44.62 +26.30 +33.41 +43.64 +25.22 +NOVA+RFS +35.33 +45.29 +26.27 +33.39 +43.26 +24.73 +LRMM +29.88 +40.96 +21.87 +19.85 +29.83 +15.15 +FDSA+LRMM +35.20 +44.50 +26.49 +33.43 +42.94 +25.18 +NOVA+LRMM +35.65 +45.50 +26.61 +34.51 +44.47 +25.51 +MIIR +40.11 +49.80 +29.64 +39.01 +48.89 +28.74 +MIIR-F +39.00 +47.76 +29.57 +38.25 +47.45 +28.75 +MIIR-R +35.80 +45.37 +26.00 +34.69 +44.30 +24.81 +Impr. (%) ++4.46∗ ++4.30∗ +3.03∗ +4.50∗ ++4.42∗ +3.23∗ +Section 4.2. Table 2, 3 and 4 list the evaluation results of +all methods on each dataset, respectively. Based on these +results, we have the following observations. +First, on all datasets, MIIR performs significantly better +than all baselines by a large margin despite the different +missing rates, in terms of HR@5, HR@10 and MRR. MIIR +has two major advantages: (i) MIIR trains the model using +MII to enhance its ability to deal with missing side infor- +mation in sequential recommendation (see detailed analysis +in Section 5.2), and (ii) MIIR employs DFSA to improve the +side information fusion in the model (see Section 5.3 for +further analysis). +Second, the item side information can help sequential +recommender systems to more accurately model the tran- +sition patterns among items. To verify this, we divide all +methods into three groups: (i) GRU4Rec and PRNN that +are based on RNNs; (ii) SASRec and FDSA that are based +on left-to-right self-attention networks; and (iii) BERT4Rec, +NOVA, and MIIR that employ bidirectional self-attention +networks and the masked item prediction task. In each +group, we see that methods that fuse side information +outperform methods that only rely on item IDs, which +illustrates that item side information does help. +Third, the performance of PRNN, FDSA, NOVA and +MIIR on the “Beauty”, “Sports and Outdoors” and “Toys +and Games” datasets is higher than that on the discarded +versions of the datasets (i.e., “Beauty D”, “Sports and Out- +doors D” and “Toys and Games D”). We see two reasons for +this difference: (i) the “Beauty D”, “Sports and Outdoors D” +and “Toys and Games D” datasets discard some side infor- +mation, so the available side information becomes less, and +(ii) using the special values (i.e., imiss, cmiss, bmiss, tmiss +and dmiss) to fill missing feature fields may be harmful to +PRNN, FDSA and NOVA. +Fourth, by comparing FDSA+RFS and NOVA+RFS with +FDSA and NOVA, we see RFS cannot consistently improve +the performance of FDSA and NOVA on all datasets. What’s +worse, RFS would degrade the performance of FDSA and +NOVA in some cases. Because RFS is to introduce more + +8 +TABLE 5 +Performance comparison of whether to exploit missing feature fields on +the “Beauty” dataset. MIIR-M and MIIR-R-M are the variants of MIIR +and MIIR-R respectively that mask missing feature fields in +self-attention (see Section 5.2). +Beauty +Beauty D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +MIIR +38.92 +48.61 +29.46 +37.30 +46.85 +27.90 +MIIR-R +35.59 +45.60 +25.85 +34.92 +44.96 +25.41 +MIIR-M +39.16 +48.67 +29.45 +37.12 +46.58 +27.83 +MIIR-R-M +36.40 +46.31 +27.11 +34.71 +45.01 +25.42 +TABLE 6 +Performance comparison of whether to exploit missing feature fields on +the “Sports and Outdoors” dataset. +Sports and Outdoors +Sports and Outdoors D +Method +HR@5 HR@10 MRR HR@5 HR@10 +MRR +MIIR +43.66 +52.63 +32.66 +40.55 +49.80 +30.04 +MIIR-R +40.01 +49.70 +29.40 +38.07 +47.82 +27.77 +MIIR-M +43.04 +52.12 +32.16 +40.36 +49.65 +29.81 +MIIR-R-M +39.71 +48.98 +29.15 +38.33 +48.10 +28.12 +missing feature values into the model training instead of +imputing missing feature fields, it cannot deal with the +missing side information problem fundamentally. +Fifth, the performance of LRMM is significantly worse +than that of the sequential recommendation models with +side +information. +LRMM +even +performs +worse +than +GRU4Rec, SASRec and BERT4Rec that neglect the item +side information. The main reason is that LRMM is not +a sequential model, so it cannot exploit the relation and +information in sequences to make recommendation and +imputation, however it is essential in the sequential rec- +ommendation task. We can also observe that FDSA+LRMM +and NOVA+LRMM outperform FDSA and NOVA in exper- +iments, which verifies the effectiveness of the imputation +results of LRMM. It also proves imputing missing feature +values is a better way to alleviate the missing side informa- +tion problem than using fixed special values and RFS. +Sixth, modeling sequential recommendation as missing +information imputation is sufficient to train a recommen- +dation model. To verify this, we conduct an experiment +that first pre-trains MIIR using the missing information +imputation loss (Eq. 13), and then fine-tunes it using the +recommendation loss (Eq. 14). We use MIIR-F to denote this +variant of MIIR. In Table 2 we see that MIIR-F performs +worse than MIIR in most cases. Fine-tuning MIIR-F with the +recommendation loss might lead to overfitting, resulting in +performance decreases. This result supports the conclusion +that with MII we can unify the sequential recommendation +task as a particular type of missing information imputation +task to train MIIR together with the other imputation task +for missing item side information. +5.2 +Benefits of MII +To answer RQ2, we analyze how MIIR benefits from training +with MII. +TABLE 7 +Performance comparison of whether to exploit missing feature fields on +the “Toys and Games” dataset. +Toys and Games +Toys and Games D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +MIIR +40.11 +49.80 +29.64 +39.01 +48.89 +28.74 +MIIR-R +35.80 +45.37 +26.00 +34.69 +44.30 +24.81 +MIIR-M +39.33 +49.22 +28.97 +37.80 +47.58 +27.82 +MIIR-R-M +35.22 +45.29 +26.28 +34.53 +44.47 +25.58 +In Table 2, 3 and 4, we report on results of a variant of +MIIR that directly trains MIIR with the recommendation loss +shown in Eq. 14. We write MIIR-R for this variant of MIIR +without the supervised signal of MII. When we compare the +performance of MIIR and MIIR-R, we see very substantial +gaps. This confirms the effectiveness of training MIIR with +MII, which accounts for the main part of the improvement +of MIIR over other methods. +To demonstrate that MIIR can mine useful information +from missing feature fields by training with MII, we design +a variant of MIIR called MIIR-M by masking missing feature +fields. In MIIR-M, we revise the attention mask M used in +Eq. 5, which is a null matrix in MIIR. The revision in M is +defined as: +Mj,y +i,x = +� −∞, +if sx +i or sy +j ∈ {cmiss, bmiss, tmiss, dmiss}, +0, +otherwise, +(15) +where the condition of sx +i or sy +j ∈ {cmiss, bmiss, tmiss, +dmiss} depends on the original input sequence instead +of the sequence after randomly masking. The purpose of +the variant is to prevent the model from attending to the +missing feature fields about item side information in the +sequence. On the one hand, MIIR-M cannot mine and fuse +any information in missing feature fields for sequential +recommendation. On the other hand, MIIR-M is unable +to exploit the information in non-missing feature fields to +impute the missing side information. Besides, we mask +missing feature fields for MIIR-R to analyze how missing +feature values affects the performance of MIIR without MII, +denoted as MIIR-R-M. +In Table 5, 6 and 7, we compare MIIR and MIIR-R with +MIIR-M and MIIR-R-M, respectively. We can find that MIIR +outperforms MIIR-M in most cases, which illustrates that +MIIR can extract useful information from missing feature +fields to improve the sequential recommendation perfor- +mance. We can also observe that MIIR-R-M performs better +than MIIR-R in some cases. This phenomenon indicates that +using fixed special values for filling missing feature fields +can suffer the model performance, therefore masking miss- +ing feature fields may be a better way without imputation. +On the “Beauty” dataset, MIIR only achieves comparable +performance with MIIR-M, and MIIR-R also performs worse +than MIIR-R-M. However, the performance gap from MIIR +to MIIR-M is smaller than that from MIIR-R to MIIR-R- +M, and we have similar observations on other datasets. +It illustrates that imputing missing feature values has the +superiority over masking them for alleviating missing side +information problem. + +9 +TABLE 8 +Performance comparison of dense and sparse attention on the +“Beauty” dataset. Sparse-MIIR and Sparse-MIIR-R are variants of MIIR +and MIIR-R, respectively, in which DFSA is replaced by SFSA (see +Section 5.3). +Beauty +Beauty D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +MIIR +38.92 +48.61 +29.46 +37.30 +46.85 +27.90 +MIIR-R +35.59 +45.60 +25.85 +34.92 +44.96 +25.41 +Sparse-MIIR +36.71 +46.60 +26.87 +36.04 +45.98 +26.34 +Sparse-MIIR-R +34.95 +45.02 +25.35 +34.61 +44.84 +25.19 +TABLE 9 +Performance comparison of dense and sparse attention on the “Sports +and Outdoors” dataset. +Sports and Outdoors +Sports and Outdoors D +Method +HR@5 HR@10 MRR HR@5 HR@10 +MRR +MIIR +43.66 +52.63 +32.66 +40.55 +49.80 +30.04 +MIIR-R +40.01 +49.70 +29.40 +38.07 +47.82 +27.77 +Sparse-MIIR +40.52 +50.04 +29.64 +39.24 +48.91 +28.67 +Sparse-MIIR-R +38.61 +48.29 +28.25 +37.56 +47.72 +27.21 +MIIR-M also outperforms all baselines on the three +datasets with different missing rates. Training MIIR with +MII helps MIIR to make use non-missing feature fields. +Imputing the masked non-missing feature values requires +the model to capture the relations between different feature +fields, so MII guides MIIR to better fuse side information +into the model for improving the sequential recommenda- +tion performance. +5.3 +Effectiveness of DFSA +To answer RQ3, we conduct an ablation study to analyze the +effectiveness of DFSA in MIIR. +We first compare MIIR-R, the variant of MIIR that is +trained with recommendation loss only, with the baselines +in Table 2, 3, 4. MIIR-R achieves better or comparable per- +formance with the baselines on most evaluation metrics of +all datasets, even without the help of MII. The main reason +is that MIIR-R has dense fusion self-attention (DFSA) to +better fuse information in the item sequence for improving +sequential recommendation. +In order to validate that it is important to model all pos- +sible pairwise relations in an item sequence for sequential +recommendation, we design another self-attention mecha- +nism called sparse fusion self-attention (SFSA). SFSA modifies +the attention mask M in Eq. 5 into: +Mj,y +i,x = +� 0, +if i == j or x == y, +−∞, +otherwise, +(16) +where the condition i == j or x == y means that SFSA +only allows to attend between the pair of feature fields +belonging to the same item or the same type. Therefore, +SFSA only models the relation between different feature +fields of the same item or the relation between the same type +of feature fields of different items in the sequence. These +relations are also modeled in some baselines, such as PRNN +and FDSA. +TABLE 10 +Performance comparison of dense and sparse attention on the “Toys +and Games” dataset. +Toys and Games +Toys and Games D +Method +HR@5 HR@10 MRR HR@5 HR@10 MRR +MIIR +40.11 +49.80 +29.64 +39.01 +48.89 +28.74 +MIIR-R +35.80 +45.37 +26.00 +34.69 +44.30 +24.81 +Sparse-MIIR +37.61 +47.77 +27.23 +37.06 +47.27 +26.79 +Sparse-MIIR-R +35.58 +45.66 +25.80 +34.46 +44.54 +24.53 +TABLE 11 +Performance comparison of LRMM and MIIR for the missing side +information imputation on the “Beauty D”, “Sports and Outdoors D” and +“Toys and Games D” datasets, where P: precision, R: recall, F1: F1 +score, ACC: accuracy, MSE: mean square error. +Dataset +Field +Metric LRMM +MIIR +Beauty D +Category +P +70.15 +79.64 +R +48.41 +36.97 +F1 +52.96 +48.61 +Brand +ACC +7.84 +5.01 +Title +MSE +0.0871 +0.0514 +Description MSE +0.1454 +0.0704 +Sports and +Outdoors D +Category +P +57.02 +74.97 +R +51.31 +35.91 +F1 +46.38 +46.40 +Brand +ACC +6.06 +4.43 +Title +MSE +0.0927 +0.0534 +Description MSE +0.1474 +0.0835 +Toys and +Games D +Category +P +72.32 +89.31 +R +51.08 +42.31 +F1 +54.11 +55.50 +Brand +ACC +18.61 +14.49 +Title +MSE +0.0858 +0.0514 +Description MSE +0.1427 +0.0777 +In Table 8, 9 and 10, we compare the performance of +DFSA and SFSA as components of MIIR and MIIR-R. We +write Sparse-MIIR and Sparse-MIIR-R for the variants of +MIIR and MIIR-R, respectively, in which DFSA is replaced +by SFSA. We can see that MIIR outperforms Sparse-MIIR +on all datasets despite different missing rates. What’s more, +MIIR-R outperforms Sparse-MIIR-R in most cases too. Mod- +eling the relations between any pair of feature fields helps to +make more effective use of item side information to improve +sequential recommendation performance. +By comparing MIIR with Sparse-MIIR, we also notice +that the improvement by DFSA on the three datasets is +higher than the improvement on the discarded versions of +the datasets. We guess that DFSA may also model more +noisy relations related to missing feature fields when the +missing rate increases, which would make the performance +degeneration. +5.4 +Imputation performance (RQ4) +To answer RQ4, we compare LRMM and MIIR based on +their imputation results for the discarded side information +of all datasets. +For the test sequences from the “Beauty D”, “Sports and +Outdoors D” and “Toys and Games D” datasets, we can +compare the imputed results with the ground-truth before +discard. For different types of feature fields, we consider + +10 +Fig. 4. (a) and (b) are two sequences with their imputed categories from the “Beauty D” dataset, (c) and (d) are two sequences with their imputed +brands from the “Toys and Games D” dataset, (e) is the sequence with its imputed categories and brands from the “Sports and Outdoors D” dataset. +different metrics: (i) for category that is corresponding to +the multi-class classification task, we calculate the preci- +sion, recall and F1 score for evaluation which bigger is +better; (ii) for brand that is corresponding to the one-class +classification task, we calculate the accuracy for evaluation +which bigger is better; (iii) for title and description that are +both corresponding to the multi-variable regression task, +we calculate the mean square error (averaged by the length +of title/description vector) for evaluation which smaller is +better. +In Table 11, we list the evaluation results for compari- +son. We can observe that MIIR achieves better imputation +performance than LRMM on the precision of category and +the MSE of title and description. Whereas, LRMM outper- +forms MIIR on the recall of category and the accuracy of +brand. Both LRMM and MIIR can infer some discarded side +information, so they can alleviate the missing side infor- +mation problem. Comparing to LRMM, MIIR can exploit +more information from the sequence to impute the missing +side information. However, MIIR may also impute some +inaccurate results due to the over-dependence on the given +context. +5.5 +Case Study (RQ5) +Finally, to answer RQ5, we sample some test cases from +datasets. +As shown in Fig.4, we list some sequences with their +imputed results. We can observe that MIIR can generate +different feature values for missing feature fields according +to different contexts (i.e., items and sequences), which is +better than using fixed predefined values. What’s more, +MIIR may infer the ground-truth missing value, including +the side information of the target next item, to give the +model with a more accurate guidance for recommendation. +For example, MIIR imputes a part of the discarded cate- +gories in the sequence (b) and the discarded brands in the +sequence (d). We can also observe that the side information +of items in the same sequence may be related, which is why +MIIR can infer the ground-truth missing value in light of +the given context. However, MIIR may be over-dependent +on the information from the sequence, leading to impute +inaccurate results. For instance, in the sequence (e), MIIR +imputes the wrong categories and brand for the item 5401. +Additionally, we visualize the attention weights from +the missing item ID (i.e., the next item ID) to all feature +fields in the given sequence in DFSA, as shown in Fig.5. We +reshape the attention weights into a matrix with the shape +of 5 × n, where 5 is the number of the feature field types +and n is the sequence length. First, we can see that MIIR +exploits the information from all feature fields of the given +sequence to predict the next item, which emphasizes the +necessity to model the relation between any pair of feature +fields. Second, we can observe that different layers focus on +different types of feature fields, where the first layer mainly +attends to ID, and the third layer mainly attends to title +and description. It illustrates MIIR gradually fuses different +types of side information into the model by different layers. +Because the information in textual feature fields is more +difficult to extract, MIIR needs more deeper layers to fuse +textual feature fields. Third, we can find different heads +in the same layers have similar attention patterns, which +means there may be some redundant parameters in MIIR. + +(a) +id: 22434 +id: 69550 +id: 52863 +id: 52866 +id: 52867 +id: 52868 +id: 65782 +id: 65790 +id: 83354 +Original +Sequence +category: 384, 487, +category: 38, 271, +category: 185, 384, +category: 185, 384, +category: 384, 487, +category: 185, 384, +category: 384, 487, +category: 384, 487, +category: 185, 384, +489 +479 +487 +487 +489 +487 +489 +489 +487 +id: 22434 +id: 69550 +id: 52863 +id: 52866 +id: 52867 +id: 52868 +id: 65782 +id: 65790 +id: 83354 +Imputed +Sequence +category: 384, 487, +category: 185, 384, +category: 185, 384, +category: 185, 384, +category: 384, 487, +category: 384, 487, +category: 185, 384, +category: 384, 487 +category: 384, 487 +489 +487 +487 +487 +489 +489 +487 +(b) +id: 19904 +id: 101600 +id: 53133 +id: 25434 +id: 89249 +id: 44195 +id: 25437 +id: 103752 +Original +id: 89561 +Sequence +category: 114, 454, +category: 108, 114, +category: 114, 367, +category: 114, 369, +category: 114, 367, +category: 43, 114, +category: 41, 47, 114, +category: 43, 114 +category: 15, 26, 56 +558 +493, 598 +598 +535, 598 +598 +267 +369, 598 +id: 19904 +id: 101600 +id: 53133 +id: 25434 +id: 44195 +id: 25437 +id: 89249 +id: 89561 +id: 103752 +Imputed +Sequence +category: 114, 454, +category: 108, 114, +category: 43, 114, +category: 43, 114 +category: 114, 598 +category: 114, 598 +category: 114, 598 +category: 15, 26, 56 +category: 114, 598 +558 +493, 598 +267 +(c) +id: 6177 +id: 96866 +id: 108189 +id: 117401 +id: 117402 +id: 134383 +id: 134384 +id: 157885 +Original +Sequence +brand: + brand: 3614 +brand: 3614 +brand: 3614 +brand: 3614 + brand: +brand: 7166 +brand: 3614 +id: 6177 +id: 96866 +id: 108189 +id: 117401 +id: 117402 +id: 134383 +id: 134384 +id: 157885 +Imputed +Sequence + brand: 3614 +brand: 3614 +brand: 3614 +brand: 3614 +brand: 3614 + brand: 3614 +brand: 7166 +brand: 3614 +(d) +id: 27109 +id: 4139 +L096 :p! +id: 63016 +id: 80413 +id: 16994 +id: 83174 +Original +id: 23768 +Sequence +brand: 11595 +brand: +brand: 11595 +brand: 6844 + brand: 12538 +brand: 2308 +brand: 11595 +brand: 5472 +id: 27109 +id: 4139 +id: 9607 +id: 63016 +id: 80413 +id: 16994 +id: 83174 +id: 23768 +Imputed +Sequence +brand: 11595 +brand: 11595 +brand: 5472 +brand: 11595 +brand: 6844 + brand: 12538 +brand: 2308 +brand: 11595 +(e) +id: 55944 +id: 1956 +id: 5401 +id: 22000 +id: 36936 +id: 173423 +id: 25085 +id: 101607 +Original +Sequence +category: 952, 1286, +category: 632, 1562, +category: 952, 1286, +category: 952, 1286, +category: 252, 1286, +category: 296, 1286, +category: 952, 1286, +category: 2078, 2299 + Discard +2191, 2479 +2533 +2191, 2479 +2191, 2479 +2191, 2479 +2230, 2479 +2233, 2479 +brand: 4696 +brand: 4087 +brand: 6636 +brand: 4087 +brand: 4087 + brand: +brand: 9545 +brand: 10778 +Impute +id: 55944 +id: 1956 +id: 5401 +id: 22000 +id: 36936 +id: 173423 +id: 25085 +id: 101607 +Imputed +Sequence +category: 952, 1286, +category: 952, 1286, +category: 952, 1286, +category: 952, 1286, + Target Item +category: 2078, 2299 +category: 1286, 2479 +category: 1286, 2479 +category: 1286, 2479 +2191, 2479 +2191, 2479 +2191, 2479 +2191, 2479 +brand: 4696 + brand: 4087 +brand: 4087 +brand: 4087 +brand: 4087 +brand: 4087 +brand: 9545 +brand: 408711 +(a) A sequence from the “Beauty D” dataset. +(b) A sequence from the “Sports and Outdoors D” dataset. +(c) A sequence from the “Toys and Games D” dataset. +Fig. 5. Visualization for the attention weights from the missing item ID field to all feature fields of all heads and layers in MIIR on three sequences +from different datasets. +6 +CONCLUSION +We have studied the missing side information problem in +sequential recommendation. We have proposed the missing +information imputation (MII) task to unify the missing side +information imputation task and the sequential recommen- +dation task. We have presented a novel sequential recom- +mendation model named missing information imputation +recommender (MIIR) to simultaneously impute missing fea- +ture values and predict the next item for a given sequence +of items. We have proposed a dense fusion self-attention +(DFSA) mechanism to model different relations in the item +sequence and to fuse side information. +Based on experiments and analyses on three datasets +with different settings of the missing rates we have found +that MIIR outperforms state-of-the-art methods for sequen- +tial recommendation with side information. We have ver- +ified that MIIR can identify useful side information from +missing feature fields by training with the MII task, and +that the DFSA mechanism improves the recommendation +effectiveness of MIIR. +As to broader implications of our work, we offer a new +perspective by revealing a correlation between missing side +information imputation and the sequential recommendation +task. They both concern the prediction of missing infor- +mation. The perspective operationalized with MIIR can be +adopted as a foundational paradigm. Other prediction tasks +related to recommendation, such as rating prediction, user +profile prediction, and next basket recommendation can also +be formulated as a MII task. +Limitations of our work include the following: (i) since +DFSA treats side information as part of the sequence (e.g., +in our case, the actual sequence length is 5x the number of +items) and models all possible pairwise relations in an item +sequence, it is computationally costly and not easy to scale +to long sequences; and (ii) we have not optimized the MII +losses on different types of feature fields in MIIR for the + +Layer 3 Head 2 +category +category + category +brand +brand +branc +itle +title +description +description + description +Layer 1 Head 3 +Layer 1 Head 4 +Layer 2 Head 3 +Layer 2 Head 4 +Layer 3 Head 3 +Layer 3 Head 4 +category +category +category +brand + branc +brand +title +itle +title +description +description +description +1 +0.000 0.009 +0.090 +0.000 +0.0900.100 +0.000 +0.190 +0.081 +0.010Layer 1 Head 1 +Layer 1 Head 2 +Layer 2 Head 1 +Layer 2 Head 2 +category +category + category +brand +brand +brand +title +title +description +description + description +Layer 2 Head 4 +Layer 1 Head 4 +Layer 2 Head 3 +Layer 3 Head 3 +Layer 3 Head 4 +d +category +category +category +brand + brand +brand +title +title +description + description +description +97899800 1469157567833 540810595m1557 +0.190 +0.000 +0.000ayer 1 Head 2 +Layer 2 Head 2 +category +category + category +brand +bran +branc +title +title +title +description +description + description +ayer 1 Head 4 +Layer 2 Head 3 +Layer 2 Head 4 +Layer 3 Head 3 +Layer 3 Head 4 +category +category +category +brand + brand +brand +title +title +description + description + description +1699 83004 90253151526203262038 5850mis5716999 3004902531515220322038 5850m557 +16999 300402531515220322038850m5716999 3009025315152203620385850mi57 +16999 8300490253151526203262038 5850m55716999 3004902531515262032620385850ms57 ++o-0.100 +0.000 +0.084 +0.000 +0.011 +0.05512 +recommendation task. +We aim to further improve MIIR in different directions. +We will assess the ability of the linear transformer [47, 48] +to reduce the computational costs of DFSA and design a +mechanism to filter out useless relations at an early stage. +We also plan to design a tailored loss for MIIR by building +on recent loss weighting methods [49, 50]. +REPRODUCIBILITY +To facilitate reproducibility of the results reported in this +paper, the code and data used in experiments are available +at https://github.com/TempSDU/MIIR. +REFERENCES +[1] +H. 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Gao, “Multi-task learning +with sample re-weighting for machine reading comprehension,” +in Proceedings of the Conference of the North American Chapter of the +Association for Computational Linguistics, 2019, pp. 2644–2655. + diff --git a/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/load_file.txt b/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..a6124a54dc6299844565b9f311f18acbbc569763 --- /dev/null +++ b/MtAzT4oBgHgl3EQfy_6c/content/tmp_files/load_file.txt @@ -0,0 +1,1530 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf,len=1529 +page_content='1 Modeling Sequential Recommendation as Missing Information Imputation Yujie Lin, Zhumin Chen, Zhaochun Ren, Chenyang Wang, Qiang Yan, Maarten de Rijke, Xiuzhen Cheng, Fellow, IEEE, and Pengjie Ren Abstract—Side information is being used extensively to improve the effectiveness of sequential recommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It is said to help capture the transition patterns among items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Most previous work on sequential recommendation that uses side information models item IDs and side information separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' This can only model part of relations between items and their side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Moreover, in real-world systems, not all values of item feature fields are available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' This hurts the performance of models that rely on side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Existing methods tend to neglect the context of missing item feature fields, and fill them with generic or special values, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', unknown, which might lead to sub-optimal performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To address the limitation of sequential recommenders with side information, we define a way to fuse side information and alleviate the problem of missing side information by proposing a unified task, namely the missing information imputation (MII), which randomly masks some feature fields in a given sequence of items, including item IDs, and then forces a predictive model to recover them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' By considering the next item as a missing feature field, sequential recommendation can be formulated as a special case of MII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We propose a sequential recommendation model, called missing information imputation recommender (MIIR), that builds on the idea of MII and simultaneously imputes missing item feature values and predicts the next item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We devise a dense fusion self-attention (DFSA) for MIIR to capture all pairwise relations between items and their side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Empirical studies on three benchmark datasets demonstrate that MIIR, supervised by MII, achieves a significantly better sequential recommendation performance than state-of-the-art baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Index Terms—Sequential recommendation, side information fusion, missing information imputation !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1 INTRODUCTION S EQUENTIAL recommendation models transition patterns among items and generates a recommendation for the next item [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Traditional sequential recommendation so- lutions use the item ID as the only item feature field [2, 3, 4, 5, 6, 7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In real-world cases, however, there is rich side information in the form of multiple types of structural feature fields, such as categories and brands, and unstructured feature fields, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', titles and descriptions, that can help to better model transitions between items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In recent years, several publications have exploited side information to improve sequential recommendation perfor- mance [9, 10, 11, 12, 13, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Most focus on designing different mechanisms to fuse side information into rec- ommendation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For example, Hidasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [9] use parallel recurrent neural networks (RNNs) [16] to encode Yujie Lin, School of Computer Science and Technology, Shandong Univer- sity, Qingdao, China, E-mail: yu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='jie.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='lin@outlook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='com Zhumin Chen, School of Computer Science and Technology, Shandong University, Qingdao, China, E-mail: chenzhumin@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='cn Zhaochun Ren, School of Computer Science and Technology, Shandong University, Qingdao, China, E-mail: zhaochun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='ren@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='cn Chenyang Wang, School of Computer Science and Technology, Shandong University, Qingdao, China, E-mail: 201900122032@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='cn Qiang Yan, WeChat, Tencent, Guangzhou, China, E-mail: rolanyan@tencent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='com Maarten de Rijke, Informatics Institute, University of Amsterdam, Ams- terdam, The Netherlands, E-mail: m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='derijke@uva.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='nl Xiuzhen Cheng, School of Computer Science and Technology, Shandong University, Qingdao, China, E-mail: xzcheng@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='cn Pengjie Ren, School of Computer Science and Technology, Shandong University, Qingdao, China, E-mail: renpengjie@sdu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='cn the information in item IDs and attributes, respectively, and then combine the outputs of RNNs for item recommenda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [10] employ two groups of self-attention blocks [17] for modeling items and features, and fuse them in the final stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Importantly, previous work for sequential recommenda- tion with side information usually regards side information as an auxiliary representation of the item, so models item IDs and side information separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As a result, such meth- ods only encode partial relations in item sequences, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', the relation between an item and its side information, while the relation between an item and the side information of other items in the sequence is not well captured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Even more importantly, previous studies often assume that all side information is available, which is rarely the case in real-world scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1(a), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', the second and third items lack category and title information, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Previous work has proposed to fill such gaps with special values, such as a general category and a padding text, to make models trainable and produce outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, for different items and item sequences, these special values are the same: they do not provide useful and specific information for recommendations and might introduce biases into the model learning instead [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As a result, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1(b), a model might recommend the wrong item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Instead, we propose to impute the missing side information, so that the recommendation model can use information from missing feature fields based on contexts, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Some recent studies address the probem of missing side information in recommendation data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [19] arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01762v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='IR] 4 Jan 2023 2 (a) Original sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (b) Existing work without imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (c) Our work with imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Sequential recommendation of items with side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Gray blocks represent missing information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' “[PAD]” (in (b)) indicates padding with generic or special values as often done in existing work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' “[Impute]” (in (c)) indicates imputation with actual values for missing feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' employ an auto-encoder (AE) with a modality dropout to recover the missing rating and side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [18] propose an adaptive feature sampling strategy to introduce more missing feature fields into the training process, which increases the robustness of the recommen- dation model against missing side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [20] define item recommendation and attribute inference in a user-item bipartite graph with attributes, and propose a graph convolutional network (GCN) [21] based model to join these two tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, the work just listed mainly targets non-sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Moreover, it treats item recommendation and side information imputation as different tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In this work, we seek to design a sequential recommen- dation model that can handle missing feature fields of items in items sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The main challenge is how to adap- tively impute missing information, including missing side information and the next item, according to the information available in the item sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' First, we propose a task, the missing information imputation (MII) task that randomly masks some non-missing feature fields, including item IDs, in the input sequence, and then asks the model to recover them in the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Since the next item to be recommended can also be seen as a missing feature field in the sequence, MII unifies the missing side information imputation task with the next item prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MII can be considered as the extension of the masked item prediction task [22] that only considers and masks item IDs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Based on the MII task, we propose a sequential recommendation model, called missing information imputation recommender (MIIR), that jointly imputes missing side information and predicts the next item for the given item sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR employs a dense fusion self-attention (DFSA) mechanism to fuse the information in IDs and other feature fields for predicting both missing side information and the next item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' DFSA captures the relation between any pair of feature fields in the input sequence, allowing it to fully fuse various types of (side) information to impute missing feature values and address the main recommendation challenge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We conduct extensive experiments on three public datasets and show that MIIR significantly outperforms state-of-the-art sequential recommendation baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We also confirm that (i) imputing missing side information and (ii) DFSA both help to improve the performance of sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The main contributions of this work are as follows: We propose to unify the missing side information imputa- tion task and the sequential recommendation task through missing information imputation (MII).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To the best of our knowledge, this is the first work of its kind in sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We present a novel sequential recommendation model, missing information imputation recommender (MIIR), that employs MII to provide the signal for simultaneously imputing the missing item side information and predict- ing the next item and dense fusion self-attention (DFSA) to fuse various information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We conduct extensive experiments to verify the effective- ness of MII, MIIR, and DFSA in sequential recommenda- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2 RELATED WORK 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1 Sequential recommendation with side information Side information fusion has been widely used in sequential recommendation because it can help to capture transition patterns among items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We classify existing work into work that uses self-attention and work that does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As to work that does not use self-attention, Hidasi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [9] employ parallel RNNs to extract the information from ID sequences of item IDs and sequences of features;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' they then examine different ways of combining the outputs of the RNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [23] propose self-supervised tasks to maximize the mutual information between an item and its attributes or between a sequence of item IDs and the sequence of their attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [24] construct a het- erogeneous graph to aggregate different types of categorical attributes, then aggregate the representations of attribute types to get item representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Inspired by the success of self-attention mechanisms [25, 26, 27], some work uses self-attention to fuse items and side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [10] first use a vanilla atten- tion mechanism to fuse different types of side information on each item, and then use two branches of self-attention blocks to model transition patterns between IDs and side information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' they then concatenate the hidden states of the two blocks for item recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [28] pro- pose a non-invasive self-attention mechanism that uses pure item ID representations as values and representations that integrate side information as queries and keys to calculate the attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Xie et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [15] decouple the non-invasive self- attention of different types of side information to get fused attention matrices for items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Although many methods have been proposed for se- quential recommendation with side information, they (i) ne- glect the missing information problem, and use fixed special values to fill missing feature fields, which might harm the performance, and (ii) hardly explore the relation between an item and the side information of other items in the same sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' These are aspects that we contribute on top of prior work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Title Title Truth Title Item Item Item Item Category Category CategoryTitle Title [PAD] Title Predict Item Item Item Item Category [PAD] Category CategoryTitle Title [Impute] Title Predict Item Item Item Item Category Category [Impute] Category3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2 Missing side information in recommendation In real-world applications, the side information of users and items may be incomplete or missing, which may hurt the performance of recommendation models that rely on side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The traditional way to solve the problem of missing side information is to fill the missing feature fields with heuristic values [29, 30, 18], such as the most frequent feature values, average values, randomized values, the value unknown, or padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As some studies have reported, these special values are independent of the context, and using them may lead to biased parameter estimation and prediction [31, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Another way to deal with missing feature fields is to impute their missing values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Early approaches use KNN-based methods [33] or auto-encoders (AEs) [34, 35] to predict the missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [19] propose an AE- based model with modality dropout, which randomly drops representations of user or item information of different modalities in hidden states and reconstructs them by an AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Cao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [36] present a translation-based recommendation model that models preferences as translations from users to items, and jointly trains it with a knowledge graph (KG) completion model that predicts the missing relations in the KG for incorporating knowledge into the recommendation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Instead of imputing the missing side information, Shi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [18] propose an adaptive feature sampling strategy, which employs layer-wise relevance propagation [37] to calculate the importance of different features and samples features to make the model more robust against unknown features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' [20] propose a GCN-based model to jointly predict users’ preferences to items and predict the missing attribute values of users or items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' What we add on top of prior work on missing infor- mation in recommendation is that we focus on missing information in the context of sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3 METHOD 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1 Overview Before going into details of the proposed MII task and MIIR model, we introduce notation used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We denote the item set as I = {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , iNi}, where Ni is the number of items and each item ID ik ∈ RNi is represented as a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In addition to IDs, items have other feature fields corresponding to their side infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In this work, we consider categorical feature fields, including category and brand, and textual feature fields, including title and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We denote the category set as C = {c1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , cNc}, where Nc is the number of categories and each category ck ∈ RNc is a one-hot vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Similarly, we denote the brand set as B = {b1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , bNb}, where Nb is the number of brands and each brand bk ∈ RNb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For titles and descriptions of items, we employ BERT [38] to encode them into fixed-length vectors of size 768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We denote all titles and all descriptions as T = {t1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , tNi} and D = {d1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , dNi}, respectively, where tk and dk ∈ R768.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We use S = [s1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , sn] to denote a sequence with n items, where sk = [si k, sc k, sb k, st k, sd k] is the sequence of features fields of the k-th item, si k ∈ I, sc k ⊆ C, sb k ∈ B, st k ∈ T, and sd k ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As an item may have multiple categories, (a) Sequential recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (b) Missing information imputation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Comparing the sequential recommendation task and the missing information imputation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (Same visual conventions as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=') we let sc k be a subset of C, which can be represented as a multi-hot vector sc k ∈ RNc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For missing item IDs, cate- gories and brands, we have special one-hot vectors denoted as imiss ∈ I, cmiss ∈ C and bmiss ∈ B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For missing titles and descriptions, we use the vector of “[CLS][SEP]” encoded by BERT to represent them, which are denoted as tmiss ∈ T and dmiss ∈ D, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' These missing representations will be used in both MIIR and the baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It is worth noting that other feature fields can be formalized and modeled in a similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The missing information imputation task is to im- pute the values of the missing feature fields in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The sequential recommendation task is to predict the next item sn+1 for S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' By appending a new item sn+1 = [imiss, cmiss, bmiss, tmiss, dmiss] to the end of S and im- puting the imiss of sn+1, we can formulate the next item prediction task as a special case of missing information imputation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2, we compare the sequential rec- ommendation task and the missing information imputation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In the sequential recommendation task, the next item is not considered as a missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In the missing information imputation task, the next item is simply a missing feature field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' A model for the missing information imputation task that follows a unified way to impute both the next item and the other missing side information can be used for sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To unify the missing side information imputation and next item recommendation tasks, we propose a sequential recommendation model called missing information imputa- tion recommender (MIIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As we illustrate in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3, MIIR consists of three main components: (i) an embedding layer, (ii) a dense fusion self-attention (DFSA) mechanims, and Fusion T ID Side information808 Missing information imputation 0808 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='084 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Architecture of the missing information imputation recommender (MIIR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR takes a sequence of randomly masked feature fields as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It transforms the input sequence into embeddings using the em- bedding layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Then it employs a dense fusion self-attention mechanism to fuse information in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Finally, MIIR uses an output layer to reconstruct the input sequence and calculate the MII loss on masked feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (Same visual conventions as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=') (iii) an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' First, the embedding layer translates the input sequence into a series of embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Then, the DFSA mechanism employs several transformer [17] layers to model the relation between any pair of feature fields in the sequence and fuse side information into the model for both imputation and recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Finally, the output layer imputes the missing feature values including item IDs in the sequence based on the output of DFSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Next, we will introduce the details of these main components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2 Embedding layer The embedding layer projects all item feature fields in the input sequence into low-dimensional dense vectors with a unified length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the k-th item sk = [si k, sc k, sb k, st k, sd k] in the given sequence S, the embedding layer uses different ways to translate different feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the high-dimensional sparse vectors of si k, sc k and sb k, we follow Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 1 to get the item embedding ei k ∈ Re, the category embedding ec k ∈ Re, and the brand embedding eb k ∈ Re: ei k = Eisi k, ec k = Ecsc k, eb k = Ebsb k, (1) where Ei ∈ Re×Ni is the item embedding matrix, Ec ∈ Re×Nc is the category embedding matrix, Eb ∈ Re×Nb is the brand embedding matrix, and e is the embedding size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the high-dimensional dense vectors of st k and sd k, we project them into low-dimensional embeddings, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', the title embedding et k ∈ Re and the description embedding ed k ∈ Re, respectively, using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2: et k = Etst k, ed k = Edsd k, (2) where Et ∈ Re×768 and Ed ∈ Re×768 are the projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In order to distinguish different types of feature fields in the same item, we learn a field embedding for each type of feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We denote the field embeddings of ID, category, brand, title and description as f i, f c, f b, f t and f d ∈ Re, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To distinguish different items in different positions in the same sequence, we also inject the position information into the model by learning position embeddings, where the k-th position embedding is denoted as pk ∈ Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Finally, we add each field embedding to the corresponding item or feature embedding of sk, and add pk to all embeddings of sk, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3: Hk = � ����� hi k hc k hb k ht k hd k � ����� = � ����� ei k + f i + pk ec k + f c + pk eb k + f b + pk et k + f t + pk ed k + f d + pk � ����� , (3) where hi k, hc k, hb k, ht k, hd k ∈ Re, and Hk ∈ R5×e is the hidden state of sk that is the stack of all embeddings of its feature fields in order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3 Dense fusion self-attention The dense fusion self-attention (DFSA) mechanism follows a unified way to impute missing feature fields, both item IDs and side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To exploit the information in a given context for imputation, we need to model the relations be- tween different feature fields and fuse the representations of various feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' DFSA calculates the attention values between any pair of feature fields and fuses the information of other feature fields based on the attention value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' By calculating the attention value, DFSA captures all possible (hence dense) pairwise relations between feature fields to facilitate missing information imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Specifically, we first stack the hidden states of all items in S in order by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 4: H = � ���� H1 H2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Hn � ���� , (4) where H ∈ R5n×e is the hidden state matrix of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Then, DFSA employs a transformer with L layers to update H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Each transformer layer Trm(·) is composed of two sub- layers: (i) multi-head self-attention MH(·) and (ii) position– wise feed-forward PFFN(·), as defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5: Hl+1 = Trm(Hl) = LN( �Hl + Dropout(PFFN( �Hl))) �Hl = LN(Hl + Dropout(MH(Hl))) MH(Hl) = [head1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' headh]WH headi = Attn(HlWQ i , HlWK i , HlWV i ) Attn(Q, K, V) = softmax(QK⊤/√e + M)V PFFN( �Hl) = GELU( �HlWF 1 + bF 1 )WF 2 + bF 2 , (5) 808 Output sequence Output layer 王 Dense fusion self-attention Position embedding + Field embedding + Item/feature embedding Embedding layer 0808 08 Randomly masked Mask Input sequence5 where LN is layer normalization [39], Dropout is dropout [40], Attn is attention, GELU is a Gaussian error linear unit activation [41], [.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='] is the concatenation operation, h is the number of heads, WH ∈ Re×e, WQ i , WK i , WV i ∈ Re×e/h, WF 1 ∈ Re×4e, WF 2 ∈ R4e×e, bF 1 ∈ R4e and bF 2 ∈ Re are trainable parameters, Hl and Hl+1 ∈ R5n×e are the output hidden state matrices in the l-th layer and the (l+1)- th layer, and H0 = H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The matrix M ∈ R5n×5n in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5 is the attention mask which is defined as: Mj,y i,x = � 0, allow to attend, −∞, prevent from attending, (6) where i and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' , n}, x and y ∈ {i, c, b, t, d}, Mj,y i,x ∈ M is the mask to control whether the feature field sy j can attend to the feature field sx i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We set all Mj,y i,x = 0,1 which means we allow to attend between any pair of feature fields in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Therefore, the DFSA can model relations and fuse information between all possible pairs of feature fields to facilitate both imputation and recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='4 Output layer The output layer reconstructs the input feature fields based on the output hidden states of DFSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' First, we split the final output hidden state matrix HL of DFSA by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 7: HL = �E = � ����� �E1 �E2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' �En � ����� , where �Ek = � ����� ˆei k ˆec k ˆeb k ˆet k ˆed k � ����� , (7) and ˆei k, ˆec k, ˆeb k, ˆet k, ˆed k ∈ Re.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Similar to the embedding layer, the output layer takes different ways to reconstruct different types of feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Specifically, for the categorical feature fields, we calculate the probability distributions pi k ∈ RNi, pc k ∈ RNc and pb k ∈ RNb of the item ID, category and brand of the k-th item sk as follows: pi k = softmax(Ei⊤ˆei k) pc k = sigmoid(Ec⊤ˆec k) pb k = softmax(Eb⊤ˆeb k), (8) where Ei ∈ Re×Ni, Ec ∈ Re×Nc, Eb ∈ Re×Nb are the re- used item embedding matrix, category embedding matrix and brand embedding matrix in the embedding layer, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Note that we see the category prediction as a series of binary classifications, because an item may contain multiple categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Then we get the reconstructed item ID ˆsi k ∈ RNi, category ˆsc k ∈ RNc and brand ˆsb k ∈ RNb based on the probability distributions, as shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 9: ˆsi k = argmax(pi k) ˆsc k = 1(pc k > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5) ˆsb k = argmax(pb k), (9) where 1(α) is the indicator function that equals 1 if α is true and 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Meanwhile, for the textual feature fields, 1Here we neglect the padding items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' we follow Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 10 to get the reconstructed title ˆst k ∈ R768 and description ˆsd k ∈ R768 directly: ˆst k = Otˆet k, ˆsd k = Odˆed k, (10) where Ot ∈ R768×e and Od ∈ R768×e are the projection matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5 Missing information imputation loss We train MIIR with MII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MII first randomly masks feature fields in the sequence with probability p, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', replacing a non-missing feature value with the corresponding missing feature value imiss, cmiss, bmiss, tmiss or dmiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the k-th item sk in the sequence S, we use mi k, mc k, mb k, mt k and md k ∈ {true, false} to denote whether its ID, category, brand, title and description are masked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Then, MIIR learns to recover the masked feature fields by MII and impute the missing feature values based on the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Specifically, there are differences in the calculation of the missing information imputation loss for different types of feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the categorical feature fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', ID, category and brand), our goal is to minimize the cross- entropy loss: Li k = −1(mi k)si k ⊤ log(pi k) Lc k = −1(mc k)(sc k ⊤ log(pc k) + (1 − sc k ⊤) log(1 − pc k))/Nc Lb k = −1(mb k)sb k ⊤ log(pb k), (11) where Li k, Lc k and Lb k are the imputation loss for the item ID, category and brand of sk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the textual feature fields (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', title and description), our goal is to minimize the mean square error loss: Lt k = 1(mt k)∥st k − ˆst k∥2 Ld k = 1(md k)∥sd k − ˆsd k∥2, (12) where Lt k and Ld k are the imputation loss for the title and description of sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The missing information imputation objective of the entire model on S is shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 13: Lmii S = 1/n n � k=1 Lmii k Lmii k = Li k + Lc k + Lb k + Lt k + Ld k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (13) Note that since the item ID is one of the feature fields and the next item prediction is a MII task, MIIR trained by MII can directly be applied to sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In experiments, we also consider to further fine-tune MIIR or directly train MIIR with the masked item prediction loss to make the model only focus on the item prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Specifically, we randomly mask some items with their all feature fields in the given sequence, and then let MIIR predict the masked item IDs only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The recommendation loss (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', the masked item prediction loss) on S is defined as: Lrec S = 1/n n � k=1 Lrec k Lrec k = Li k = −1(mi k)si k ⊤log(pi k), (14) where Lrec k is the recommendation loss for sk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 6 TABLE 1 Summary of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The missing rate is the percentage of missing feature fields in all feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Especially, “Missing rate D” is the missing rate on the dataset after discarding side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Dataset Beauty Sports and Outdoors Toys and Games #items 121,291 194,715 164,978 #sequences 52,374 84,368 58,314 Average length 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='97 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='50 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='99 #categories 656 3,035 957 #brands 13,188 14,163 14,135 Missing rate 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='54% 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11% 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='20% Missing rate D 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='32% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='12% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='51% 4 EXPERIMENTAL SETUP 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1 Research questions In this paper, we seek to answer the following research questions: (RQ1) How does MIIR perform on the sequential recom- mendation task compared to state-of-the-art meth- ods?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (RQ2) What are the benefits of training MIIR with MII?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (RQ3) Does modeling the relation between any pair of feature fields in item sequences help sequential rec- ommendation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (RQ4) How about the performance of MIIR on imputing the missing side information?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (RQ5) What can we find about MIIR by the case study?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2 Datasets There are many public datasets for experimenting with sequential recommendation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' see [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, we need sequential recommendation datasets that come with side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We conduct experiments on three public datasets: “Beauty”, “Sports and Outdoors” and “Toys and Games” [42], as they have rich item side information, in- cluding category, brand, title and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We follow common practices [10, 28] to process the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We sort each user’s records in chronological order to construct an item sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We filter out item sequences whose length is less than 5 to avoid noise from the cold- start problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For each item sequence, we use the last item for test, the second last item for validation, and the rest items for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For each test or validation item, we randomly sample 99 negative items for ranking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We randomly discard side information of items with probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We use “Beauty D”, “Sports and Outdoors D” and “Toys and Games D” to denote the datasets after discarding side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The statistics of the datasets after pre- processing are summarized in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3 Baselines We compare MIIR with the following recommendation baselines, which can be grouped into (i) methods without side information fusion, (ii) methods with side information fusion and (iii) methods with missing feature values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Methods without side information fusion: – GRU4Rec employs RNNs to capture sequential pat- terns between items for sequential recommendation [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' – SASRec uses the self-attention mechanism to model item sequences for next item recommendations [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' – BERT4Rec uses a bidirectional self-attention network train-ed by a masked item prediction task for sequential recommendation [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Methods with side information fusion: – PRNN employs parallel RNNs to process items and their side information respectively, then combines the hidden states of the RNNs for next item prediction [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' – FDSA leverages two separate self-attention networks to model the ID transition patterns and the feature transition patterns respectively, then concatenates the outputs of two networks for next item prediction [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' – NOVA adopts a non-invasive self-attention mecha- nism to leverage side information under the BERT4Rec framework for sequential recommendation [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Methods with missing feature values: – RFS randomly samples feature fields to introduce more missing information when training [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' RFS is to make the model more robust with missing feature values instead of imputing missing feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We combine RFS with FDSA and NOVA, and denote the variants as FDSA+RFS and NOVA+RFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' – LRMM designs an auto-encoder with the modality dropout to impute both user ratings and missing side information for each item [19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Note that LRMM is not a sequential model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Furthermore, we use the imputed missing side information by LRMM to train FDSA and NOVA, and denote them as FDSA+LRMM and NOVA+LRMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Other methods with side information fusion, such as [23, 24], can only model categorical item side information;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' for a fair comparison, we do not consider them as baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In addition to the baselines listed above, we compare MIIR against four variants, namely MIIR-F, MIIR-R, MIIR-M, and Sparse-MIIR, to be defined in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We unify the sequential recommendation loss in all base- lines, MIIR, and its variants to the cross-entropy loss, rather than the pairwise loss [43], to avoid noise due to negative sampling in the pairwise loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='4 Metrics and implementation To evaluate the performance of sequential recommendation methods, we employ two widely used evaluation metrics: HR@k (hit ratio) and MRR (mean reciprocal rank) [1], where k ∈ {5, 10}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' HR measures the proportion of the sequences whose ground-truth items are amongst the top ranked items in all test sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MRR is the average of reciprocal ranks of the ground- truth items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For all baselines and our proposed model, we initialize the trainable parameters randomly with the Xavier method [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We train all methods with the Adam optimizer [45] for 100 epochs, with a batch size of 128 and a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We also apply gradient clipping [46] with range [−5, 5] during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' According to the average length in Table 1, we set the maximum sequence length to 20 for both datasets for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 7 TABLE 2 Performance comparison of MIIR, variants, and the baselines on the “Beauty” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR-F is a variant of MIIR that is fine-tuned using the recommendation loss (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1) and MIIR-R is a variant trained using the recommendation loss only (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The highest overall performance is denoted in bold face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The highest performance among the baselines is underlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (%) is the performance gain of MIIR against the best baseline method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' ∗ indicates that an improvement is statistically significant based on a two-sided paired t-test with p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='05.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Beauty Beauty D Method HR@5 HR@10 MRR HR@5 HR@10 MRR GRU4Rec 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='47 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='50 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='47 SASRec 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='83 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='83 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16 BERT4Rec 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='22 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='22 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 PRNN 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='27 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='70 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='08 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='55 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='23 FDSA 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='22 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='83 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='63 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='55 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 MIIR-F 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 51.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='40 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='07 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 Impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (%) +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='18∗ +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11∗ +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='25∗ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='09∗ +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='13∗ +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80∗ All hyper-parameters of the baselines are set following the suggestions from the original papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the hyper- parameters of MIIR, we set the embedding size e to 64, the number of heads h to 4, and the number of layers L to 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We set the dropout rate in DFSA and the mask probability p in MII to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5 EXPERIMENTAL RESULTS 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1 Overall performance To answer RQ1, we compare MIIR against the recommenda- tion models listed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3 on the three datasets from TABLE 4 Performance comparison of MIIR, variants, and the baselines on the “Toys and Games” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Toys and Games Toys and Games D Method HR@5 HR@10 MRR HR@5 HR@10 MRR GRU4Rec 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='19 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='15 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='90 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='19 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='15 21.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='64 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='89 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='74 MIIR-F 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='00 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='76 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='57 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='25 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='45 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='75 MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='37 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='00 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='69 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='81 Impr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (%) +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='46∗ +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30∗ +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='03∗ +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='50∗ +4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='42∗ +3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='23∗ Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Table 2, 3 and 4 list the evaluation results of all methods on each dataset, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Based on these results, we have the following observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' First, on all datasets, MIIR performs significantly better than all baselines by a large margin despite the different missing rates, in terms of HR@5, HR@10 and MRR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR has two major advantages: (i) MIIR trains the model using MII to enhance its ability to deal with missing side infor- mation in sequential recommendation (see detailed analysis in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2), and (ii) MIIR employs DFSA to improve the side information fusion in the model (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3 for further analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Second, the item side information can help sequential recommender systems to more accurately model the tran- sition patterns among items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To verify this, we divide all methods into three groups: (i) GRU4Rec and PRNN that are based on RNNs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (ii) SASRec and FDSA that are based on left-to-right self-attention networks;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' and (iii) BERT4Rec, NOVA, and MIIR that employ bidirectional self-attention networks and the masked item prediction task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In each group, we see that methods that fuse side information outperform methods that only rely on item IDs, which illustrates that item side information does help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Third, the performance of PRNN, FDSA, NOVA and MIIR on the “Beauty”, “Sports and Outdoors” and “Toys and Games” datasets is higher than that on the discarded versions of the datasets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', “Beauty D”, “Sports and Out- doors D” and “Toys and Games D”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We see two reasons for this difference: (i) the “Beauty D”, “Sports and Outdoors D” and “Toys and Games D” datasets discard some side infor- mation, so the available side information becomes less, and (ii) using the special values (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', imiss, cmiss, bmiss, tmiss and dmiss) to fill missing feature fields may be harmful to PRNN, FDSA and NOVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fourth, by comparing FDSA+RFS and NOVA+RFS with FDSA and NOVA, we see RFS cannot consistently improve the performance of FDSA and NOVA on all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' What’s worse, RFS would degrade the performance of FDSA and NOVA in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Because RFS is to introduce more 8 TABLE 5 Performance comparison of whether to exploit missing feature fields on the “Beauty” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR-M and MIIR-R-M are the variants of MIIR and MIIR-R respectively that mask missing feature fields in self-attention (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Beauty Beauty D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='92 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='46 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='85 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='90 MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='59 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='60 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='85 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='92 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='96 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='41 MIIR-M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='67 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='45 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='12 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='83 MIIR-R-M 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='40 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='31 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='71 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='42 TABLE 6 Performance comparison of whether to exploit missing feature fields on the “Sports and Outdoors” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Sports and Outdoors Sports and Outdoors D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='63 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='55 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 MIIR-R 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='70 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='40 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='07 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 MIIR-M 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='12 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='36 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='65 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='81 MIIR-R-M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='71 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='98 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='15 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='33 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='10 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='12 missing feature values into the model training instead of imputing missing feature fields, it cannot deal with the missing side information problem fundamentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fifth, the performance of LRMM is significantly worse than that of the sequential recommendation models with side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' LRMM even performs worse than GRU4Rec, SASRec and BERT4Rec that neglect the item side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The main reason is that LRMM is not a sequential model, so it cannot exploit the relation and information in sequences to make recommendation and imputation, however it is essential in the sequential rec- ommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can also observe that FDSA+LRMM and NOVA+LRMM outperform FDSA and NOVA in exper- iments, which verifies the effectiveness of the imputation results of LRMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It also proves imputing missing feature values is a better way to alleviate the missing side informa- tion problem than using fixed special values and RFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Sixth, modeling sequential recommendation as missing information imputation is sufficient to train a recommen- dation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To verify this, we conduct an experiment that first pre-trains MIIR using the missing information imputation loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 13), and then fine-tunes it using the recommendation loss (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We use MIIR-F to denote this variant of MIIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In Table 2 we see that MIIR-F performs worse than MIIR in most cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fine-tuning MIIR-F with the recommendation loss might lead to overfitting, resulting in performance decreases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' This result supports the conclusion that with MII we can unify the sequential recommendation task as a particular type of missing information imputation task to train MIIR together with the other imputation task for missing item side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='2 Benefits of MII To answer RQ2, we analyze how MIIR benefits from training with MII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' TABLE 7 Performance comparison of whether to exploit missing feature fields on the “Toys and Games” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Toys and Games Toys and Games D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='64 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='89 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='74 MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='37 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='00 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='69 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='81 MIIR-M 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='33 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='22 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='97 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='82 MIIR-R-M 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='22 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='29 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='28 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='53 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='47 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 In Table 2, 3 and 4, we report on results of a variant of MIIR that directly trains MIIR with the recommendation loss shown in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We write MIIR-R for this variant of MIIR without the supervised signal of MII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' When we compare the performance of MIIR and MIIR-R, we see very substantial gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' This confirms the effectiveness of training MIIR with MII, which accounts for the main part of the improvement of MIIR over other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' To demonstrate that MIIR can mine useful information from missing feature fields by training with MII, we design a variant of MIIR called MIIR-M by masking missing feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In MIIR-M, we revise the attention mask M used in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5, which is a null matrix in MIIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The revision in M is defined as: Mj,y i,x = � −∞, if sx i or sy j ∈ {cmiss, bmiss, tmiss, dmiss}, 0, otherwise, (15) where the condition of sx i or sy j ∈ {cmiss, bmiss, tmiss, dmiss} depends on the original input sequence instead of the sequence after randomly masking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The purpose of the variant is to prevent the model from attending to the missing feature fields about item side information in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' On the one hand, MIIR-M cannot mine and fuse any information in missing feature fields for sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' On the other hand, MIIR-M is unable to exploit the information in non-missing feature fields to impute the missing side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Besides, we mask missing feature fields for MIIR-R to analyze how missing feature values affects the performance of MIIR without MII, denoted as MIIR-R-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In Table 5, 6 and 7, we compare MIIR and MIIR-R with MIIR-M and MIIR-R-M, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can find that MIIR outperforms MIIR-M in most cases, which illustrates that MIIR can extract useful information from missing feature fields to improve the sequential recommendation perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can also observe that MIIR-R-M performs better than MIIR-R in some cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' This phenomenon indicates that using fixed special values for filling missing feature fields can suffer the model performance, therefore masking miss- ing feature fields may be a better way without imputation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' On the “Beauty” dataset, MIIR only achieves comparable performance with MIIR-M, and MIIR-R also performs worse than MIIR-R-M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, the performance gap from MIIR to MIIR-M is smaller than that from MIIR-R to MIIR-R- M, and we have similar observations on other datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It illustrates that imputing missing feature values has the superiority over masking them for alleviating missing side information problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 9 TABLE 8 Performance comparison of dense and sparse attention on the “Beauty” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Sparse-MIIR and Sparse-MIIR-R are variants of MIIR and MIIR-R, respectively, in which DFSA is replaced by SFSA (see Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Beauty Beauty D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='92 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='46 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='85 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='90 MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='59 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='60 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='85 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='92 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='96 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='41 Sparse-MIIR 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='71 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='60 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='87 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='98 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='34 Sparse-MIIR-R 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='95 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='02 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='35 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='84 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='19 TABLE 9 Performance comparison of dense and sparse attention on the “Sports and Outdoors” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Sports and Outdoors Sports and Outdoors D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='63 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='55 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 MIIR-R 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='70 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='40 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='07 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='82 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 Sparse-MIIR 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='52 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='04 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='64 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='24 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='91 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='67 Sparse-MIIR-R 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='29 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='25 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='56 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='72 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='21 MIIR-M also outperforms all baselines on the three datasets with different missing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Training MIIR with MII helps MIIR to make use non-missing feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Imputing the masked non-missing feature values requires the model to capture the relations between different feature fields, so MII guides MIIR to better fuse side information into the model for improving the sequential recommenda- tion performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='3 Effectiveness of DFSA To answer RQ3, we conduct an ablation study to analyze the effectiveness of DFSA in MIIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We first compare MIIR-R, the variant of MIIR that is trained with recommendation loss only, with the baselines in Table 2, 3, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' MIIR-R achieves better or comparable per- formance with the baselines on most evaluation metrics of all datasets, even without the help of MII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The main reason is that MIIR-R has dense fusion self-attention (DFSA) to better fuse information in the item sequence for improving sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In order to validate that it is important to model all pos- sible pairwise relations in an item sequence for sequential recommendation, we design another self-attention mecha- nism called sparse fusion self-attention (SFSA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' SFSA modifies the attention mask M in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5 into: Mj,y i,x = � 0, if i == j or x == y, −∞, otherwise, (16) where the condition i == j or x == y means that SFSA only allows to attend between the pair of feature fields belonging to the same item or the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Therefore, SFSA only models the relation between different feature fields of the same item or the relation between the same type of feature fields of different items in the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' These relations are also modeled in some baselines, such as PRNN and FDSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' TABLE 10 Performance comparison of dense and sparse attention on the “Toys and Games” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Toys and Games Toys and Games D Method HR@5 HR@10 MRR HR@5 HR@10 MRR MIIR 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='64 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='89 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='74 MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='37 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='00 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='69 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='30 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='81 Sparse-MIIR 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='77 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='23 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='06 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='27 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='79 Sparse-MIIR-R 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='58 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='66 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='80 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='46 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='54 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='53 TABLE 11 Performance comparison of LRMM and MIIR for the missing side information imputation on the “Beauty D”, “Sports and Outdoors D” and “Toys and Games D” datasets, where P: precision, R: recall, F1: F1 score, ACC: accuracy, MSE: mean square error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Dataset Field Metric LRMM MIIR Beauty D Category P 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='15 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='64 R 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='41 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='97 F1 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='96 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 Brand ACC 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='01 Title MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0871 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0514 Description MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1454 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0704 Sports and Outdoors D Category P 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='02 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='97 R 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='31 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='91 F1 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='38 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='40 Brand ACC 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='06 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='43 Title MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0927 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0534 Description MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1474 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0835 Toys and Games D Category P 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='32 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='31 R 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='08 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='31 F1 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='11 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='50 Brand ACC 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='61 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='49 Title MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0858 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0514 Description MSE 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='0777 In Table 8, 9 and 10, we compare the performance of DFSA and SFSA as components of MIIR and MIIR-R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We write Sparse-MIIR and Sparse-MIIR-R for the variants of MIIR and MIIR-R, respectively, in which DFSA is replaced by SFSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can see that MIIR outperforms Sparse-MIIR on all datasets despite different missing rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' What’s more, MIIR-R outperforms Sparse-MIIR-R in most cases too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Mod- eling the relations between any pair of feature fields helps to make more effective use of item side information to improve sequential recommendation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' By comparing MIIR with Sparse-MIIR, we also notice that the improvement by DFSA on the three datasets is higher than the improvement on the discarded versions of the datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We guess that DFSA may also model more noisy relations related to missing feature fields when the missing rate increases, which would make the performance degeneration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='4 Imputation performance (RQ4) To answer RQ4, we compare LRMM and MIIR based on their imputation results for the discarded side information of all datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For the test sequences from the “Beauty D”, “Sports and Outdoors D” and “Toys and Games D” datasets, we can compare the imputed results with the ground-truth before discard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For different types of feature fields, we consider 10 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (a) and (b) are two sequences with their imputed categories from the “Beauty D” dataset, (c) and (d) are two sequences with their imputed brands from the “Toys and Games D” dataset, (e) is the sequence with its imputed categories and brands from the “Sports and Outdoors D” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' different metrics: (i) for category that is corresponding to the multi-class classification task, we calculate the preci- sion, recall and F1 score for evaluation which bigger is better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (ii) for brand that is corresponding to the one-class classification task, we calculate the accuracy for evaluation which bigger is better;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (iii) for title and description that are both corresponding to the multi-variable regression task, we calculate the mean square error (averaged by the length of title/description vector) for evaluation which smaller is better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' In Table 11, we list the evaluation results for compari- son.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can observe that MIIR achieves better imputation performance than LRMM on the precision of category and the MSE of title and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Whereas, LRMM outper- forms MIIR on the recall of category and the accuracy of brand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Both LRMM and MIIR can infer some discarded side information, so they can alleviate the missing side infor- mation problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Comparing to LRMM, MIIR can exploit more information from the sequence to impute the missing side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, MIIR may also impute some inaccurate results due to the over-dependence on the given context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5 Case Study (RQ5) Finally, to answer RQ5, we sample some test cases from datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='4, we list some sequences with their imputed results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can observe that MIIR can generate different feature values for missing feature fields according to different contexts (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', items and sequences), which is better than using fixed predefined values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' What’s more, MIIR may infer the ground-truth missing value, including the side information of the target next item, to give the model with a more accurate guidance for recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For example, MIIR imputes a part of the discarded cate- gories in the sequence (b) and the discarded brands in the sequence (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We can also observe that the side information of items in the same sequence may be related, which is why MIIR can infer the ground-truth missing value in light of the given context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' However, MIIR may be over-dependent on the information from the sequence, leading to impute inaccurate results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' For instance, in the sequence (e), MIIR imputes the wrong categories and brand for the item 5401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Additionally, we visualize the attention weights from the missing item ID (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', the next item ID) to all feature fields in the given sequence in DFSA, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We reshape the attention weights into a matrix with the shape of 5 × n, where 5 is the number of the feature field types and n is the sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' First, we can see that MIIR exploits the information from all feature fields of the given sequence to predict the next item, which emphasizes the necessity to model the relation between any pair of feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Second, we can observe that different layers focus on different types of feature fields, where the first layer mainly attends to ID, and the third layer mainly attends to title and description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' It illustrates MIIR gradually fuses different types of side information into the model by different layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Because the information in textual feature fields is more difficult to extract, MIIR needs more deeper layers to fuse textual feature fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Third, we can find different heads in the same layers have similar attention patterns, which means there may be some redundant parameters in MIIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (a) id: 22434 id: 69550 id: 52863 id: 52866 id: 52867 id: 52868 id: 65782 id: 65790 id: 83354 Original Sequence category: 384,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 487,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' category: 38,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 271,' metadata={'source': 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category: 2078,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2299 category: 1286,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 category: 1286,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 category: 1286,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 2191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 2191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 2191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 2191,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 2479 brand: 4696 brand: 4087 brand: 4087 brand: 4087 brand: 4087 brand: 4087 brand: 9545 brand: 408711 (a) A sequence from the “Beauty D” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (b) A sequence from the “Sports and Outdoors D” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' (c) A sequence from the “Toys and Games D” dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Visualization for the attention weights from the missing item ID field to all feature fields of all heads and layers in MIIR on three sequences from different datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' 6 CONCLUSION We have studied the missing side information problem in sequential recommendation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We have proposed the missing information imputation (MII) task to unify the missing side information imputation task and the sequential recommen- dation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We have presented a novel sequential recom- mendation model named missing information imputation recommender (MIIR) to simultaneously impute missing fea- ture values and predict the next item for a given sequence of items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We have proposed a dense fusion self-attention (DFSA) mechanism to model different relations in the item sequence and to fuse side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Based on experiments and analyses on three datasets with different settings of the missing rates we have found that MIIR outperforms state-of-the-art methods for sequen- tial recommendation with side information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We have ver- ified that MIIR can identify useful side information from missing feature fields by training with the MII task, and that the DFSA mechanism improves the recommendation effectiveness of MIIR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' As to broader implications of our work, we offer a new perspective by revealing a correlation between missing side information imputation and the sequential recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' They both concern the prediction of missing infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' The perspective operationalized with MIIR can be adopted as a foundational paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Other prediction tasks related to recommendation, such as rating prediction, user profile prediction, and next basket recommendation can also be formulated as a MII task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' Limitations of our work include the following: (i) since DFSA treats side information as part of the sequence (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=', in our case, the actual sequence length is 5x the number of items) and models all possible pairwise relations in an item sequence, it 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+page_content='description ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='1699 83004 90253151526203262038 5850mis5716999 3004902531515220322038 5850m557 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16999 300402531515220322038850m5716999 3009025315152203620385850mi57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='16999 8300490253151526203262038 5850m55716999 3004902531515262032620385850ms57 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='+o-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='084 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='011 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='05512 recommendation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We aim to further improve MIIR in different directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We will assess the ability of the linear transformer [47, 48] to reduce the computational costs of DFSA and design a mechanism to filter out useless relations at an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' We also plan to design a tailored loss for MIIR by building on recent loss weighting methods [49, 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content=' REPRODUCIBILITY To facilitate reproducibility of the results reported in this paper, the code and data used in experiments are available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtAzT4oBgHgl3EQfy_6c/content/2301.01762v1.pdf'} +page_content='com/TempSDU/MIIR.' metadata={'source': 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-0,0 +1,1735 @@ +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +1 +Self-Supervised RGB-T Tracking with Cross-Input +Consistency +Xingchen Zhang∗, Member, IEEE, Yiannis Demiris, Senior Member, IEEE +Abstract—In this paper, we propose a self-supervised RGB-T +tracking method. Different from existing deep RGB-T trackers +that are using a large number of annotated RGB-T image pairs +for training, our RGB-T tracker is trained using unlabeled +RGB-T video pairs in a self-supervised manner. We propose +a novel cross-input consistency-based self-supervised training +strategy based on the idea that tracking can be performed using +different inputs. Specifically, we construct two distinct inputs +using unlabeled RGB-T video pairs. We then track objects using +these two inputs to generate results, based on which we construct +our cross-input consistency loss. Meanwhile, we propose a re- +weighting strategy to make our loss function robust to low-quality +training samples. We build our tracker on a Siamese correlation +filter network. To the best of our knowledge, our tracker is +the first self-supervised RGB-T tracker. Extensive experiments +on two public RGB-T tracking benchmarks demonstrate that +the proposed training strategy is effective. Remarkably, despite +training only with a corpus of unlabeled RGB-T video pairs, our +tracker outperforms seven supervised RGB-T trackers on the +GTOT dataset. +Index Terms—RGBT tracking, object tracking, thermal im- +ages, image fusion, information fusion +I. INTRODUCTION +O +BJECT tracking is an important task and has many +applications in areas such as robots and surveillance. In +recent years, many tracking algorithms have been proposed, +and tracking performance has witnessed a significant improve- +ment. However, most visual trackers operate on RGB im- +ages. The performance of these trackers degrades significantly +when RGB images are not reliable (e.g., under poor lighting +conditions), limiting their practical applications. +To improve tracking performance, researchers have used +thermal images and RGB images together to perform RGB-T +tracking [1–7]. This is based on the fact that thermal images +are insensitive to illumination changes while RGB images con- +tain more texture details [8]. Although many efforts have been +put into developing deep learning-based RGB-T trackers and +RGB-T tracking performance has been significantly improved, +existing deep RGB-T trackers [6, 9–11] require a large number +of annotated RGB-T image pairs, as shown in Fig. 1. +It is well-known that annotation is time-consuming and +expensive. Some unsupervised single object trackers have been +proposed to avoid the need for annotations. For example, Wang +et al. [15, 16] and Zhu et al. [17] use a cycle consistency +based on forward-backward tracking to train trackers. There +X. Zhang and Y. Demiris are with the Personal Robotics Laboratory, +Department of Electrical and Electronic Engineering, Imperial College +London, London SW7 2AZ, U.K. (e-mail: xingchen.zhang@imperial.ac.uk, +y.demiris@imperial.ac.uk) +∗ Corresponding author: Xingchen Zhang +Fig. 1. +Performance v.s. required labeled RGB-T image pairs in train- +ing. Although our RGB-T tracker does not need any labeled training data, it +outperforms seven supervised RGB-T trackers on the GTOT dataset [12]. Note +that the exact number of labeled data for TCNN [13] and JCDA-InvSR [14] +(both are supervised trackers) is not mentioned in their papers. +are also some trackers using very spare annotation in training, +e.g., annotation in the initial frame [18–20]. However, all +unsupervised trackers use only single-modal images, namely, +either use RGB images [15, 16, 18–24] or thermal images +[25]. +In this paper, we propose a self-supervised RGB-T tracker +that does not need any manual annotations in training. To +achieve this, we propose a cross-input consistency-based train- +ing strategy to exploit temporal information in unlabeled RGB- +T videos. Our intuition resides on the observation that object +tracking can be performed using different inputs. As shown +in Fig. 2, given a target at frame t, we can track it to obtain +its position at frame (t + 1) using different inputs (e.g., RGB +images, thermal images, or a combination of them). Ideally, if +all tracking are successful, the tracking results in frame (t+1) +should be consistent. +We integrate our self-supervised training strategy into a +Siamese-based discriminative correlation filter (DCF) frame- +work. In implementation, we construct two distinct inputs for +tracking to build cross-input consistency. This cross-input con- +sistency, which is based on temporal information in unlabeled +RGB-T video pairs, can be used to guide the training of our +RGB-T tracker. In addition, because we do not want to use +any manual annotations, we randomly initialize a bounding +box in our training data. Therefore, the training samples are +usually noisy or have bad quality. We propose a re-weighting +strategy to re-weight our loss function to make our training +easier and more effective. In summary, the main contributions +arXiv:2301.11274v1 [cs.CV] 26 Jan 2023 + +75 +CMPP(1.3FPS) +HMFT (30.2 FPS) +APFNet +dataset +73 +JMMAC (4FPS) +DMCNet (2.4 FPS) +MAcNet (0.8 FPS) +71 +-Ours +MFGNet (3.4 FPS), +MSR (%) on the GTOT ( +DAPNet (23 FPS) +69 +mfDiMP (10.3 FPS) +DAPNet (2 FPS) +LTDA (0.4 FPS) +67 +65 +63 +TCNN (15 FPS) +DuSiamRT (116 FPS) +61 +JCDA-InvSR (1.6 FPS) +59 +0 +10 +100 +1000 +Number of manually-annotated RGB-T image pairs (K)JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +2 +Fig. 2. Our cross-input consistency is based on the observation that tracking +can be performed using different inputs. Ideally, the tracking results in frame +(t + 1) obtained from different inputs should be close enough. +of this paper include: +• We propose a self-supervised RGB-T tracker trained +using RGB-T video pairs without human annotations. +• We propose a cross-input consistency-based strategy to +achieve self-supervised training. We use RGB images +and thermal images to construct different inputs for +tracking, based on which a cross-input consistency loss +is constructed to guide training. +• We propose a re-weighting scheme to re-weight our loss +function to make the training more effective. +• Extensive experiments on two RGB-T tracking bench- +marks demonstrate the favorable performance of the pro- +posed method and the potential of self-supervised RGB-T +tracking. +The rest of this paper is organized as follows. Section +II introduces related work. Then, Section III introduces the +proposed method in detail, followed by the introduction to +training data processing in Section IV. Then, Section V +presents results and Section VI gives discussions. Finally, +Section VII concludes this paper. +II. RELATED WORK +A. Single object tracking +Single +object +tracking +methods +mainly +include +deep +learning-based methods [16, 26, 27] and discriminative cor- +relation filter (DCF)-based methods [28, 29]. Most trackers +use RGB images as input and have a high requirement for +good lighting conditions. To make trackers insensitive to light +conditions, some researchers performed tracking using thermal +images [30, 31]. However, thermal images do not have enough +texture details, leading to worse performance than RGB-based +trackers when lighting conditions are good. +B. RGB-T tracking +To alleviate the issue of RGB-based and thermal-based +trackers, researchers performed RGB-T tracking [1, 2, 9, +32]. For example, Zhang et al. [32] proposed a pixel-level +fusion-based RGB-T tracker. In contrast, some RGB-T trackers +are based on feature-level fusion [7, 33, 34] or decision-level +[35] or combine several fusion levels [9]. The performance of +RGB-T trackers have been significantly improved. However, +existing deep RGB-T trackers need a large number of RGB-T +image pairs for training. +C. Unsupervised object tracking +Researchers have proposed unsupervised trackers to allevi- +ate the need for annotations. For example, Vondrick et al. [21] +proposed to train an RGB tracker by colorizing videos. Wang +et al. [15, 16] and Shen et al. [23] proposed to use cycle +consistency to train an RGB tracker. Yuan et al. [18] and +Shen et al. [23] further used region proposal network in the +cycle consistency framework. Some other unsupervised RGB +trackers have also been proposed based on different ideas, such +as cycle memory learning [20], crop-transform-paste operation +[22], and training using images and their cropped regions +[36]. In addition, unsupervised thermal tracker based on cycle +consistency has also been proposed [25]. However, existing +unsupervised trackers are limited to one single modality, i.e., +based on only RGB images or only thermal images. +D. Self-supervised training +Some self-supervised learning methods, e.g., BYOL [37] +and SimCLR [38], first use different data augmentations to +generate two correlated views and then maximize similarity +to learn representations for downstream tasks. Our idea is +inspired by these self-supervised learning methods. However, +in our work, we use images from different modalities to +replace traditional data augmentation. Moreover, we use an +object tracking framework with different inputs to exploit +temporal information in RGB-T video pairs and construct +cross-input consistency to guide training. Furthermore, we +do not use a pretext task and downstream tasks like many +studies. Instead, we only have one task (RGB-T tracking). We +directly use the proposed training method to obtain an RGB-T +tracker. +E. Cross-input consistency +Cross-input consistency has been rarely utilized in track- +ing. Bastani et al. [39] applied cross-input consistency to +develop a self-supervised multi-object tracker. Our work is +inspired by [39] and aims to train an RGB-T single object +tracker. We utilize RGB-T video pairs as different inputs to +build cross-input consistency. +III. PROPOSED METHOD +The basic idea of this work (see Fig. 2) is that object +tracking can be performed with different inputs to generate +consistent results. Specifically, in this paper, we construct two +distinct inputs, i.e., RGB images and RGB-T image pairs, +to build cross-input consistency, as shown in Fig. 3(a). The +RGB input is handled by an RGB tracker, and the RGB-T +input is handled by our RGB-T tracker. We implement our +cross-input consistency self-supervised training strategy in a +Siamese-based DCF tracking framework. +A. Background: Siamese-based DCF tracker +Given two consecutive frames from an unlabeled video, we +first crop the template patch T and the search patch S. In +Siamese-based DCF trackers [15, 40], CNNs are first used to +extract features from T and S. Then, a filter W is learned, + +RGB +Thermal +2 +Close enough +frame t +frame t+1 +RGB-TJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +3 +Fig. 3. An overview of our method. (a) The basic idea of cross-input consistency using RGB and RGB-T images as inputs. (b) The training pipeline based +on Siamese DCF tracking frameworks. Loss function is computed based on response maps. Feature-level fusion is used to obtain fused template and search +features. The blue part in (b) is used for object tracking after training. +which can be used to generate a response map by convolving +W with the feature of a search patch S. The response map is +used for target localization. Specifically, the filter W for RGB +images can be obtained as +WTRGB = F −1 +� +F(φRGB(TRGB)) ⊙ F ⋆(YTRGB) +F ⋆(φRGB(TRGB) ⊙ F(φRGB(TRGB)) + λ +� +, +(1) +where ⊙ is element-wise produce, F is the Discrete Fourier +Transform (DFT), F−1 is inverse DFT, ⋆ means the complex- +conjugate operation, φRGB() is the CNN used to extract RGB +features, YTRGB is the label of the RGB template patch, which +is a Gaussian response map centered at the bounding box +region. Once the filter WTRGB is obtained, the response map +of an RGB search patch SRGB is +RSRGB = F−1(F⋆(WTRGB) ⊙ F(φRGB(SRGB))). +(2) +The main advantage of using CNNs in DCF-based trackers +is that CNNs and the CF layer are integrated into an end- +to-end framework. Therefore, the CNNs can learn to extract +more suitable features for tracking. Both our RGB tracker and +RGB-T tracker use Siamese-based DCF framework, but RGB +CNN and thermal CNN have different weights. +B. Cross-input consistency +As can be seen from Fig. 3, our framework uses two distinct +inputs to construct cross-input consistency. The first input is +RGB images, and the second input is RGB-T image pairs. The +key idea of our self-supervised training strategy is that we can +arrive the location of our target in frame (t + 1) from frame t +by tracking with either input if both the RGB tracker and the +RGB-T tracker work well. +In the training process, a Gaussian response map centered +at the bounding box region is used as the initial label for both +the RGB tracker and the RGB-T tracker. We use a cross-input +consistency loss to guide the training of the RGB tracker and +the RGB-T tracker together. The main objective is to learn the +CNN models in the trackers to learn features that are suitable +for tracking. In the inference stage, we only use the RGB-T +tracker to perform tracking by using RGB and thermal +images as input, as shown in the blue part in Fig. 3(b). +Our cross-input consistency is generic. In this study, we +use videos of different modalities to construct cross-input +consistency. There may be other schemes that can construct +cross-input consistency and give comparable or better perfor- +mance. Also, as we will show in the experiments, we can also +construct cross-input consistency between thermal images and +RGB-T image pairs, or between RGB images, thermal images, +and RGB-T image pairs. +C. Our RGB-T tracker +The architecture of our RGB-T tracker is shown in the blue +part of Fig. 3(b). As can be seen, our RGB-T tracker consists +of two RGB CNNs and two thermal CNNs. The two RGB +CNNs are used to extract RGB template and search features, +and two thermal CNNs are used to extract thermal template +and search features. The RGB template feature and thermal +template feature are fused to give fused template feature, while +the RGB search feature and the thermal search feature are +fused to give fused search feature. Then, following [41], the +fused template feature and fused search feature are used to +generate response map through correlation filter and circular +convolution operations, i.e., +RSRGBT = F −1(F ⋆(WTRGBT)⊙F(φRGB(SRGB)⊕φT(ST))), (3) + +RGB Tracking using RGB images +Feature +RGB Tracking +RGB +Correlation +t +Response +t+1 +filter +CNN +Ter +* +t+1 +RGB +CNN +Initial label +← +ate +t +RGB-T Tracking using RGB-T image pairs +RGB +Cross-input +CNN + consistency +t+1 +t +Initial label +！ +t+1 +RGB +Correlation +RGB-T Tracking +Feature +Response +CNN +filter +Fusion +* +Feature +Template +Thermal +Fusion +CNN +t+1 +Thermal +Search +Cross-input consistency computation at +CNN +Frame t+1 +(a) Idea of cross-input consistencyJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +4 +where ⊕ means feature fusion. Tracking result can then be +obtained based on the response map. +1) Feature fusion: Feature fusion can be performed in +various ways. In this study, to make our RGB-T tracker +lightweight so that it can run fast, we do not employ compli- +cated feature fusion modules. Instead, we concatenate the RGB +feature and thermal feature to generate the fused feature. This +is simple but effective as we will show in Section V-B. +2) Online object tracking: We first run offline training to +train our CNNs. Then, we perform online tracking using the +RGB-T tracker. During tracking, all CNNs are fixed. Following +previous studies [15, 16, 41], we update the DCF parameters +in the RGB-T tracker to make the tracker more robust, i.e., +Wt +RGBT = (1 − αt)Wt−1 +RGBT + αtWRGBT, +(4) +where αt is the parameter controlling the update speed. +D. Cross-input consistency loss function +Ideally, the tracking results from different inputs should be +the same if all trackers work well. We formulate the loss +function to minimize the difference between the response maps +obtained using different inputs. Specifically, our cross-input +consistency loss is +L = ||RSRGB − RSRGBT||, +(5) +where RSRGB is the response map generated by the RGB +tracker and RSRGBT is the response map generated by the +RGB-T tracker. +IV. TRAINING DATA PROCESSING AND LOSS FUNCTION +RE-WEIGHTING +A. Training data processing +We do not want to use any human labels in training. It is +thus essential to obtain good initial bounding boxes (pseudo +labels) in self-supervised training. In this work, we cropped the +center patch from RGB-T video pairs to generate our training +data, as done by Wang et al. [15]. In this way, we track the +objects appear in the center of the cropped region. Note that +the object in the center may be just a part of the object. Some +examples of the cropped images are shown in Fig. 4. As can +be seen, some cropped images contain useful moving objects, +while some images only contain background information. In +this study, we propose several ways to improve the usage of +these training data, inspired by Wang et al. [15]. +Noisy sample dropping. The cropped center patches con- +tain noisy samples that provide very large loss values. These +noisy samples make the training unstable and less effec- +tive. We assign a weight value Di +noisy to each training pair +to exclude 10% of train pairs that provide very high loss val- +ues. Based on our observation, these samples usually contain +sudden camera movement or sharp appearance change. Unlike +the method of Wang et al. [15] which plays with response +maps, we use the difference between the RGB template patch +and RGB search patch, i.e., +Di = ||Ti +RGB − Si +RGB||2 +2 +H × W +, +(6) +Fig. 4. +Examples of cropped patches from RGBT234 [1]. Top: good +examples. Bottom: bad examples. +where H and W are the height and width of training samples, +respectively. We sort the elements in D. Then, we assign a +weight value Di +noisy to each training pair. 10% of elements in +the weight vector Dnoisy corresponding to noisy samples are +0. In this way, we exclude 10% of training pairs that produce +large difference values. +Background sample dropping. As shown in Fig. 4, some +cropped center patches contain only background or still ob- +jects. These background samples make little contribution to +model training. To exclude these background training samples, +we set a value Di +background to each training pair. 25% of the +elements in the weight vector Dbackground corresponding to +the 25% lowest values in D are zero. Combining Dnoisy and +Dbackground, we can normalize the weight of each training pair +to ensure the sum of useful weights in one mini-batch is 1, +i.e., +Di +norm = +Di +noisy · Di +background +�n +i=1 Di +noisy · Di +background +, +(7) +where n is the number of training pairs in a mini-batch. +B. Loss function re-weighting +After computing the re-weighting weight vector using +Eq. (7), we use it to re-weight the loss obtained from training +samples of various quality, i.e., +Lfinal = 1 +n +n +� +i=1 +Di +norm · Li, +(8) +where Li is computed using Eq. (5). Using this loss can make +the training more effective and avoid overfitting. +V. EXPERIMENTS +Implementation +details. Following [15, 16], we use +lightweight CNNs in our trackers. Specifically, the filter sizes +of the two convolutional layers in our CNN are 3×3×3×32 +and 3 × 3 × 32 × 32. All experiments were performed using a +desktop equipped with two NVIDIA RTX3090 GPUs and an +i9-10900X CPU. The batch size is 32. We change the learning +rate from 10−4 to 10−6 from epoch 0 to epoch 30. The weight +decay is 5 × 10−5. + +Good examples +RGB +Thermal +RGB +Bad examples +ThermalJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +5 +Test set. We report results on the GTOT dataset [12] +and the RGBT234 dataset, which have been widely used in +RGB-T tracking studies [9, 10, 42]. GTOT consists of 50 +RGB-T videos (15.8K frames). Moreover, seven attributes are +annotated for each sequence, including occlusion (OCC), large +scale variation (LSV), fast motion (FM), low illumination (LI), +thermal crossover (TC), small object (SO), and deformation +(DEF). RGBT234 contains 234 RGB-T video pairs (around +233.8K frames) and 12 attributes are annotated. Compared to +GTOT, RGBT234 is more challenging by having longer frames +in videos and more challenging attributes. +Training data. When testing on the GTOT dataset, we use +the RGBT234 dataset [1] as training data. 10,000 RGB-T pairs +are randomly chosen as the validation set in training. When +testing on the RGBT234 dataset, we use the GTOT dataset as +training data, and 1000 RGB-T pairs are randomly chosen as +the validation set in training. +Evaluation +metrics. +In +this +work, +we +utilize +two +commonly-used evaluation metrics in RGB-T tracking, maxi- +mum precision rate (MPR) and maximum success rate (MSR) +[1, 34], to evaluate the performance of our tracker. Following +previous studies [1, 6, 12], the threshold of MPR is set to 5 +pixels for GTOT (because the targets in GTOT are relatively +small) and 20 pixels for RGBT234. +A. Self-supervised v.s. supervised training +To show the effectiveness of our self-supervised training +strategy, we use the ground truth of the RGBT234 dataset to +train a supervised RGB-T tracker. Specifically, we only train +the RGB-T tracker shown in the blue part of Fig. 3(b). The +comparison between the supervised RGB-T tracker and our +self-supervised RGB-T tracker is shown in Table I. As can +be seen, our self-supervised RGB-T tracker achieves better +performance than the supervised one on GTOT. This is inter- +esting and supervising, as training using ground truth labels +is usually more effective. A possible reason is that by using +center-cropped regions from RGBT234 (contains 110K RGB- +T image pairs) as training data, the training set has more cat- +egories of targets than the ground truth labels. Similar pattern +has been observed in some unsupervised RGB tracking studies +[43], where the unsupervised DCFNet performs slightly better +than supervised DCFNet. We also use the ground truth of +the GTOT dataset to train a supervised RGB-T tracker an +test it on RGBT234. As can be seen from Table I, our self- +supervised RGB-T tracker is slightly worse than its supervised +counterpart. This may because the GTOT dataset is smaller +(7.9K RGB-T image pairs) and does not provide enough high- +quality training data for our self-supervised training. +TABLE I +COMPARISON OF THE PROPOSED SELF-SUPERVISED TRAINING AND +SUPERVISED TRAINING. BETTER RESULTS ARE MAKED IN BOLD. +Variant +GTOT +RGBT234 +MPR(↑) +MSR(↑) +MPR(↑) +MSR(↑) +Supervised +76.7 +66.1 +59.5 +43.7 +Ours +85.6 +70.5 +56.2 +41.6 +TABLE II +EFFECT OF INPUT IN THE FIRST BRANCH. +First input +Second input +MPR(↑) +MSR(↑) +RGB +RGB-T +85.6 +70.5 +Thermal +RGB-T +75.4 +64.9 +4-channel RGBT +RGB-T +81.6 +67.8 +B. Ablation studies and analysis +The GTOT dataset is used in ablation studies unless other- +wise specified. +Different cross input combinations. We can use different +combinations of inputs to construct cross-input consistency. In +this study, we keep the second input (RGB-T image pairs) +and change the first input to different variants. Specifically, +we trained two variants. In the first variant, we use thermal +images as the first input. In the second variant, we use 4- +channel RGB-T images as the first input. When using 4- +channel RGB-T images, we change the dimension of the +first convolution layer to adapt to these 4-channel images. As +shown in Table II, all these combinations can be used to guide +our cross-input consistency-based training and gives useful +RGB-T trackers. Among these combinations, when the first +input is RGB image and the second input is RGB-T image +pairs, our RGB-T tracker shows the best tracking performance. +Using more branches in cross-input consistency. Previously, +we used two branches (different inputs) to build cross-input +consistency, as shown in Fig. 3. We can extend our idea to +use more branches. For example, we can use three inputs, i.e., +RGB images, thermal images, and RGB-T image pairs. In this +case, we can add another two loss terms, i.e., LRGB−T and +LT−RGBT, to compute the difference of every two outputs. Us- +ing this idea, we obtain an RGB-T tracker with MPR of 83.7 +and MSR of 70.3, which is better than using 4-channel RGBT +images and RGB-T image pairs. However, the performance +is slightly worse than using RGB images and RGB-T image +pairs as distinct inputs. +Impact of loss function. Mean square error (MSE) loss is +usually used to train unsupervised trackers [15, 16, 25]. In this +study, we use L1 loss instead. Table III shows the performance +comparison of using MSE loss and L1 loss in our cross- +input consistency-based self-supervised training. As can be +seen, L1 loss gives better performance in all cases. This may +because our training samples are noisy (although we use re- +weighting strategy), MSE loss will amplify the errors due to +noisy training samples, making the training less effective. +Impact of loss re-weighting. We proposed two components +to generate weight vectors based on cropped patches, i.e., +noisy sample dropping and background sample dropping. In +this section, we report the results of removing one of the +components in Table IV. As can be seen, after we remove any +comment, the tracking performance will drop slightly, showing +that our loss re-weighting strategy is helpful. +Impact of unlabeled training data size. We use different +portions of RGBT234 as the training set. As can be seen, in +general, the proposed self-supervised training strategy benefits + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +6 +TABLE III +IMPACT OF LOSS FUNCTION. IN THESE VARIANTS, L1 LOSS GIVES BETTER +PERFORMANCE THAN MSE LOSS. +Variant +MPR(↑) +MSR(↑) +Three branches +MSE loss +72.5 +63.1 +L1 loss +75.3 +65.6 +RGB, RGB-T +MSE loss +78.8 +64.9 +L1 loss +85.6 +70.5 +Thermal, RGB-T +MSE loss +65.7 +58.2 +L1 loss +75.4 +64.9 +TABLE IV +IMPACT OF LOSS RE-WEIGHTING SCHEME. BETTER RESULTS ARE +OBTAINED AFTER RE-WEIGHTING THE LOSS FUNCTION. +Loss re-weighting +MPR(↑) +MSR(↑) +Noisy sample +dropping +Background sample +dropping +82.6 +69.1 +� +83.7 +69.5 +� +� +85.6 +70.5 +from training using more unlabeled RGB-T video pairs. Be- +cause unlabeled RGB-T pairs are much easier to obtain than +annotated ones, our method infers the great potential of +unsupervised RGB-T tracking. +TABLE V +ABLATION STUDIES ON TRAINING DATA SIZE. WITH MORE UNLABELED +RGB-T VIDEOS FOR TRAINING, THE PROPOSED RGB-T TRACKER +ACHIEVES BETTER RESULTS ON THE GTOT DATASET. +Size +MPR(↑) +MSR(↑) +Size +MPR(↑) +MSR(↑) +RGBT234 (90%) +85.6 +70.5 +RGBT234 (50%) +81.5 +67.5 +RGBT234 (70%) +81.2 +67.3 +RGBT234 (20%) +73.6 +61.3 +Impact of feature fusion methods. We trained three variants +of RGB-T trackers using three feature-level fusion methods, +i.e., element-wise average, concatenation, and the DFF fusion +module proposed by Zhang et al. [9]. Specifically, we keep the +RGB tracker in Fig. 3(b) and change the feature fusion method +in the RGB-T tracker. We compare different fusion methods +in Table VI, which shows the fusion level has a significant +impact on the performance of RGB-T trackers. Specifically, by +concatenating RGB features and thermal features, we obtain +the best tracking performance. +TABLE VI +IMPACT OF FEATURE FUSION METHODS. CONCATENATION IS SIMPLE YET +GIVES THE BEST PERFORMANCE. +Variant +MPR(↑) +MSR(↑) +Feature-level (Average) +83.7 +69.1 +Feature-level (Concat.) +85.6 +70.5 +Feature-level (DFF [9]) +79.1 +66.0 +Impact of sharing weights between RGB CNN and thermal +CNN. In our experiments, we find that whether the weights +are shared between RGB CNN and thermal CNN (please see +Fig. 3(b) of the paper) or not affects the performance of our +tracker. We designed a variant of our method, i.e., the weights +of the RGB CNN are shared with the thermal CNN. The results +Fig. 5. Training using two frames (a) and three frames (b). Red arrow: the +cross-input consistency. +are shown in Table VII. As can be seen, our RGB-T tracker +gives better performance than the variant where the weights +are shared between the RGB CNN and the thermal CNN. +TABLE VII +IMPACT OF SHARING WEIGHTS BETWEEN RGB CNN AND THERMAL CNN +ON TRACKING PERFORMANCE. THE GTOT DATASET IS USED. +Variant +MPR(↑) +MSR(↑) +Sharing +82.2 +68.2 +Not Sharing +85.6 +70.5 +Impact of tracking sequence length. In this paper, we +build our cross-input consistency-based training strategy using +two frames, i.e., frame t and frame (t + 1), as shown in +Fig. 5(a). Indeed, our cross-input consistency-based training +strategy can be extended to more frames, e.g., frames t, (t+1) +and (t+2), as shown in Fig. 5. In this case, the response map +generated at frame (t + 1) will be used as the pseudo label of +frame (t + 1). Based on the pseudo label, our tracker tracks +the target from (t + 1) to frame (t + 2). +When more frames are used, the cross-input consistency loss +is computed in the final frame, as shown by the red dash arrow +in Fig. 5(b). We show the impact of tracking sequence length +on the tracking performance in Table VIII. From the results, +we can see that using three frames can also train a good RGB- +T tracker. However, the performance is slightly worse than +using two frames in training. These results indicate that our +self-supervised training strategy can train our RGB-T tracker +effectively using only two frames in each training pair. +TABLE VIII +IMPACT OF TRACKING SEQUENCE LENGTH ON TRACKING +PERFORMANCE. THE GTOT DATASET IS USED. +Variant +MPR(↑) +MSR(↑) +Using three frames +84.3 +69.5 +Using two frames +85.6 +70.5 +C. Compared with cycle consistency +Cycle consistency is commonly used in training unsu- +pervised RGB-based trackers [15–19, 23] or thermal-based +trackers [25]. In this section, we compare our cross-input +consistency with cycle consistency. Specifically, we train +our RGB-T tracker (the blue part of Fig. 3(b)) using cycle +consistency. We trained two variants, one with two frames +and one with three frames, as shown in Fig. 6. All other + +RGB +Consistency I +Consistency i +RGB-T +(a) Tracking using two frames +(b) Tracking using three framesJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +7 +Fig. 6. Cycle consistency using two frames (left) and three frames (right). +settings are kept the same as our cross-input consistency- +based training. As can be seen from Table IX, our cross- +input consistency-based self-supervised training strategy is +more effective than cycle consistency. The attributed-based +performance given in Table XI also indicates that our cross- +input consistency-based training is more effective than cycle +consistency-based training. Moreover, using three frames in +cycle consistency-based training is more effective than using +two frames. In addition to the RGB-T tracker, we also use +our trained CNNs to run an RGB tracker and a thermal +tracker based on the red part of Fig. 3(b). As can be seen, +our training strategy provides consistent better performance +in RGB tracker, thermal tracker and RGB-T tracker. The +reason is that the forward-backward tracking-based cycle con- +sistency may not hold in some challenging real world tracking +scenarios [22]. In addition, it should be mentioned that the +cycle consistency-based strategy, e.g., UDT [15], has only +been applied to uni-modal object tracking, while the proposed +cross-input consistency-based self-supervised training strategy +is applied to multi-modal tracking. +TABLE IX +PERFORMANCE COMPARISON WITH CYCLE CONSISTENCY. +Variant +RGB tracker +T tracker +RGB-T tracker +PR +SR +PR +SR +MPR +MSR +Cycle consistency (2 frames) +67.6 +56.7 +62.4 +56.0 +74.1 +64.5 +Cycle consistency (3 frames) +64.5 +55.6 +62.8 +56.2 +75.4 +65.6 +Ours +68.1 +57.8 +64.8 +57.3 +85.6 +70.5 +D. Compared with SOTA RGB-T tackers +Compared methods. There are no existing unsupervised +deep RGB-T trackers. We selected the following supervised +RGB-T trackers for comparison, i.e., HMFT [9], CMPP +[10], DMCNet [11], JMMAC [6], CBPNet [46], MaCNet +[2], MANet [47], CMP [48], MFGNet [49], DAFNet [50], +DAPNet [51], mfDiMP [52], LTDA [43], DuSiamRT [53], +TCNN [13], JCDA-InvSR [14], CFNet [40]+RGBT, SiamDW +[54]+RGBT. A transformer-based method, namely APFNet +[45], is also selected for comparison. We also selected some +non-deep RGB-T trackers, including CMCF [55], NRCMR +[44], LGMG [42], CMR [56], SGT [57], the method of +Li et al. [58], CSR [12], [59], MEEF[60]+RGBT and KCF +[61]+RGBT. Almost all categories of RGB-T methods are +covered. +Results on GTOT. The tracking results on the GTOT dataset +are shown in Table X. As can be seen, our RGB-T tracker is +better than all non-learning-based RGB-T trackers in terms of +both metrics. Furthermore, our self-supervised RGB-T tracker +is better than seven supervised RGB-T trackers (DAFNet, +DAPNet, mfDiMP, LTDA, DuSiamRT, TCNN, JCDA-InvSR) +and the RGB-T version of CFNet. Our tracker also shows +comparable performance with several supervised trackers, such +as MFGNet, CMP and MANet. Although there are some gaps +between our performance and the state-of-the-art supervised +RGB-T trackers, it is understandable because those supervised +RGB-T trackers use large-scale annotated RGB-T image pairs +for training. In contrast, our method does not use any anno- +tations. Moreover, those trackers use more complex models, +as indicated by their tracking speed. For example, the FPS of +CMPP and DMCNet are 1.3 and 2.4, respectively. +Attribute-based results. Attribute-based performance on +the GTOT dataset is shown in Table XI. As can be seen, as +a very lightweight RGB-T tracker trained without any ground +truth labels, our tracker achieves very competitive performance +in terms of LSV, LI, TC and SO. Especially, our RGB- +T tracker achieves better performance than most supervised +RGB-T trackers in terms of LSV, indicating that our tracker +can well handle scale variation of targets. +Results on RGBT234. The results on RGBT234 dataset are +shown in Table X. From the table, we can see that the +proposed RGB-T tracker outperforms four supervised trackers +and three non-learning-based RGB-T trackers in terms of +MSR. However, our RGB-T tracker shows worse performance +on RGBT234 than GTOT. This is because RGBT234 is much +more challenging than GTOT by having more images (223.8K +frames v.s. 15.8K frames) and challenging scenarios (12 chal- +lenging attributes v.s. 7 attributes). Most deep RGB-T trackers +achieve good performance on RGBT234 by using complex +model architectures and a large number of annotated RGB-T +image pairs for training. In contrast, our tracker is trained using +6,900 unlabeled RGB-T image pairs. In our future work, we +will aim to narrow this gap by using better backbone trackers +or larger training sets. However, it is worth mentioning that the +performance gap between our self-supervised RGB-T tracker +and the state-of-the-art supervised RGB-T tracker (HMFT), +i.e., 22.6% in MPR and 15.2% in MSR, is acceptable. This +level of gaps also exist between state-of-the-art unsupervised +RGB trackers and supervised RGB trackers. For example, the +ULAST [23] achieves 59.2% in precision and 65.4% in success +rate on the TrackingNet, while SwinTrack [62] achieves 82.8% +and 84.0%, respectively. +E. Qualitative results +1) Compared with SOTA trackers: We first show qualitative +comparison of our RGB-T tracker with other RGB-T trackers +in Fig. 7. Several RGB-T trackers are selected, including SGT +[57], LGMG [42], DAPNet [51], MaCNet [2]. As can be +seen, our RGB-T tracker obtains better performance on these +four sequences, namely, BlackCar, Exposure2, LightOcc, and +carNig. +2) Challenging cases: In this section, we show qualitative +results of some challenging cases in Fig. 8. In these cases, +tracking using RGB images and tracking using thermal images +fail, while our RGB-T tracker can correctly track targets. These + +t+1 +t +t +RGB-T +t+1 +RGB-T +t+2 +RGB-T +RGB-T +RGB-TJOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +8 +TABLE X +COMPARISON WITH EXISTING RGB-T TRACKERS ON GTOT DATASET AND RGBT234 DATASET. THE RESULTS MARKED WITH † ARE COMPUTED BY US +USING RAW TRACKING RESULTS. THE RESULTS MARKED WITH § ARE COPIED FROM [1, 12, 44]. OTHER RESULTS ARE EXTRACTED FROM +CORRESPONDING PAPERS. ‘-’ MEANS NOT MENTIONED IN THE CORRESPONDING PAPER. VALUES WORSE THAN OUR METHOD ARE MARKED IN +GREEN . +Tracker +GTOT +RGBT234 +FPS (↑) +Category +Supervised +Venue +MPR (↑) +MSR (↑) +MPR (↑) +MSR (↑) +HMFT† [9] +90.6 +74.2 +78.8 +56.8 +30.2 +DL-based +Yes +CVPR 2022 +CMPP [10] +92.6 +73.8 +82.3 +57.5 +1.3 +DL-based +Yes +CVPR 2020 +APFNet [45] +90.5 +73.7 +82.7 +57.9 +- +DL-based +Yes +AAAI 2022 +DMCNet [11] +90.9 +73.3 +83.9 +59.3 +2.4 +DL-based +Yes +IEEE TNNLS 2022 +JMMAC† [6] +89.3 +73.1 +79.0 +57.3 +4 +DL-based +Yes +IEEE TIP 2021 +CBPNet [46] +88.5 +71.6 +79.4 +54.1 +3.7 +DL-based +Yes +IEEE TMM 2021 +MaCNet [2] +88.0 +71.4 +79.0 +55.4 +0.8 +DL-based +Yes +Sensors 2020 +MANet† [47] +88.9 +71.1 +77.7 +53.9 +3.1 +DL-based +Yes +ICCVW 2019 +CMP [48] +86.9 +71.1 +75.1 +49.1 +35 +DL-based +Yes +Neurocomputing 2021 +MFGNet [49] +88.9 +70.7 +78.3 +53.5 +3.4 +DL-based +Yes +IEEE TMM 2022 +DAFNet† [50] +88.6 +69.9 +79.6 +54.4 +23 +DL-based +Yes +ICCVW 2019 +DAPNet† [51] +87.4 +68.9 +76.6 +53.7 +2 +DL-based +Yes +ACM MM 2019 +mfDiMP§ [52] +84.1 +69.3 +78.5 +55.9 +10.3 +DL-based +Yes +ICCVW 2019 +LTDA [43] +84.3 +67.7 +78.7 +54.5 +0.4 +DL-based +Yes +ICIP 2019 +DuSiamRT [53] +76.6 +62.8 +56.7 +38.4 +116 +DL-based +Yes +The Visual Computer 2022 +TCNN [13] +85.2 +62.6 +- +- +15 +DL-based +Yes +Neurocomputing 2018 +JCDA-InvSR [14] +- +60.5 +60.6 +41.4 +1.6 +ML-based +Yes +IEEE TIP 2019 +CFNet [40] + RGBT§ +- +- +55.1 +39.0 +- +DL-based +Yes +CVPR 2017 +SiamDW [54] + RGBT§ +68.0 +56.5 +60.4 +39.7 +- +DL-based +Yes +CVPR 2019 +CMCF [55] +77.0 +63.2 +- +- +227 +Non-DL (CF-based) +Neurocomputing 2019 +NRCMR [44] +83.7 +66.4 +72.9 +50.2 +7 +Non-DL (Graph-based) +IEEE TNNLS 2021 +LGMG [42] +83.7 +65.8 +- +- +7 +Non-DL (Graph-based) +IEEE TCSVT 2019 +CMR [56] +82.7 +64.3 +- +- +8 +Non-DL (Graph-based) +ECCV 2018 +SGT§ [57] +85.1 +62.8 +72.0 +47.2 +5 +Non-DL (Graph-based) +ACM MM 2017 +[58] +84.2 +62.2 +- +- +7 +Non-DL (Graph-based) +SPIC 2018 +CSR [12] +74.5 +61.5 +46.3 +32.8 +1.6 +Non-DL (SR-based) +IEEE TIP 2016 +[59] +77.3 +61.2 +72.9 +48.6 +1.2 +Non-DL (Graph-based) +Neurocomputing 2022 +MEEF[60] + RGBT§ +- +52.0 +63.6 +40.5 +4.9 +Non-DL +ECCV 2014 +KCF [61] + RGBT§ +- +42.0 +46.3 +30.5 +124.1 +Non-DL (CF-based) +IEEE TPAMI 2014 +Ours +85.6 +70.5 +56.2 +41.6 +179.3 +DL-based +No +TABLE XI +ATTRIBUTE-BASED PERFORMANCE (MSR) ON GTOT. VALUES WORSE +THAN OUR METHOD ARE MARKED IN +GREEN . +Method +OCC +LSV +FM +LI +TC +SO +DEF +HMFT +72.1 +74.0 +75.2 +75.6 +72.3 +70.8 +72.3 +MANet +68.9 +69.4 +69.2 +71.9 +69.1 +68.6 +73.4 +CBPNet +68.6 +68.8 +67.5 +73.1 +69.3 +69.0 +75.4 +MaCNet +68.7 +67.3 +65.9 +73.1 +69.7 +69.5 +76.5 +DAPNet +67.4 +64.8 +61.9 +72.2 +69.0 +69.2 +77.1 +DuSiamRT +57.7 +64.5 +58.0 +62.3 +61.4 +64.2 +62.9 +TCNN +55.6 +64.6 +51.7 +64.2 +59.5 +59.1 +73.4 +SGT +56.7 +54.7 +55.9 +65.1 +61.5 +61.8 +73.3 +Cycle-2 +59.1 +67.4 +60.3 +63.9 +65.7 +61.9 +61.3 +Cycle-3 +62.0 +67.1 +62.9 +67.9 +65.6 +61.9 +65.0 +Ours +65.9 +72.2 +65.2 +73.0 +68.8 +66.9 +67.7 +examples demonstrate that our self-supervised RGB-T tracker +can well fuse RGB and thermal information to improve +tracking performance. For example, in the LightOcc case, both +the RGB tracker and thermal tracker has problems when the +car is heavily occluded by the trees. In contrast, our self- +supervised RGB-T tracker can successfully track the car by +fusing RGB and thermal features. Note that the RGB tracker +uses our RGB CNN, and the thermal tracker uses our thermal +CNN. +F. Failure cases +Our self-supervised RGB-T tracker fails in some very +challenging cases. Fig. 9 shows failure cases of our method. In +these cases, all the RGB tracker, thermal tracker, and RGB-T +tracker fail. In the Pool case, the occlusion of the tree and the +similar color between tree and pedestrian’s cloth are the main +reasons of failure. In the RainyCar2 case, the occlusion of the +trees and the similar color between the white car and road are +the main reasons of failure. +VI. DISCUSSION +Benefits of our self-supervised training strategy. First, the +proposed training strategy allows us to train an RGB-T tracker +without any human annotations. Some self-supervised tracking +methods [18, 19, 22] still require some annotations (very +sparse). Some self-supervised tracking methods use unsuper- +vised optical flow models [20, 23] to generate pseudo labels +or use EdgeBox [24] to generate object proposals. Our meth- +ods totally remove the need for any annotations and optical +flow models while achieving reasonable tracking performance, +which is very beneficial. Second, our cross-input consistency +is flexible. Specifically, we can use different modalities as dif- +ferent inputs, which is very suitable for visual data of different +modalities, such as RGB images and thermal images. The +idea has the potential to be used in other modalities, e.g., +RGB-D tracking. We can also easily add or remove branches + +JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +9 +Fig. 7. Qualitative comparison of our RGB-T tracker with other RGB-T trackers. The four sequences are BlackCar, Exposure2, LightOcc, and carNig from +the GTOT dataset. We visualize results on both RGB (1st and 3rd rows) and thermal images (2nd and 4th rows). +Fig. 8. +Tracking using RGB images and tracking using thermal images +fail. Our RGB-T tracker can track targets successfully. From top to bot- +tom: LightOcc, Quarreling, WalkingOcc. Left: RGB tracker. Middle: thermal +tracker. Right: our self-supervised RGB-T tracker. +(inputs). Third, the proposed self-supervised training strategy +is generic. In theory, different backbone trackers can be used +in the framework. We will explore this in the future. +Model collapsing. A collapsed solution may exist in +our cross-input consistency-based method, that is, different +Fig. 9. Failure cases of our RGB-T tracker. Results on Pool (top) and Rainy- +Car2 (bottom) are shown. Left: RGB tracker. Middle: thermal tracker. Right: +our self-supervised RGB-T tracker. +branches always generate the same wrong results. However, +this collapsed solution never appeared in our experiments. Our +tracker avoids this issue safely. A possible reason is that our +inputs come from different modalities, so it is unlikely that +different branches generate the same features. +Limitations: There is a clear performance gap between our +method and state-of-the-art supervised RGB-T trackers. More +efforts should be made to narrow this gap in the future. A +possible solution is to apply our self-supervised training +strategy to better backbone trackers, such as SiamAtt [63], +SiamDW [64], and ATOM [65]. Another promising way is to +use larger RGB-T tracking datasets for training. In addition, + +RGB +口 +口 +115 +口 +Thermal +口 +回 +BlackCar +Exposure2 +回 +RGB +Thermal +口 +日 +carNig +LightOcc +GT +Ours + SGT +LGMG +DAPNet +MACNetRGB-234 +T-234 +Fused-234 +口 +RGB-157 +T-157 +Fused-157 +A +RGB-112 +T-112 +Fused-112 +0RGB-123 +-123 +Fused-123 +口 +口 +RGB-114 +T-114 +Fused-114 +口JOURNAL OF LATEX CLASS FILES, VOL. XX, NO. XX, 2023 +10 +we do not learn an IoU net for bounding box regression. This +is a common issue in unsupervised tracking [15–17, 19]. We +will also solve this in our future study. However, it is worth +mentioning that although a performance gap exists between +the proposed self-supervised method and state-of-the-art super- +vised methods, the proposed method has a very good potential +because it does not require any manual annotations. +VII. CONCLUSIONS +In this paper, we propose a self-supervised training strategy +based on cross-input consistency to train RGB-T trackers. We +construct two distinct inputs using RGB images and ther- +mal images. Then, we compute the cross-input consistency +loss based on the tracking results obtained using these two +inputs. We also propose a loss re-weighting scheme to im- +prove training. The main benefit of the proposed method +is that only unlabeled RGB-T video pairs are needed for +training. Our experiments show that the proposed method +achieves favorable performance. Specifically, with very simple +CNNs and a simple feature fusion method, our RGB-T tracker +outperforms several supervised RGB-T trackers on the GTOT +dataset. We also show that the tracking performance can be +further improved by using more unlabeled RGB-T videos. In +the future, we will use a more complex network and add a +bounding box regression component to the framework. +ACKNOWLEDGMENTS +This study has received funding from the European +Union’s Horizon 2020 research and innovation programme +under the Marie Skłodowska-Curie grant agreement No. +101025274. This work is also funded by a Royal Academy +of Engineering Chair in Emerging Technologies to YD. +REFERENCES +[1] C. Li, X. Liang, Y. Lu, N. Zhao, and J. Tang, “RGB-T object +tracking: Benchmark and baseline,” Pattern Recognition, p. +106977, 2019. +[2] H. Zhang, L. Zhang, L. Zhuo, and J. 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Felsberg, “Atom: +Accurate tracking by overlap maximization,” in Proceedings +of the IEEE/CVF conference on computer vision and pattern +recognition, 2019, pp. 4660–4669. +Xingchen Zhang (M’21) received the B.Sc. de- +gree from the Huazhong University of Science and +Technology in 2012, and the Ph.D. degree from the +Queen Mary University of London in 2018. He is +currently a Marie Skłodowska-Curie Individual Fel- +low at the Personal Robotics Laboratory, Department +of Electrical and Electronic Engineering, Imperial +College London. Prior to this, he was a Teaching Fel- +low and Research Associate at the same department. +His main research interests include human intention +prediction, image fusion, and object tracking. He is +a recipient of the Best Paper Honourable Mention Award of the 9th Chinese +Conference on Information fusion. He is a co-author of the book Image +Fusion that has been awarded the National Science and Technology Academic +Publications Fund of China (2019). He is a reviewer for UKRI Future Leaders +Fellowship, EPSRC New Investigator Award and EPSRC Open Fellowship. +Yiannis +Demiris +(SM’03) +received +the +B.Sc. +(Hons.) degree in artificial intelligence and com- +puter science and the Ph.D. degree in intelligent +robotics from the Department of Artificial Intelli- +gence, University of Edinburgh, Edinburgh, U.K., in +1994 and 1999,respectively. He is a Professor with +the Department of Electrical and Electronic Engi- +neering, Imperial College London, London, U.K., +where he is the Royal Academy of Engineering +Chair in Emerging Technologies, and the Head of the +Personal Robotics Laboratory. His current research +interests include human-robot interaction, machine learning, user modeling, +and assistive robotics. Prof. Demiris was a recipient of the Rector’s Award +for Teaching Excellence in 2012 and the FoE Award for Excellence in Engi- +neering Education in 2012. He is a Fellow of the Institution of Engineering +and Technology (IET), and the British Computer Society (BCS). + diff --git a/O9FIT4oBgHgl3EQfeSvC/content/tmp_files/load_file.txt b/O9FIT4oBgHgl3EQfeSvC/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..be9897acb6fda05a6a3e6a0b9041204d21cac4a7 --- /dev/null +++ b/O9FIT4oBgHgl3EQfeSvC/content/tmp_files/load_file.txt @@ -0,0 +1,1353 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf,len=1352 +page_content='JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 1 Self-Supervised RGB-T Tracking with Cross-Input Consistency Xingchen Zhang∗, Member, IEEE, Yiannis Demiris, Senior Member, IEEE Abstract—In this paper, we propose a self-supervised RGB-T tracking method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Different from existing deep RGB-T trackers that are using a large number of annotated RGB-T image pairs for training, our RGB-T tracker is trained using unlabeled RGB-T video pairs in a self-supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We propose a novel cross-input consistency-based self-supervised training strategy based on the idea that tracking can be performed using different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we construct two distinct inputs using unlabeled RGB-T video pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We then track objects using these two inputs to generate results, based on which we construct our cross-input consistency loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Meanwhile, we propose a re- weighting strategy to make our loss function robust to low-quality training samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We build our tracker on a Siamese correlation filter network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' To the best of our knowledge, our tracker is the first self-supervised RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Extensive experiments on two public RGB-T tracking benchmarks demonstrate that the proposed training strategy is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Remarkably, despite training only with a corpus of unlabeled RGB-T video pairs, our tracker outperforms seven supervised RGB-T trackers on the GTOT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Index Terms—RGBT tracking, object tracking, thermal im- ages, image fusion, information fusion I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' INTRODUCTION O BJECT tracking is an important task and has many applications in areas such as robots and surveillance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In recent years, many tracking algorithms have been proposed, and tracking performance has witnessed a significant improve- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, most visual trackers operate on RGB im- ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The performance of these trackers degrades significantly when RGB images are not reliable (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', under poor lighting conditions), limiting their practical applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' To improve tracking performance, researchers have used thermal images and RGB images together to perform RGB-T tracking [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This is based on the fact that thermal images are insensitive to illumination changes while RGB images con- tain more texture details [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Although many efforts have been put into developing deep learning-based RGB-T trackers and RGB-T tracking performance has been significantly improved, existing deep RGB-T trackers [6, 9–11] require a large number of annotated RGB-T image pairs, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' It is well-known that annotation is time-consuming and expensive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Some unsupervised single object trackers have been proposed to avoid the need for annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [15, 16] and Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [17] use a cycle consistency based on forward-backward tracking to train trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' There X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Zhang and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Demiris are with the Personal Robotics Laboratory, Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (e-mail: xingchen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='zhang@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='uk, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='demiris@imperial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='uk) ∗ Corresponding author: Xingchen Zhang Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Performance v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' required labeled RGB-T image pairs in train- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Although our RGB-T tracker does not need any labeled training data, it outperforms seven supervised RGB-T trackers on the GTOT dataset [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Note that the exact number of labeled data for TCNN [13] and JCDA-InvSR [14] (both are supervised trackers) is not mentioned in their papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' are also some trackers using very spare annotation in training, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', annotation in the initial frame [18–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, all unsupervised trackers use only single-modal images, namely, either use RGB images [15, 16, 18–24] or thermal images [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this paper, we propose a self-supervised RGB-T tracker that does not need any manual annotations in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' To achieve this, we propose a cross-input consistency-based train- ing strategy to exploit temporal information in unlabeled RGB- T videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our intuition resides on the observation that object tracking can be performed using different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 2, given a target at frame t, we can track it to obtain its position at frame (t + 1) using different inputs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', RGB images, thermal images, or a combination of them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Ideally, if all tracking are successful, the tracking results in frame (t+1) should be consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We integrate our self-supervised training strategy into a Siamese-based discriminative correlation filter (DCF) frame- work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In implementation, we construct two distinct inputs for tracking to build cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This cross-input con- sistency, which is based on temporal information in unlabeled RGB-T video pairs, can be used to guide the training of our RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In addition, because we do not want to use any manual annotations, we randomly initialize a bounding box in our training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Therefore, the training samples are usually noisy or have bad quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We propose a re-weighting strategy to re-weight our loss function to make our training easier and more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In summary, the main contributions arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='11274v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='CV] 26 Jan 2023 75 CMPP(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3FPS) HMFT (30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 FPS) APFNet dataset 73 JMMAC (4FPS) DMCNet (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 FPS) MAcNet (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 FPS) 71 Ours MFGNet (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 FPS), MSR (%) on the GTOT ( DAPNet (23 FPS) 69 mfDiMP (10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 FPS) DAPNet (2 FPS) LTDA (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 FPS) 67 65 63 TCNN (15 FPS) DuSiamRT (116 FPS) 61 JCDA-InvSR (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 FPS) 59 0 10 100 1000 Number of manually-annotated RGB-T image pairs (K)JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our cross-input consistency is based on the observation that tracking can be performed using different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Ideally, the tracking results in frame (t + 1) obtained from different inputs should be close enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' of this paper include: We propose a self-supervised RGB-T tracker trained using RGB-T video pairs without human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We propose a cross-input consistency-based strategy to achieve self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We use RGB images and thermal images to construct different inputs for tracking, based on which a cross-input consistency loss is constructed to guide training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We propose a re-weighting scheme to re-weight our loss function to make the training more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Extensive experiments on two RGB-T tracking bench- marks demonstrate the favorable performance of the pro- posed method and the potential of self-supervised RGB-T tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Section II introduces related work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, Section III introduces the proposed method in detail, followed by the introduction to training data processing in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, Section V presents results and Section VI gives discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Finally, Section VII concludes this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' RELATED WORK A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Single object tracking Single object tracking methods mainly include deep learning-based methods [16, 26, 27] and discriminative cor- relation filter (DCF)-based methods [28, 29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Most trackers use RGB images as input and have a high requirement for good lighting conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' To make trackers insensitive to light conditions, some researchers performed tracking using thermal images [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, thermal images do not have enough texture details, leading to worse performance than RGB-based trackers when lighting conditions are good.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' RGB-T tracking To alleviate the issue of RGB-based and thermal-based trackers, researchers performed RGB-T tracking [1, 2, 9, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [32] proposed a pixel-level fusion-based RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In contrast, some RGB-T trackers are based on feature-level fusion [7, 33, 34] or decision-level [35] or combine several fusion levels [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The performance of RGB-T trackers have been significantly improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, existing deep RGB-T trackers need a large number of RGB-T image pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Unsupervised object tracking Researchers have proposed unsupervised trackers to allevi- ate the need for annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, Vondrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [21] proposed to train an RGB tracker by colorizing videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [15, 16] and Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [23] proposed to use cycle consistency to train an RGB tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Yuan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [18] and Shen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [23] further used region proposal network in the cycle consistency framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Some other unsupervised RGB trackers have also been proposed based on different ideas, such as cycle memory learning [20], crop-transform-paste operation [22], and training using images and their cropped regions [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In addition, unsupervised thermal tracker based on cycle consistency has also been proposed [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, existing unsupervised trackers are limited to one single modality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', based on only RGB images or only thermal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Self-supervised training Some self-supervised learning methods, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', BYOL [37] and SimCLR [38], first use different data augmentations to generate two correlated views and then maximize similarity to learn representations for downstream tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our idea is inspired by these self-supervised learning methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, in our work, we use images from different modalities to replace traditional data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Moreover, we use an object tracking framework with different inputs to exploit temporal information in RGB-T video pairs and construct cross-input consistency to guide training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Furthermore, we do not use a pretext task and downstream tasks like many studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Instead, we only have one task (RGB-T tracking).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We directly use the proposed training method to obtain an RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Cross-input consistency Cross-input consistency has been rarely utilized in track- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Bastani et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [39] applied cross-input consistency to develop a self-supervised multi-object tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our work is inspired by [39] and aims to train an RGB-T single object tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We utilize RGB-T video pairs as different inputs to build cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' PROPOSED METHOD The basic idea of this work (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 2) is that object tracking can be performed with different inputs to generate consistent results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, in this paper, we construct two distinct inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', RGB images and RGB-T image pairs, to build cross-input consistency, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The RGB input is handled by an RGB tracker, and the RGB-T input is handled by our RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We implement our cross-input consistency self-supervised training strategy in a Siamese-based DCF tracking framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Background: Siamese-based DCF tracker Given two consecutive frames from an unlabeled video, we first crop the template patch T and the search patch S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In Siamese-based DCF trackers [15, 40], CNNs are first used to extract features from T and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, a filter W is learned, RGB Thermal 2 Close enough frame t frame t+1 RGB-TJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 3 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' An overview of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (a) The basic idea of cross-input consistency using RGB and RGB-T images as inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (b) The training pipeline based on Siamese DCF tracking frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Loss function is computed based on response maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Feature-level fusion is used to obtain fused template and search features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The blue part in (b) is used for object tracking after training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' which can be used to generate a response map by convolving W with the feature of a search patch S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The response map is used for target localization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, the filter W for RGB images can be obtained as WTRGB = F −1 � F(φRGB(TRGB)) ⊙ F ⋆(YTRGB) F ⋆(φRGB(TRGB) ⊙ F(φRGB(TRGB)) + λ � , (1) where ⊙ is element-wise produce, F is the Discrete Fourier Transform (DFT), F−1 is inverse DFT, ⋆ means the complex- conjugate operation, φRGB() is the CNN used to extract RGB features, YTRGB is the label of the RGB template patch, which is a Gaussian response map centered at the bounding box region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Once the filter WTRGB is obtained, the response map of an RGB search patch SRGB is RSRGB = F−1(F⋆(WTRGB) ⊙ F(φRGB(SRGB))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (2) The main advantage of using CNNs in DCF-based trackers is that CNNs and the CF layer are integrated into an end- to-end framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Therefore, the CNNs can learn to extract more suitable features for tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Both our RGB tracker and RGB-T tracker use Siamese-based DCF framework, but RGB CNN and thermal CNN have different weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Cross-input consistency As can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3, our framework uses two distinct inputs to construct cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The first input is RGB images, and the second input is RGB-T image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The key idea of our self-supervised training strategy is that we can arrive the location of our target in frame (t + 1) from frame t by tracking with either input if both the RGB tracker and the RGB-T tracker work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the training process, a Gaussian response map centered at the bounding box region is used as the initial label for both the RGB tracker and the RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We use a cross-input consistency loss to guide the training of the RGB tracker and the RGB-T tracker together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The main objective is to learn the CNN models in the trackers to learn features that are suitable for tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the inference stage, we only use the RGB-T tracker to perform tracking by using RGB and thermal images as input, as shown in the blue part in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our cross-input consistency is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this study, we use videos of different modalities to construct cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' There may be other schemes that can construct cross-input consistency and give comparable or better perfor- mance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Also, as we will show in the experiments, we can also construct cross-input consistency between thermal images and RGB-T image pairs, or between RGB images, thermal images, and RGB-T image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our RGB-T tracker The architecture of our RGB-T tracker is shown in the blue part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our RGB-T tracker consists of two RGB CNNs and two thermal CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The two RGB CNNs are used to extract RGB template and search features, and two thermal CNNs are used to extract thermal template and search features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The RGB template feature and thermal template feature are fused to give fused template feature, while the RGB search feature and the thermal search feature are fused to give fused search feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, following [41], the fused template feature and fused search feature are used to generate response map through correlation filter and circular convolution operations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', RSRGBT = F −1(F ⋆(WTRGBT)⊙F(φRGB(SRGB)⊕φT(ST))), (3) RGB Tracking using RGB images Feature RGB Tracking RGB Correlation t Response t+1 filter CNN Ter t+1 RGB CNN Initial label ← ate t RGB-T Tracking using RGB-T image pairs RGB Cross-input CNN consistency t+1 t Initial label ！' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' t+1 RGB Correlation RGB-T Tracking Feature Response CNN filter Fusion Feature Template Thermal Fusion CNN t+1 Thermal Search Cross-input consistency computation at CNN Frame t+1 (a) Idea of cross-input consistencyJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 4 where ⊕ means feature fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Tracking result can then be obtained based on the response map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 1) Feature fusion: Feature fusion can be performed in various ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this study, to make our RGB-T tracker lightweight so that it can run fast, we do not employ compli- cated feature fusion modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Instead, we concatenate the RGB feature and thermal feature to generate the fused feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This is simple but effective as we will show in Section V-B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 2) Online object tracking: We first run offline training to train our CNNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, we perform online tracking using the RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' During tracking, all CNNs are fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Following previous studies [15, 16, 41], we update the DCF parameters in the RGB-T tracker to make the tracker more robust, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', Wt RGBT = (1 − αt)Wt−1 RGBT + αtWRGBT, (4) where αt is the parameter controlling the update speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Cross-input consistency loss function Ideally, the tracking results from different inputs should be the same if all trackers work well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We formulate the loss function to minimize the difference between the response maps obtained using different inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, our cross-input consistency loss is L = ||RSRGB − RSRGBT||, (5) where RSRGB is the response map generated by the RGB tracker and RSRGBT is the response map generated by the RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TRAINING DATA PROCESSING AND LOSS FUNCTION RE-WEIGHTING A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Training data processing We do not want to use any human labels in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' It is thus essential to obtain good initial bounding boxes (pseudo labels) in self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this work, we cropped the center patch from RGB-T video pairs to generate our training data, as done by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this way, we track the objects appear in the center of the cropped region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Note that the object in the center may be just a part of the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Some examples of the cropped images are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, some cropped images contain useful moving objects, while some images only contain background information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this study, we propose several ways to improve the usage of these training data, inspired by Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Noisy sample dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The cropped center patches con- tain noisy samples that provide very large loss values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' These noisy samples make the training unstable and less effec- tive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We assign a weight value Di noisy to each training pair to exclude 10% of train pairs that provide very high loss val- ues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Based on our observation, these samples usually contain sudden camera movement or sharp appearance change.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Unlike the method of Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [15] which plays with response maps, we use the difference between the RGB template patch and RGB search patch, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', Di = ||Ti RGB − Si RGB||2 2 H × W , (6) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Examples of cropped patches from RGBT234 [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Top: good examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Bottom: bad examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' where H and W are the height and width of training samples, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We sort the elements in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, we assign a weight value Di noisy to each training pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 10% of elements in the weight vector Dnoisy corresponding to noisy samples are 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this way, we exclude 10% of training pairs that produce large difference values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Background sample dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 4, some cropped center patches contain only background or still ob- jects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' These background samples make little contribution to model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' To exclude these background training samples, we set a value Di background to each training pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 25% of the elements in the weight vector Dbackground corresponding to the 25% lowest values in D are zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Combining Dnoisy and Dbackground, we can normalize the weight of each training pair to ensure the sum of useful weights in one mini-batch is 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', Di norm = Di noisy · Di background �n i=1 Di noisy · Di background , (7) where n is the number of training pairs in a mini-batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Loss function re-weighting After computing the re-weighting weight vector using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (7), we use it to re-weight the loss obtained from training samples of various quality, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', Lfinal = 1 n n � i=1 Di norm · Li, (8) where Li is computed using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Using this loss can make the training more effective and avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' EXPERIMENTS Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Following [15, 16], we use lightweight CNNs in our trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, the filter sizes of the two convolutional layers in our CNN are 3×3×3×32 and 3 × 3 × 32 × 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' All experiments were performed using a desktop equipped with two NVIDIA RTX3090 GPUs and an i9-10900X CPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The batch size is 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We change the learning rate from 10−4 to 10−6 from epoch 0 to epoch 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The weight decay is 5 × 10−5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Good examples RGB Thermal RGB Bad examples ThermalJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 5 Test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We report results on the GTOT dataset [12] and the RGBT234 dataset, which have been widely used in RGB-T tracking studies [9, 10, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' GTOT consists of 50 RGB-T videos (15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8K frames).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Moreover, seven attributes are annotated for each sequence, including occlusion (OCC), large scale variation (LSV), fast motion (FM), low illumination (LI), thermal crossover (TC), small object (SO), and deformation (DEF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' RGBT234 contains 234 RGB-T video pairs (around 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8K frames) and 12 attributes are annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Compared to GTOT, RGBT234 is more challenging by having longer frames in videos and more challenging attributes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' When testing on the GTOT dataset, we use the RGBT234 dataset [1] as training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 10,000 RGB-T pairs are randomly chosen as the validation set in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' When testing on the RGBT234 dataset, we use the GTOT dataset as training data, and 1000 RGB-T pairs are randomly chosen as the validation set in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Evaluation metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this work, we utilize two commonly-used evaluation metrics in RGB-T tracking, maxi- mum precision rate (MPR) and maximum success rate (MSR) [1, 34], to evaluate the performance of our tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Following previous studies [1, 6, 12], the threshold of MPR is set to 5 pixels for GTOT (because the targets in GTOT are relatively small) and 20 pixels for RGBT234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Self-supervised v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' supervised training To show the effectiveness of our self-supervised training strategy, we use the ground truth of the RGBT234 dataset to train a supervised RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we only train the RGB-T tracker shown in the blue part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The comparison between the supervised RGB-T tracker and our self-supervised RGB-T tracker is shown in Table I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our self-supervised RGB-T tracker achieves better performance than the supervised one on GTOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This is inter- esting and supervising, as training using ground truth labels is usually more effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A possible reason is that by using center-cropped regions from RGBT234 (contains 110K RGB- T image pairs) as training data, the training set has more cat- egories of targets than the ground truth labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Similar pattern has been observed in some unsupervised RGB tracking studies [43], where the unsupervised DCFNet performs slightly better than supervised DCFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We also use the ground truth of the GTOT dataset to train a supervised RGB-T tracker an test it on RGBT234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen from Table I, our self- supervised RGB-T tracker is slightly worse than its supervised counterpart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This may because the GTOT dataset is smaller (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9K RGB-T image pairs) and does not provide enough high- quality training data for our self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE I COMPARISON OF THE PROPOSED SELF-SUPERVISED TRAINING AND SUPERVISED TRAINING.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' BETTER RESULTS ARE MAKED IN BOLD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant GTOT RGBT234 MPR(↑) MSR(↑) MPR(↑) MSR(↑) Supervised 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 Ours 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 TABLE II EFFECT OF INPUT IN THE FIRST BRANCH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' First input Second input MPR(↑) MSR(↑) RGB RGB-T 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Thermal RGB-T 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 4-channel RGBT RGB-T 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Ablation studies and analysis The GTOT dataset is used in ablation studies unless other- wise specified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Different cross input combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We can use different combinations of inputs to construct cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this study, we keep the second input (RGB-T image pairs) and change the first input to different variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we trained two variants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the first variant, we use thermal images as the first input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the second variant, we use 4- channel RGB-T images as the first input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' When using 4- channel RGB-T images, we change the dimension of the first convolution layer to adapt to these 4-channel images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As shown in Table II, all these combinations can be used to guide our cross-input consistency-based training and gives useful RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Among these combinations, when the first input is RGB image and the second input is RGB-T image pairs, our RGB-T tracker shows the best tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Using more branches in cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Previously, we used two branches (different inputs) to build cross-input consistency, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We can extend our idea to use more branches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, we can use three inputs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', RGB images, thermal images, and RGB-T image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this case, we can add another two loss terms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', LRGB−T and LT−RGBT, to compute the difference of every two outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Us- ing this idea, we obtain an RGB-T tracker with MPR of 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 and MSR of 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3, which is better than using 4-channel RGBT images and RGB-T image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, the performance is slightly worse than using RGB images and RGB-T image pairs as distinct inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Impact of loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Mean square error (MSE) loss is usually used to train unsupervised trackers [15, 16, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this study, we use L1 loss instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Table III shows the performance comparison of using MSE loss and L1 loss in our cross- input consistency-based self-supervised training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, L1 loss gives better performance in all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This may because our training samples are noisy (although we use re- weighting strategy), MSE loss will amplify the errors due to noisy training samples, making the training less effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Impact of loss re-weighting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We proposed two components to generate weight vectors based on cropped patches, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', noisy sample dropping and background sample dropping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this section, we report the results of removing one of the components in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, after we remove any comment, the tracking performance will drop slightly, showing that our loss re-weighting strategy is helpful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Impact of unlabeled training data size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We use different portions of RGBT234 as the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, in general, the proposed self-supervised training strategy benefits JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 6 TABLE III IMPACT OF LOSS FUNCTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' IN THESE VARIANTS, L1 LOSS GIVES BETTER PERFORMANCE THAN MSE LOSS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant MPR(↑) MSR(↑) Three branches MSE loss 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 L1 loss 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 RGB, RGB-T MSE loss 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 L1 loss 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Thermal, RGB-T MSE loss 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 L1 loss 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 TABLE IV IMPACT OF LOSS RE-WEIGHTING SCHEME.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' BETTER RESULTS ARE OBTAINED AFTER RE-WEIGHTING THE LOSS FUNCTION.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Loss re-weighting MPR(↑) MSR(↑) Noisy sample dropping Background sample dropping 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 � 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 � � 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 from training using more unlabeled RGB-T video pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Be- cause unlabeled RGB-T pairs are much easier to obtain than annotated ones, our method infers the great potential of unsupervised RGB-T tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE V ABLATION STUDIES ON TRAINING DATA SIZE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' WITH MORE UNLABELED RGB-T VIDEOS FOR TRAINING, THE PROPOSED RGB-T TRACKER ACHIEVES BETTER RESULTS ON THE GTOT DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Size MPR(↑) MSR(↑) Size MPR(↑) MSR(↑) RGBT234 (90%) 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 RGBT234 (50%) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 RGBT234 (70%) 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 RGBT234 (20%) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 Impact of feature fusion methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We trained three variants of RGB-T trackers using three feature-level fusion methods, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', element-wise average, concatenation, and the DFF fusion module proposed by Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we keep the RGB tracker in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b) and change the feature fusion method in the RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We compare different fusion methods in Table VI, which shows the fusion level has a significant impact on the performance of RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, by concatenating RGB features and thermal features, we obtain the best tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE VI IMPACT OF FEATURE FUSION METHODS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' CONCATENATION IS SIMPLE YET GIVES THE BEST PERFORMANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant MPR(↑) MSR(↑) Feature-level (Average) 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 Feature-level (Concat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=') 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Feature-level (DFF [9]) 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0 Impact of sharing weights between RGB CNN and thermal CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In our experiments, we find that whether the weights are shared between RGB CNN and thermal CNN (please see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b) of the paper) or not affects the performance of our tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We designed a variant of our method, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', the weights of the RGB CNN are shared with the thermal CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The results Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Training using two frames (a) and three frames (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Red arrow: the cross-input consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' are shown in Table VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our RGB-T tracker gives better performance than the variant where the weights are shared between the RGB CNN and the thermal CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE VII IMPACT OF SHARING WEIGHTS BETWEEN RGB CNN AND THERMAL CNN ON TRACKING PERFORMANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' THE GTOT DATASET IS USED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant MPR(↑) MSR(↑) Sharing 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 Not Sharing 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Impact of tracking sequence length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this paper, we build our cross-input consistency-based training strategy using two frames, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', frame t and frame (t + 1), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 5(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Indeed, our cross-input consistency-based training strategy can be extended to more frames, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', frames t, (t+1) and (t+2), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this case, the response map generated at frame (t + 1) will be used as the pseudo label of frame (t + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Based on the pseudo label, our tracker tracks the target from (t + 1) to frame (t + 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' When more frames are used, the cross-input consistency loss is computed in the final frame, as shown by the red dash arrow in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 5(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We show the impact of tracking sequence length on the tracking performance in Table VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' From the results, we can see that using three frames can also train a good RGB- T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, the performance is slightly worse than using two frames in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' These results indicate that our self-supervised training strategy can train our RGB-T tracker effectively using only two frames in each training pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE VIII IMPACT OF TRACKING SEQUENCE LENGTH ON TRACKING PERFORMANCE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' THE GTOT DATASET IS USED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant MPR(↑) MSR(↑) Using three frames 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Using two frames 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Compared with cycle consistency Cycle consistency is commonly used in training unsu- pervised RGB-based trackers [15–19, 23] or thermal-based trackers [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In this section, we compare our cross-input consistency with cycle consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we train our RGB-T tracker (the blue part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b)) using cycle consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We trained two variants, one with two frames and one with three frames, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' All other RGB Consistency I Consistency i RGB-T (a) Tracking using two frames (b) Tracking using three framesJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 7 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Cycle consistency using two frames (left) and three frames (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' settings are kept the same as our cross-input consistency- based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen from Table IX, our cross- input consistency-based self-supervised training strategy is more effective than cycle consistency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The attributed-based performance given in Table XI also indicates that our cross- input consistency-based training is more effective than cycle consistency-based training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Moreover, using three frames in cycle consistency-based training is more effective than using two frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In addition to the RGB-T tracker, we also use our trained CNNs to run an RGB tracker and a thermal tracker based on the red part of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our training strategy provides consistent better performance in RGB tracker, thermal tracker and RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The reason is that the forward-backward tracking-based cycle con- sistency may not hold in some challenging real world tracking scenarios [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In addition, it should be mentioned that the cycle consistency-based strategy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', UDT [15], has only been applied to uni-modal object tracking, while the proposed cross-input consistency-based self-supervised training strategy is applied to multi-modal tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' TABLE IX PERFORMANCE COMPARISON WITH CYCLE CONSISTENCY.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Variant RGB tracker T tracker RGB-T tracker PR SR PR SR MPR MSR Cycle consistency (2 frames) 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 Cycle consistency (3 frames) 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 Ours 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Compared with SOTA RGB-T tackers Compared methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' There are no existing unsupervised deep RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We selected the following supervised RGB-T trackers for comparison, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', HMFT [9], CMPP [10], DMCNet [11], JMMAC [6], CBPNet [46], MaCNet [2], MANet [47], CMP [48], MFGNet [49], DAFNet [50], DAPNet [51], mfDiMP [52], LTDA [43], DuSiamRT [53], TCNN [13], JCDA-InvSR [14], CFNet [40]+RGBT, SiamDW [54]+RGBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A transformer-based method, namely APFNet [45], is also selected for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We also selected some non-deep RGB-T trackers, including CMCF [55], NRCMR [44], LGMG [42], CMR [56], SGT [57], the method of Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' [58], CSR [12], [59], MEEF[60]+RGBT and KCF [61]+RGBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Almost all categories of RGB-T methods are covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Results on GTOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The tracking results on the GTOT dataset are shown in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our RGB-T tracker is better than all non-learning-based RGB-T trackers in terms of both metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Furthermore, our self-supervised RGB-T tracker is better than seven supervised RGB-T trackers (DAFNet, DAPNet, mfDiMP, LTDA, DuSiamRT, TCNN, JCDA-InvSR) and the RGB-T version of CFNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our tracker also shows comparable performance with several supervised trackers, such as MFGNet, CMP and MANet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Although there are some gaps between our performance and the state-of-the-art supervised RGB-T trackers, it is understandable because those supervised RGB-T trackers use large-scale annotated RGB-T image pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In contrast, our method does not use any anno- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Moreover, those trackers use more complex models, as indicated by their tracking speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, the FPS of CMPP and DMCNet are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Attribute-based results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Attribute-based performance on the GTOT dataset is shown in Table XI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, as a very lightweight RGB-T tracker trained without any ground truth labels, our tracker achieves very competitive performance in terms of LSV, LI, TC and SO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Especially, our RGB- T tracker achieves better performance than most supervised RGB-T trackers in terms of LSV, indicating that our tracker can well handle scale variation of targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Results on RGBT234.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The results on RGBT234 dataset are shown in Table X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' From the table, we can see that the proposed RGB-T tracker outperforms four supervised trackers and three non-learning-based RGB-T trackers in terms of MSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, our RGB-T tracker shows worse performance on RGBT234 than GTOT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This is because RGBT234 is much more challenging than GTOT by having more images (223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8K frames v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8K frames) and challenging scenarios (12 chal- lenging attributes v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 7 attributes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Most deep RGB-T trackers achieve good performance on RGBT234 by using complex model architectures and a large number of annotated RGB-T image pairs for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In contrast, our tracker is trained using 6,900 unlabeled RGB-T image pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In our future work, we will aim to narrow this gap by using better backbone trackers or larger training sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, it is worth mentioning that the performance gap between our self-supervised RGB-T tracker and the state-of-the-art supervised RGB-T tracker (HMFT), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6% in MPR and 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2% in MSR, is acceptable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This level of gaps also exist between state-of-the-art unsupervised RGB trackers and supervised RGB trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, the ULAST [23] achieves 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2% in precision and 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='4% in success rate on the TrackingNet, while SwinTrack [62] achieves 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8% and 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Qualitative results 1) Compared with SOTA trackers: We first show qualitative comparison of our RGB-T tracker with other RGB-T trackers in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Several RGB-T trackers are selected, including SGT [57], LGMG [42], DAPNet [51], MaCNet [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' As can be seen, our RGB-T tracker obtains better performance on these four sequences, namely, BlackCar, Exposure2, LightOcc, and carNig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 2) Challenging cases: In this section, we show qualitative results of some challenging cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In these cases, tracking using RGB images and tracking using thermal images fail, while our RGB-T tracker can correctly track targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' These t+1 t t RGB-T t+1 RGB-T t+2 RGB-T RGB-T RGB-TJOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 8 TABLE X COMPARISON WITH EXISTING RGB-T TRACKERS ON GTOT DATASET AND RGBT234 DATASET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' THE RESULTS MARKED WITH † ARE COMPUTED BY US USING RAW TRACKING RESULTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' THE RESULTS MARKED WITH § ARE COPIED FROM [1, 12, 44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' OTHER RESULTS ARE EXTRACTED FROM CORRESPONDING PAPERS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' ‘-’ MEANS NOT MENTIONED IN THE CORRESPONDING PAPER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' VALUES WORSE THAN OUR METHOD ARE MARKED IN GREEN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Tracker GTOT RGBT234 FPS (↑) Category Supervised Venue MPR (↑) MSR (↑) MPR (↑) MSR (↑) HMFT† [9] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 DL-based Yes CVPR 2022 CMPP [10] 92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='3 DL-based Yes CVPR 2020 APFNet [45] 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='5 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 DL-based Yes AAAI 2022 DMCNet [11] 90.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='1 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='6 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0 Ours 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='2 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='0 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='8 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='9 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='7 examples demonstrate that our self-supervised RGB-T tracker can well fuse RGB and thermal information to improve tracking performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' For example, in the LightOcc case, both the RGB tracker and thermal tracker has problems when the car is heavily occluded by the trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In contrast, our self- supervised RGB-T tracker can successfully track the car by fusing RGB and thermal features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Note that the RGB tracker uses our RGB CNN, and the thermal tracker uses our thermal CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Failure cases Our self-supervised RGB-T tracker fails in some very challenging cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 9 shows failure cases of our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In these cases, all the RGB tracker, thermal tracker, and RGB-T tracker fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the Pool case, the occlusion of the tree and the similar color between tree and pedestrian’s cloth are the main reasons of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the RainyCar2 case, the occlusion of the trees and the similar color between the white car and road are the main reasons of failure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' DISCUSSION Benefits of our self-supervised training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' First, the proposed training strategy allows us to train an RGB-T tracker without any human annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Some self-supervised tracking methods [18, 19, 22] still require some annotations (very sparse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Some self-supervised tracking methods use unsuper- vised optical flow models [20, 23] to generate pseudo labels or use EdgeBox [24] to generate object proposals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our meth- ods totally remove the need for any annotations and optical flow models while achieving reasonable tracking performance, which is very beneficial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Second, our cross-input consistency is flexible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, we can use different modalities as dif- ferent inputs, which is very suitable for visual data of different modalities, such as RGB images and thermal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The idea has the potential to be used in other modalities, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', RGB-D tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We can also easily add or remove branches JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 9 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Qualitative comparison of our RGB-T tracker with other RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The four sequences are BlackCar, Exposure2, LightOcc, and carNig from the GTOT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We visualize results on both RGB (1st and 3rd rows) and thermal images (2nd and 4th rows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Tracking using RGB images and tracking using thermal images fail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our RGB-T tracker can track targets successfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' From top to bot- tom: LightOcc, Quarreling, WalkingOcc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Left: RGB tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Middle: thermal tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Right: our self-supervised RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (inputs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Third, the proposed self-supervised training strategy is generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In theory, different backbone trackers can be used in the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We will explore this in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Model collapsing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A collapsed solution may exist in our cross-input consistency-based method, that is, different Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Failure cases of our RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Results on Pool (top) and Rainy- Car2 (bottom) are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Left: RGB tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Middle: thermal tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Right: our self-supervised RGB-T tracker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' branches always generate the same wrong results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, this collapsed solution never appeared in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our tracker avoids this issue safely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A possible reason is that our inputs come from different modalities, so it is unlikely that different branches generate the same features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Limitations: There is a clear performance gap between our method and state-of-the-art supervised RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' More efforts should be made to narrow this gap in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' A possible solution is to apply our self-supervised training strategy to better backbone trackers, such as SiamAtt [63], SiamDW [64], and ATOM [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Another promising way is to use larger RGB-T tracking datasets for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In addition, RGB 口 口 115 口 Thermal 口 回 BlackCar Exposure2 回 RGB Thermal 口 日 carNig LightOcc GT Ours SGT LGMG DAPNet MACNetRGB-234 T-234 Fused-234 口 RGB-157 T-157 Fused-157 A RGB-112 T-112 Fused-112 0RGB-123 123 Fused-123 口 口 RGB-114 T-114 Fused-114 口JOURNAL OF LATEX CLASS FILES, VOL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, NO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' XX, 2023 10 we do not learn an IoU net for bounding box regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This is a common issue in unsupervised tracking [15–17, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We will also solve this in our future study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' However, it is worth mentioning that although a performance gap exists between the proposed self-supervised method and state-of-the-art super- vised methods, the proposed method has a very good potential because it does not require any manual annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' CONCLUSIONS In this paper, we propose a self-supervised training strategy based on cross-input consistency to train RGB-T trackers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We construct two distinct inputs using RGB images and ther- mal images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Then, we compute the cross-input consistency loss based on the tracking results obtained using these two inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We also propose a loss re-weighting scheme to im- prove training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' The main benefit of the proposed method is that only unlabeled RGB-T video pairs are needed for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Our experiments show that the proposed method achieves favorable performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Specifically, with very simple CNNs and a simple feature fusion method, our RGB-T tracker outperforms several supervised RGB-T trackers on the GTOT dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' We also show that the tracking performance can be further improved by using more unlabeled RGB-T videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' In the future, we will use a more complex network and add a bounding box regression component to the framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' ACKNOWLEDGMENTS This study has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 101025274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' This work is also funded by a Royal Academy of Engineering Chair in Emerging Technologies to YD.' metadata={'source': 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+page_content=' [65] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Danelljan, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Bhat, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Khan, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Felsberg, “Atom: Accurate tracking by overlap maximization,” in Proceedings of the IEEE/CVF conference on computer vision and pattern recognition, 2019, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' 4660–4669.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Xingchen Zhang (M’21) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' de- gree from the Huazhong University of Science and Technology in 2012, and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' degree from the Queen Mary University of London in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is currently a Marie Skłodowska-Curie Individual Fel- low at the Personal Robotics Laboratory, Department of Electrical and Electronic Engineering, Imperial College London.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Prior to this, he was a Teaching Fel- low and Research Associate at the same department.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' His main research interests include human intention prediction, image fusion, and object tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is a recipient of the Best Paper Honourable Mention Award of the 9th Chinese Conference on Information fusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is a co-author of the book Image Fusion that has been awarded the National Science and Technology Academic Publications Fund of China (2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is a reviewer for UKRI Future Leaders Fellowship, EPSRC New Investigator Award and EPSRC Open Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Yiannis Demiris (SM’03) received the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='Sc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' (Hons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=') degree in artificial intelligence and com- puter science and the Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' degree in intelligent robotics from the Department of Artificial Intelli- gence, University of Edinburgh, Edinburgh, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', in 1994 and 1999,respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is a Professor with the Department of Electrical and Electronic Engi- neering, Imperial College London, London, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=', where he is the Royal Academy of Engineering Chair in Emerging Technologies, and the Head of the Personal Robotics Laboratory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' His current research interests include human-robot interaction, machine learning, user modeling, and assistive robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' Demiris was a recipient of the Rector’s Award for Teaching Excellence in 2012 and the FoE Award for Excellence in Engi- neering Education in 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} +page_content=' He is a Fellow of the Institution of Engineering and Technology (IET), and the British Computer Society (BCS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/O9FIT4oBgHgl3EQfeSvC/content/2301.11274v1.pdf'} diff --git a/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/2301.00125v1.pdf.txt b/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/2301.00125v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..3de066e9f8c59acc75cb4675cb79eb849a71b6b3 --- /dev/null +++ b/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/2301.00125v1.pdf.txt @@ -0,0 +1,2287 @@ +arXiv:2301.00125v1 [math.CO] 31 Dec 2022 +Generating Function for Pinsky’s Combinatorial Second Moment +Formula for the Generalized Ulam Problem +Samen Hossein +The Bronx High School of Science +75 Bronx Science Bvd +Bronx, NY 10468 +Shannon Starr∗ +Department of Mathematics +University of Alabama at Birmingham +University Hall, Room 4005 +1402 Tenth Avenue South +Birmingham, AL 35294-1241 +December 30, 2022 +Abstract +In two papers, Ross Pinsky considered the generalized Ulam’s problem for the distribution of the +number of increasing subsequences of length k in a random permutation of {1, . . . , n}. The probability +that there is at least one increasing subsequence of length k equals 1 − F(k), where F is the cdf of the +length of the longest increasing subsequence. Hence we call it the “generalized Ulam problem.” Pinsky +found various interesting results, but we consider just his combinatorial formula for the second moment, +which is the first step towards calculating all higher moments. Pinsky obtained rigorous general bounds +sufficient to determine applicability of the Paley-Zygmund second moment method. Spin glass theorists +would still like the precise asymptotics. We formulate the generating function for his A(N, j) sequence +using the complex analysis approach as in the monograph of Pemantle and Wilson. +1 +Introduction +Ulam’s problem is one of the problems that maximizes a combination of tools and techniques applicable +to the problem and yet also difficulty of the results known for the problem. Let us resist the temptation +to give details for the previous sentence. The depth of the results related to Ulam’s problem is well- +known, and any review would be better done by others. (See, for example, the old review article of +Aldous and Diaconis [2] although many new and important results have been discovered since then, +including relatively recently as in [5].) But we will state the actual problem known as Ulam’s problem. +Let π be a random permutation in Sn, chosen according to the uniform measure, so all permutations +have equal probability of 1/n!. Given a cardinality k subset of {1, . . . , n}, we may write the set as +{i1, . . . , ik} for 1 ≤ i1 < · · · < ik ≤ n. Then we say that (i1, . . . , ik) is an increasing subsequence for +π if πi1 < · · · < πik. If we fix (i1, . . . , ik) and vary over π, then the probability to have (i1, . . . , ik) to +be an increasing subsequence for a uniform, random π is equal to 1/k!. This is because if we are given +k numbers 1 ≤ j1 < · · · < jk ≤ n and we are told that {πi1, . . . , πik} is equal to {j1, . . . , jk} there are +still k! possible permutations σ = (σ1, . . . , σk) such that πir = jσr for all r ∈ {1, . . ., k} but we want +πir = jr for all r which is just one of the k! possible permutations for σ. +Various authors of textbooks, such as J. Michael Steele for [14] have used this to note that the +expected number of cardinality k subsequences which are increasing for π is equal to +�n +k +� +· 1 +k!. Ulam’s +∗Partially funded by Simons Collaboration Grant + +problem is to let L(π) be the length of the longest increasing subsequence for π: so the largest k such +that there is at least one subsequence (i1, . . . , ik) which is increasing for π. Then in Ulam’s problem, +one wishes to know the distribution of L(π), the marginal on L, under the uniform distribution for +π ∈ Sn. Steele would note that the probability that L is at least k is no larger than the expected +number of cardinality k subsequences which is +�n +k +� +· 1 +k!. So the median for L cannot be larger than +the smallest k making this quantity less than 1/2. It has a cutoff at approximately e√n. So that +places an upper bound on the average length of the longest increasing subsequence. The actual mean +is approximately 2√n by a result of Vershik and Kerov [16], and a partial result of Logan and Shepp +[9]. Moreover, there is a complete characterization of the fluctuations for large n, according to the +Baik-Deift-Johannson theorem [3]. +Suppose that instead one asks for the distribution of Zn,k(π) where Zn,k is the number of cardinality +k subsequences of which are increasing for π ∈ Sn. Then the probability that Zn,k > 0 is exactly equal +to the probability that L ≥ k. The expectation of Zn,k gives bounds on the probability of {Zn,k ≥ 1} +according to Markov’s inequality. But more precise bounds could in principle be obtained if we knew +higher moments of Zn,k. In principle, the Bonferroni inequalities could even lead to precise bounds. See +for example Charalambides [4]. (We have an unpublished article reviewing for example Charalambides’s +results on this topic, which we may incorporate as an appendix to a future version of this paper.) So +it is interesting to know the distribution of Zn,k, beyond just its mean. +In [12], Pinsky investigated the second moment. He obtained an exact combinatorial formula, albeit +a difficult one. He then rigorously analyzed the formula obtaining upper and lower bounds of matching +growth rates (at the most important, leading term, the exponential term) although with different +constant multipliers. Using this, he could determine exactly the criteria for the Paley-Zygmund second +moment method to guarantee that Zn,k satisfies a weak law of large numbers type result. Note that +just because the Paley-Zygmund method does not imply a weak law, that does not mean that a weak +law may not still be true (since it does not necessarily require vanishing of the variance of the rescaled +quantity). So in [13], Pinsky proved a threshold to be able to disprove a weak law for Zn,k. He also +has stated interesting equivalent formulations of his questions, and he has interesting conjectures which +remain as open problems. The pair of papers is fascinating. +In theoretical physics and in the mathematical physics of spin glasses, even though the second +moment of Zn,k may be too large to prove a weak law, one is still interested in the precise form of the +second moment, and indeed of all higher moments. In principle, spin glass physicists have methods +which sometimes allow one to go from formulas for the moments, asymptotically, to information about +the actual distribution. This is in cases where the usual mathematical tools do not apply, such as +where the moment problem is indeterminate or in certain types of limits in asymptotics where finite n +methods may not apply. An excellent reference for the limitations of classical mathematical tools when +trying to apply them to spin glass techniques was written by van Hemmen and Palmer [17]. However, +the spin glass techniques have proved effective in illuminating the Sherrington-Kirkpatrick mean field +spin glass problem, and may be useful beyond that. See for example, the classical physics textbook [10]. +The first rigorous proof of Parisi’s ansatz for the SK model, given by Talagrand, seemed to proceed by +very different techniques [15], with a starting point the beautiful approach of Guerra [7] and Guerra +and Toninelli [8]. +So we are interested in the more general moments of Zn,k, starting with the second moment. We call +this the generalize Ulam problem. Technically, the generalized Ulam problem should be the problem of +stating as much as possible about the distribution of Zn,k for the doubly-indexed sequence of numbers +for n and k. But we are most hopeful to be able to calculate the positive integer moments, in some +asymptotic sense. So far as we know, this has not been done, except by Pinsky. We begin, in the next +section, by stating Pinsky’s precise combinatorial formula for the second moment. Then we calculate a +generating function for the quantities that appear in Pinsky’s decomposition, with hopes of returning +later and extracting the precise asymptotics from the generating function. For us, the calculation of +the generating function was already difficult. +2 + +2 +Pinsky’s combinatorial second moment formula +In his paper [12], Pinsky calculated +E +� +Z2 +n,k +� += +k +� +j=0 +� +n +2k − j +� +1 +(2k − j)! A(k − j, j) , +(1) +where +A(N, j) = +� +ℓ0,...,ℓj∈{0,1,... } +ℓ0+···+ℓj=N +� +m0,...,mj∈{0,1,... } +m0+···+mj=N +j� +r=0 +��ℓr + mr +ℓr, mr +��2 +. +(2) +See Pinsky’s Proposition 2. We will not reproduce his proof, but we mention that his argument is +elegant. +Defining the new quantity +B(L, M, j) = +� +ℓ0,...,ℓj∈{0,1,... } +ℓ0+···+ℓj=L +� +m0,...,mj∈{0,1,... } +m0+···+mj=M +j� +r=0 +��ℓr + mr +ℓr, mr +��2 +, +(3) +we have A(N, j) = B(N, N, j). Defining the generating function +β(w, x, y) = +∞ +� +j=0 +∞ +� +L=0 +∞ +� +M=0 +wjxLyMB(L, M, j) , +(4) +for x, y, w ∈ C, we have +β(w, x, y) = +∞ +� +j=0 +wj +� ∞ +� +ℓ=0 +∞ +� +m=0 +xℓym +��ℓ + m +ℓ, m +��2�j+1 +, +(5) +by the Tonelli theorem when x, y, w are in [0, ∞) since all the coefficients are nonnegative. (See for +example, Section 2.5 of Folland [6].) So the left-hand-side diverges if and only if the right-hand-side +diverges. If neither diverges, the equality is true. Also, if neither diverges for a choice of x = x1, y = y1, +w = w1 for x1, y1, w1 ∈ [0, ∞), then for any complex numbers x, y, w ∈ C such that |x| ≤ x1, |y| ≤ y1 +and |z| ≤ z1, the two sides of (5) are convergent and there is equality of the two sides by Fubini’s +theorem. +Lemma 2.1 Given |x|, |y| ∈ [0, 1/4), defining +γ(2)(x, y) = +∞ +� +ℓ,m=0 +xℓym +��ℓ + m +ℓ, m +��2 +, +(6) +the series converges and +γ(2)(x, y) = +�� +(1 − (x + y))2 − 4xy +�−1 +. +(7) +Remark 2.2 We use the standard branch cut for the natural logarithm along the non-positive real +axis. So the domain is D1 = {reiθ : r > 0 , −π < θ < π} and for each z ∈ D1, the formula is +log(z) = +� 1 +0 (z − 1)(1 + (z − 1)t)−1 dt. Then, for each z ∈ D1, we use √z = exp((1/2) log(z)). See +Figure 1 for the schematic. +Note that it is elementary that for |x|, |y| ∈ [0, 1/2), defining +γ(x, y) = +∞ +� +ℓ,m=0 +�ℓ + m +ℓ, m +� +xℓym , +(8) +3 + +z = reiθ +r > 0 and +−π < θ < π +√z = √r eiθ/2 = exp((1/2) log(z)) +Figure 1: Schematic for the standard square-root function on the open domain of C with branch cut +along the non-positive real axis: D1 = {reiθ : r > 0 , −π < θ < π}. For z ∈ D1 we use the formula +√z = exp((1/2) +� 1 +0 (z − 1)(1 + t(z − 1))−1 dt). +using the multinomial notation for the binomial coefficient +�ℓ+m +ℓ,m +� += (ℓ+m)! +ℓ! m! , we have +γ(x, y) = +∞ +� +n=0 +(x + y)n = +1 +1 − (x + y) . +(9) +This is a first example in many textbooks: see for example Pemantle and Wilson, Example 2.2.2 [11]. +Then γ(2) may be calculated from γ using the residue method: +γ(xξ, yζ)γ +�u +ξ , v +ζ +� += +∞ +� +ℓ,m,k,n=0 +�ℓ + m +ℓ, m +��k + n +k, n +� +xℓymukvnξℓ−kζm−n . +(10) +Therefore, integrating over unit circles, we have +γ(2)(xu, yv) = +� +C(0;1) +� +C(0;1) +γ(xξ, yζ)γ +�u +ξ , v +ζ +� +dξ +2πiξ · +dζ +2πiζ . +(11) +It is an instance related to the fact that the diagonal of a rational bivariate generating function is +an algebraic univariate generating function. This is proved in Section 2.4 of Pemantle and Wilson, +credited to Furstenberg and to Hautus and Klarner. +The residue method may be used to derive formulas when they are unknown. But once known, +such formulas may often be proved by easier combinatorial methods. +Proof of Lemma 2.1: Let the right-hand-side of equation (7) be denoted by g(2)(x, y). Then we +may rewrite +g(2)(x, y) = +1 +1 − (x + y) +� +1 − +4xy +(1 − (x + y))2 +�−1/2 +. +(12) +Therefore, by Newton’s version of the binomial theorem, this gives +g(2)(x, y) = +1 +1 − (x + y) +∞ +� +n=0 +(−1/2)n +n! +(−1)n4nxnyn +(1 − (x + y))2n +(13) += +∞ +� +n,k=0 +(−1/2)n +n! +· (−2n − 1)k +k! +(−1)n+k4nxnyn(x + y)k , +(14) +4 + +where the Pochhammer symbol, falling factorial, is +(z)n = z(z − 1)(z − 2) · · · (z − n + 1) = +n−1 +� +k=0 +(z − k) . +(15) +Then, by the regular binomial formula, we have +g(2)(x, y) = +∞ +� +n,k=0 +k +� +j=0 +(−1/2)n +n! +· (−2n − 1)k +k! +�k +j +� +(−1)n+k4nxn+jyn+k−j . +(16) +Therefore, the lemma will be proved if we check that +∞ +� +n=0 +(−1/2)n +n! +· (−2n − 1)k +k! +�k +j +� +(−1)n+k4n +����� +j=ℓ−n +k−j=m−n += +��ℓ + m +ℓ, m +��2 +, +(17) +for each ℓ, m ∈ {0, 1, . . .}. But using well-known combinatorial formulas for the Pochhammer symbols +of −1/2 and of negative integers, which we review in Appendix A, this is equivalent to checking the +multinomial identity: +∞ +� +n=0 +� +ℓ + m +n, n, ℓ − n, m − n +� += +��ℓ + m +ℓ, m +��2 +. +(18) +(We use the standard convention that if one of the indices in the subscript of a binomial or multinomial +coefficient is outside the range of 0 to the superscript index, then that coefficient equals 0 by definition.) +Equation (18) may be proved as follows. +Draw ℓ + m black dots on the number line from 1 to ℓ + m. Above each dot place a blue or a red +dot such that if A is the subset of those r ∈ {1, . . . , ℓ + m} with a blue dot above it, then |A| = ℓ +and |A∁| = m. Below each black dot, place a green or a yellow dot, such that if B is the set of those +r ∈ {1, . . . , ℓ + m} with a green dot below it, then |B| = m and |B∁| = ℓ. +Then, if we let n = |A∩B| we have |A∩B∁| = ℓ−n and |A∁ ∩B| = m−n, as well. Therefore, |A∁ ∩B∁| +is equal to ℓ + m − |A ∩ B| − |A ∩ B∁| − |A∁ ∩ B|, which is n. The number of ways of assigning pairs of +colors to each dot – (blue,green), (blue,yellow), (red,green), (red,yellow) – with a prescribed number +of each n, ℓ − n, m − n, n is the multinomial coefficient +� +ℓ+m +n,ℓ−n,m−n,n +� +. Summing over all choices of n +gives all possibilities for choosing the two sets A and B. Since they are chosen independently, that is +��ℓ+m +ℓ,m +��2 +. This proves equation (18). +□ +Because of Lemma 2.1 and equation (5), we have +β(w, x, y) = +γ(2)(x, y) +1 − wγ(2)(x, y) = +1 +� +(1 − (x + y))2 − 4xy +�1/2 +− w +. +(19) +Now define another generating function +α(w, z) = +∞ +� +j,N=0 +wjzNA(N, j) = +∞ +� +j,N=0 +wjzNB(N, N, j) , +(20) +which is the generating function for the combinatorial numbers we want, A(N, j) for N, j ∈ {0, 1, . . .}. +Then, by applying the diagonal method, we obtain +� +C(0;1) +β +� +w , xξ , y +ξ +� +dξ +2πiξ = α(w, xy) , +(21) +5 + +for sufficiently small x, y and w. +Let us digress briefly to consider the type of equation which appears in the square-root in the +denominator in the expression of β from (19). For a quadratic equation of the form +ξ2 + 1 − kξ = 0 , +(22) +the quadratic formula for the roots gives +ξ± = k ± +√ +k2 − 4 +2 +, +(23) +which in turn can be written as +ξ± = 1 +4 +�√ +k − 2 ± +√ +k + 2 +�2 +, +(24) +if we know that k − 2 and k + 2 are in the domain D1 for the square-root function from Remark 2.2. +The reason this type of quantity appears for us is that we are considering β(w, xξ, y/ξ), and we may +also simplify by taking y = x. This is because in (21) we get a formula for α(w, xy) which we may +specialize to a formula for α(w, x2), taking y = x, with no loss of utility. Then we see that +β(w, xξ, y/ξ) = +1 +�� +1 − x(ξ + ξ−1) +�2 − 4x2 +�1/2 +− w += +1 +((x2/ξ2) ((ξ2 + 1) − (x−1 + 2) ξ) ((ξ2 + 1) − (x−1 − 2) ξ))1/2 − w +, +(25) +containing several terms of the form of the left-hand-side of (22) for different choices of k. For the +purpose of doing contour deformation, we will need to factorize these terms. +In fact, they arise twice, in two rounds, because we rationalize the denominator in order to more +easily calculate the residues and the contribution due to the branch-cut in (21). That means we must +consider the argument of the square-root function in order to handle the branch-cut contribution. But +we must also consider the poles of the denominator, after we have rationalized it, which is a slightly +different quartic. +Remark 2.3 We frequently refer to the argument of the square-root meaning that if we have +� +f(x) +this means f(x). In complex analysis “argument” may also refer to the imaginary part of the logarithm +function. (See for example, Section 3.4 of [1].) But we will instead use the word “phase” to mean eiϕ +when we decompose a complex number as z = Reiϕ for R ≥ 0 and ϕ ∈ R. +Because of equation (24) we will assign variable names to some of the terms such as +√ +k − 2 or +√ +k + 2 or related algebraic functions that arise. The extra variable names have helped us to organize +our calculations, and we hope they are not too much of a distraction for an interested reader. +Theorem 2.4 Suppose we have numbers t, u ∈ R, such that 0 ≤ t < 1/4 and 0 ≤ u < 1. Then +α +� +2t +� +1 − u2 , t2u2� += 1 +π +� c2 +c1 +� +−Q1(r) +−Q2(r) +dr ++ +2t +√ +1 − u2 +�� +1 − (2t + 2tu)2 + +� +1 − (2t − 2tu)2 +� �� +(1 + 2tu)2 − 4t2 + +� +(1 − 2tu)2 − 4t2 +� +� +1 − (1 + u2)t2 + +� +1 − 8(1 + u2)t2 + 16(1 − u2)2t4 +� � +1 − 8(1 + u2)t2 + 16(1 − u2)2t4 +. +(26) +where +Q1(r) = +� +(r2 + 1)tu − (1 − 2tu)r +�� +(r2 + 1)tu − (1 + 2tu)r +� +, +(27) +Q2(r) = +� +(r2 + 1)tu − (1 − 2t)r +�� +(r2 + 1)tu − (1 + 2t)r +� +, +(28) +are two quartic polynomals, and where +c1 = 1 + 2tu − √1 + 4tu +2tu +and c2 = 1 − 2tu − √1 − 4tu +2tu +(29) +are the two roots of Q1 in (0, 1). Both Q1(r) and Q2(r) are negative on (c1, c2). +6 + +c1 +c2 +a1 +a2 +d1 +d2 +b1 +b2 +> +Figure 2: A schematic picture – it is not to scale but it does accurately depict the order of points – for +the branch cuts and poles of the integral in (38). The configuration of points and branch cuts is shown +assuming u, t ∈ (0, 1) are sufficiently small. The poles are at a1, a2, b1 and b2. The branch cuts are the +intervals [c1, c2] and [d1, d2]. In other words, c1, c2, d1, d2 are roots of Q1 and a1, a2, b1, b2 are roots of Q2 +for the quartic polynomials Q1 and Q2 from Theorem 2.4. +We will prove the theorem in the next section. For now let us note that this is a potentially useful +representation for the sequence A(N, j). We note that for x > 0 and w > 0 we can find t and u to +match them. If we did have x = t2u2 and w = 2t +√ +1 − u2, then we would have +w2 + 4x = +� +2t +� +1 − u2 +�2 ++ 4t2u2 = 4t2 . +(30) +Therefore, we may take +t = 1 +2 +� +w2 + 4x . +(31) +So |t| < 1 as long as 4x + w2 < 1 +4. We can then take u = t−1√x = +√ +4x/ +√ +w2 + 4x. (If x and w are +both 0 then t = 0 and the value of u is irrelevant.) So u is in [0, 1] as long as x, w are both nonnegative. +In principle, calculating the generating function for all x and w in say [0, 1/32] does determine all +the coefficients. Note that all the coefficients are nonnegative, since they are combinatorial (either 0 +or positive integers). So if the series is convergent for 0 ≤ x < 1/32 and 0 ≤ w < 1/32 then the +complex series is absolutely convergent for |x| < 1/32 and |w| < 1/32. This is the same, usual fact that +lies behind the Fubini-Tonelli theorem (see for example [6]). Perhaps more importantly, if we want to +obtain asymptotics for A(N, j), then one often uses techniques related to positivity such as Chebyshev’s +inequality. Certainly Chebyshev’s inequality gives one-sided bounds. Those could potentially be turned +into asymptotics if the correct Tauberian-type theorem is found and used. +The integral appearing in equation (26) is an example of an elliptic integral, although we do not +know anything beyond that about a more precise characterization of this integral. +3 +Proof of Theorem 2.4: generating function formula +If we take x, y, z ∈ C such that +2|x| + 2|y| + |w|2 < 1 , +(32) +7 + +then we have +α(w, xy) = +1 +2πi +� +C(0;1) +1 +ξ · +� +(1 − (xξ + yξ−1))2 − 4xy +�1/2 +− wξ +dξ , +(33) +where we have used equation (19). The argument of the square-root function in the denominator is +contained in the open disk C centered at 1 with radius 1, since 2|x| + 2|y| < 1. So the square-root is +unambiguously defined in the sense of Remark 2.2. Then the entire denominator may be written as ξ +times a quantity which is within the open disk C centered at 1 with radius 1, since 2|x|+2|y|+|w|2 < 1 +by equation (32). +Next we rationalize the denominator, in order to simplify the calculation of the residues. So equation +(33) can be rewritten as +α(w, xy) = +1 +2πi +� +C(0;1) +ξ · +�� +1 − (xξ + yξ−1) +�2 − 4xy +�1/2 ++ wξ +� +(ξ − (xξ2 + y))2 − 4xyξ2 +� +− w2ξ2 +dξ . +(34) +Now, we will specialize to simplify the calculations going forward. Firstly, let us take y = x. +α(w, x2) = +1 +2πi +� +C(0;1) +ξ · +�� +1 − x(ξ + ξ−1) +�2 − 4x2�1/2 ++ wξ +� +(ξ − x(ξ2 + 1))2 − 4x2ξ2 +� +− w2ξ2 +dξ , +(35) +Next, let us assume that x and w are real variables for the remainder of the proof. +In fact, we change variables. Let us assume that we have real numbers t, u ∈ R which are sufficiently +small, which effectively amounts to −1/4 < t < 1/4 and −1 < u < 1, and let us take w = 2t +√ +1 − u2 +and x = tu. Then we have +α +� +2t +� +1 − u2 , t2u2� += +1 +2πi +� +C(0;1) +ξ · +�� +1 − (ξ + ξ−1)tu +�2 − 4t2u2�1/2 ++ 2tξ +√ +1 − u2 +� +(ξ − (ξ2 + 1)tu)2 − 4t2u2ξ2 +� +− 4(t2 − t2u2)ξ2 +dξ . +(36) +Collecting terms in the denominator, this may be simplified to give +α +� +2t +� +1 − u2 , t2u2� += +1 +2πi +� +C(0;1) +ξ · +�� +1 − (ξ + ξ−1)tu +�2 − 4t2u2�1/2 ++ 2tξ +√ +1 − u2 +(ξ − (ξ2 + 1)tu)2 − 4t2ξ2 +dξ . +(37) +Now we factorize the denominator to obtain +α +� +2t +� +1 − u2 , t2u2� += +1 +2πi +� +C(0;1) +ξ · +�� +1 − (ξ + ξ−1)tu +�2 − 4t2u2�1/2 ++ 2tξ +√ +1 − u2 +((1 − 2t)ξ − (ξ2 + 1)tu) ((1 + 2t)ξ − (ξ2 + 1)tu) dξ . +(38) +The two factors of the denominator are quadratic polynomials, and they may each be further factorized +as (multiplying each one by −1 first) +� +(ξ2 + 1)tu − (1 − 2t)ξ +� += tu · (ξ − a2)(ξ − b1) , +(39) +� +(ξ2 + 1)tu − (1 + 2t)ξ +� += tu · (ξ − a1)(ξ − b2) , +(40) +by direct appeal to the quadratic formula, where +a1 = 1 + 2t − +� +(1 + 2t − 2tu)(1 + 2t + 2tu) +2tu +, +(41) +a2 = 1 − 2t − +� +(1 − 2t − 2tu)(1 − 2t + 2tu) +2tu +, +(42) +b1 = 1 − 2t + +� +(1 − 2t − 2tu)(1 − 2t + 2tu) +2tu +, +(43) +b2 = 1 + 2t + +� +(1 + 2t − 2tu)(1 + 2t + 2tu) +2tu +. +(44) +8 + +A power series expansion in the small parameter gives +a1 = tu − 2t2u + O(t3) , +a2 = tu + 2t2u + O(t3) , +(45) +b1 = 1 +tu − 2 +u + O(t) , +b2 = 1 +tu + 2 +u + O(t) . +(46) +This means that 0 < a1 < a2 < 1 < b1 < b2, when t ∈ (0, 1) is sufficiently small. For future reference, +we note that +Q2(ξ) = +� +(1 − 2t)ξ − (ξ2 + 1)tu +� +· +� +(1 + 2t)ξ − (ξ2 + 1)tu +� += t2u2 · (ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) . +(47) +Now let us introduce some extra notation that helps to organize the calculations to come. Let us define, +for each σ1, σ2 ∈ {1, −1} the quantity +ρ(σ1, σ2) = +√ +1 + 2σ1t + 2σ2tu . +(48) +Since we assumed that −1/4 < t < 1/4 and −1 < u < 1 we know that the argument of the square-root +function is in the open disk in C, centered at 1 with radius 1. So the standard square-root function +may be applied (as in Remark 2.2). Let us abbreviate the notation to +ρ(+) ++ += ρ(1, 1) , +ρ(+) +− += ρ(1, −1) , +ρ(−) ++ += ρ(−1, 1) , +ρ(−) +− += ρ(−1, −1) . +(49) +Then we may write +a1 = +1 +4tu +� +ρ(+) ++ +− ρ(+) +− +�2 +, +a2 = +1 +4tu +� +ρ(−) ++ +− ρ(−) +− +�2 +, +(50) +b1 = +1 +4tu +� +ρ(−) ++ ++ ρ(−) +− +�2 +, +b2 = +1 +4tu +� +ρ(+) ++ ++ ρ(+) +− +�2 +. +(51) +This is a rewriting of the roots of these particular quadratic equations similar to what we described in +equation (24). We will return to further calculations involving these quantities, briefly. +First, let us perform a similar algebraic simplification for the polynomial in the argument of the +numerator in equation (38). This may be factorized as +� +1 − (ξ + ξ−1)tu +�2 − 4t2u2 = +� +1 − 2tu − (ξ + ξ−1)tu +� � +1 + 2tu − (ξ + ξ−1)tu +� +. +(52) +Applying the quadratic formula, we have +� +1 − 2tu − (ξ + ξ−1)tu +� += tuξ−1 · (ξ − c2)(ξ − d1) , +(53) +� +1 + 2tu − (ξ + ξ−1)tu +� += tuξ−1 · (ξ − c1)(ξ − d2) , +(54) +where +c1 = 1 + 2tu − √1 + 4tu +2tu +, +c2 = 1 − 2tu − √1 − 4tu +2tu +, +(55) +d1 = 1 − 2tu + √1 − 4tu +2tu +, +d2 = 1 + 2tu + √1 + 4tu +2tu +. +(56) +A power series expansion in the small parameter gives +c1 = tu − 2t2u2 + O(t3) , +c2 = tu + 2t2u2 + O(t3) , +(57) +d1 = 1 +tu − 2 + O(t) , +d2 = 1 +tu + 2 + O(t) . +(58) +This means that 0 < a1 < c1 < c2 < a2 < 1 < b1 < d1 < d2 < b2, when t ∈ (0, 1) and u ∈ (0, 1) are +sufficiently small. In other words, the ordering of the poles and branch cut endpoints are as depicted +in Figure 2. For future reference, we note that +Q1(ξ) = +� +(1 − 2tu)ξ − (ξ2 + 1)tu +� +· +� +(1 + 2tu)ξ − (ξ2 + 1)tu +� += t2u2 · (ξ − c1)(ξ − c2)(ξ − d1)(ξ − d2) , +(59) +9 + +c1 +> +c2 +< +a1 +Γ1(δ) +> +a2 +Γ2(δ) +> +Γ3(δ) +Figure 3: A schematic of the contours after deformation. The two positively oriented circular contours – +of radius δ for 0 < δ ≪ 1 – are centered at the poles a1 and a2. They are Γ1(δ) and Γ2(δ). The branch cut +along the real axis interval from c1 to c2 is enclosed by the contour Γ3(δ) which is the rectangle joining +c1 − iδ, c2 − iδ, c2 + iδ and c1 + iδ, traversed in the counter-clockwise orientation. +although now we will be more concerned with the interpretation of +� +Q1(ξ) or ξ +� +Q1(ξ)/ξ2 as analytic +functions in prescribed domains. Now we deform the unit circle contour C(0; 1) into the unit disk, but +avoiding cutting through the poles a1, a2 or the branch cut interval [c1, c2] inside the unit disk. See +Figure 3. +Just before doing that, let us make a precise declaration of how we consider +� +Q1(ξ) in the domain +that we deform the contours through. +3.1 +Analytic extension of +� +Q1(ξ) outside [c1, c2] and [d1, d2] +Let us define that domain +D2 = C \ +� +[c1, c2] ∪ [d1, d2] +� +. +(60) +Note that for ξ ∈ C with |ξ| close to 1, the usual power series expansion of +� +Q1(ξ)/ξ2, expanding in +t, may be written +�� +1 − (ξ + ξ−1)tu +�2 − 4t2u2�1/2 += +∞ +� +n=0 +(1/2)n +n! +· +� +−2 +� +ξ + ξ−1� +tu + +� +ξ − ξ−1�2t2u2�n +. +(61) +It is convergent if (1 + 2 max(|ξ|, |ξ|−1)|tu|)2 − 1 < 1. We may then say that, as long |ξ| and |ξ|−1 are +both contained in a compact set, we have +� +Q1(ξ)/ξ2 = 1 + O(|t|) , +as t → 0 . +(62) +Let U(z0; ρ) denotes the open disk in C centered at z0, with radius ρ, and let U(z0; ρ) denote the +closure, the closed disk. Then, for each ǫ ∈ (0, 1), if ξ is in U(0; 1 + ǫ) \ U(0; 1 − ǫ) then the power series +defined by the right-hand-side of equation (61) is what we mean by +� +Q1(ξ)/ξ2. +We now describe a second version of the same function on D2. Initially, consider another domain +D3 equal to C \ +� +(−∞, c2] ∪ [d1, ∞) +� +, which is the domain depicted on the right-hand-side of Figure +5, the complex plane minus any point in any of the branch cuts for √ξ − c1, √ξ − c2, √d1 − ξ and +√d2 − ξ. We note that 1 is in D2 and with the standard definition of the square-root (from Remark +2.2) we have +√ +1 − c1 , +√ +1 − c2 , +� +d1 − 1 , +� +d2 − 1 > 0 . +(63) +Therefore, for some ǫ > 0, and for ξ ∈ U(1, ǫ), we have +ξ +� +Q1(ξ)/ξ2 = t2u2 � +ξ − c1 +� +ξ − c2 +� +d1 − ξ +� +d2 − ξ . +(64) +Note that both sides have the property that their square equals Q1(ξ). A priori the right-hand-side of +(64) is defined on all of D3 including the subset of U(0; 1 + ǫ) \ U(0; 1 − ǫ) equal to all points except a +neighborhood of −1, say U(−1; ǫ). But in fact, we may see that it is defined on all of C\ +� +[c1, c2]∪[d1, d2] +� +. +10 + +c1 +c2 +d1 +d2 +c1 +c2 +d2 +d1 +Figure 4: The branch cuts for the function +� +Q1(ξ) are shown on the left. In the figure on the right +we show the a priori overlapping branch-cuts if we take (d1 − ξ)1/2(d2 − ξ)1/2(ξ − c1)1/2(ξ − c2)1/2 as a +proposal to extend the “power series formula” of ξ +� +Q1(ξ)/ξ2 when ξ is in the vicinity of the unit circle +C(0; 1), meaning the power series in the variable t. For the picture on the right, wherever two branch +cuts overlap (or for more general problems wherever an even number overlap) the branch cuts may be +removed because each branch cut is a square-root branch cut. So they are counted modulo 2. If we check +the limits from the upper half-plane ξ = x + iy, y > 0, they match the limits from the lower half-plane at +points where 2 branch cuts overlap. +Given a number r ∈ R \ {c1, c2, d1, d2} we may define two limits ζ+(r) and ζ−(r) as +ζ±(r) = +lim +ǫ→0± +√r + iǫ − c1 +√r + iǫ − cd +� +d1 − (r + iǫ) +� +d2 − (r + iǫ−) +��� +√r + iǫ − c1 +√r + iǫ − cd +� +d1 − (r + iǫ) +� +d2 − (r + iǫ−) +��� +. +(65) +This is the phase which is also the source of the discontinuities if we try to define +� +Q1(ξ) everywhere in +C as a single-valued function. Or in other words, if we analytically continued +� +Q1(ξ) to a holomorphic +(single-valued) function on a Riemann surface projecting down to C, then the phases could be used to +denote the sheets of the Riemann surface. We may easily calculate +ζ±(r) = + + + + + + + + + +1 +for r ∈ (c2, d1), +±i +for r ∈ (c1, c2), +−1 +for r ∈ (−∞, c1) ∪ (d2, ∞), +∓i +for r ∈ (d1, d2). +(66) +Let us consider one example case: r ∈ (c1, c2) and ǫ > 0. We may refer to Figure 5 to determine +the phase of +� +Q1(ξ) for ξ = r + iǫ. In this example case, we have +ξ − c1 = +� +(r − c1)2 + ǫ2eiθ1 and ξ − c2 = +� +(c2 − r)2 + ǫ2eiθ2 , +(67) +where θ1 ∈ (0, π/2) and θ1 ∈ (π/2, π). But as ǫ approaches 0 from the positive side, we have θ1 → 0 +while θ2 → π. So we have +lim +ǫ→0+ +ξ − c1 +|ξ − c1| = 1 and +lim +ǫ→0+ +ξ − c2 +|ξ − c2| = −1 . +(68) +But then taking the square-root, we have eiθ1/2 converges to 1 while eiθ2/2 converges to i. Multiplying +these together, we get ζ+(r) = i. Note that d1 − r > 0 and d2 − r > 0 for r throughout [−1, 1], which +is why we did not consider those factors. Of course if we replace ǫ by −ǫ then the angles are reflected: +θ1 ← −θ1 and θ2 ← −θ2 using replacement notation (as in APL). So the phase changes from ζ+(r) = i +to its reciprocal 1/i = −i. +11 + +c1 +c2 +• +< +< +ξ = r + iǫ +θ1 +θ2 +Figure 5: Consideration of the phase of +� +Q∞(ξ), in the vicinity of [−1, 1]. If we write the output of +the square root as eiϕR for R ≥ 0 and ϕ ∈ R then by “phase” we mean eiϕ. Near the real axis, except +in neighborhoods of {c1, c2, d1, d2}, in the upper half-plane and lower half-plane, the phase eiϕ is well +approximated by one of the numbers in {1, i, −1, −i}. +The point is that for r ∈ (−∞, c1) and r ∈ (d2, ∞), the limit from the upper half-plane and the +limit from the lower half-plane coincide. They are both −1. Therefore, the analytic function, +z ∈ C \ +� +(−∞, c2] ∪ [d1, ∞) +� +�→ t2u2 � +ξ − c1 +� +ξ − c2 +� +d1 − ξ +� +d2 − ξ , +(69) +may be extended to z ∈ C\ +� +[c1, c2]∪[d1, d2] +� +just by taking limits from the upper half-plane and lower +half-plane since the two limits agree. +In particular, note that this formula is negative in [−1, c1]. Reconsider +� +Q1(ξ)/ξ2, defined by the +power series in t for small t and for |ξ| near to 1. For that quantity, we know +� +Q1(ξ)/ξ2 = 1 + O(t) +as t → 0. But then that means that ξ +� +Q1(ξ)/ξ2 = ξ + O(t) as t → 0. In particular, near ξ = −1 +the phase is approximated by −1. That matches the behavior determined by +� +Q1(ξ) on D2 in a +neighborhood of −1. +3.2 +Resumption of the proof +First, let us quickly rewrite the formulas for the roots, as we did for the denominator. Let us define, +for each σ1, σ2 ∈ {1, −1} the quantity +�ρ(σ1, σ2) = +√ +1 + 2σ1tu + 2σ2tu , +(70) +for σ1, σ2 ∈ {1, −1} and let us abbreviate the notation to +�ρ(+) ++ += �ρ(1, 1) , +�ρ(+) +− += �ρ(1, −1) , +�ρ(−) ++ += �ρ(−1, 1) , +�ρ(−) +− += �ρ(−1, −1) . +(71) +Of course, two of these quantities �ρ(+) +− +and �ρ(−) ++ +are each equal to just 1, and the formulas in (55) and +(56) are not terribly complex as compared to the equationsfor a1, a2, b1, b2, which did get simpler in +(50) and (51). But using all of these, we may write +c1 = +1 +4tu +� +�ρ(+) ++ +− �ρ(+) +− +�2 +, +c2 = +1 +4tu +� +�ρ(−) ++ +− �ρ(−) +− +�2 +, +(72) +d1 = +1 +4tu +� +�ρ(−) ++ ++ �ρ(−) +− +�2 +, +d2 = +1 +4tu +� +�ρ(+) ++ ++ �ρ(+) +− +�2 +. +(73) +Now let us finally consider the contour deformation. We note that we may rewrite the integral in +(38) as +α +� +2t +� +1 − u2 , t2u2� += +1 +2πi +� +C(0;1) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +Q2(ξ) +dξ , +(74) +12 + +using the notation for the quartic polynomials of (27) and (28). Referring to the three contours shown +in Figure 3, this means +α +� +2t +� +1 − u2 , t2u2� += +1 +2πi +� +j∈{1,2,3} +� +Γj(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +Q2(ξ) +dξ , +(75) +but where we know the contour integrals along the contours Γ1(δ) and Γ2(δ), for δ sufficiently small, +gives the residues at a1 and a2, respectively. We will state what the δ → 0+ limit of the Γ2(δ) contour +gives, later. We know that the branch cut is along [c1, c2], and the poles are at a1 and a2. But we +must calculate the residues at a1 and a2, including the numerator which includes the algebraic term. +We may also simplify the integral a bit for the portion of the contour that comes from the two sides of +the branch cut. +Let us first consider the residues at ξ = a1 and ξ = a2. When taking the square-root of Q(ξ)/ξ2 for +ξ ∈ {a1, a2} ⊂ R we need to check the sign of the square-root output. Note that we wrote +� +Q1(ξ)/ξ2 +because if |ξ| ∈ (1 − ǫ, 1 + ǫ) for ǫ ∈ [0, 1) fixed, then we have +Q1(ξ)/ξ2 = 1 + O(t2) , +(76) +as t → 0. Therefore, it is in the domain D1. The algebraic function ξ +� +Q1(ξ)/ξ2 is analytic for ξ ∈ C +satisfying |ξ| < 1 and ξ ̸∈ [c1, c2]. Moreover for ξ ∈ (−1, −1 + ǫ) the function is negative, while for +ξ ∈ (1 − ǫ, 1) the function is positive. But then, by inspection, we see that for r ∈ (−1, 1) \ [c1, c2], the +function r +� +Q1(r)/r2, specializing the analytic function, changes sign at r = 0. So it is positive for +r ∈ (0, c1) ∪ (c2, 1). At r = a1 and r = a2 we have +Q2(r) = Q1(r) − 4t2(1 − u2)r2 = 0 . +(77) +Therefore, since 0 < a1 < c1 < c2 < a2 < 1, we do have +∀r ∈ {a1, a2} , +we have r · (Q1(r)/r2)1/2 = 2tr . +(78) +This can also be seen to be consistent with the analysis in Subsection 3.1. Therefore, for the integrand +of (74) at the poles we have the value of the numerator being as follows: +∀r ∈ {a1, a2} , +we have r · (Q1(r)/r2)1/2 + 2tr +� +1 − u2 = 4tr +� +1 − u2 . +(79) +Therefore, since +Q2(ξ) = t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) , +(80) +we have the residues for the two contours +1 +2πi +� +Γ1(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ = +4ta1 +√ +1 − u2 +t2u2(a1 − a2)(a1 − b1)(a1 − b2) , +(81) +1 +2πi +� +Γ2(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ = +4ta2 +√ +1 − u2 +t2u2(a2 − a1)(a2 − b1)(a2 − b2) , +(82) +if δ > 0 is small enough. Combining these two residue integrals, and performing some simple algebra, +13 + +we have (for sufficiently small δ > 0) +1 +2πi +� +j∈{1,2} +� +Γj(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ += 4t +√ +1 − u2 +t2u2 +� +− +a1 +(a2 − a1)(b1 − a1)(b2 − a1) + +a2 +(a2 − a1)(b1 − a2)(b2 − a2) +� += 4 +√ +1 − u2 +tu2 +· +� +− a1(b1 − a2)(b2 − a2) + a2(b1 − a1)(b2 − a1) +� +(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) += 4 +√ +1 − u2 +tu2 +· +� +− a1b1b2 + a1a2(b1 + b2) − a1a2 +2 + a2b1b2 − a2a1(b1 + b2) + a2 +1a2 +� +(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) += 4 +√ +1 − u2 +tu2 +· +(−a1 + a2)(b1b2 − a1a2) +(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) += 4 +√ +1 − u2 +tu2 +· +(b1b2 − a1a2) +(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) . +(83) +Now let us use the formulas from (50) and (51). We have +b1 − a1 = +1 +4tu +�� +ρ(−) ++ ++ ρ(−) +− +�2 +− +� +ρ(+) ++ +− ρ(+) +− +�2� += +1 +4tu +� +ρ(+) ++ +− ρ(+) +− ++ ρ(−) ++ ++ ρ(−) +− +� � +ρ(+) +− +− ρ(+) ++ ++ ρ(−) ++ ++ ρ(−) +− +� +, +(84) +b2 − a1 = +1 +4tu +�� +ρ(+) ++ ++ ρ(+) +− +�2 +− +� +ρ(+) ++ +− ρ(+) +− +�2� += +1 +4tu +� +2ρ(+) ++ +� � +2ρ(+) +− +� +, +(85) +b1 − a2 = +1 +4tu +�� +ρ(−) ++ ++ ρ(−) +− +�2 +− +� +ρ(−) ++ +− ρ(−) +− +�2� += +1 +4tu +� +2ρ(−) ++ +� � +2ρ(−) +− +� +, +(86) +b2 − a2 = +1 +4tu +�� +ρ(+) ++ ++ ρ(+) +− +�2 +− +� +ρ(−) ++ +− ρ(−) +− +�2� += +1 +4tu +� +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) ++ +− ρ(−) +− +� � +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) +− +− ρ(−) ++ +� +, +(87) +and finally +b1b2 − a1a2 = +1 +16t2u2 +�� +ρ(+) ++ ++ ρ(+) +− +�2 � +ρ(−) ++ ++ ρ(−) +− +�2 +− +� +ρ(+) ++ +− ρ(+) +− +�2 � +ρ(−) ++ +− ρ(−) +− +�2� += +1 +16t2u2 +�� +ρ(+) ++ ++ ρ(+) +− +� � +ρ(−) ++ ++ ρ(−) +− +� +− +� +ρ(+) ++ +− ρ(+) +− +� � +ρ(−) ++ +− ρ(−) +− +�� +· +�� +ρ(+) ++ ++ ρ(+) +− +� � +ρ(−) ++ ++ ρ(−) +− +� ++ +� +ρ(+) ++ +− ρ(+) +− +� � +ρ(−) ++ +− ρ(−) +− +�� += +1 +16t2u2 +� +2ρ(+) ++ ρ(−) +− ++ 2ρ(+) +− ρ(−) ++ +� � +2ρ(+) ++ ρ(−) ++ ++ 2ρ(+) +− ρ(−) +− +� +. +(88) +We will simplify this a bit more. But first, let us state the small algebraic simplifications available for +the contour integral +1 +2πi +� +Γ3(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 + 2tξ +√ +1 − u2 +Q2(ξ) +dξ , +(89) +from (75). +14 + +3.3 +Simplifying the integral around the contour Γ3(δ) +We may rewrite the contour integral as +1 +2πi +� +Γ3(δ) +tu √ξ − c1 +√ξ − c2 +√d1 − ξ √d2 − ξ + 2tξ +√ +1 − u2 +Q2(ξ) +dξ , +(90) +by rewriting ξ · +� +Q1(ξ)/ξ2�1/2 as in Subsection 3.1. We know that for r ∈ (c1, c2) we have +� +ξ − c1 +� +ξ − c2 ≈ i +� +(r − c1)(c2 − r) for ξ = r + iδ, as δ → 0+. +(91) +Again, we refer to the analysis of Subsection 3.1. But for r ∈ (c1, c2), we have +� +ξ − c1 +� +ξ − c2 ≈ −i +� +(r − c1)(c2 − r) for ξ = r − iδ, as δ → 0+. +(92) +Also, by analyticity, we know +1 +2πi +� +Γ3(δ) +2tξ +√ +1 − u2 +Q2(ξ) +dξ = 0 . +(93) +It is also easily seen that the vertical edge contributions to the contour integral may be bounded by +terms which are vanishingly small when δ → 0. So +lim +δ→0 +1 +2πi +� +Γ3(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 +Q2(ξ) +dξ = −2i +2πi +� c2 +c1 +tu +� +(r − c1)(c2 − r)(d1 − r)(d2 − r) +Q2(r) +dr , +(94) +which we obtained also by using the fact that for the top edge face of the rectangle Γ3, the contribution +to the contour integral has the opposite orientation as a typical real integral, while for the bottom +edge face it has the positive orientation. But, of course, Q2(r) is also negative on the whole interval +[c1, c2]. This is how we obtain a positive integral contribution from the third contour integral around +the branch cut: +lim +δ→0 +1 +2πi +� +Γ3(δ) +ξ · +� +Q1(ξ)/ξ2�1/2 +Q2(ξ) +dξ = +1 +2π +� c2 +c1 +� +−Q1(r) +−Q2(r) +dr . +(95) +3.4 +Resumption of simplification of residues +Based on equations (83) and (84), (85), (86), (87) and (88), the sum of the two residues is given by +4 +√ +1 − u2 +tu2 +· +(b1b2 − a1a2) +(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) = 16t +� +1 − u2 +· +� +ρ(+) ++ ρ(−) +− ++ ρ(+) +− ρ(−) ++ +� � +ρ(+) ++ ρ(−) ++ ++ ρ(+) +− ρ(−) +− +� +1 +· +1 +� +ρ(+) ++ +− ρ(+) +− ++ ρ(−) ++ ++ ρ(−) +− +� � +ρ(+) +− +− ρ(+) ++ ++ ρ(−) ++ ++ ρ(−) +− +� +· +1 +ρ(+) ++ ρ(+) +− ρ(−) ++ ρ(−) +− +· +1 +� +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) ++ +− ρ(−) +− +� � +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) +− +− ρ(−) ++ +� +(96) +We note that, since +ρ(+) ++ += +√ +1 + 2t + 2tu , +ρ(+) +− += +√ +1 + 2t − 2tu , +ρ(−) ++ += +√ +1 − 2t + 2tu , +ρ(−) +− += +√ +1 − 2t − 2tu , +(97) +15 + +we may more easily express +ρ(+) ++ ρ(−) +− += +√ +1 + 2t + 2tu +√ +1 − 2t − 2tu = +� +1 − (2t + 2tu)2 , +(98) +ρ(+) +− ρ(−) ++ += +√ +1 + 2t − 2tu +√ +1 − 2t + 2tu = +� +1 − (2t − 2tu)2 , +(99) +ρ(+) ++ ρ(−) ++ += +√ +1 + 2t + 2tu +√ +1 − 2t + 2tu = +� +(1 + 2tu)2 − 4t2 , +(100) +ρ(+) +− ρ(−) +− += +√ +1 + 2t − 2tu +√ +1 − 2t − 2tu = +� +(1 − 2tu)2 − 4t2 . +(101) +Hence, we may rewrite (96) as +4 +√ +1 − u2 +tu2 +· +(b1b2 − a1a2) +(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) = 16t +� +1 − u2 +· +�� +1 − (2t + 2tu)2 + +� +1 − (2t − 2tu)2 +� �� +(1 + 2tu)2 − 4t2 + +� +(1 − 2tu)2 − 4t2 +� +1 +· +1 +� +ρ(+) ++ +− ρ(+) +− ++ ρ(−) ++ ++ ρ(−) +− +� � +ρ(+) +− +− ρ(+) ++ ++ ρ(−) ++ ++ ρ(−) +− +� +· +1 +� +(1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) +· +1 +� +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) ++ +− ρ(−) +− +� � +ρ(+) ++ ++ ρ(+) +− ++ ρ(−) +− +− ρ(−) ++ +� +(102) +For the remaining factors involving ρ(+) ++ , ρ(+) +− , ρ(−) ++ +and ρ(−) +− , we note that collecting the terms the +denominator is of the form +(a+b+c−d)(a+b−c+d)(a−b+c+d)(−a+b+c+d) , for {a, b, c, d} = +� +ρ(+) ++ , ρ(+) +− , ρ(−) ++ , ρ(−) +− +� +. (103) +That can be seen to equal (a2 + b2 + c2 + d2)2 − 2(a4 + b4 + c4 + d4) + 8abcd. But we may see that +� +ρ(+) ++ +�2 ++ +� +ρ(+) +− +�2 ++ +� +ρ(−) ++ +�2 ++ +� +ρ(−) +− +�2 += 4 , +(104) +while +� +ρ(+) ++ +�4 ++ +� +ρ(+) +− +�4 ++ +� +ρ(−) ++ +�4 ++ +� +ρ(−) +− +�4 += 4 + 4t2 + 4t2u2 . +(105) +So, we obtain that the product of the remaining unsimplified denominators (meaning still expressed in +terms of ρ(+) ++ , ρ(+) +− , ρ(−) ++ +and ρ(−) +− ) as +16 − 2 +� +4 + 4t2 + 4t2u2� ++ 8 +� +(1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) . +(106) +We may also notice that +(1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) = 1 − 8(1 + u2)t2 + 16(1 − u2)2t4 . +(107) +So the expression in (106) may be rewritten as +8 − 8(1 + u2)t2 + 8 +� +1 − 8(1 + u2)t2 + 16(1 − u2)2t4 . +(108) +A +Pochhamer identities for −1/2 and negative integers +Here we state what needs to be used to derive the equivalence of (17) and (18). Note that +(−1/2)n = +� +−1 +2 +� � +−3 +2 +� +· · · +� +−1 + 2(n − 1) +2 +� +, +(109) +16 + +which equals +(−1/2)n = (−1)n +2n +(1) (3) · · · (2n − 1) , +(110) +which in turn may be rewritten as +(−1/2)n = (−1)n +2n +· +(2n)! +(2)(4) · · · (2n) = (−1)n +22n +· (2n)! +n! +. +(111) +This well-known fact is easily found, for example on Wikipedia it can be deduced from the formula +Γ(1/2) = √π and Γ( 1 +2 − n) = (−1)n22nn!√π/(2n)!. +We also know that for the Pochhammer symbol at a negative integer we have +(−n)k = (−n)(−n − 1) · · · (−n − k + 1) = (−1)k(n)(n + 1)(n + k − 1) = (−1)k (n + k − 1)! +(n − 1)! +. (112) +Hence, in particular replacing n by 2n + 1 we have +(−2n − 1)k = (−1)k (2n + k)! +(2n)! +. +(113) +Therefore, putting these two together, we see that the summands in (17) are simplified by using the +formula +(−1/2)n +n! +· (−2n − 1)k +k! += (−1)n +22n +· (2n)! +(n!)2 · (2n + k)! +(2n)! k! += (−1)n+k +22n +�2n + k +n, n, k +� +. +(114) +Hence, the left-hand-side of equals (17) +∞ +� +n=0 +(−1/2)n +n! +· (−2n − 1)k +k! +�k +j +� +(−1)n+k4n = +∞ +� +n=0 +(−1)n+k +22n +�2n + k +n, n, k +� �k +j +� +(−1)n+k4n , +(115) +which may be further simplified using +�2n + k +n, n, k +� �k +j +� += +� +2n + k +n, n, j, k − j +� +. +(116) +So, the left-hand-side of equals (17) may be rewritten as +∞ +� +n=0 +(−1/2)n +n! +· (−2n − 1)k +k! +�k +j +� +(−1)n+k4n = +∞ +� +n=0 +� +2n + k +n, n, j, k − j +� +. +(117) +In (17) we wanted to verify that, when we replace j ← ℓ − n and k − j ← m − n (using replacement +notation as in APL) then we get +�ℓ+m +ℓ,m +��ℓ+m +ℓ,m +� +. So this equation may be rewritten as +∞ +� +n=0 +� +2n + ℓ + m +n, n, ℓ − n, m − n +� += +�ℓ + m +ℓ, m +��ℓ + m +ℓ, m +� +. +(118) +We then take account of the fact that in the multinomial coefficient notation, the multinomial coefficient +equals 0 unless all the terms in the bottom are nonnegative integers. So we require ℓ − n ≥ 0 and +m − n ≥ 0. Then the infinite sum for n ∈ {0, 1, . . .} may be restricted to n ∈ {0, . . ., min(ℓ, m)}. That +is how we reduce equation (17) to (18) or its even more direct version +min(m,n) +� +n=0 +� +2n + ℓ + m +n, n, ℓ − n, m − n +� += +�ℓ + m +ℓ, m +��ℓ + m +ℓ, m +� +. +(119) +17 + +References +[1] Lars V. Ahlfors. Complex Analysis. McGraw-Hill Inc., New York, USA, 1979. +[2] David Aldous and Persi Diaconis. Longest increasing subsequences: from patience sorting to the +Baik-Deift-Johansson theorem. Bull Amer. Math. Soc. 36, no. 4, 413–432 (1999). +[3] Jinho Baik, Percy Deift and Kurt Johansson. On the distribution of the length of the longest +increasing subsequence of random permutations. J. Amer. Math. Soc. 12, no. 4, 1119–1178 (1999). +[4] Charalambos A. Charalambides. Enumerative Combinatorics. CRC Press, Taylor & Francis group, +Boca Raton, FL, 2002. +[5] Duncan Dauvergne, Janosch Ortmann, and B´alint Vir´ag. +The directed landscape. +Preprint, +https://arxiv.org/abs/1812.00309 +[6] Gerald B. Folland. Real Analysis: Modern Techniques and Their Applications. 2nd Ed. John +Wiley & Sons, Inc., New York, USA, 1999. +[7] Francesco Guerra. Broken replica symmetry bounds in the mean field spin glass model. Com- +mun. Math. Phys. 233 1–12 (2003). +[8] Francesco Guerra and Fabio-Lucio Toninelli. Quadratic replica coupling in the SherringtonKirk- +patrick mean field spin glass model. J. Math. Phys. 43, 3704–3716 (2002). +[9] B. F. Logan and L. A. Shepp. A variational problem for random Young tableaux. Advances in +Math. 26 206–222 (1977). +[10] Marc Mezard, Giorgio Parisi and Miguel-Angel Virasoro. Spin Glass Theory and Beyond: An +Introduction to the Replica Method and Its Applications. World Scientific Publishing Company, +Singapore, 1987. +[11] Robin Pemantle and Mark C. Wilson. Analytic Combinatorics in Several Variables. Cambridge +University Press, New York, USA, 2013. +[12] Ross G. Pinsky. +Law of large numbers for increasing subsequences of random permutations. +Random Structures Algorithms 29, no. 3, 277–295 (2006). +[13] Ross G. Pinsky. +When the law of large numbers fails for increasing subsequences of random +permutations. Ann. Probab. 35, no. 2, 758–772 (2007). +[14] J. Michael Steele. Probability theory and combinatorial optimization. Society for Industrial and +Applied Mathematics, Philadelphia, PA, 1997. +[15] Michel Talagrand. The Parisi formula. Ann. Math. 163, 221–263 (2006). +[16] A. M. Vershik and S. V. Kerov. Asymptotics of the Plancherel measure of the symmetric group +and the limiting form of Young tables. Soviet Math. Dokl., 18, 527–531 (1977). +[17] J. L. van Hemmen and R. G. Palmer. J. Phys. A: Math. Gen. 12 563–580 (1979). +18 + diff --git a/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/load_file.txt b/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7342902c9f095eb4facb04bdd59aee3eca15c215 --- /dev/null +++ b/ONAyT4oBgHgl3EQfUfdD/content/tmp_files/load_file.txt @@ -0,0 +1,680 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf,len=679 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='00125v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='CO] 31 Dec 2022 Generating Function for Pinsky’s Combinatorial Second Moment Formula for the Generalized Ulam Problem Samen Hossein The Bronx High School of Science 75 Bronx Science Bvd Bronx,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' NY 10468 Shannon Starr∗ Department of Mathematics University of Alabama at Birmingham University Hall,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Room 4005 1402 Tenth Avenue South Birmingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' AL 35294-1241 December 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 2022 Abstract In two papers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Ross Pinsky considered the generalized Ulam’s problem for the distribution of the number of increasing subsequences of length k in a random permutation of {1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The probability that there is at least one increasing subsequence of length k equals 1 − F(k), where F is the cdf of the length of the longest increasing subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Hence we call it the “generalized Ulam problem.” Pinsky found various interesting results, but we consider just his combinatorial formula for the second moment, which is the first step towards calculating all higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Pinsky obtained rigorous general bounds sufficient to determine applicability of the Paley-Zygmund second moment method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Spin glass theorists would still like the precise asymptotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We formulate the generating function for his A(N, j) sequence using the complex analysis approach as in the monograph of Pemantle and Wilson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1 Introduction Ulam’s problem is one of the problems that maximizes a combination of tools and techniques applicable to the problem and yet also difficulty of the results known for the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us resist the temptation to give details for the previous sentence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The depth of the results related to Ulam’s problem is well- known, and any review would be better done by others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (See, for example, the old review article of Aldous and Diaconis [2] although many new and important results have been discovered since then, including relatively recently as in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') But we will state the actual problem known as Ulam’s problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let π be a random permutation in Sn, chosen according to the uniform measure, so all permutations have equal probability of 1/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='. Given a cardinality k subset of {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , n}, we may write the set as {i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ik} for 1 ≤ i1 < · · · < ik ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then we say that (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ik) is an increasing subsequence for π if πi1 < · · · < πik.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' If we fix (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ik) and vary over π, then the probability to have (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ik) to be an increasing subsequence for a uniform, random π is equal to 1/k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='. This is because if we are given k numbers 1 ≤ j1 < · · · < jk ≤ n and we are told that {πi1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , πik} is equal to {j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , jk} there are still k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' possible permutations σ = (σ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , σk) such that πir = jσr for all r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=', k} but we want πir = jr for all r which is just one of the k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' possible permutations for σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Various authors of textbooks, such as J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Michael Steele for [14] have used this to note that the expected number of cardinality k subsequences which are increasing for π is equal to �n k � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='. Ulam’s ∗Partially funded by Simons Collaboration Grant problem is to let L(π) be the length of the longest increasing subsequence for π: so the largest k such that there is at least one subsequence (i1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ik) which is increasing for π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then in Ulam’s problem, one wishes to know the distribution of L(π), the marginal on L, under the uniform distribution for π ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Steele would note that the probability that L is at least k is no larger than the expected number of cardinality k subsequences which is �n k � 1 k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='. So the median for L cannot be larger than the smallest k making this quantity less than 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' It has a cutoff at approximately e√n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So that places an upper bound on the average length of the longest increasing subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The actual mean is approximately 2√n by a result of Vershik and Kerov [16], and a partial result of Logan and Shepp [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Moreover, there is a complete characterization of the fluctuations for large n, according to the Baik-Deift-Johannson theorem [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Suppose that instead one asks for the distribution of Zn,k(π) where Zn,k is the number of cardinality k subsequences of which are increasing for π ∈ Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then the probability that Zn,k > 0 is exactly equal to the probability that L ≥ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The expectation of Zn,k gives bounds on the probability of {Zn,k ≥ 1} according to Markov’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But more precise bounds could in principle be obtained if we knew higher moments of Zn,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In principle, the Bonferroni inequalities could even lead to precise bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' See for example Charalambides [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (We have an unpublished article reviewing for example Charalambides’s results on this topic, which we may incorporate as an appendix to a future version of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') So it is interesting to know the distribution of Zn,k, beyond just its mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In [12], Pinsky investigated the second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' He obtained an exact combinatorial formula, albeit a difficult one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' He then rigorously analyzed the formula obtaining upper and lower bounds of matching growth rates (at the most important, leading term, the exponential term) although with different constant multipliers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Using this, he could determine exactly the criteria for the Paley-Zygmund second moment method to guarantee that Zn,k satisfies a weak law of large numbers type result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that just because the Paley-Zygmund method does not imply a weak law, that does not mean that a weak law may not still be true (since it does not necessarily require vanishing of the variance of the rescaled quantity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So in [13], Pinsky proved a threshold to be able to disprove a weak law for Zn,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' He also has stated interesting equivalent formulations of his questions, and he has interesting conjectures which remain as open problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The pair of papers is fascinating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In theoretical physics and in the mathematical physics of spin glasses, even though the second moment of Zn,k may be too large to prove a weak law, one is still interested in the precise form of the second moment, and indeed of all higher moments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In principle, spin glass physicists have methods which sometimes allow one to go from formulas for the moments, asymptotically, to information about the actual distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This is in cases where the usual mathematical tools do not apply, such as where the moment problem is indeterminate or in certain types of limits in asymptotics where finite n methods may not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' An excellent reference for the limitations of classical mathematical tools when trying to apply them to spin glass techniques was written by van Hemmen and Palmer [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' However, the spin glass techniques have proved effective in illuminating the Sherrington-Kirkpatrick mean field spin glass problem, and may be useful beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' See for example, the classical physics textbook [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The first rigorous proof of Parisi’s ansatz for the SK model, given by Talagrand, seemed to proceed by very different techniques [15], with a starting point the beautiful approach of Guerra [7] and Guerra and Toninelli [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So we are interested in the more general moments of Zn,k, starting with the second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We call this the generalize Ulam problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Technically, the generalized Ulam problem should be the problem of stating as much as possible about the distribution of Zn,k for the doubly-indexed sequence of numbers for n and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But we are most hopeful to be able to calculate the positive integer moments, in some asymptotic sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So far as we know, this has not been done, except by Pinsky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We begin, in the next section, by stating Pinsky’s precise combinatorial formula for the second moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then we calculate a generating function for the quantities that appear in Pinsky’s decomposition, with hopes of returning later and extracting the precise asymptotics from the generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For us, the calculation of the generating function was already difficult.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 2 2 Pinsky’s combinatorial second moment formula In his paper [12], Pinsky calculated E � Z2 n,k � = k � j=0 � n 2k − j � 1 (2k − j)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' A(k − j, j) , (1) where A(N, j) = � ℓ0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=',ℓj∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' } ℓ0+···+ℓj=N � m0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=',mj∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' } m0+···+mj=N j� r=0 ��ℓr + mr ℓr, mr ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (2) See Pinsky’s Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We will not reproduce his proof, but we mention that his argument is elegant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Defining the new quantity B(L, M, j) = � ℓ0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=',ℓj∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' } ℓ0+···+ℓj=L � m0,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=',mj∈{0,1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' } m0+···+mj=M j� r=0 ��ℓr + mr ℓr, mr ��2 , (3) we have A(N, j) = B(N, N, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Defining the generating function β(w, x, y) = ∞ � j=0 ∞ � L=0 ∞ � M=0 wjxLyMB(L, M, j) , (4) for x, y, w ∈ C, we have β(w, x, y) = ∞ � j=0 wj � ∞ � ℓ=0 ∞ � m=0 xℓym ��ℓ + m ℓ, m ��2�j+1 , (5) by the Tonelli theorem when x, y, w are in [0, ∞) since all the coefficients are nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (See for example, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='5 of Folland [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') So the left-hand-side diverges if and only if the right-hand-side diverges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' If neither diverges, the equality is true.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Also, if neither diverges for a choice of x = x1, y = y1, w = w1 for x1, y1, w1 ∈ [0, ∞), then for any complex numbers x, y, w ∈ C such that |x| ≤ x1, |y| ≤ y1 and |z| ≤ z1, the two sides of (5) are convergent and there is equality of the two sides by Fubini’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 Given |x|, |y| ∈ [0, 1/4), defining γ(2)(x, y) = ∞ � ℓ,m=0 xℓym ��ℓ + m ℓ, m ��2 , (6) the series converges and γ(2)(x, y) = �� (1 − (x + y))2 − 4xy �−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (7) Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2 We use the standard branch cut for the natural logarithm along the non-positive real axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So the domain is D1 = {reiθ : r > 0 , −π < θ < π} and for each z ∈ D1, the formula is log(z) = � 1 0 (z − 1)(1 + (z − 1)t)−1 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then, for each z ∈ D1, we use √z = exp((1/2) log(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' See Figure 1 for the schematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that it is elementary that for |x|, |y| ∈ [0, 1/2), defining γ(x, y) = ∞ � ℓ,m=0 �ℓ + m ℓ, m � xℓym , (8) 3 z = reiθ r > 0 and −π < θ < π √z = √r eiθ/2 = exp((1/2) log(z)) Figure 1: Schematic for the standard square-root function on the open domain of C with branch cut along the non-positive real axis: D1 = {reiθ : r > 0 , −π < θ < π}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For z ∈ D1 we use the formula √z = exp((1/2) � 1 0 (z − 1)(1 + t(z − 1))−1 dt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' using the multinomial notation for the binomial coefficient �ℓ+m ℓ,m � = (ℓ+m)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ℓ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' m!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , we have γ(x, y) = ∞ � n=0 (x + y)n = 1 1 − (x + y) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (9) This is a first example in many textbooks: see for example Pemantle and Wilson, Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2 [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then γ(2) may be calculated from γ using the residue method: γ(xξ, yζ)γ �u ξ , v ζ � = ∞ � ℓ,m,k,n=0 �ℓ + m ℓ, m ��k + n k, n � xℓymukvnξℓ−kζm−n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (10) Therefore, integrating over unit circles, we have γ(2)(xu, yv) = � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) γ(xξ, yζ)γ �u ξ , v ζ � dξ 2πiξ · dζ 2πiζ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (11) It is an instance related to the fact that the diagonal of a rational bivariate generating function is an algebraic univariate generating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This is proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4 of Pemantle and Wilson, credited to Furstenberg and to Hautus and Klarner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The residue method may be used to derive formulas when they are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But once known, such formulas may often be proved by easier combinatorial methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1: Let the right-hand-side of equation (7) be denoted by g(2)(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then we may rewrite g(2)(x, y) = 1 1 − (x + y) � 1 − 4xy (1 − (x + y))2 �−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (12) Therefore, by Newton’s version of the binomial theorem, this gives g(2)(x, y) = 1 1 − (x + y) ∞ � n=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−1)n4nxnyn (1 − (x + y))2n (13) = ∞ � n,k=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−1)n+k4nxnyn(x + y)k , (14) 4 where the Pochhammer symbol, falling factorial, is (z)n = z(z − 1)(z − 2) · · · (z − n + 1) = n−1 � k=0 (z − k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (15) Then, by the regular binomial formula, we have g(2)(x, y) = ∞ � n,k=0 k � j=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' �k j � (−1)n+k4nxn+jyn+k−j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (16) Therefore, the lemma will be proved if we check that ∞ � n=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' �k j � (−1)n+k4n ����� j=ℓ−n k−j=m−n = ��ℓ + m ℓ, m ��2 , (17) for each ℓ, m ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But using well-known combinatorial formulas for the Pochhammer symbols of −1/2 and of negative integers, which we review in Appendix A, this is equivalent to checking the multinomial identity: ∞ � n=0 � ℓ + m n, n, ℓ − n, m − n � = ��ℓ + m ℓ, m ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (18) (We use the standard convention that if one of the indices in the subscript of a binomial or multinomial coefficient is outside the range of 0 to the superscript index, then that coefficient equals 0 by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') Equation (18) may be proved as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Draw ℓ + m black dots on the number line from 1 to ℓ + m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Above each dot place a blue or a red dot such that if A is the subset of those r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ℓ + m} with a blue dot above it, then |A| = ℓ and |A∁| = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Below each black dot, place a green or a yellow dot, such that if B is the set of those r ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' , ℓ + m} with a green dot below it, then |B| = m and |B∁| = ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then, if we let n = |A∩B| we have |A∩B∁| = ℓ−n and |A∁ ∩B| = m−n, as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Therefore, |A∁ ∩B∁| is equal to ℓ + m − |A ∩ B| − |A ∩ B∁| − |A∁ ∩ B|, which is n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The number of ways of assigning pairs of colors to each dot – (blue,green), (blue,yellow), (red,green), (red,yellow) – with a prescribed number of each n, ℓ − n, m − n, n is the multinomial coefficient � ℓ+m n,ℓ−n,m−n,n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Summing over all choices of n gives all possibilities for choosing the two sets A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Since they are chosen independently, that is ��ℓ+m ℓ,m ��2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This proves equation (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' □ Because of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 and equation (5), we have β(w, x, y) = γ(2)(x, y) 1 − wγ(2)(x, y) = 1 � (1 − (x + y))2 − 4xy �1/2 − w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (19) Now define another generating function α(w, z) = ∞ � j,N=0 wjzNA(N, j) = ∞ � j,N=0 wjzNB(N, N, j) , (20) which is the generating function for the combinatorial numbers we want, A(N, j) for N, j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then, by applying the diagonal method, we obtain � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) β � w , xξ , y ξ � dξ 2πiξ = α(w, xy) , (21) 5 for sufficiently small x, y and w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us digress briefly to consider the type of equation which appears in the square-root in the denominator in the expression of β from (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For a quadratic equation of the form ξ2 + 1 − kξ = 0 , (22) the quadratic formula for the roots gives ξ± = k ± √ k2 − 4 2 , (23) which in turn can be written as ξ± = 1 4 �√ k − 2 ± √ k + 2 �2 , (24) if we know that k − 2 and k + 2 are in the domain D1 for the square-root function from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The reason this type of quantity appears for us is that we are considering β(w, xξ, y/ξ), and we may also simplify by taking y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This is because in (21) we get a formula for α(w, xy) which we may specialize to a formula for α(w, x2), taking y = x, with no loss of utility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then we see that β(w, xξ, y/ξ) = 1 �� 1 − x(ξ + ξ−1) �2 − 4x2 �1/2 − w = 1 ((x2/ξ2) ((ξ2 + 1) − (x−1 + 2) ξ) ((ξ2 + 1) − (x−1 − 2) ξ))1/2 − w , (25) containing several terms of the form of the left-hand-side of (22) for different choices of k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For the purpose of doing contour deformation, we will need to factorize these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In fact, they arise twice, in two rounds, because we rationalize the denominator in order to more easily calculate the residues and the contribution due to the branch-cut in (21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' That means we must consider the argument of the square-root function in order to handle the branch-cut contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But we must also consider the poles of the denominator, after we have rationalized it, which is a slightly different quartic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='3 We frequently refer to the argument of the square-root meaning that if we have � f(x) this means f(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In complex analysis “argument” may also refer to the imaginary part of the logarithm function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (See for example, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') But we will instead use the word “phase” to mean eiϕ when we decompose a complex number as z = Reiϕ for R ≥ 0 and ϕ ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Because of equation (24) we will assign variable names to some of the terms such as √ k − 2 or √ k + 2 or related algebraic functions that arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The extra variable names have helped us to organize our calculations, and we hope they are not too much of a distraction for an interested reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4 Suppose we have numbers t, u ∈ R, such that 0 ≤ t < 1/4 and 0 ≤ u < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then α � 2t � 1 − u2 , t2u2� = 1 π � c2 c1 � −Q1(r) −Q2(r) dr + 2t √ 1 − u2 �� 1 − (2t + 2tu)2 + � 1 − (2t − 2tu)2 � �� (1 + 2tu)2 − 4t2 + � (1 − 2tu)2 − 4t2 � � 1 − (1 + u2)t2 + � 1 − 8(1 + u2)t2 + 16(1 − u2)2t4 � � 1 − 8(1 + u2)t2 + 16(1 − u2)2t4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (26) where Q1(r) = � (r2 + 1)tu − (1 − 2tu)r �� (r2 + 1)tu − (1 + 2tu)r � , (27) Q2(r) = � (r2 + 1)tu − (1 − 2t)r �� (r2 + 1)tu − (1 + 2t)r � , (28) are two quartic polynomals, and where c1 = 1 + 2tu − √1 + 4tu 2tu and c2 = 1 − 2tu − √1 − 4tu 2tu (29) are the two roots of Q1 in (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Both Q1(r) and Q2(r) are negative on (c1, c2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 6 c1 c2 a1 a2 d1 d2 b1 b2 > Figure 2: A schematic picture – it is not to scale but it does accurately depict the order of points – for the branch cuts and poles of the integral in (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The configuration of points and branch cuts is shown assuming u, t ∈ (0, 1) are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The poles are at a1, a2, b1 and b2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The branch cuts are the intervals [c1, c2] and [d1, d2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In other words, c1, c2, d1, d2 are roots of Q1 and a1, a2, b1, b2 are roots of Q2 for the quartic polynomials Q1 and Q2 from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We will prove the theorem in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For now let us note that this is a potentially useful representation for the sequence A(N, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We note that for x > 0 and w > 0 we can find t and u to match them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' If we did have x = t2u2 and w = 2t √ 1 − u2, then we would have w2 + 4x = � 2t � 1 − u2 �2 + 4t2u2 = 4t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (30) Therefore, we may take t = 1 2 � w2 + 4x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (31) So |t| < 1 as long as 4x + w2 < 1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We can then take u = t−1√x = √ 4x/ √ w2 + 4x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (If x and w are both 0 then t = 0 and the value of u is irrelevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=') So u is in [0, 1] as long as x, w are both nonnegative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In principle, calculating the generating function for all x and w in say [0, 1/32] does determine all the coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that all the coefficients are nonnegative, since they are combinatorial (either 0 or positive integers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So if the series is convergent for 0 ≤ x < 1/32 and 0 ≤ w < 1/32 then the complex series is absolutely convergent for |x| < 1/32 and |w| < 1/32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This is the same, usual fact that lies behind the Fubini-Tonelli theorem (see for example [6]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Perhaps more importantly, if we want to obtain asymptotics for A(N, j), then one often uses techniques related to positivity such as Chebyshev’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Certainly Chebyshev’s inequality gives one-sided bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Those could potentially be turned into asymptotics if the correct Tauberian-type theorem is found and used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The integral appearing in equation (26) is an example of an elliptic integral, although we do not know anything beyond that about a more precise characterization of this integral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 3 Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4: generating function formula If we take x, y, z ∈ C such that 2|x| + 2|y| + |w|2 < 1 , (32) 7 then we have α(w, xy) = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) 1 ξ · � (1 − (xξ + yξ−1))2 − 4xy �1/2 − wξ dξ , (33) where we have used equation (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The argument of the square-root function in the denominator is contained in the open disk C centered at 1 with radius 1, since 2|x| + 2|y| < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So the square-root is unambiguously defined in the sense of Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then the entire denominator may be written as ξ times a quantity which is within the open disk C centered at 1 with radius 1, since 2|x|+2|y|+|w|2 < 1 by equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Next we rationalize the denominator, in order to simplify the calculation of the residues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So equation (33) can be rewritten as α(w, xy) = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · �� 1 − (xξ + yξ−1) �2 − 4xy �1/2 + wξ � (ξ − (xξ2 + y))2 − 4xyξ2 � − w2ξ2 dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (34) Now, we will specialize to simplify the calculations going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Firstly, let us take y = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' α(w, x2) = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · �� 1 − x(ξ + ξ−1) �2 − 4x2�1/2 + wξ � (ξ − x(ξ2 + 1))2 − 4x2ξ2 � − w2ξ2 dξ , (35) Next, let us assume that x and w are real variables for the remainder of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In fact, we change variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us assume that we have real numbers t, u ∈ R which are sufficiently small, which effectively amounts to −1/4 < t < 1/4 and −1 < u < 1, and let us take w = 2t √ 1 − u2 and x = tu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then we have α � 2t � 1 − u2 , t2u2� = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · �� 1 − (ξ + ξ−1)tu �2 − 4t2u2�1/2 + 2tξ √ 1 − u2 � (ξ − (ξ2 + 1)tu)2 − 4t2u2ξ2 � − 4(t2 − t2u2)ξ2 dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (36) Collecting terms in the denominator, this may be simplified to give α � 2t � 1 − u2 , t2u2� = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · �� 1 − (ξ + ξ−1)tu �2 − 4t2u2�1/2 + 2tξ √ 1 − u2 (ξ − (ξ2 + 1)tu)2 − 4t2ξ2 dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (37) Now we factorize the denominator to obtain α � 2t � 1 − u2 , t2u2� = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · �� 1 − (ξ + ξ−1)tu �2 − 4t2u2�1/2 + 2tξ √ 1 − u2 ((1 − 2t)ξ − (ξ2 + 1)tu) ((1 + 2t)ξ − (ξ2 + 1)tu) dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (38) The two factors of the denominator are quadratic polynomials,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' and they may each be further factorized as (multiplying each one by −1 first) � (ξ2 + 1)tu − (1 − 2t)ξ � = tu · (ξ − a2)(ξ − b1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (39) � (ξ2 + 1)tu − (1 + 2t)ξ � = tu · (ξ − a1)(ξ − b2) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (40) by direct appeal to the quadratic formula,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' where a1 = 1 + 2t − � (1 + 2t − 2tu)(1 + 2t + 2tu) 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (41) a2 = 1 − 2t − � (1 − 2t − 2tu)(1 − 2t + 2tu) 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (42) b1 = 1 − 2t + � (1 − 2t − 2tu)(1 − 2t + 2tu) 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (43) b2 = 1 + 2t + � (1 + 2t − 2tu)(1 + 2t + 2tu) 2tu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (44) 8 A power series expansion in the small parameter gives a1 = tu − 2t2u + O(t3) , a2 = tu + 2t2u + O(t3) , (45) b1 = 1 tu − 2 u + O(t) , b2 = 1 tu + 2 u + O(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (46) This means that 0 < a1 < a2 < 1 < b1 < b2, when t ∈ (0, 1) is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For future reference, we note that Q2(ξ) = � (1 − 2t)ξ − (ξ2 + 1)tu � � (1 + 2t)ξ − (ξ2 + 1)tu � = t2u2 · (ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (47) Now let us introduce some extra notation that helps to organize the calculations to come.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us define, for each σ1, σ2 ∈ {1, −1} the quantity ρ(σ1, σ2) = √ 1 + 2σ1t + 2σ2tu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (48) Since we assumed that −1/4 < t < 1/4 and −1 < u < 1 we know that the argument of the square-root function is in the open disk in C, centered at 1 with radius 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So the standard square-root function may be applied (as in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us abbreviate the notation to ρ(+) + = ρ(1, 1) , ρ(+) − = ρ(1, −1) , ρ(−) + = ρ(−1, 1) , ρ(−) − = ρ(−1, −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (49) Then we may write a1 = 1 4tu � ρ(+) + − ρ(+) − �2 , a2 = 1 4tu � ρ(−) + − ρ(−) − �2 , (50) b1 = 1 4tu � ρ(−) + + ρ(−) − �2 , b2 = 1 4tu � ρ(+) + + ρ(+) − �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (51) This is a rewriting of the roots of these particular quadratic equations similar to what we described in equation (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We will return to further calculations involving these quantities, briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' First, let us perform a similar algebraic simplification for the polynomial in the argument of the numerator in equation (38).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This may be factorized as � 1 − (ξ + ξ−1)tu �2 − 4t2u2 = � 1 − 2tu − (ξ + ξ−1)tu � � 1 + 2tu − (ξ + ξ−1)tu � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (52) Applying the quadratic formula, we have � 1 − 2tu − (ξ + ξ−1)tu � = tuξ−1 · (ξ − c2)(ξ − d1) , (53) � 1 + 2tu − (ξ + ξ−1)tu � = tuξ−1 · (ξ − c1)(ξ − d2) , (54) where c1 = 1 + 2tu − √1 + 4tu 2tu , c2 = 1 − 2tu − √1 − 4tu 2tu , (55) d1 = 1 − 2tu + √1 − 4tu 2tu , d2 = 1 + 2tu + √1 + 4tu 2tu .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (56) A power series expansion in the small parameter gives c1 = tu − 2t2u2 + O(t3) , c2 = tu + 2t2u2 + O(t3) , (57) d1 = 1 tu − 2 + O(t) , d2 = 1 tu + 2 + O(t) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (58) This means that 0 < a1 < c1 < c2 < a2 < 1 < b1 < d1 < d2 < b2, when t ∈ (0, 1) and u ∈ (0, 1) are sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In other words, the ordering of the poles and branch cut endpoints are as depicted in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For future reference, we note that Q1(ξ) = � (1 − 2tu)ξ − (ξ2 + 1)tu � � (1 + 2tu)ξ − (ξ2 + 1)tu � = t2u2 · (ξ − c1)(ξ − c2)(ξ − d1)(ξ − d2) , (59) 9 c1 > c2 < a1 Γ1(δ) > a2 Γ2(δ) > Γ3(δ) Figure 3: A schematic of the contours after deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The two positively oriented circular contours – of radius δ for 0 < δ ≪ 1 – are centered at the poles a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' They are Γ1(δ) and Γ2(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The branch cut along the real axis interval from c1 to c2 is enclosed by the contour Γ3(δ) which is the rectangle joining c1 − iδ, c2 − iδ, c2 + iδ and c1 + iδ, traversed in the counter-clockwise orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' although now we will be more concerned with the interpretation of � Q1(ξ) or ξ � Q1(ξ)/ξ2 as analytic functions in prescribed domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Now we deform the unit circle contour C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1) into the unit disk, but avoiding cutting through the poles a1, a2 or the branch cut interval [c1, c2] inside the unit disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' See Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Just before doing that, let us make a precise declaration of how we consider � Q1(ξ) in the domain that we deform the contours through.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 Analytic extension of � Q1(ξ) outside [c1, c2] and [d1, d2] Let us define that domain D2 = C \\ � [c1, c2] ∪ [d1, d2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (60) Note that for ξ ∈ C with |ξ| close to 1, the usual power series expansion of � Q1(ξ)/ξ2, expanding in t, may be written �� 1 − (ξ + ξ−1)tu �2 − 4t2u2�1/2 = ∞ � n=0 (1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' � −2 � ξ + ξ−1� tu + � ξ − ξ−1�2t2u2�n .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (61) It is convergent if (1 + 2 max(|ξ|, |ξ|−1)|tu|)2 − 1 < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We may then say that, as long |ξ| and |ξ|−1 are both contained in a compact set, we have � Q1(ξ)/ξ2 = 1 + O(|t|) , as t → 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (62) Let U(z0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ) denotes the open disk in C centered at z0, with radius ρ, and let U(z0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ) denote the closure, the closed disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then, for each ǫ ∈ (0, 1), if ξ is in U(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1 + ǫ) \\ U(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1 − ǫ) then the power series defined by the right-hand-side of equation (61) is what we mean by � Q1(ξ)/ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We now describe a second version of the same function on D2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Initially, consider another domain D3 equal to C \\ � (−∞, c2] ∪ [d1, ∞) � , which is the domain depicted on the right-hand-side of Figure 5, the complex plane minus any point in any of the branch cuts for √ξ − c1, √ξ − c2, √d1 − ξ and √d2 − ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We note that 1 is in D2 and with the standard definition of the square-root (from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2) we have √ 1 − c1 , √ 1 − c2 , � d1 − 1 , � d2 − 1 > 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (63) Therefore, for some ǫ > 0, and for ξ ∈ U(1, ǫ), we have ξ � Q1(ξ)/ξ2 = t2u2 � ξ − c1 � ξ − c2 � d1 − ξ � d2 − ξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (64) Note that both sides have the property that their square equals Q1(ξ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' A priori the right-hand-side of (64) is defined on all of D3 including the subset of U(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1 + ǫ) \\ U(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1 − ǫ) equal to all points except a neighborhood of −1, say U(−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But in fact, we may see that it is defined on all of C\\ � [c1, c2]∪[d1, d2] � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 10 c1 c2 d1 d2 c1 c2 d2 d1 Figure 4: The branch cuts for the function � Q1(ξ) are shown on the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In the figure on the right we show the a priori overlapping branch-cuts if we take (d1 − ξ)1/2(d2 − ξ)1/2(ξ − c1)1/2(ξ − c2)1/2 as a proposal to extend the “power series formula” of ξ � Q1(ξ)/ξ2 when ξ is in the vicinity of the unit circle C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 1), meaning the power series in the variable t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For the picture on the right, wherever two branch cuts overlap (or for more general problems wherever an even number overlap) the branch cuts may be removed because each branch cut is a square-root branch cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So they are counted modulo 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' If we check the limits from the upper half-plane ξ = x + iy, y > 0, they match the limits from the lower half-plane at points where 2 branch cuts overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Given a number r ∈ R \\ {c1, c2, d1, d2} we may define two limits ζ+(r) and ζ−(r) as ζ±(r) = lim ǫ→0± √r + iǫ − c1 √r + iǫ − cd � d1 − (r + iǫ) � d2 − (r + iǫ−) ��� √r + iǫ − c1 √r + iǫ − cd � d1 − (r + iǫ) � d2 − (r + iǫ−) ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (65) This is the phase which is also the source of the discontinuities if we try to define � Q1(ξ) everywhere in C as a single-valued function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Or in other words, if we analytically continued � Q1(ξ) to a holomorphic (single-valued) function on a Riemann surface projecting down to C, then the phases could be used to denote the sheets of the Riemann surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We may easily calculate ζ±(r) = \uf8f1 \uf8f4 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f4 \uf8f3 1 for r ∈ (c2, d1), ±i for r ∈ (c1, c2), −1 for r ∈ (−∞, c1) ∪ (d2, ∞), ∓i for r ∈ (d1, d2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (66) Let us consider one example case: r ∈ (c1, c2) and ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We may refer to Figure 5 to determine the phase of � Q1(ξ) for ξ = r + iǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In this example case, we have ξ − c1 = � (r − c1)2 + ǫ2eiθ1 and ξ − c2 = � (c2 − r)2 + ǫ2eiθ2 , (67) where θ1 ∈ (0, π/2) and θ1 ∈ (π/2, π).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But as ǫ approaches 0 from the positive side, we have θ1 → 0 while θ2 → π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So we have lim ǫ→0+ ξ − c1 |ξ − c1| = 1 and lim ǫ→0+ ξ − c2 |ξ − c2| = −1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (68) But then taking the square-root, we have eiθ1/2 converges to 1 while eiθ2/2 converges to i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Multiplying these together, we get ζ+(r) = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that d1 − r > 0 and d2 − r > 0 for r throughout [−1, 1], which is why we did not consider those factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Of course if we replace ǫ by −ǫ then the angles are reflected: θ1 ← −θ1 and θ2 ← −θ2 using replacement notation (as in APL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So the phase changes from ζ+(r) = i to its reciprocal 1/i = −i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 11 c1 c2 < < ξ = r + iǫ θ1 θ2 Figure 5: Consideration of the phase of � Q∞(ξ), in the vicinity of [−1, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' If we write the output of the square root as eiϕR for R ≥ 0 and ϕ ∈ R then by “phase” we mean eiϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Near the real axis, except in neighborhoods of {c1, c2, d1, d2}, in the upper half-plane and lower half-plane, the phase eiϕ is well approximated by one of the numbers in {1, i, −1, −i}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The point is that for r ∈ (−∞, c1) and r ∈ (d2, ∞), the limit from the upper half-plane and the limit from the lower half-plane coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' They are both −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Therefore, the analytic function, z ∈ C \\ � (−∞, c2] ∪ [d1, ∞) � �→ t2u2 � ξ − c1 � ξ − c2 � d1 − ξ � d2 − ξ , (69) may be extended to z ∈ C\\ � [c1, c2]∪[d1, d2] � just by taking limits from the upper half-plane and lower half-plane since the two limits agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In particular, note that this formula is negative in [−1, c1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Reconsider � Q1(ξ)/ξ2, defined by the power series in t for small t and for |ξ| near to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' For that quantity, we know � Q1(ξ)/ξ2 = 1 + O(t) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But then that means that ξ � Q1(ξ)/ξ2 = ξ + O(t) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' In particular, near ξ = −1 the phase is approximated by −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' That matches the behavior determined by � Q1(ξ) on D2 in a neighborhood of −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2 Resumption of the proof First, let us quickly rewrite the formulas for the roots, as we did for the denominator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us define, for each σ1, σ2 ∈ {1, −1} the quantity �ρ(σ1, σ2) = √ 1 + 2σ1tu + 2σ2tu , (70) for σ1, σ2 ∈ {1, −1} and let us abbreviate the notation to �ρ(+) + = �ρ(1, 1) , �ρ(+) − = �ρ(1, −1) , �ρ(−) + = �ρ(−1, 1) , �ρ(−) − = �ρ(−1, −1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (71) Of course, two of these quantities �ρ(+) − and �ρ(−) + are each equal to just 1, and the formulas in (55) and (56) are not terribly complex as compared to the equationsfor a1, a2, b1, b2, which did get simpler in (50) and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But using all of these, we may write c1 = 1 4tu � �ρ(+) + − �ρ(+) − �2 , c2 = 1 4tu � �ρ(−) + − �ρ(−) − �2 , (72) d1 = 1 4tu � �ρ(−) + + �ρ(−) − �2 , d2 = 1 4tu � �ρ(+) + + �ρ(+) − �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (73) Now let us finally consider the contour deformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We note that we may rewrite the integral in (38) as α � 2t � 1 − u2 , t2u2� = 1 2πi � C(0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1) ξ · � Q1(ξ)/ξ2�1/2 + 2tξ √ 1 − u2 Q2(ξ) dξ , (74) 12 using the notation for the quartic polynomials of (27) and (28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Referring to the three contours shown in Figure 3, this means α � 2t � 1 − u2 , t2u2� = 1 2πi � j∈{1,2,3} � Γj(δ) ξ · � Q1(ξ)/ξ2�1/2 + 2tξ √ 1 − u2 Q2(ξ) dξ , (75) but where we know the contour integrals along the contours Γ1(δ) and Γ2(δ), for δ sufficiently small, gives the residues at a1 and a2, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We will state what the δ → 0+ limit of the Γ2(δ) contour gives, later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We know that the branch cut is along [c1, c2], and the poles are at a1 and a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But we must calculate the residues at a1 and a2, including the numerator which includes the algebraic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We may also simplify the integral a bit for the portion of the contour that comes from the two sides of the branch cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Let us first consider the residues at ξ = a1 and ξ = a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' When taking the square-root of Q(ξ)/ξ2 for ξ ∈ {a1, a2} ⊂ R we need to check the sign of the square-root output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that we wrote � Q1(ξ)/ξ2 because if |ξ| ∈ (1 − ǫ, 1 + ǫ) for ǫ ∈ [0, 1) fixed, then we have Q1(ξ)/ξ2 = 1 + O(t2) , (76) as t → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Therefore, it is in the domain D1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' The algebraic function ξ � Q1(ξ)/ξ2 is analytic for ξ ∈ C satisfying |ξ| < 1 and ξ ̸∈ [c1, c2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Moreover for ξ ∈ (−1, −1 + ǫ) the function is negative, while for ξ ∈ (1 − ǫ, 1) the function is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But then, by inspection, we see that for r ∈ (−1, 1) \\ [c1, c2], the function r � Q1(r)/r2, specializing the analytic function, changes sign at r = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So it is positive for r ∈ (0, c1) ∪ (c2, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' At r = a1 and r = a2 we have Q2(r) = Q1(r) − 4t2(1 − u2)r2 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (77) Therefore, since 0 < a1 < c1 < c2 < a2 < 1, we do have ∀r ∈ {a1, a2} , we have r · (Q1(r)/r2)1/2 = 2tr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (78) This can also be seen to be consistent with the analysis in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Therefore, for the integrand of (74) at the poles we have the value of the numerator being as follows: ∀r ∈ {a1, a2} , we have r · (Q1(r)/r2)1/2 + 2tr � 1 − u2 = 4tr � 1 − u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (79) Therefore, since Q2(ξ) = t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) , (80) we have the residues for the two contours 1 2πi � Γ1(δ) ξ · � Q1(ξ)/ξ2�1/2 + 2tξ √ 1 − u2 t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ = 4ta1 √ 1 − u2 t2u2(a1 − a2)(a1 − b1)(a1 − b2) , (81) 1 2πi � Γ2(δ) ξ · � Q1(ξ)/ξ2�1/2 + 2tξ √ 1 − u2 t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ = 4ta2 √ 1 − u2 t2u2(a2 − a1)(a2 − b1)(a2 − b2) , (82) if δ > 0 is small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Combining these two residue integrals,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' and performing some simple algebra,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 13 we have (for sufficiently small δ > 0) 1 2πi � j∈{1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2} ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='Γj(δ) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='ξ · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='Q1(ξ)/ξ2�1/2 + 2tξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='t2u2(ξ − a1)(ξ − a2)(ξ − b1)(ξ − b2) dξ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='= 4t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='t2u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='a1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(a2 − a1)(b1 − a1)(b2 − a1) + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(a2 − a1)(b1 − a2)(b2 − a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='= 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='tu2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− a1(b1 − a2)(b2 − a2) + a2(b1 − a1)(b2 − a1) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='= 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='tu2 � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− a1b1b2 + a1a2(b1 + b2) − a1a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='2 + a2b1b2 − a2a1(b1 + b2) + a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1a2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='= 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='tu2 (−a1 + a2)(b1b2 − a1a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(a2 − a1)(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='= 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='tu2 (b1b2 − a1a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (83) Now let us use the formulas from (50) and (51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We have b1 − a1 = 1 4tu �� ρ(−) + + ρ(−) − �2 − � ρ(+) + − ρ(+) − �2� = 1 4tu � ρ(+) + − ρ(+) − + ρ(−) + + ρ(−) − � � ρ(+) − − ρ(+) + + ρ(−) + + ρ(−) − � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (84) b2 − a1 = 1 4tu �� ρ(+) + + ρ(+) − �2 − � ρ(+) + − ρ(+) − �2� = 1 4tu � 2ρ(+) + � � 2ρ(+) − � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (85) b1 − a2 = 1 4tu �� ρ(−) + + ρ(−) − �2 − � ρ(−) + − ρ(−) − �2� = 1 4tu � 2ρ(−) + � � 2ρ(−) − � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (86) b2 − a2 = 1 4tu �� ρ(+) + + ρ(+) − �2 − � ρ(−) + − ρ(−) − �2� = 1 4tu � ρ(+) + + ρ(+) − + ρ(−) + − ρ(−) − � � ρ(+) + + ρ(+) − + ρ(−) − − ρ(−) + � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (87) and finally b1b2 − a1a2 = 1 16t2u2 �� ρ(+) + + ρ(+) − �2 � ρ(−) + + ρ(−) − �2 − � ρ(+) + − ρ(+) − �2 � ρ(−) + − ρ(−) − �2� = 1 16t2u2 �� ρ(+) + + ρ(+) − � � ρ(−) + + ρ(−) − � − � ρ(+) + − ρ(+) − � � ρ(−) + − ρ(−) − �� �� ρ(+) + + ρ(+) − � � ρ(−) + + ρ(−) − � + � ρ(+) + − ρ(+) − � � ρ(−) + − ρ(−) − �� = 1 16t2u2 � 2ρ(+) + ρ(−) − + 2ρ(+) − ρ(−) + � � 2ρ(+) + ρ(−) + + 2ρ(+) − ρ(−) − � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (88) We will simplify this a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But first, let us state the small algebraic simplifications available for the contour integral 1 2πi � Γ3(δ) ξ · � Q1(ξ)/ξ2�1/2 + 2tξ √ 1 − u2 Q2(ξ) dξ , (89) from (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='3 Simplifying the integral around the contour Γ3(δ) We may rewrite the contour integral as 1 2πi � Γ3(δ) tu √ξ − c1 √ξ − c2 √d1 − ξ √d2 − ξ + 2tξ √ 1 − u2 Q2(ξ) dξ , (90) by rewriting ξ · � Q1(ξ)/ξ2�1/2 as in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' We know that for r ∈ (c1, c2) we have � ξ − c1 � ξ − c2 ≈ i � (r − c1)(c2 − r) for ξ = r + iδ, as δ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (91) Again, we refer to the analysis of Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But for r ∈ (c1, c2), we have � ξ − c1 � ξ − c2 ≈ −i � (r − c1)(c2 − r) for ξ = r − iδ, as δ → 0+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (92) Also, by analyticity, we know 1 2πi � Γ3(δ) 2tξ √ 1 − u2 Q2(ξ) dξ = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (93) It is also easily seen that the vertical edge contributions to the contour integral may be bounded by terms which are vanishingly small when δ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So lim δ→0 1 2πi � Γ3(δ) ξ · � Q1(ξ)/ξ2�1/2 Q2(ξ) dξ = −2i 2πi � c2 c1 tu � (r − c1)(c2 − r)(d1 − r)(d2 − r) Q2(r) dr , (94) which we obtained also by using the fact that for the top edge face of the rectangle Γ3, the contribution to the contour integral has the opposite orientation as a typical real integral, while for the bottom edge face it has the positive orientation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But, of course, Q2(r) is also negative on the whole interval [c1, c2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' This is how we obtain a positive integral contribution from the third contour integral around the branch cut: lim δ→0 1 2πi � Γ3(δ) ξ · � Q1(ξ)/ξ2�1/2 Q2(ξ) dξ = 1 2π � c2 c1 � −Q1(r) −Q2(r) dr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (95) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4 Resumption of simplification of residues Based on equations (83) and (84),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (85),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (86),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (87) and (88),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' the sum of the two residues is given by 4 √ 1 − u2 tu2 (b1b2 − a1a2) (b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) = 16t � 1 − u2 � ρ(+) + ρ(−) − + ρ(+) − ρ(−) + � � ρ(+) + ρ(−) + + ρ(+) − ρ(−) − � 1 1 � ρ(+) + − ρ(+) − + ρ(−) + + ρ(−) − � � ρ(+) − − ρ(+) + + ρ(−) + + ρ(−) − � 1 ρ(+) + ρ(+) − ρ(−) + ρ(−) − 1 � ρ(+) + + ρ(+) − + ρ(−) + − ρ(−) − � � ρ(+) + + ρ(+) − + ρ(−) − − ρ(−) + � (96) We note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' since ρ(+) + = √ 1 + 2t + 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(+) − = √ 1 + 2t − 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(−) + = √ 1 − 2t + 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(−) − = √ 1 − 2t − 2tu ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (97) 15 we may more easily express ρ(+) + ρ(−) − = √ 1 + 2t + 2tu √ 1 − 2t − 2tu = � 1 − (2t + 2tu)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (98) ρ(+) − ρ(−) + = √ 1 + 2t − 2tu √ 1 − 2t + 2tu = � 1 − (2t − 2tu)2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (99) ρ(+) + ρ(−) + = √ 1 + 2t + 2tu √ 1 − 2t + 2tu = � (1 + 2tu)2 − 4t2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (100) ρ(+) − ρ(−) − = √ 1 + 2t − 2tu √ 1 − 2t − 2tu = � (1 − 2tu)2 − 4t2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (101) Hence,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' we may rewrite (96) as ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='tu2 (b1b2 − a1a2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(b1 − a1)(b2 − a1)(b1 − a2)(b2 − a2) = 16t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − u2 �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − (2t + 2tu)2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 − (2t − 2tu)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� �� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(1 + 2tu)2 − 4t2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(1 − 2tu)2 − 4t2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='1 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='− ρ(−) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='(102) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='For the remaining factors involving ρ(+) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='+ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(+) − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(−) + and ρ(−) − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' we note that collecting the terms the denominator is of the form (a+b+c−d)(a+b−c+d)(a−b+c+d)(−a+b+c+d) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' for {a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' c,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' d} = � ρ(+) + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(+) − ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(−) + ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' ρ(−) − � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (103) That can be seen to equal (a2 + b2 + c2 + d2)2 − 2(a4 + b4 + c4 + d4) + 8abcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' But we may see that � ρ(+) + �2 + � ρ(+) − �2 + � ρ(−) + �2 + � ρ(−) − �2 = 4 , (104) while � ρ(+) + �4 + � ρ(+) − �4 + � ρ(−) + �4 + � ρ(−) − �4 = 4 + 4t2 + 4t2u2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (105) So, we obtain that the product of the remaining unsimplified denominators (meaning still expressed in terms of ρ(+) + , ρ(+) − , ρ(−) + and ρ(−) − ) as 16 − 2 � 4 + 4t2 + 4t2u2� + 8 � (1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (106) We may also notice that (1 + 2t + 2tu)(1 + 2t − 2tu)(1 − 2t + 2tu)(1 − 2t − 2tu) = 1 − 8(1 + u2)t2 + 16(1 − u2)2t4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (107) So the expression in (106) may be rewritten as 8 − 8(1 + u2)t2 + 8 � 1 − 8(1 + u2)t2 + 16(1 − u2)2t4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (108) A Pochhamer identities for −1/2 and negative integers Here we state what needs to be used to derive the equivalence of (17) and (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Note that (−1/2)n = � −1 2 � � −3 2 � · · � −1 + 2(n − 1) 2 � , (109) 16 which equals (−1/2)n = (−1)n 2n (1) (3) · · · (2n − 1) , (110) which in turn may be rewritten as (−1/2)n = (−1)n 2n (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (2)(4) · · · (2n) = (−1)n 22n (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (111) This well-known fact is easily found, for example on Wikipedia it can be deduced from the formula Γ(1/2) = √π and Γ( 1 2 − n) = (−1)n22nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='√π/(2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='. We also know that for the Pochhammer symbol at a negative integer we have (−n)k = (−n)(−n − 1) · · · (−n − k + 1) = (−1)k(n)(n + 1)(n + k − 1) = (−1)k (n + k − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (112) Hence, in particular replacing n by 2n + 1 we have (−2n − 1)k = (−1)k (2n + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (113) Therefore, putting these two together, we see that the summands in (17) are simplified by using the formula (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' = (−1)n 22n (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' )2 · (2n + k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (2n)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' = (−1)n+k 22n �2n + k n, n, k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (114) Hence, the left-hand-side of equals (17) ∞ � n=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' �k j � (−1)n+k4n = ∞ � n=0 (−1)n+k 22n �2n + k n, n, k � �k j � (−1)n+k4n , (115) which may be further simplified using �2n + k n, n, k � �k j � = � 2n + k n, n, j, k − j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (116) So, the left-hand-side of equals (17) may be rewritten as ∞ � n=0 (−1/2)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (−2n − 1)k k!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' �k j � (−1)n+k4n = ∞ � n=0 � 2n + k n, n, j, k − j � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (117) In (17) we wanted to verify that, when we replace j ← ℓ − n and k − j ← m − n (using replacement notation as in APL) then we get �ℓ+m ℓ,m ��ℓ+m ℓ,m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So this equation may be rewritten as ∞ � n=0 � 2n + ℓ + m n, n, ℓ − n, m − n � = �ℓ + m ℓ, m ��ℓ + m ℓ, m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (118) We then take account of the fact that in the multinomial coefficient notation, the multinomial coefficient equals 0 unless all the terms in the bottom are nonnegative integers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' So we require ℓ − n ≥ 0 and m − n ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Then the infinite sum for n ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content='} may be restricted to n ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=', min(ℓ, m)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' That is how we reduce equation (17) to (18) or its even more direct version min(m,n) � n=0 � 2n + ℓ + m n, n, ℓ − n, m − n � = �ℓ + m ℓ, m ��ℓ + m ℓ, m � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' (119) 17 References [1] Lars V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Ahlfors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Complex Analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' McGraw-Hill Inc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=', New York, USA, 1979.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' [2] David Aldous and Persi Diaconis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Longest increasing subsequences: from patience sorting to the Baik-Deift-Johansson theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Bull Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 36, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 4, 413–432 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' [3] Jinho Baik, Percy Deift and Kurt Johansson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' On the distribution of the length of the longest increasing subsequence of random permutations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 12, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' 4, 1119–1178 (1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' [4] Charalambos A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Charalambides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' Enumerative Combinatorics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ONAyT4oBgHgl3EQfUfdD/content/2301.00125v1.pdf'} +page_content=' CRC Press, Taylor & Francis group, Boca Raton, FL, 2002.' metadata={'source': 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-0,0 +1,2324 @@ +MNRAS 000, 1–14 (2023) +Preprint 6 January 2023 +Compiled using MNRAS LATEX style file v3.0 +Mapping Circumgalactic Medium Observations to Theory +Using Machine Learning +Sarah Appleby1⋆, Romeel Davé1,2,3, Daniele Sorini1,4,5, Christopher Lovell6, Kevin Lo1 +1 SUPA†, Institute for Astronomy, University of Edinburgh, Royal Observatory, Edinburgh EH9 3HJ, UK +2 University of the Western Cape, Bellville, Cape Town 7535, South Africa +3 South African Astronomical Observatories, Observatory, Cape Town 7925, South Africa +4 Département de Physique Théorique, Université de Genève, 24 quai Ernest Ansermet, 1211 Genève 4, Switzerland +5 Institute for Computational Cosmology, Department of Physics, Durham University, South Road, Durham, DH1 3LE, UK +6 Institute of Cosmology and Gravitation, University of Portsmouth, Burnaby Road, Portsmouth, PO1 3FX, UK +Accepted XXX. Received YYY; in original form ZZZ +ABSTRACT +We present a random forest framework for predicting circumgalactic medium (CGM) physical conditions from quasar +absorption line observables, trained on a sample of Voigt profile-fit synthetic absorbers from the Simba cosmolog- +ical simulation. Traditionally, extracting physical conditions from CGM absorber observations involves simplifying +assumptions such as uniform single-phase clouds, but by using a cosmological simulation we bypass such assumptions +to better capture the complex relationship between CGM observables and underlying gas conditions. We train random +forest models on synthetic spectra for H i and selected metal lines around galaxies across a range of star formation +rates, stellar masses, and impact parameters, to predict absorber overdensities, temperatures, and metallicities. The +models reproduce the true values from Simba well, with transverse standard deviations of 0.2−0.3 dex in overdensity, +0.14 − 0.2 dex in temperature, and 0.16 − 0.2 dex in metallicity predicted from metal lines (not H i), across all ions. +Examining the feature importance, the random forest indicates that the overdensity is most informed by the absorber +column density, the temperature is driven by the line width, and the metallicity is most sensitive to the specific star +formation rate. Alternatively examining feature importance by removing one observable at a time, the overdensity +and metallicity appear to be more driven by the impact parameter. We introduce a normalising transform approach in +order to ensure the scatter in the true physical conditions is accurately spanned by the network. The trained models +are available online. +Key words: galaxies: general – galaxies: haloes – galaxies: evolution – quasars: absorption lines +1 INTRODUCTION +Over recent years, there has been much effort to characterise +the CGM via quasar absorption line studies (see reviews by +Putman et al. 2012; Tumlinson et al. 2017; Péroux & Howk +2020). Many of the studies probe the strong transitions that +exist in the rest ultra-violet (UV) regime and which trace cool +or warm gas. Such studies are motivated by the wish to un- +derstand the baryon cycle of gas flows in the CGM: accretion +onto galaxies from the IGM and satellite galaxies; expulsion +of gas via stellar winds and AGN feedback; recycling of pre- +viously ejected material back onto galaxies. +The physical conditions of the CGM are studied by retriev- +ing kinematics, spatial distributions, metallicities, densities, +and temperatures from the absorption features (e.g. Stocke +et al. 2013; Savage et al. 2014; Werk et al. 2014; Lehner +et al. 2014, 2018, 2019; Wotta et al. 2016, 2019; Keeney +⋆ E-mail: sarahappleby20@gmail.com +† Scottish Universities Physics Alliance +et al. 2017; Prochaska et al. 2017; Qu et al. 2022). To extract +physical conditions, absorption systems are commonly fitted +with Voigt profiles to model each absorption component and +obtain column densities, linewidths and redshift-space posi- +tions. By running ionisation models (typically using Cloudy, +Ferland et al. 2017) and varying the input physical param- +eters, a Bayesian search can be performed across parameter +space for the physical conditions of each absorber component +using the ensemble of absorption properties as constraints. In +such models the clouds are often modelled as plane-parallel +slabs of gas with an ionising flux incident on one face, making +the (simplifying) assumption that each cloud is spatially iso- +lated with single-valued properties (e.g. Churchill et al. 2003; +Tripp et al. 2008; Werk et al. 2014; Fumagalli et al. 2016; +Keeney et al. 2017; Prochaska et al. 2017). +The analysis and interpretation of CGM observations poses +many challenges owing to the complex nature of the halo en- +vironment. The shapes of absorption profiles are sensitive to +the underlying phase structure and likely contain contribu- +tions from different phases, for example due to the motion of +© 2023 The Authors +arXiv:2301.02001v1 [astro-ph.GA] 5 Jan 2023 + +2 +S. Appleby et al. +gas within the halo and the clumpy gas structure. Even within +individual absorber systems the metallicity of the absorbing +gas can vary and multiple gas phases may be present (Lehner +et al. 2019; Zahedy et al. 2019; Sankar et al. 2020; Chen et al. +2020; Haislmaier et al. 2021; Sameer et al. 2021). Detailed +analysis of absorption systems can give relative abundances +of different ions that constrain the physical conditions, but +this requires high resolution spectroscopy. Studies that use +this technique have moved away from the assumption of a +single cloud, by modelling the high and low excitation ions +separately (Zahedy et al. 2019, 2021; Haislmaier et al. 2021; +Qu et al. 2022), or by modelling the absorption components +as arising from multiple clouds (Cooper et al. 2021; Sameer +et al. 2021; Nielsen et al. 2022). Interpreting the observational +picture is further complicated due to the sensitivity of den- +sity and metallicity estimates to the shape of the UVB (Op- +penheimer & Schaye 2013; Acharya & Khaire 2022; Gibson +et al. 2022). Furthermore, particular ions are not necessarily +produced by the same structures and processes at different +redshifts due to the evolving UVB (Haardt & Madau 2012; +Faucher-Giguère 2020). +Galaxy formation simulations provide a valuable theoreti- +cal perspective on these problems as they offer complete par- +ticle data and physical properties for the gas that makes up +the CGM, making it possible to directly interpret observa- +tions. A range of UV metal lines have been used to probe the +cool and warm ionised CGM in simulations, testing specific +stellar wind implementations (Ford et al. 2013, 2014, 2016; +Hummels et al. 2013), the NIHAO simulation suite (Gutcke +et al. 2017), EAGLE (Oppenheimer et al. 2016, 2018), Illus- +trisTNG (Nelson et al. 2020; DeFelippis et al. 2021), FIRE-2 +(Li et al. 2021) and Simba (Appleby et al. 2021, 2022). +Such simulations can be useful for examining the impact +of different line analysis methods on the retrieved CGM gas +conditions (e.g. Churchill et al. 2015; Liang et al. 2018). In a +recent analysis of a sample of synthetic absorption lines from +a cosmological simulation, Marra et al. (2021) tested the ac- +curacy of the single cloud ionisation modelling method of re- +trieving physical gas conditions. The authors find that while +there is general agreement between intrinsic conditions and +those derived from ionisation modelling, such methods cap- +ture the average properties of absorbing gas cells, consistent +with observational tests by Sameer et al. (2021) comparing +single-phase and multiphase modelling. Marra et al. (2022) +followed up by testing the assumption of single spatially- +isolated absorbing clouds in the CGM, showing that several +distinct absorbing clouds may be present within a single ab- +sorption component. The distinct clouds may arise from gas +of different phases that happen to be aligned kinematically. +These results demonstrate that the CGM is a complex envi- +ronment, with non-linear relationships between the underly- +ing CGM conditions and the resulting absorption observables. +Machine learning (ML) algorithms have the capacity to +learn complex, non-linear relationships and as such they have +been widely applied to astrophysical problems (see review +by Fluke & Jacobs 2020). In this paper, we explore a novel +approach for cosmological simulations to aid in interpreting +CGM absorption observations using ML models. We present +a framework for Random Forest (RF) mapping between syn- +thetic CGM absorption observables from the Simba simula- +tion (Davé et al. 2019) and the underlying absorber condi- +tions from particle data. Such a mapping has the potential +to be employed as a useful tool in retrieving physical con- +ditions from real, multi-component absorption observations. +This approach eliminates the need for simplifying assump- +tions about the structure and state of the gas, i.e. whether +absorption arise from single or multiple gas phases. Instead +the RF mappings implicitly assume the veracity of the Simba +galaxy formation model and our choice of UVB (Faucher- +Giguère 2020) to produce its predictions. +The Simba simulations accurately reproduce a variety of +observational galaxy properties. At low redshift, these in- +clude the star-forming main sequence, black hole-galaxy co- +evolution, radio galaxy populations, dust properties, cold gas +properties, and the baryonic Tully-Fisher relation (Davé et al. +2019, 2020; Thomas et al. 2019, 2020; Li et al. 2019; Lovell +et al. 2021; Glowacki et al. 2020; Appleby et al. 2020). On +larger mass scales, Simba reproduces X-ray scaling relations +for massive halos (Robson & Davé 2020) and low redshift Lyα +absorption statistics of the IGM (Christiansen et al. 2020). +In previous work we have shown that Simba also broadly +reproduces the observed absorption properties of H i (Sorini +et al. 2020) and selected metal lines in the CGM (Appleby +et al. 2021), and that such absorption arises from physically +reasonable gaseous conditions (Appleby et al. 2022); there- +fore Simba is a reasonable choice of simulation with which to +explore the capabilities of ML methods to learn relationships +in the CGM. +Nonetheless, there is no guarantee Simba yields fully accu- +rate and representative circum-galactic media. Indeed, CGM +zoom simulations suggest that Simba’s resolution may be too +poor to capture finer details of multi-phase gas, particularly +for stronger absorbers (e.g. van de Voort et al. 2019; Suresh +et al. 2019 though see Nelson et al. 2020). This drawback +could be explored via comparing the results of this frame- +work applied to other simulations. We leave this aspect for +future work, and here focus on presenting the general frame- +work and its results when applied to the Simba model. +In this paper we train Random Forest (RF) machine learn- +ing networks on the low-redshift Simba CGM absorber sam- +ple presented in Appleby et al. (2022) to produce predictions +for the underlying gas conditions in the CGM. This paper is +organised as follows. In §2 we present the Simba simulations. +In §3 we describe the galaxy selection, spectrum generation +and fitting processes. In §4 we describe the Random Forest +(RF) model and training process. In §5 we examine the accu- +racy of the RF models. In §6 we examine the feature impor- +tance of the RF models. In §7 we present the RF predictions +in phase space. Finally in §8 we conclude and summarise. +2 SIMULATIONS +Simba (Davé et al. 2019) is a suite of state-of-the-art cosmo- +logical simulations that is the successor to the Mufasa sim- +ulations (Davé et al. 2016), with the major additions being +the inclusion of two-mode black hole growth and three-mode +black hole feedback, along with an on-the-fly dust evolution +model. The main simulation, and the one employed in this +work, contains 10243 gas cells and the same number of dark +matter particles within a (100h−1Mpc)3 volume. This yields +a particle mass resolution of 1.8 × 107M⊙ per gas cell, and +9.6 × 107M⊙ per dark matter particle, with a spatial resolu- +tion of ≈ 1h−1kpc in the densest regions. Since Simba has +MNRAS 000, 1–14 (2023) + +Machine learning and the CGM +3 +been extensively described in many previous works, and since +the primary goal on this work is to present and explore our +machine learning framework that is not crucially dependent +on which simulation it is applied to, for brevity we do not +present all of Simba’s input physics, but rather refer read- +ers to Davé et al. (2019), Thomas et al. (2019) and Li et al. +(2019) for full details. +3 ABSORBER SAMPLE +In this work we use the sample of z = 0 absorbers from our in- +vestigation into the physical conditions of absorbing halo gas +in Appleby et al. (2022). Here we summarise the procedure +for generating the absorber sample. We select a sample of +central galaxies within the fiducial Simba volume that evenly +sample a range of global galaxy properties. The galaxies fall +into three categories based on their star formation rates: star +forming, green valley, and quenched. We define star forming +galaxies as with log(sSFR/Gyr−1) > −1.8 + 0.3z for consis- +tency with previous work with the Simba simulation (e.g. +Thomas et al. 2019), define green valley galaxies as within +1 dex below the star forming galaxy threshold, and define +quenched galaxies as those having zero star formation. We +further define six M⋆ bins of width 0.25 dex, with a mini- +mum of M⋆ > 1010 M⊙ to ensure well-resolved systems. In +Appleby et al. (2022) we selected 12 galaxies from each of +the 18 M⋆ − SFR bin. Here we select a further 12 galaxies in +each to double the underlying galaxy sample (except in the +highest mass star forming and green valley bins, which have +only 23 and 8 galaxies respectively) to increase the sample +available for training a machine learning mapping. +For each central galaxy, we generate synthetic line of sight +(LOS) absorption spectra through the simulation volume at +a range of r200-normalised impact parameters (r⊥), probing +both the inner and outer CGM (r⊥/r200 = 0.25, 0.5, 0.75, +1.0, 1.25). In addition, for each r⊥, we select 8 equally-spaced +LOS in a circle around the galaxy. Thus, for each galaxy in +our sample we generate 40 LOS spectra for each of the fol- +lowing ions, selected to probe a range of excitation energies: +H i 1215Å, Mg ii 2796Å, C ii 1334Å, Si iii 1206Å, C iv 1548Å +and O vi 1031Å. This results in a total sample of 17280 lines +of sight. +The spectra are generated along the z-axis of the simula- +tion using the Pygad analysis package (Röttgers et al. 2020); +the procedure is as follows. Gas elements whose smoothing +lengths intersect with the LOS are identified and their ioni- +sation fractions obtained, using look up tables that are gen- +erated with version 17.01 of the Cloudy cloud simulation +code (Ferland et al. 2017) using Cloudy Cooling Tools1. We +assume a spatially uniform Faucher-Giguère (2020) photoion- +ising UV background spectrum, since it was shown in Chris- +tiansen et al. (2020) to provide the best match to low-redshift +Lyα absorption. Self-shielding for H i is applied during the +simulation run, but for generating the metal lines we employ +the Rahmati et al. (2013) prescription to attenuate the ion- +ising background strength based on the local density. +Ion densities for each gas element are obtained by multiply- +ing the gas densities by each species’ ionisation fractions. The +1 https://github.com/brittonsmith/cloudy_cooling_tools +mass fractions of each element are individually tracked within +Simba, based on yields from Type II and Ia supernovae and +stellar evolution. Metals are carried out into the CGM pri- +marily by stellar feedback processes, since winds are mass +and metal-loaded (Appleby et al. 2021). The ion densities +are smoothed along the LOS into pixels of width 2.5km s−1, +using the same spline kernel used in the Gizmo simulation +code and the gas elements’ individual smoothing lengths and +metal masses (for metal lines). Optical depths are then com- +puted from the column densities at a pixel scale, using the +oscillator strength for each species. We exclude wind parti- +cles since those gas elements are hydrodynamically decoupled +from the surrounding gas, which represent a very small frac- +tion of the CGM mass (Appleby et al. 2021). Pygad also +computes column density-weighted physical density, temper- +ature, metallicity, and peculiar velocity in the same manner +within the LOS pixels. +We identify regions of absorption within a ±600 km s−1 +window centered on the galaxy by computing the de- +tection significance ratio of each pixel, defined as the +Gaussian-smoothed flux equivalent width (EW) divided by +the Gaussian-smoothed noise EW. Regions are identified as +contiguous intervals where the flux drops below the level of +the continuum with an overall significance ratio of > 4σ, en- +suring that the edges of the regions begin at the continuum +and merging nearby regions within 2 pixels of one another. +We fit a superposition of Voigt profiles to each absorption +region in order to extract the absorption line observables: the +column density N, the Doppler b parameter, the wavelength +(or velocity) location along the LOS, and the EW. For the fit- +ting, absorption lines are added to the model fit one at a time, +with initial guesses for the line parameters that depend on +whether or not the absorption is saturated. For non-saturated +absorption, the line is placed at the position of lowest flux, +and the initial N and b is based on the depth and velocity +width of the local flux minimum. For saturated absorption, +the line is placed in the middle of the saturated trough, and +the initial N and b are chosen from a coarse grid in order +to minimise the χ2 +r. This procedure broadly follows that in +AutoVP (Davé et al. 1997). The best-fit Voigt parameters +that minimise χ2 +r are then found using the scipy.optimize +subpackage2. Loose prior bounds on N and b are set based +on typical H i and metal line column densities and thermal +line widths from 104 − 107K. +If the fit has χ2 +r < 2.5 then the model is accepted; oth- +erwise we identify the next strongest area of absorption by +subtracting the model from the data and place a line at the +residual minimum. We repeat the process until an accept- +able model is found, up to a maximum of 10 absorption lines +per region. Each line must improve the χ2 +r of the model by +at least 5%; the process is halted if 2 consecutive additional +lines do not improve the χ2 +r by at least this margin. If af- +ter 10 lines an acceptable model is not found then we adopt +the model with the number of lines that performed best. We +again check that each line improves the χ2 +r of the model by +iteratively recomputing the χ2 +r with each line removed; if the +χ2 +r acceptance threshold is reached, or the χ2 +r increases by less +than 5% then the line is removed from the solution. In this +2 https://docs.scipy.org/doc/scipy/reference/optimize. +html +MNRAS 000, 1–14 (2023) + +4 +S. Appleby et al. +Species +n +log(Nmin/cm−2) +χ2,90 +r +Median χ2 +r +E(eV) +H i +17750 +12.7 +3.5 +0.7 +13.60 +Mg ii +5306 +11.5 +39.8 +1.0 +15.04 +C ii +11062 +12.8 +15.8 +1.3 +24.38 +Si iii +14119 +11.7 +35.5 +1.9 +33.49 +C iv +17463 +12.8 +6.3 +1.2 +64.49 +O vi +17463 +13.2 +4.0 +1.2 +138.12 +Table 1. Absorber sample properties for the RF models: the number of absorbers below the χ2 +r limit for each species; the column density +completeness limit; the χ2 +r below which we recover 90% of the total EW; the median χ2 +r of all absorbers; the excitation energy of the +species. +way we attempt to obtain a satisfactory fit with the fewest +number of absorption lines. +The galaxy selection, spectrum generation and LOS fit- +ting pipeline results in our sample of absorbers. We find that +adopting the same strict χ2 +r limit for all ions results in an +incomplete sample. As such we compute the EW directly for +each LOS and adopt an upper χ2 +r threshold for each ion such +that we recover 90% of the total EW across all LOS for each +species. The sample size and χ2 +r upper limits (χ2 +r,90) for each +ion are given in Table 1. In practice the typical χ2 +r for a given +region is much lower than these upper limits; the median χ2 +r +of absorption lines in our sample is also shown in Table 1. +We also adopt the column density completeness limits from +Appleby et al. (2022), which are computed by fitting the +power law portion of the column density distribution func- +tion (CDDF) for each ion and identifying where the CDDF +falls below 50% of the expectation at low column densities. +The completeness limits are given in Table 1. We note that +routines to do the spectrum generation, absorption region +identification, and Voigt profile fitting are all contained with +the publicly-available Pygad package (Röttgers et al. 2020). +4 RANDOM FOREST METHODS +4.1 Random Forest Regression +Random Forest (RF) regression (Breiman 2001) is a super- +vised, decision tree-based, ensemble method of machine learn- +ing. The term ‘ensemble method’ refers to the process of com- +bining predictions from several machine learning runs (in this +case, individual decision trees) in order to more accurately +predict the output. Decision trees work in a top-down man- +ner, in which the best split for the data is found by minimising +a cost function. They have the advantage of being easy to in- +terpret and have low bias in their predictions for the training +data. However, individual decision trees are prone to over- +fitting to the training data, hence their predictions for new +data have high variance. +The RF algorithm counteracts this effect by construct- +ing many decision trees, each trained on a subset of the +data. Random forests may be used for both classification +and regression problems; in this work we use RF in its re- +gression mode to deal with our continuous target predic- +tors. The training data subsets are randomly chosen with +replacement, and their outputs averaged for an overall pre- +diction in a process known as bootstrap aggregation (‘bag- +ging’, see Breiman 1996). In this way, RF models retain the +low bias of a decision tree, while also minimising the vari- +ance on predictions for new data. Training a single decision +tree is considerably faster, however such models are less re- +liable, particularly when trained on non-linear data (such +as the absorber data used here). In this work, we use the +Scikit-Learn (Pedregosa et al. 2011) module’s RF imple- +mentation, RandomForestRegressor. +RF models are widely used in a range of astronomical ap- +plications, and have been remarkably successful given the rel- +ative simplicity of the approach. The advantage of RF mod- +els over other methods (for example Neural Network based +algorithms) is in the interpretability of the output models, +as they indicate the relative importance of the input vari- +ables in reaching a prediction. In galaxy formation, RFs (and +related tree-based methods) have been widely used for re- +gression problems using both simulation and observational +data, for example in predicting the properties of large scale +structure (Lucie-Smith et al. 2018; Lovell et al. 2022; Li et al. +2022) and the properties of galaxies and haloes (Ucci et al. +2017; Nadler et al. 2018; Rafieferantsoa et al. 2018; Cohn & +Battaglia 2020; Moews et al. 2021; Mucesh et al. 2021; Del- +gado et al. 2022; McGibbon & Khochfar 2022). +4.2 Input features and target predictors +For each of the ions in our selection, we train a RF model +on the dataset of simulated CGM absorbers to predict their +underlying physical gas conditions. We do this separately for +each of the 6 ions we consider, such that the usefulness of this +pipeline is not contingent on having line information simulta- +neously for all 6 ions. We exclude absorbers where the quality +of the Voigt profile fit is low (i.e. the fit has a χ2 +r above the +acceptable threshold for that ion) and the column density is +below the completeness limit. +For each ion, we use the same set of input features and tar- +get predictors. The input features are chosen from among the +properties of the CGM absorbers and their central galaxies. +Included features which describe the absorbers themselves +are: the column density (N), the equivalent width (EW), the +linewidth (b), the velocity separation from the host galaxy +(dv), and the impact parameter, expressed as a fraction of +halo virial radius (fr200 = r⊥/r200). Properties of the central +galaxy that are also included as input features are the stellar +mass (M⋆), the specific star formation rate (sSFR), and the +fraction of kinetic energy contained in rotation (κrot, Sales +et al. 2012), which Kraljic et al. (2020) found is a reasonable +proxy for visual morphology. +From these 8 input features we predict 3 target gas pre- +dictors: the overdensity (δ = ρ/¯ρm), temperature (T) and +MNRAS 000, 1–14 (2023) + +Machine learning and the CGM +5 +metallicity (Z). Each of these is a column density-weighted +average at the nearest LOS pixel to the absorber, computed +at the time of spectral generation and binned along the LOS +(see §3). +4.3 Training +Each of the features is transformed into log space; Jo & Kim +(2019) showed that transforming quantities into log space im- +proves the accuracy of machine learning predictions for as- +tronomy problems, owing to the wide range of physical scales +present in astronomical data. Exceptions to this are dv, fr200 +and κrot; dv and κrot have nearly uniform intrinsic distri- +butions, while fr200 consists of 5 specific values due to our +choices of LOS. In addition, we standardize the input and +output data by subtracting the mean of the distribution and +scaling the variance to unity in each case. Where there are +zeros in our dataset, we set them to a small non-zero value. +For each ion, we divide the absorber dataset into 80% train- +ing data used to build the RF model, and 20% test data used +to evaluate the performance of the model. Where multiple +absorbers arise from the same LOS these can be separated +into the training and test datasets; this mitigates over-fitting +in the model due to galaxy or LOS properties. +We train the RF model separately for each target feature, +as we find that this improves the accuracy of the predic- +tion. We separately tune the hyperparameters of each RF +model to optimize the model accuracy, using Scikit-Learn’s +GridSearchCV method to perform an exhaustive grid search +over hyperparameter space. The hyperparameters are the +number of trees, the minimum number of data points required +in order to split the data, and the minimum number of data +points in each resulting split. For each set of hyperparam- +eters, a k-fold cross validation is performed with k = 5, in +which the training data is split into k ‘folds’, and k RF models +are iteratively constructed using k − 1 folds of the data; the +overall score for each set of hyperparameters is the average of +each of the k RF models. The coefficient of determination R2 +is used internally to evaluate the performance of each model, +given n data points and true and predicted quantities Xtrue +and Xpredicted: +R2 = 1 − RSS +TSS +(1) +where RSS is the residual sum of squares: +RSS = +n +� +i=1 +(Xi +true − Xi +predicted)2 +(2) +and TSS is the total sum of squares: +TSS = +n +� +i=1 +(Xi +true − ⟨Xtrue⟩)2 +(3) +The mean squared error, MSE = RSS/n, is the cost function +used to determine the best decision tree splits. An MSE of +zero represents a perfectly accurate prediction. +5 PREDICTIVE ACCURACY +Here we assess the performance of each of the RF models. Fig- +ure 1 shows the test data RF predictions for the H i absorber +physical conditions (δ, T and Z) against the true values. The +color scale of the hexagonal bins indicates the number of data +points in each bin. The black diagonal dashed line represents +the 1:1 case of a perfectly predicting model. In each panel +the 1D histograms for the true and predicted values lie along +the top and right, respectively. The accuracy of the model is +summarised in each panel with three quantities: 1) the scatter +σ⊥, which we define perpendicular to the perfect 1:1 relation; +2) the Pearson correlation coefficient ρr, given by: +ρr = cov(Xtrue, Xpredicted) +σXtrueσXpredicted +, +(4) +where σXi is the standard deviation of Xi; and 3) the MSE. +High correlation is preferred, but does not necessarily indicate +an accurate prediction as the outputs could have a systematic +offset. +Beginning with the predictions for H i absorbers, den- +sity and temperature are well-predicted by the ML model. +True values are highly correlated with the predictions, and +the model predictions have low scatter and error. Density +and temperature are physically correlated with one another +and have Gaussian distributions. Of the two, temperature +(σ⊥ = 0.2 dex, MSE = 0.08) is predicted more accurately +than overdensity (σ⊥ = 0.3 dex, MSE = 0.18). The RF mod- +els for HI density and temperature perform particularly well +considering the models’ relative simplicity (compared with +e.g. a NN-based model). Aside from transforming the features +into log space and using the k-fold hyperparameter cross val- +idation, the model has not been extensively tuned by hand. +As such, these results represent a basic model which demon- +strate the capability of RF models to predict gas conditions, +which could be improved upon with further tuning. We have +also explored alternative ML approaches such as NNs and +CNNs, and found that such models do not offer a substantial +improvement in terms of predictive accuracy and take consid- +erably longer to run. This has lead us to favour the RF model +for its simplicity, speed, and the degree of interpretability in +the form of feature ‘importances’ (see §6). +The predictions for HI metallicities are less accurate (σ⊥ = +0.5 dex, MSE = 0.51). In general, points with logZ/Z⊙ < −1 +are overpredicted, while points with logZ/Z⊙ > −1 are un- +derpredicted. This points to the general tendency of our ML +models to output a narrower predicted distribution than in +the input dataset (this behaviour is also seen to a lesser ex- +tent in the density and temperature predictions). This means +that the tails of the original distributions are not well cap- +tured in the ML model, perhaps as a result of sparse training +data at the extremes. Perhaps the poor prediction is unsur- +prising since HI absorption is not Z-dependent, unlike metal +lines which by necessity arise from metal-enriched gas. There- +fore it was not obvious that any relationship between H i ab- +sorption and metallicity could have been learned from the +data. The learned mapping in the metallicity RF model likely +arises from the provided galaxy properties and H i absorption +strength; we will explore the input feature importance later +(§6). +The metal line absorber physical conditions are also rea- +sonably well predicted. Figures 2 and 3 show the perfor- +mance of the RF models for predicting C ii and C iv ab- +sorber conditions, using the same plot structure as above. +The performance for Mg ii, Si iii and O vi absorbers are +shown in Appendix A. The RF models perform similarly +well among all the metal lines, with the same tendency to +MNRAS 000, 1–14 (2023) + +6 +S. Appleby et al. +Figure 1. Hexagonal joint histogram of the predicted H i physical conditions from the RF mapping and the true H i physical conditions, +including only data in the test set. The number of data points in each bin is shown using colorbars. From left to right, the panels show +overdensity, temperature and metallicity. The diagonal line represents the case where the RF model makes a perfect prediction. The +accuracy of the predictions in each panel is summarised by the inset displaying the transverse scatter σ⊥, the correlation coefficient ρr +and the mean square error MSE. The 1D histograms of the true and predicted values are shown on the top and side of each panel, +respectively. +Figure 2. As in Figure 1, showing the predictions and true values for C ii absorbers. +Figure 3. As in Figure 1, showing the predictions and true values for C iv absorbers. +MNRAS 000, 1–14 (2023) + +0.1 +0.1 +0.1 - +4.0 - +6.0 +102 +1 +102 +3.5 +101 +5.5 - +10'm +n +0. +3.0 +100 + 5.0 +/K)Pred +pald(z/z) +2.5 +4.5 +1.5 +4.0 +1.0 +01 = 0.30 +01 = 0.20 +-3 - +3.5 +01 = 0.50 +0.5 - +pr = 0.84 +p, = 0.88 +MSE = 0.18 +MSE = 0.08 +MSE = 0.51 +0.0 + +2 +3.0+ +0 +1 +3 +0.1 +3.0 +3.5 +4.0 +4.5 +5.0 +5.5 +6.0 +0.1 +-4 +-3 +-2 +-1 +0 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)True0.1 +0.1 +0.1 / +4.00- +5.0 +101 +101 +0.6 +3.75 +101 +n +4.8- +n +n +3.50 +0.4 +100 +100 +3.25 +0.2 +号 3.00 +0.0 +2.75 +2.50 +0.4 +01 = 0.24 +4.2 +01 = 0.15 +01 = 0.19 +2.25- +Pr = 0.67 +P, = 0.75 +Or += 0.68 +MSE = 0.04 +-0.6 +MSE = 0.11 +MSE = 0.07 +2.0 +2.5 +3.0 +3.5 +4.0 +0.1 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +0.1 +-0.5 +0.0 +0.5 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)True0.1 +0.1 +0.1 +3.5 - +5.2 +0.6 +101 +101 +101 +3.0 +n +T +n +5.0 - +0.4 +100 +0.2 +0.0 +E4.6 +2.0 +4.4 +0.4 +1.5 +1 = 0.21 +01 = 0.14 +01 = 0.16 +4.2 - +P, = 0.72 +p, = 0.78 +Or += 0.73 +MSE = 0.09 +MSE = 0.04 +-0.6 +MSE = 0.05 +1.0 + +4.0- +1.0 +1.5 +2.0 +2.5 +3.0 +3.50.1 +4.00 +4.25 +4.50 +4.75 +5.00 +5.25 +0.1 +-0.5 +0.0 +0.5 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)TrueMachine learning and the CGM +7 +predict a more concentrated distribution of values than in +the original data. In general, for each metal line the pre- +dictions are less well correlated with the truth values than +for H i; metal line absorber δ, T and Z have median corre- +lation coefficients of ρr = 0.69, 0.7, 0.68, respectively, com- +pared with ρr = 0.84, 0.88, 0.81 for H i. However, the errors +are in general lower for the metal line RF models, with me- +dian MSE = 0.11, 0.04, 0.06 and median σ⊥ = 0.23, 0.15, 0.17 +for δ, T and Z, compared with MSE = 0.18, 0.08, 0.51 and +σ⊥ = 0.2, 0.3, 0.5 for H i. The metal line RF models also have +lower scatter, although we emphasise that this is partly due to +the reduced range in physical conditions traced by the metal +lines. Overall the RF models give reasonable predictions for +the physical conditions, and again were not extensively tuned +to achieve this. +An interesting feature of the original absorber dataset is bi- +modal metallicity distributions at logZ/Z⊙ ∼ −0.25 and 0.25, +which have not been reported in earlier Simba CGM work. +The bimodality is apparent in every metal line apart from +Mg ii, and is broadly reproduced by the RF models. Popu- +lations of absorbers in the cool CGM of low redshift Lyman +Limit Systems (LLSs) have also been observed to have bi- +modal metallicity distributions, with both metal-poor and +metal-rich absorbers (albeit shifted to lower metallicities, +Lehner et al. 2013, 2018, 2019; Wotta et al. 2016, 2019; Berg +et al. 2022), suggesting multiple origins for the cool CGM +gas, although the metallicities of the observed metal-poor +absorbers are much lower than that seen in Simba. In future +work we will investigate the origin of the bimodal absorber +metallicity distribution in Simba. +6 FEATURE IMPORTANCE +In this section, we seek insights into the physical origin of the +ML-probed correlations by assessing which input features are +most useful in predicting the physical conditions. +6.1 RF model importance +An advantage of the RF method over other ML algorithms +(such as neural networks) is it allows some degree of inter- +pretability in the form of the ‘importance’ of each feature, +which arise ‘for free’ from the structure of the RF model. For +an individual decision tree, a feature’s importance is com- +puted from the number of times it is used to split the data +and how close to the top of the tree the splits are. For an RF +model, the importances are the normalised average over all +decision trees. However, importance metrics are biased if the +input features are highly correlated with one another (Strobl +et al. 2007, 2008) and so they should be treated with caution. +Thus we prefer not to use the importance directly reported +by RF, but instead compute it more empirically. +To do so, we determine each input feature’s importance by +iteratively building the RF model and removing each of the +features in turn, using the same optimized hyperparameters +as in the full feature model. We then retrieve the importance +of the remaining features for each model. This process de- +termines whether a feature is genuinely important, or merely +defined as such through a fluke of feature combinations. When +the most important features are removed, identifying which +features take its place as the most important gives an indi- +cation of what the RF model is learning. +Figure 4 shows the feature importance values for predict- +ing H i absorber conditions, against the feature removed from +the training data. For predicting overdensity, N is most im- +portant feature. When column density is removed, the most +important feature is EW; N and EW are correlated with one +another and both are correlated with physical density. When +predicting temperature, b is the most important feature since +the linewidths of individual absorbers in the original spec- +trum are set in partly by thermal Doppler broadening (with +the additional effect of bulk gas motions). When predicting +metallicity, the velocity separation is the most important fea- +ture. It is not intuitively obvious why this is the case; perhaps +due to a dependence on halo velocity dispersion, which is cor- +related with M⋆ and thus the metallicity of the host galaxy +that is predominantly responsible for enriching its CGM. +Figures 5 and 6 likewise show the feature importance val- +ues for predicting C ii and C iv absorber properties. We have +examined feature importance for all metals and found that +these are representative cases. For the low ion C ii, the feature +importance rankings for δ and T are similar to that of H i. +There is a slightly reduced relative importance of N in pre- +dicting δ in favour of b (N and b are correlated features due +to their underlying dependence on δ and T). For both ions, +the importance of b in predicting T is enhanced compared +with H i. In contrast to H i, the most important feature for +predicting Z for metal lines is sSFR; when sSFR is removed, +the RF model learns from M⋆ and κrot instead, indicating +that the RF model predicts Z from the galaxy properties. +The picture is similar for the high ion C iv, except that in +predicting δ the most important features are instead fr200 +and sSFR. fr200 is perhaps less useful for C ii since most of +the low ion absorption arises from the inner CGM (Appleby +et al. 2022). Galaxies with high star formation have denser +gas in their CGM, although this is also the case for H i and +C ii, so it is unclear why sSFR specifically is an important +feature for C iv absorbers. +6.2 Change in predictive accuracy +Arguably the most meaningful measure of ‘importance’ to the +model is in which features add the most useful information +in terms of predictive accuracy. We assess this by iteratively +removing each of the features in turn from the training data, +and running the RF model as before (with the hyperparame- +ters optimized for the full feature case). In contrast with §6.1, +we now compute the scatter σ⊥ for each new model; if the +quality of the predictions are significantly degraded in the ab- +sence of a particular feature, this necessarily indicates that +this feature encodes crucial information about the physical +conditions. +Figure 7 shows the change in σ⊥ resulting from the removal +of each input feature in predicting the physical conditions, +where a positive change indicates an increase in scatter. Each +of the six panels shows the models for a different ion; the rows +within each panel show the models for each of δ, T and Z. The +upper plots show H i (left), Mg ii (middle) and C ii (right) +absorber models, while the lower plots show Si iii (left), C iv +(middle) and O vi (right). +In contrast to the feature importance values, the largest +increase in scatter when predicting H i absorber overdensity +MNRAS 000, 1–14 (2023) + +8 +S. Appleby et al. +Figure 4. Importance values in predicting H i physical conditions for each remaining input feature, against the input feature removed +from the training data. From left to right, the target predictors are δ, T and Z. +Figure 5. Feature importance values as in Figure 4, for CII absorbers. +Figure 6. Feature importance values as in Figure 4, for CIV absorbers. +comes from removing dv and fr200. The loss of accuracy from +removing fr200 suggests that the RF model is learning the ra- +dial density profile; this is also the case for the metal line δ +predictions. It is less clear why dv is necessary for an accu- +rate prediction, since halo absorbers can appear at any ve- +locity separation depending on their kinematics. For all ions, +removing b causes an increase in scatter in T predictions, +confirming that the high feature importance of b reflects the +genuine physical relationship with temperature. +For H i, the metallicity predictions are degraded by remov- +MNRAS 000, 1–14 (2023) + +log +log T +log Z +-1.0 +logN- +0.43 +0.48 +0.46 +0.45/0.43 +0.42 +10.42 +logN. +0.280.24 +0.21 +0.21 +0.21 +0.2 +0.21 +logN- +0.0670.089|0.11 0.064 0.062 0.059 0.062 +b-0.12 +0.14 +0.12 +0.13 +0.1 +0.1 +0.1 +b -0.56 +0.540.56 0.54 +0.55 +0.54 +0.54 +b - 0.13 +0.14 +0.2 +0.14 +0.14 +0.14 +0.14 + 0.8 +logEW- +0.5 +0.16 +0.13 +10.12 +10.12 +0.12 +0.12 +log EW - 0.220.29 +0.062 b.055 0.058 l0.056 b.055 1og EW - 0.09 l0.075 +0.12 0.062 0.064 0.061 0.062 +importano +0.14 +du - 0.39|0.41|0.39 + 0.6 +d - 0.14 0.15 +0.150.14 |0.14 +0.14 +du-0.070.120.074 +0.0710.0760.0720.074 +0.410.410.4 +0.4 +Feature +fr200 -0.11 +0.097 +0.095 +0.1 +0.0970.096b.096 +f-200 -0.013|0.042|0.014|0.017 +0.0150.013b.014 +fr200 -0.078 0.083|0.078 +0.1 +0.080.079 +0.08 +-0.4 +logM+ -( +0.052b.061 +0.0540.07 0.061 +0.059b.061 +logM+-0 +0.058/ 0.13 0.061 0.069 0.058 +0.061b.066 +logM+ +0.12 +0.11 +0.17|0.11 +0.1 +0.12 +0.13 +sSFR -0.044|0.05 +0.0440.057|0.040.049 +0.054 +sSFR-0.029|0.0620.0290.036b.0290.038 +0.04 +sSFR -0.11 +0.12 +0.11 +0.16 +10.12 +0.13 +0.13 + 0.2 +0.045 b.053 0.045 0.063 0.055 0.054 0.059 +Kirot - +Kirot - +0.092|0.11 +0.09 +0.14 0.097 +0.12 +0.13 +log.N- +EW- +-ap +log.N +-ap +log.N +EW- +0.0 +6 +00-f +log M+- +sSFR +6 +EW- +fr200 +log M+ +sSFR- +Kirot +fr200 +log.M* +sSFR. +log +Removed feature +Removed feature +Removed featurelog +log T +log Z +-1.0 +logN. +0.35 +0.35 +0.35 +0.36 0.330.34 +0.33 +logN. +0.290.150.140.13 +0.14 +0.130.13 +logN- +0.035 0.043 b0.041 0.028 0.031 b.028 0.031 +b -0.25 +0.22 +0.23 +0.24 +10.22 +0.22 +0.22 +b -0.68 +0.670.69 +0.68 +0.69 +0.67 +10.68 +b - 0.06 +0.0610.0810.0740.069 b.058 0.064 + 0.8 +logEW- +0.3 +0.11 +0.037 b.033 0.034 0.033|0.03 +1og EW -0.0410.031 +0.0370.0240.0290.0240.026 +importan + 0.6 +d -0.098|0.13 +0.099 +0.11 +0.1 +0.11 +0.1 +du -0.041/0.13 0.042 +0.0460.0460.045b0.044 +d-0.160.17 +0.16 +0.170.15 0.16 +0.16 +Feature +fr200 +0.1 +0.1 +0.089 0.092 +0.09 0.084|0.09 +f-200 -0.016|0.037/0.019|0.019 +0.0160.019b.019 +0.0620.072 +0.06 +-0.4 +1ogM+-0.057b.066 +6 0.056 0.069 b.063 +0.076b.063 +1ogM-0.051/0.150.0530.055|0.05 +0.056b.055 +logM+- +0.18/0.19 +10.18 +0.220.18 +0.29 +0.22 +sSFR - 0.14 0.17 +0.13 +0.14 +0.096/0.14 +0.14 +sSFR-0.0330.0720.0310.035b.0340.045 +0.041 +sSFR -0.41 +0.42 +0.41 +0.45 +10.43 +0.51 +0.44 + 0.2 +0.0590.07 +0.058 0.066 b.065 0.065/0.12 +0-301y +0.0280.0590.0290.03 +0.0290.037|0.04 +Kirot - +0.0890.0960.085 +0.1 +0.099 +0.15 +0.36 +log.N- +EW- +-ap +log.N +EW- +-ap +log.N +EW- +0.0 +6 +fr200 +log M+ +sSFR +6 +fr200 +log M+ +sSFR +fr200 +log.M* +sSFR. +log +Removed feature +Removed feature +Removed featurelog +log T +log Z +-1.0 +logN. +0.079 +9|0.11 0.0820.17 0.0780.078 b.073 +logN. +0.29 0.078 0.058 b.054 0.053 0.055 b.054 +logN- +0.083 +30.1 +0.097 0.084 0.084 0.076 0.078 +b-0.063 +0.0630.07 +0.130.0720.0750.063 +0.720.730.72 +0.73 +0.720.72 +b -0.14 +0.14 +0.17 +10.15 +0.15 +0.16 +0.15 + 0.8 +0.065b.094|0.070.074 b.0651og EW +0.052 0.048 0.048 0.048 | 0.05 +1og EW -0.098|0.079 +0.073 0.055 0.059 0.056 0.057 +importan +du - 0.05 0.081|0.051 + 0.6 +dv - 0.08 b.087 +0.08 +0.120.09 +0.09b.082 +0.0520.057|0.051 0.052 +d-0.150.18 +0.14 +0.150.15 |0.15 +0.14 +Feature +fr200 +0.3 +0.31 +0.3 +0.31 +0.3 +0.31 +fr200 -0.0190.023 +30.020.021 +0.019|0.020.02 +fr200 -0.049|0.051 +1 0.049 b.052 +0.0470.051 +0.05 +-0.4 +logM+- +0.1210.13 +0.12 +0.130.14 +0.14 0.14 +logM+-0.0470.1 +0.046|0.050.046 +0.051b.052 +logM+ - +0.14/0.16 +0.15 +0.17|0.15 +0.21 +0.17 +sSFR -0.25 +10.26 +0.25 +0.26 +0.23 +0.29 +0.27 +sSFR -0.0390.052 +2/0.04 +0.042 0.0370.046 +0.051 +sSFR - 0.35 +10.36 +0.34 +0.36 +10.35 +0.38 +0.36 + 0.2 +0.0690.075 +50.0710.078 +0.11 +0.091 +0.24 +0-301y +0.0410.0960.0420.0460.043 +0.05g +0.056 +Kirot - +0.0730.0860.0710.0820.074 +0.13 +0.3 +log.N- +EW- +-ap +log.N +EW- +-ap +log.N +EW- +0.0 +6 +00-f +log M+ +sSFR +3oLy +6 +fr200 +log M+ +sSFR. +fr200 +log.M* - +sSFR +log +Removed feature +Removed feature +Removed featureMachine learning and the CGM +9 +Figure 7. The change in σ⊥ of the RF models when removing each feature iteratively. Each of the panels shows results for a different +ion; the three rows of each panel represent results for each of δ, T and Z. +ing dv, fr200 or any galaxy property; interestingly the model +accuracy improves when absorption-related features are re- +moved. In other words, gas metallicity in the CGM of a given +galaxy can be predicted with reasonable accuracy from only +LOS and radial absorber position. The metal lines broadly +show the same changes in scatter with removed features, al- +though the changes to the model accuracy are more marginal. +7 PHASE SPACE +Having examined the predictive accuracy and inner work- +ings of the RF models on individual properties, we now ask +whether the RF models can reproduce the two-dimensional +(δ, T) phase space structure of the absorbers. Although the +RF models can separately predict δ and T, this does not +guarantee that they reproduce the relationship between these +quantities - particularly since the models are trained sepa- +rately for δ and T, so each RF model has no knowledge of +the other target quantities. +Figure 8 shows the predicted temperature against predicted +overdensity for the 6 species we consider. The distributions +for the truth and predicted data are shown along the top and +right hand side of each panel. Note that the plot limits are +different for each ion. The points are colour coded by the +fractional distance from the true value in δ − T phase space: +σphase = +��δtrue − δpredict +δtrue +�2 ++ +�Ttrue − Tpredict +Ttrue +�2 +(5) +The colour scale indicates that in general the predicted points +lie near their truth values in phase space; the metal line ab- +sorbers in particular have a low displacement. The contours +show the true distribution in phase space for the test dataset. +The upper plots show H i (left), Mg ii (middle) and C ii (right) +absorbers, while the lower plots show Si iii (left), C iv (mid- +dle) and O vi (right). The original structure in phase space +between overdensity and temperature is reproduced well by +the RF models for each of the ions we consider, a success of +the ML approach which (since there was no specific tuning +of the model to achieve this) arises because temperature in- +formation is encoded in the overdensity data and vice versa. +This is an important test of the RF models which verifies +that accurate predictions can be made for multiple physical +properties per observation. +That said, although the RF models succeed in predicting +the phase space structure, the predictions are in general too +concentrated near the mean of the data. By comparing the +predicted distribution with the contours from the original +data, it is clear that the predictions ought to be more spread +in phase space. This appears to be a generic feature of the +RF models, which is also apparent in the 1D distributions +of each feature - in general, the RF models produce pre- +dicted distributions that are too concentrated towards the +mean. As mentioned in §5, this likely arises from sparse train- +ing data at the extremes. Thus, the predicted distributions +do not capture the important information described by the +intrinsic scatter in the original data, which biases the use- +fulness of these models for observational analysis. Therefore +some additional step beyond the basic ML model is required +in order to capture the full structure in phase space. Em- +ploying an oversampling technique (such as Synthetic Minor- +ity Over-Sampling Technique for Regression with Gaussian +Noise, SMOGN) can boost under-sampled regions of phase +space and as such mitigate issues that arise as a result of +imbalanced training datasets (de Santi et al. 2022). +In order to extend the RF network to also properly capture +the full 2-D phase space distribution, we develop a new ap- +proach based on a normalising transform. By this, we mean +that the predicted and truth data for each feature are mapped +onto standard normal distributions, and then the predicted +distribution is transformed back onto the shape of the truth +data distribution. +To accomplish this we use the quantile transform non- +MNRAS 000, 1–14 (2023) + +-0.04 +-0.02 +0.00 +0.02 +0.04 +HI 1215 +MgII 2796 +CII 1334 +log d - +log Z. +SilII 1206 +CIV 1548 +OVI 1031 +log d. +log Z - +logN- +EW +ap +sSFR +Krot +log.N- +EW +ap +fr200 +sSFR +log.N- +3oy +6 +EW. +ap +fr200 +log M +sSFR +log +log +log +Removed feature +Removed feature +Removed feature10 +S. Appleby et al. +Figure 8. Predicted temperature against predicted overdensity for each of the 6 ions we consider, coloured by overall phase space fractional +error. The 1D truth (blue curve) and predicted (pink curve) distributions are shown along the top and right of each panel. The contours +show the true distribution in phase space for the test dataset. The limits of the plots differ for each ion. +parametric +method +implemented +in +Scikit-Learn +(Pe- +dregosa et al. 2011), QuantileTransformer. The method first +maps the cumulative distribution of the data onto a stan- +dard Gaussian, and then computes the transformed values +using a quantile function. The function also provides the in- +verse mapping that transforms a distribution back into the +original coordinates. The inverse mapping for the truth data +distribution (computed from the training dataset) is used to +reconstruct the predicted distribution. In this way, we can +reproduce the larger variance in the truth data without as- +suming a shape for the predicted data. +Figure 9 shows the the predicted temperature against pre- +dicted overdensity, using the above normalising transform ap- +proach to map the shape of the truth data. This can be seen +in the 1D distributions along the top and right hand side, +which in most cases closely follow the truth data distribu- +tions. Crucially, the transformed data also retains the phase +space structure of the original predictions. In addition, trans- +forming the predictions onto the shape of the truth data re- +sults in no loss of accuracy for the predictions; the MSE, ρ +and σ⊥ for the transformed test datasets are very similar to +those for the original test dataset predictions. As such, this is +our preferred method for reproducing the scatter in the truth +CGM conditions data. +We initially explored a simpler approach where we added +scatter directly to the predicted data. The results in phase +space for the additional scatter approach is shown in Ap- +pendix B (Figure B1). We found that this approach substan- +tially washed out non-Gaussian structure in the predicted +distribution, such as the anti-correlation between δ and T at +δ > 103. By instead using the normalising transform method, +we are preserving these structures in phase space as much as +possible. +Figure 10 directly compares the distributions for the truth +data (dark purple), the predictions from the RF models (light +purple), the transformed predictions (orange), and the pre- +dictions with additional scatter (yellow) for the H i absorber +overdensities and temperatures. The predicted physical con- +ditions from the RF models are clearly too closely concen- +trated towards the mean. In contrast, applying the normalis- +ing transform approach results in a predicted dataset which +closely matches the truth data. When additional scatter is +instead added to the predictions, the resulting distribution +also more closely matches the truth data, although not so +precisely. Quantitatively, a two-sample Kolmogorov–Smirnov +test with respect to the truth data gives p-values of > 0.95 for +the transformed overdensity and temperature distributions, +but ∼ 0.25 for the distributions with added scatter. +There are pros and cons to including the normalising trans- +form approach to ensure that the phase space scatter is well +reproduced. If one wanted to compute distribution functions +for physical quantities inferred from absorption line data, not +MNRAS 000, 1–14 (2023) + +logo phase +6.0 +5.0 - +5.0 +Truth +4.8 +Prediction +5.5 +4.8 +4.6 +5.0 +log(T/K) +4.6 +4.5 +4.4 +4.0 +4.0 +3.8 +4.2 +3.5 +[MgII 2796 +[CII 1334 +HI 1215 +3.6 +3.0 +4.0 +2.5 +3.0 +2.0 +3.5 +4.0 +2.0 +2.5 +0 +3.0 +3.5 +4.0 +logd +logd +logd +5.0 +6.00 - +5.4 +5.75 +4.8 +5.2 +5.50 +5.0 +5.25 +4.6 +/K) +log(T/ +5.00 +4.6 +4.75 +4.4 +4.50 +4.2 +4.2 +4.25 +[SilII 1206. +CIV 1548 +OVI 1031 +. +4.0. +4.0 +4.00 +2.0 +3.0 +3.5 +2.5 +3.0 +3 +1.5 +2.5 +4.0 +1.0 +1.5 +2.0 +3.5 +2 +logd +logd +logdMachine learning and the CGM +11 +Figure 9. Predicted temperature against predicted overdensity for each of the 6 ions we consider, mapped to the shape of the truth 1D +distributions using a quantile transformer. Points are coloured by overall phase space fractional error. The 1D distributions for the truth +data (blue curve) and transformed predictions (pink curve) are shown along the top and right of each panel. The contours show the true +distribution in phase space for the test dataset. +Figure 10. Histograms showing the direct comparison between truth data (dark purple), the predictions (light purple), the transformed +predictions (orange) and the predictions with added scatter (yellow) for H i overdensities and temperatures. +MNRAS 000, 1–14 (2023) + +1.0 +0.8 +Prediction+Scatter +Transformed +Prediction +0.8 +0.6 +Truth +requeno +0.6 +0.4 +0.4 +0.2 +0.2 +0.0 +0.0 +0 +2 +3 +4 +3 +4 +5 +6 +log d +log (T/K)logO phase +6.0 +5.0 - +5.0 +Truth +4.8 +Transformed Prediction +5.5 +4.8 +4.6 +5.0 +log(T/K) +4.6 +log(T/K) +4.5 +4.0 +4.0 +3.8 +4.2 +3.5 +[HI 1215 +[CII 1334 +MgII 2796 +3.6 +3.0 +4.0. +2.0 +2.5 +3.0 +3.5 +4.0 +2.0 +2.5 +3.0 +3.5 +4.0 +0 +logd +logd +logd +5.0 +6.00 +5.4 +5.75 +4.8 +5.2 +5.50 +5.0 +5.25 +4.6 +/K) +4.8 +log(T/ +5.00 +4.6 +4.75 +4.4 +4.50 +4.2 +4.2 +4.25 +[SilII 1206 +CIV 1548 +OVI 1031 +4.0. +4.0. +4.00 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +2 +3 +logd +logd +logd12 +S. Appleby et al. +including this post-processing step would result in the distri- +bution functions improperly capturing the tails, which may +be important for some applications. However, the additional +step necessarily degrades the σ⊥ and MSE of the predictions, +albeit only marginally. The correlation coefficient does not +change since we are only scaling the predictions. For exam- +ple, in the case of H i overdensity, the σ⊥ and MSE increase +from 0.30 → 0.31 and 0.18 → 0.20 respectively after apply- +ing the normalising transform. For H i temperature, the σ⊥ +increases from 0.20 → 0.21, while the MSE does not increase +(at this level of precision). Whether or not to employ the +above method thus depends on the application. +8 CONCLUSIONS +We have produced machine-learnt mappings between CGM +absorption observables and the underlying gas conditions for +H i and selected metal lines using a Random Forest approach. +RF models are preferred over other ML techniques for their +relative simplicity and interpretability. These mappings rep- +resent a proof of concept for using ML models as part of +an analysis pipeline for observational CGM data, which cru- +cially does not make simplifying assumptions about the phase +or composition of the absorbing gas. We identify a general +tendency of the RF models to output a narrower predicted +distribution than in the input data. We demonstrate two +methods of reproducing the scatter of the input data: first +by adding random Gaussian noise to the predictions, and +second by transforming the predictions to the shape of the +truth data. Our main results are as follows: +• The RF models predict reasonable H i overdensities +(σ⊥ = 0.2 dex, MSE = 0.08) and temperatures (σ⊥ = 0.2 +dex, MSE = 0.08). The predictions of overdensity and tem- +perature are highly correlated with their truth values. Metal- +licity is less well predicted (σ⊥ = 0.5 dex, MSE = 0.51); +metallicity is not directly traced by HI, therefore the learned +relationship likely arises from the correlation with density. +• The RF models also predict reasonable metal absorber +conditions and perform to a similar accuracy among all +metal lines, with median ρr = 0.69, 0.7, 0.68, median σ⊥ = +0.23, 0.15, 0.17 and median MSE = 0.11, 0.04, 0.06 for the +overdensity, temperature and metallicity predictions, respec- +tively. Lower σ⊥ compared with H i predictions are partly due +to the smaller dynamic ranges probed for metals. +• We report a bimodality in the absorber metallicity dis- +tributions for four of the five metal lines (C ii, Si iii, C iv and +O vi), suggesting multiple origins for the CGM gas in the +Simba model. +• In terms of feature importances, the RF models learn +H i absorber overdensity from column density and equiva- +lent width, temperature from the Doppler parameter, and +metallicity from the LOS velocity separation. Low ion fea- +ture importances are similar to H i, except that metallicities +are learned from sSFR and κrot. High ion feature importances +are similar to the low ions, except that the overdensities are +learned from radial distance and galaxy properties. +• In terms of predictive accuracy, the radial distance and +LOS separation provide the most useful information for pre- +dicting H i overdensity; the radial distance is also most useful +for the metal line overdensities. The Doppler parameter is +again the most important feature for predicting temperature +for all lines. The LOS separation provides the most useful +information for predicting H i metallicity; the predictions for +all lines are degraded by removing galaxy properties. +• The predictions for overdensity and temperature repro- +duce the phase space structure seen in the original data for +all six ions, despite being trained for separately in the RF +models. This verifies that accurate predictions can be made +for multiple physical properties per observation. +• By mapping the predicted data distributions onto the +shape of the input distributions using a quantile normalis- +ing transformer, we can reproduce the intrinsic scatter in the +CGM phase space conditions with no loss of predictive accu- +racy or phase space structure. +Although we have considered H i and the metal ions sep- +arately, future work on this topic could explore RF models +using combinations of absorption lines to assess whether pre- +dictions may be improved by using information from multi- +ple ion species. A shortcoming of the ML models presented +here is the that the predictions are too concentrated towards +the mean of the distribution. Further development would be +needed on the pipeline in order to reproduce the scatter in +the original data without losing information in phase space. +The motivation of this project is to develop a useful analy- +sis tool for the astronomical community to aid in interpreting +absorption observations of the CGM, assuming the galaxy +formation model of the Simba simulations and the Faucher- +Giguère (2020) UVB. The next phase of this work is to test +the method by applying the ML mappings to real observa- +tional data and comparing to results derived from ionisation +modelling. As such, the trained models produced for this work +are available online3 and we encourage others to test the RF +models on their own observational data. +A natural extension of this project will be to develop addi- +tional ML mappings using absorber data from other simula- +tions such as EAGLE and IllustrisTNG to assess the impact +of galaxy formation models on the predicted conditions for +the CGM. Training the RF models on data from one sim- +ulation and testing on data from another would provide a +robust test of the impact of galaxy formation model on our re- +sults. In addition, developing mappings using absorbers from +the CAMELS project (Villaescusa-Navarro et al. 2020) would +test the dependence of our results on both astrophysical and +cosmological models. +ACKNOWLEDGEMENTS +We acknowledge helpful discussions with Arif Babul. We +thank Philip Hopkins for making Gizmo public, Robert +Thompson for developing Caesar and Horst Foidl, Thorsten +Naab and Bernhard Roettgers for developing Pygad. SA +is supported by a Science & Technology Facilities Council +(STFC) studentship through the Scottish Data-Intensive Sci- +ence Triangle (ScotDIST). RD acknowledges support from +the Wolfson Research Merit Award program of the U.K. +Royal Society. Throughout this work, DS was supported by +the STFC consolidated grant no. RA5496 and by the Swiss +National Science Foundation (SNSF) Professorship grant no. +202671. CCL acknowledges support from a Dennis Sciama +3 https://github.com/sarahappleby/cgm_ml +MNRAS 000, 1–14 (2023) + +Machine learning and the CGM +13 +fellowship funded by the University of Portsmouth for the +Institute of Cosmology and Gravitation. +Simba was run on the DiRAC@Durham facility man- +aged by the Institute for Computational Cosmology on be- +half of the STFC DiRAC HPC Facility. The equipment was +funded by BEIS (Department for Business, Energy & In- +dustrial Strategy) capital funding via STFC capital grants +ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham +University +and +STFC +operations +grant +ST/R000832/1. +DiRAC is part of the National e-Infrastructure. +For the purpose of open access, the author has applied a +Creative Commons Attribution (CC BY) licence to any Au- +thor Accepted Manuscript version arising from this submis- +sion. +DATA AND SOFTWARE AVAILABILITY +The simulation data and galaxy catalogs underlying this ar- +ticle are publicly available4. 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The metallicity bimodality is still +slightly present for O vi, although at a much lower signifi- +cance than for the lower ions. Si iii has the largest scatter in +the recovered T, and also shows some bias such that high-T +absorbers are under-predicted while low-T ones are overpre- +dicted. This may be because Si iii seems to have absorbers +spanning the widest range in temperatures from among the +metal ions considered. In terms of the RF performance, how- +ever, these ions tell a similar story, which is encouraging since +it means the RF methodology is widely applicable with sim- +ilar efficacy across a range of commonly observed low-z UV +ions. +APPENDIX B: DIRECT GAUSSIAN APPROACH +TO ADDING PHASE SPACE SCATTER +In §7 we described our normalising transform-based approach +to adding scatter to the RF predictions in order to match the +2-D truth distributions in phase space. A more straightfor- +ward approach is to directly add Gaussian scatter to the pre- +dicted δ and T distributions to match the truth without first +applying a normalising transformation. However, the results +were less satisfactory. +Figure B1 shows the results, which can be compared to +Figure 9. It is clear that the simpler approach causes fea- +tures within the true phase space to be more washed out, +and substantially degrades the predictive accuracy. Thus we +prefer the normalising transform approach presented in §7. +This paper has been typeset from a TEX/LATEX file prepared by +the author. +MNRAS 000, 1–14 (2023) + +Machine learning and the CGM +15 +Figure A1. As in Figure 1, showing the predictions and true values for Mg ii absorbers. +Figure A2. As in Figure 1, showing the predictions and true values for Si iii absorbers. +Figure A3. As in Figure 1, showing the predictions and true values for O vi absorbers. +MNRAS 000, 1–14 (2023) + +0.1 +0.1 +0.1 +4.00 +101 +101 +0.6 +101 +3.75 +4.6 - +n +3.50 +0.4 +4.4 +1100 +3.25 +0.2 +号 3.00 +0.0 +2.75 +2.50 +3.8 +0.4 +1 = 0.25 +01 = 0.13 +01 = 0.22 +2.25- +pr = 0.69 +p, = 0.90 +Or += 0.66 +-0.6 +MSE = 0.12 +3.6 +MSE = 0.03 +MSE = 0.09 +2.00+ +2.0 +2.5 +3.0 +3.5 +4.00.1 +3.50 +3.75 +4.00 +4.25 +4.50 +4.75 0.1 +-0.5 +0.0 +0.5 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)True0.1 +0.1 +0.1 +3.75 +5.0 - +101 +0.6 +3.50 +101 +101 +n +4.8- +n +n +0.4 +3.25 +poid(z/ z) +0.2 +0.0 +2.50 +2.25 +0.4 +4.2 +01= 0.22 +01 = 0.16 +01 = 0.17 +-0.6 +Or += 0.68 +2.00 +MSE = 0.09 +MSE = 0.05 +MSE = 0.06 +4.0 + +2.0 +2.5 +3.0 +3.5 +0.1 +4.0 +4.2 +4.4 +4.6 +4.8 +5.0 +0.1 +-0.5 +0.0 +0.5 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)True0.1 - +0.1 +6.0 - +102 +3.0 - +5.8 - +0.6 +101 +101 +101 +n +5.6 - +0.4 +2.5 +100 +10° +100 +5.4 +0.2 +0.0 +5.0 +1.5 +4.8- +0.4 +1.0 +4.6 +01 = 0.23 +01 = 0.15 +01 = 0.16 +pr = 0.71 +Pr = 0.79 +Or += 0.71 +4.4 +MSE = 0.05 +-0.6 +MSE = 0.11 +MSE = 0.05 +0.5 + +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +0.1 +4.5 +5.0 +5.5 +6.0 0.1 +-0.5 +0.0 +0.5 +0.1 +log OTrue +log (T/K)True +log (Z/Zo)True16 +S. Appleby et al. +Figure B1. Predicted temperature against predicted overdensity for each of the 6 ions we consider, with added random Gaussian noise +to reproduce the original 1D distributions. Points are coloured by overall phase space fractional error. The 1D truth and predicted +distributions are shown along the top and right of each panel. The contours show the true distribution in phase space for the test dataset. +The widths of the random noise Gaussians for δ and T are shown in the bottom right of each panel. Compared to the normalising transform +results shown in Figure 9, this approach washes out features in phase space substantially more. +MNRAS 000, 1–14 (2023) + +logOphase +6.0 +5.0 - +5.0 +0 +Truth +4.8 +Prediction+Scatter +5.5 +4.8 +4.6 +5.0 +log(T/K) +4.6 +4.5 +4.0 +4.0 +3.8 +4.2 +3.5 +0.49 +0.32 +0.27 +[MgII 2796 +0.08 +[CII 1334 +HI 1215 +0.25 +0.15 +3.6 +LO +LO +3.0 +4.0 +2.0 +2.5 +3.0 +3.5 +4.0 +2.0 +2.5 +3.0 +3.5 +4.0 +3 +logd +logd +logd +5.0 +6.00 +5.4 +5.75 +4.8 +5.2 +5.50 +5.0 +5.25 +4.6 +4.8 +log(T/ +5.00 +log(1 +4.6 +4.75 +4.4 +4.50 +4.2 +4.2 +0s = 0.35 +0.34 +4.25 +[SilII 1206] +0.18 +CIV 1548 +0.21 +OVI 1031 += 0.13 +Lo +OT += Lo +4.0. +1. +4.0 +4.00 +3 +1.5 +2.0 +2.5 +3.0 +3.5 +4.0 +1.0 +2.0 +2.5 +3.0 +3.5 +2 +1.5 +logd +logd +logd \ No newline at end of file diff --git a/P9A0T4oBgHgl3EQfDP-k/content/tmp_files/load_file.txt b/P9A0T4oBgHgl3EQfDP-k/content/tmp_files/load_file.txt new file mode 100644 index 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' DH1 3LE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' UK 6 Institute of Cosmology and Gravitation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Burnaby Road,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' PO1 3FX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' UK Accepted XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Received YYY;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' in original form ZZZ ABSTRACT We present a random forest framework for predicting circumgalactic medium (CGM) physical conditions from quasar absorption line observables, trained on a sample of Voigt profile-fit synthetic absorbers from the Simba cosmolog- ical simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Traditionally, extracting physical conditions from CGM absorber observations involves simplifying assumptions such as uniform single-phase clouds, but by using a cosmological simulation we bypass such assumptions to better capture the complex relationship between CGM observables and underlying gas conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We train random forest models on synthetic spectra for H i and selected metal lines around galaxies across a range of star formation rates, stellar masses, and impact parameters, to predict absorber overdensities, temperatures, and metallicities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The models reproduce the true values from Simba well, with transverse standard deviations of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2−0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3 dex in overdensity, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='14 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 dex in temperature, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='16 − 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 dex in metallicity predicted from metal lines (not H i), across all ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Examining the feature importance, the random forest indicates that the overdensity is most informed by the absorber column density, the temperature is driven by the line width, and the metallicity is most sensitive to the specific star formation rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Alternatively examining feature importance by removing one observable at a time, the overdensity and metallicity appear to be more driven by the impact parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We introduce a normalising transform approach in order to ensure the scatter in the true physical conditions is accurately spanned by the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The trained models are available online.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Key words: galaxies: general – galaxies: haloes – galaxies: evolution – quasars: absorption lines 1 INTRODUCTION Over recent years, there has been much effort to characterise the CGM via quasar absorption line studies (see reviews by Putman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Tumlinson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Péroux & Howk 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Many of the studies probe the strong transitions that exist in the rest ultra-violet (UV) regime and which trace cool or warm gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Such studies are motivated by the wish to un- derstand the baryon cycle of gas flows in the CGM: accretion onto galaxies from the IGM and satellite galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' expulsion of gas via stellar winds and AGN feedback;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' recycling of pre- viously ejected material back onto galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The physical conditions of the CGM are studied by retriev- ing kinematics, spatial distributions, metallicities, densities, and temperatures from the absorption features (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Stocke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Savage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2014, 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Wotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2016, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Keeney ⋆ E-mail: sarahappleby20@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='com † Scottish Universities Physics Alliance et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Qu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' To extract physical conditions, absorption systems are commonly fitted with Voigt profiles to model each absorption component and obtain column densities, linewidths and redshift-space posi- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' By running ionisation models (typically using Cloudy, Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017) and varying the input physical param- eters, a Bayesian search can be performed across parameter space for the physical conditions of each absorber component using the ensemble of absorption properties as constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In such models the clouds are often modelled as plane-parallel slabs of gas with an ionising flux incident on one face, making the (simplifying) assumption that each cloud is spatially iso- lated with single-valued properties (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Churchill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Tripp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Werk et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Fumagalli et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Keeney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Prochaska et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The analysis and interpretation of CGM observations poses many challenges owing to the complex nature of the halo en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The shapes of absorption profiles are sensitive to the underlying phase structure and likely contain contribu- tions from different phases, for example due to the motion of © 2023 The Authors arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='02001v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='GA] 5 Jan 2023 2 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' gas within the halo and the clumpy gas structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Even within individual absorber systems the metallicity of the absorbing gas can vary and multiple gas phases may be present (Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Zahedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Sankar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Chen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Haislmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Sameer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Detailed analysis of absorption systems can give relative abundances of different ions that constrain the physical conditions, but this requires high resolution spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Studies that use this technique have moved away from the assumption of a single cloud, by modelling the high and low excitation ions separately (Zahedy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Haislmaier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Qu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022), or by modelling the absorption components as arising from multiple clouds (Cooper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Sameer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Interpreting the observational picture is further complicated due to the sensitivity of den- sity and metallicity estimates to the shape of the UVB (Op- penheimer & Schaye 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Acharya & Khaire 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Gibson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Furthermore, particular ions are not necessarily produced by the same structures and processes at different redshifts due to the evolving UVB (Haardt & Madau 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Faucher-Giguère 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Galaxy formation simulations provide a valuable theoreti- cal perspective on these problems as they offer complete par- ticle data and physical properties for the gas that makes up the CGM, making it possible to directly interpret observa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' A range of UV metal lines have been used to probe the cool and warm ionised CGM in simulations, testing specific stellar wind implementations (Ford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2013, 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Hummels et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2013), the NIHAO simulation suite (Gutcke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017), EAGLE (Oppenheimer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2016, 2018), Illus- trisTNG (Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' DeFelippis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021), FIRE-2 (Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021) and Simba (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Such simulations can be useful for examining the impact of different line analysis methods on the retrieved CGM gas conditions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Churchill et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Liang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In a recent analysis of a sample of synthetic absorption lines from a cosmological simulation, Marra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2021) tested the ac- curacy of the single cloud ionisation modelling method of re- trieving physical gas conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The authors find that while there is general agreement between intrinsic conditions and those derived from ionisation modelling, such methods cap- ture the average properties of absorbing gas cells, consistent with observational tests by Sameer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2021) comparing single-phase and multiphase modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Marra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2022) followed up by testing the assumption of single spatially- isolated absorbing clouds in the CGM, showing that several distinct absorbing clouds may be present within a single ab- sorption component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The distinct clouds may arise from gas of different phases that happen to be aligned kinematically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' These results demonstrate that the CGM is a complex envi- ronment, with non-linear relationships between the underly- ing CGM conditions and the resulting absorption observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Machine learning (ML) algorithms have the capacity to learn complex, non-linear relationships and as such they have been widely applied to astrophysical problems (see review by Fluke & Jacobs 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this paper, we explore a novel approach for cosmological simulations to aid in interpreting CGM absorption observations using ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We present a framework for Random Forest (RF) mapping between syn- thetic CGM absorption observables from the Simba simula- tion (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019) and the underlying absorber condi- tions from particle data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Such a mapping has the potential to be employed as a useful tool in retrieving physical con- ditions from real, multi-component absorption observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This approach eliminates the need for simplifying assump- tions about the structure and state of the gas, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' whether absorption arise from single or multiple gas phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Instead the RF mappings implicitly assume the veracity of the Simba galaxy formation model and our choice of UVB (Faucher- Giguère 2020) to produce its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The Simba simulations accurately reproduce a variety of observational galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' At low redshift, these in- clude the star-forming main sequence, black hole-galaxy co- evolution, radio galaxy populations, dust properties, cold gas properties, and the baryonic Tully-Fisher relation (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Glowacki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' On larger mass scales, Simba reproduces X-ray scaling relations for massive halos (Robson & Davé 2020) and low redshift Lyα absorption statistics of the IGM (Christiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In previous work we have shown that Simba also broadly reproduces the observed absorption properties of H i (Sorini et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020) and selected metal lines in the CGM (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021), and that such absorption arises from physically reasonable gaseous conditions (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' there- fore Simba is a reasonable choice of simulation with which to explore the capabilities of ML methods to learn relationships in the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Nonetheless, there is no guarantee Simba yields fully accu- rate and representative circum-galactic media.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Indeed, CGM zoom simulations suggest that Simba’s resolution may be too poor to capture finer details of multi-phase gas, particularly for stronger absorbers (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' van de Voort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Suresh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019 though see Nelson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This drawback could be explored via comparing the results of this frame- work applied to other simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We leave this aspect for future work, and here focus on presenting the general frame- work and its results when applied to the Simba model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this paper we train Random Forest (RF) machine learn- ing networks on the low-redshift Simba CGM absorber sam- ple presented in Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2022) to produce predictions for the underlying gas conditions in the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This paper is organised as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §2 we present the Simba simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §3 we describe the galaxy selection, spectrum generation and fitting processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §4 we describe the Random Forest (RF) model and training process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §5 we examine the accu- racy of the RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §6 we examine the feature impor- tance of the RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In §7 we present the RF predictions in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Finally in §8 we conclude and summarise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2 SIMULATIONS Simba (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019) is a suite of state-of-the-art cosmo- logical simulations that is the successor to the Mufasa sim- ulations (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2016), with the major additions being the inclusion of two-mode black hole growth and three-mode black hole feedback, along with an on-the-fly dust evolution model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The main simulation, and the one employed in this work, contains 10243 gas cells and the same number of dark matter particles within a (100h−1Mpc)3 volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This yields a particle mass resolution of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 × 107M⊙ per gas cell, and 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 × 107M⊙ per dark matter particle, with a spatial resolu- tion of ≈ 1h−1kpc in the densest regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Since Simba has MNRAS 000, 1–14 (2023) Machine learning and the CGM 3 been extensively described in many previous works, and since the primary goal on this work is to present and explore our machine learning framework that is not crucially dependent on which simulation it is applied to, for brevity we do not present all of Simba’s input physics, but rather refer read- ers to Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2019), Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2019) and Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2019) for full details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 3 ABSORBER SAMPLE In this work we use the sample of z = 0 absorbers from our in- vestigation into the physical conditions of absorbing halo gas in Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Here we summarise the procedure for generating the absorber sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We select a sample of central galaxies within the fiducial Simba volume that evenly sample a range of global galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The galaxies fall into three categories based on their star formation rates: star forming, green valley, and quenched.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We define star forming galaxies as with log(sSFR/Gyr−1) > −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3z for consis- tency with previous work with the Simba simulation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thomas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2019), define green valley galaxies as within 1 dex below the star forming galaxy threshold, and define quenched galaxies as those having zero star formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We further define six M⋆ bins of width 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 dex, with a mini- mum of M⋆ > 1010 M⊙ to ensure well-resolved systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2022) we selected 12 galaxies from each of the 18 M⋆ − SFR bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Here we select a further 12 galaxies in each to double the underlying galaxy sample (except in the highest mass star forming and green valley bins, which have only 23 and 8 galaxies respectively) to increase the sample available for training a machine learning mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For each central galaxy, we generate synthetic line of sight (LOS) absorption spectra through the simulation volume at a range of r200-normalised impact parameters (r⊥), probing both the inner and outer CGM (r⊥/r200 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='75, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In addition, for each r⊥, we select 8 equally-spaced LOS in a circle around the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thus, for each galaxy in our sample we generate 40 LOS spectra for each of the fol- lowing ions, selected to probe a range of excitation energies: H i 1215Å, Mg ii 2796Å, C ii 1334Å, Si iii 1206Å, C iv 1548Å and O vi 1031Å.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This results in a total sample of 17280 lines of sight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The spectra are generated along the z-axis of the simula- tion using the Pygad analysis package (Röttgers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the procedure is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Gas elements whose smoothing lengths intersect with the LOS are identified and their ioni- sation fractions obtained, using look up tables that are gen- erated with version 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='01 of the Cloudy cloud simulation code (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017) using Cloudy Cooling Tools1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We assume a spatially uniform Faucher-Giguère (2020) photoion- ising UV background spectrum, since it was shown in Chris- tiansen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2020) to provide the best match to low-redshift Lyα absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Self-shielding for H i is applied during the simulation run, but for generating the metal lines we employ the Rahmati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2013) prescription to attenuate the ion- ising background strength based on the local density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Ion densities for each gas element are obtained by multiply- ing the gas densities by each species’ ionisation fractions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The 1 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='com/brittonsmith/cloudy_cooling_tools mass fractions of each element are individually tracked within Simba, based on yields from Type II and Ia supernovae and stellar evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Metals are carried out into the CGM pri- marily by stellar feedback processes, since winds are mass and metal-loaded (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The ion densities are smoothed along the LOS into pixels of width 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5km s−1, using the same spline kernel used in the Gizmo simulation code and the gas elements’ individual smoothing lengths and metal masses (for metal lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Optical depths are then com- puted from the column densities at a pixel scale, using the oscillator strength for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We exclude wind parti- cles since those gas elements are hydrodynamically decoupled from the surrounding gas, which represent a very small frac- tion of the CGM mass (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Pygad also computes column density-weighted physical density, temper- ature, metallicity, and peculiar velocity in the same manner within the LOS pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We identify regions of absorption within a ±600 km s−1 window centered on the galaxy by computing the de- tection significance ratio of each pixel, defined as the Gaussian-smoothed flux equivalent width (EW) divided by the Gaussian-smoothed noise EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Regions are identified as contiguous intervals where the flux drops below the level of the continuum with an overall significance ratio of > 4σ, en- suring that the edges of the regions begin at the continuum and merging nearby regions within 2 pixels of one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We fit a superposition of Voigt profiles to each absorption region in order to extract the absorption line observables: the column density N, the Doppler b parameter, the wavelength (or velocity) location along the LOS, and the EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For the fit- ting, absorption lines are added to the model fit one at a time, with initial guesses for the line parameters that depend on whether or not the absorption is saturated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For non-saturated absorption, the line is placed at the position of lowest flux, and the initial N and b is based on the depth and velocity width of the local flux minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For saturated absorption, the line is placed in the middle of the saturated trough, and the initial N and b are chosen from a coarse grid in order to minimise the χ2 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This procedure broadly follows that in AutoVP (Davé et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 1997).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The best-fit Voigt parameters that minimise χ2 r are then found using the scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='optimize subpackage2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Loose prior bounds on N and b are set based on typical H i and metal line column densities and thermal line widths from 104 − 107K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' If the fit has χ2 r < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 then the model is accepted;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' oth- erwise we identify the next strongest area of absorption by subtracting the model from the data and place a line at the residual minimum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We repeat the process until an accept- able model is found, up to a maximum of 10 absorption lines per region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Each line must improve the χ2 r of the model by at least 5%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the process is halted if 2 consecutive additional lines do not improve the χ2 r by at least this margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' If af- ter 10 lines an acceptable model is not found then we adopt the model with the number of lines that performed best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We again check that each line improves the χ2 r of the model by iteratively recomputing the χ2 r with each line removed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' if the χ2 r acceptance threshold is reached, or the χ2 r increases by less than 5% then the line is removed from the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this 2 https://docs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='scipy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='org/doc/scipy/reference/optimize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' html MNRAS 000, 1–14 (2023) 4 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Species n log(Nmin/cm−2) χ2,90 r Median χ2 r E(eV) H i 17750 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='7 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='7 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='60 Mg ii 5306 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04 C ii 11062 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='38 Si iii 14119 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='7 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='9 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='49 C iv 17463 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='49 O vi 17463 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='12 Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Absorber sample properties for the RF models: the number of absorbers below the χ2 r limit for each species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the column density completeness limit;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the χ2 r below which we recover 90% of the total EW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the median χ2 r of all absorbers;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the excitation energy of the species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' way we attempt to obtain a satisfactory fit with the fewest number of absorption lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The galaxy selection, spectrum generation and LOS fit- ting pipeline results in our sample of absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We find that adopting the same strict χ2 r limit for all ions results in an incomplete sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As such we compute the EW directly for each LOS and adopt an upper χ2 r threshold for each ion such that we recover 90% of the total EW across all LOS for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The sample size and χ2 r upper limits (χ2 r,90) for each ion are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In practice the typical χ2 r for a given region is much lower than these upper limits;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the median χ2 r of absorption lines in our sample is also shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We also adopt the column density completeness limits from Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2022), which are computed by fitting the power law portion of the column density distribution func- tion (CDDF) for each ion and identifying where the CDDF falls below 50% of the expectation at low column densities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The completeness limits are given in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We note that routines to do the spectrum generation, absorption region identification, and Voigt profile fitting are all contained with the publicly-available Pygad package (Röttgers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 4 RANDOM FOREST METHODS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 Random Forest Regression Random Forest (RF) regression (Breiman 2001) is a super- vised, decision tree-based, ensemble method of machine learn- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The term ‘ensemble method’ refers to the process of com- bining predictions from several machine learning runs (in this case, individual decision trees) in order to more accurately predict the output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Decision trees work in a top-down man- ner, in which the best split for the data is found by minimising a cost function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' They have the advantage of being easy to in- terpret and have low bias in their predictions for the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, individual decision trees are prone to over- fitting to the training data, hence their predictions for new data have high variance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The RF algorithm counteracts this effect by construct- ing many decision trees, each trained on a subset of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Random forests may be used for both classification and regression problems;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' in this work we use RF in its re- gression mode to deal with our continuous target predic- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The training data subsets are randomly chosen with replacement, and their outputs averaged for an overall pre- diction in a process known as bootstrap aggregation (‘bag- ging’, see Breiman 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this way, RF models retain the low bias of a decision tree, while also minimising the vari- ance on predictions for new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Training a single decision tree is considerably faster, however such models are less re- liable, particularly when trained on non-linear data (such as the absorber data used here).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this work, we use the Scikit-Learn (Pedregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2011) module’s RF imple- mentation, RandomForestRegressor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' RF models are widely used in a range of astronomical ap- plications, and have been remarkably successful given the rel- ative simplicity of the approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The advantage of RF mod- els over other methods (for example Neural Network based algorithms) is in the interpretability of the output models, as they indicate the relative importance of the input vari- ables in reaching a prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In galaxy formation, RFs (and related tree-based methods) have been widely used for re- gression problems using both simulation and observational data, for example in predicting the properties of large scale structure (Lucie-Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Lovell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Li et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022) and the properties of galaxies and haloes (Ucci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Nadler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Rafieferantsoa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Cohn & Battaglia 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Moews et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Mucesh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Del- gado et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' McGibbon & Khochfar 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 Input features and target predictors For each of the ions in our selection, we train a RF model on the dataset of simulated CGM absorbers to predict their underlying physical gas conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We do this separately for each of the 6 ions we consider, such that the usefulness of this pipeline is not contingent on having line information simulta- neously for all 6 ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We exclude absorbers where the quality of the Voigt profile fit is low (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the fit has a χ2 r above the acceptable threshold for that ion) and the column density is below the completeness limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For each ion, we use the same set of input features and tar- get predictors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The input features are chosen from among the properties of the CGM absorbers and their central galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Included features which describe the absorbers themselves are: the column density (N), the equivalent width (EW), the linewidth (b), the velocity separation from the host galaxy (dv), and the impact parameter, expressed as a fraction of halo virial radius (fr200 = r⊥/r200).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Properties of the central galaxy that are also included as input features are the stellar mass (M⋆), the specific star formation rate (sSFR), and the fraction of kinetic energy contained in rotation (κrot, Sales et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2012), which Kraljic et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' (2020) found is a reasonable proxy for visual morphology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' From these 8 input features we predict 3 target gas pre- dictors: the overdensity (δ = ρ/¯ρm), temperature (T) and MNRAS 000, 1–14 (2023) Machine learning and the CGM 5 metallicity (Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Each of these is a column density-weighted average at the nearest LOS pixel to the absorber, computed at the time of spectral generation and binned along the LOS (see §3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3 Training Each of the features is transformed into log space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Jo & Kim (2019) showed that transforming quantities into log space im- proves the accuracy of machine learning predictions for as- tronomy problems, owing to the wide range of physical scales present in astronomical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Exceptions to this are dv, fr200 and κrot;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' dv and κrot have nearly uniform intrinsic distri- butions, while fr200 consists of 5 specific values due to our choices of LOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In addition, we standardize the input and output data by subtracting the mean of the distribution and scaling the variance to unity in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Where there are zeros in our dataset, we set them to a small non-zero value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For each ion, we divide the absorber dataset into 80% train- ing data used to build the RF model, and 20% test data used to evaluate the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Where multiple absorbers arise from the same LOS these can be separated into the training and test datasets;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' this mitigates over-fitting in the model due to galaxy or LOS properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We train the RF model separately for each target feature, as we find that this improves the accuracy of the predic- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We separately tune the hyperparameters of each RF model to optimize the model accuracy, using Scikit-Learn’s GridSearchCV method to perform an exhaustive grid search over hyperparameter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The hyperparameters are the number of trees, the minimum number of data points required in order to split the data, and the minimum number of data points in each resulting split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For each set of hyperparam- eters, a k-fold cross validation is performed with k = 5, in which the training data is split into k ‘folds’, and k RF models are iteratively constructed using k − 1 folds of the data;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the overall score for each set of hyperparameters is the average of each of the k RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The coefficient of determination R2 is used internally to evaluate the performance of each model, given n data points and true and predicted quantities Xtrue and Xpredicted: R2 = 1 − RSS TSS (1) where RSS is the residual sum of squares: RSS = n � i=1 (Xi true − Xi predicted)2 (2) and TSS is the total sum of squares: TSS = n � i=1 (Xi true − ⟨Xtrue⟩)2 (3) The mean squared error, MSE = RSS/n, is the cost function used to determine the best decision tree splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' An MSE of zero represents a perfectly accurate prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 5 PREDICTIVE ACCURACY Here we assess the performance of each of the RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Fig- ure 1 shows the test data RF predictions for the H i absorber physical conditions (δ, T and Z) against the true values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The color scale of the hexagonal bins indicates the number of data points in each bin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The black diagonal dashed line represents the 1:1 case of a perfectly predicting model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In each panel the 1D histograms for the true and predicted values lie along the top and right, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The accuracy of the model is summarised in each panel with three quantities: 1) the scatter σ⊥, which we define perpendicular to the perfect 1:1 relation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2) the Pearson correlation coefficient ρr, given by: ρr = cov(Xtrue, Xpredicted) σXtrueσXpredicted , (4) where σXi is the standard deviation of Xi;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' and 3) the MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' High correlation is preferred, but does not necessarily indicate an accurate prediction as the outputs could have a systematic offset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Beginning with the predictions for H i absorbers, den- sity and temperature are well-predicted by the ML model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' True values are highly correlated with the predictions, and the model predictions have low scatter and error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Density and temperature are physically correlated with one another and have Gaussian distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Of the two, temperature (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='08) is predicted more accurately than overdensity (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The RF mod- els for HI density and temperature perform particularly well considering the models’ relative simplicity (compared with e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' a NN-based model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Aside from transforming the features into log space and using the k-fold hyperparameter cross val- idation, the model has not been extensively tuned by hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As such, these results represent a basic model which demon- strate the capability of RF models to predict gas conditions, which could be improved upon with further tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We have also explored alternative ML approaches such as NNs and CNNs, and found that such models do not offer a substantial improvement in terms of predictive accuracy and take consid- erably longer to run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This has lead us to favour the RF model for its simplicity, speed, and the degree of interpretability in the form of feature ‘importances’ (see §6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The predictions for HI metallicities are less accurate (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='51).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In general, points with logZ/Z⊙ < −1 are overpredicted, while points with logZ/Z⊙ > −1 are un- derpredicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This points to the general tendency of our ML models to output a narrower predicted distribution than in the input dataset (this behaviour is also seen to a lesser ex- tent in the density and temperature predictions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This means that the tails of the original distributions are not well cap- tured in the ML model, perhaps as a result of sparse training data at the extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Perhaps the poor prediction is unsur- prising since HI absorption is not Z-dependent, unlike metal lines which by necessity arise from metal-enriched gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' There- fore it was not obvious that any relationship between H i ab- sorption and metallicity could have been learned from the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The learned mapping in the metallicity RF model likely arises from the provided galaxy properties and H i absorption strength;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' we will explore the input feature importance later (§6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The metal line absorber physical conditions are also rea- sonably well predicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figures 2 and 3 show the perfor- mance of the RF models for predicting C ii and C iv ab- sorber conditions, using the same plot structure as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The performance for Mg ii, Si iii and O vi absorbers are shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The RF models perform similarly well among all the metal lines, with the same tendency to MNRAS 000, 1–14 (2023) 6 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Hexagonal joint histogram of the predicted H i physical conditions from the RF mapping and the true H i physical conditions, including only data in the test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The number of data points in each bin is shown using colorbars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' From left to right, the panels show overdensity, temperature and metallicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The diagonal line represents the case where the RF model makes a perfect prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The accuracy of the predictions in each panel is summarised by the inset displaying the transverse scatter σ⊥, the correlation coefficient ρr and the mean square error MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The 1D histograms of the true and predicted values are shown on the top and side of each panel, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As in Figure 1, showing the predictions and true values for C ii absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As in Figure 1, showing the predictions and true values for C iv absorbers.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content="5 - 10'm n 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 /K)Pred pald(z/z) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='30 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='20 3 - 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 - pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='84 p, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='88 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='18 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='08 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 + 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0+ 0 1 3 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 号 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='75 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='24 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='15 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25- Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='67 P, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='75 Or = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='68 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='11 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='07 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 log OTrue log (T/K)True log (Z/Zo)True0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 E4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='21 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='14 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 - P, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='72 p, = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='78 Or = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='73 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='09 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 + 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0- 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='00 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='50 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='75 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='00 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 log OTrue log (T/K)True log (Z/Zo)TrueMachine learning and the CGM 7 predict a more concentrated distribution of values than in the original data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In general, for each metal line the pre- dictions are less well correlated with the truth values than for H i;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' metal line absorber δ, T and Z have median corre- lation coefficients of ρr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='69, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='68, respectively, com- pared with ρr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='84, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='88, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='81 for H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, the errors are in general lower for the metal line RF models, with me- dian MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='06 and median σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='17 for δ, T and Z, compared with MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='18, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='08, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='51 and σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='3, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 for H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The metal line RF models also have lower scatter, although we emphasise that this is partly due to the reduced range in physical conditions traced by the metal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Overall the RF models give reasonable predictions for the physical conditions, and again were not extensively tuned to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' An interesting feature of the original absorber dataset is bi- modal metallicity distributions at logZ/Z⊙ ∼ −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25, which have not been reported in earlier Simba CGM work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The bimodality is apparent in every metal line apart from Mg ii, and is broadly reproduced by the RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Popu- lations of absorbers in the cool CGM of low redshift Lyman Limit Systems (LLSs) have also been observed to have bi- modal metallicity distributions, with both metal-poor and metal-rich absorbers (albeit shifted to lower metallicities, Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2013, 2018, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Wotta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2016, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Berg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022), suggesting multiple origins for the cool CGM gas, although the metallicities of the observed metal-poor absorbers are much lower than that seen in Simba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In future work we will investigate the origin of the bimodal absorber metallicity distribution in Simba.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 6 FEATURE IMPORTANCE In this section, we seek insights into the physical origin of the ML-probed correlations by assessing which input features are most useful in predicting the physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 RF model importance An advantage of the RF method over other ML algorithms (such as neural networks) is it allows some degree of inter- pretability in the form of the ‘importance’ of each feature, which arise ‘for free’ from the structure of the RF model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For an individual decision tree, a feature’s importance is com- puted from the number of times it is used to split the data and how close to the top of the tree the splits are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For an RF model, the importances are the normalised average over all decision trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, importance metrics are biased if the input features are highly correlated with one another (Strobl et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2007, 2008) and so they should be treated with caution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thus we prefer not to use the importance directly reported by RF, but instead compute it more empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' To do so, we determine each input feature’s importance by iteratively building the RF model and removing each of the features in turn, using the same optimized hyperparameters as in the full feature model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We then retrieve the importance of the remaining features for each model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This process de- termines whether a feature is genuinely important, or merely defined as such through a fluke of feature combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' When the most important features are removed, identifying which features take its place as the most important gives an indi- cation of what the RF model is learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 4 shows the feature importance values for predict- ing H i absorber conditions, against the feature removed from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For predicting overdensity, N is most im- portant feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' When column density is removed, the most important feature is EW;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' N and EW are correlated with one another and both are correlated with physical density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' When predicting temperature, b is the most important feature since the linewidths of individual absorbers in the original spec- trum are set in partly by thermal Doppler broadening (with the additional effect of bulk gas motions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' When predicting metallicity, the velocity separation is the most important fea- ture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' It is not intuitively obvious why this is the case;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' perhaps due to a dependence on halo velocity dispersion, which is cor- related with M⋆ and thus the metallicity of the host galaxy that is predominantly responsible for enriching its CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figures 5 and 6 likewise show the feature importance val- ues for predicting C ii and C iv absorber properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We have examined feature importance for all metals and found that these are representative cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For the low ion C ii, the feature importance rankings for δ and T are similar to that of H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' There is a slightly reduced relative importance of N in pre- dicting δ in favour of b (N and b are correlated features due to their underlying dependence on δ and T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For both ions, the importance of b in predicting T is enhanced compared with H i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In contrast to H i, the most important feature for predicting Z for metal lines is sSFR;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' when sSFR is removed, the RF model learns from M⋆ and κrot instead, indicating that the RF model predicts Z from the galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The picture is similar for the high ion C iv, except that in predicting δ the most important features are instead fr200 and sSFR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' fr200 is perhaps less useful for C ii since most of the low ion absorption arises from the inner CGM (Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Galaxies with high star formation have denser gas in their CGM, although this is also the case for H i and C ii, so it is unclear why sSFR specifically is an important feature for C iv absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 Change in predictive accuracy Arguably the most meaningful measure of ‘importance’ to the model is in which features add the most useful information in terms of predictive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We assess this by iteratively removing each of the features in turn from the training data, and running the RF model as before (with the hyperparame- ters optimized for the full feature case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In contrast with §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1, we now compute the scatter σ⊥ for each new model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' if the quality of the predictions are significantly degraded in the ab- sence of a particular feature, this necessarily indicates that this feature encodes crucial information about the physical conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 7 shows the change in σ⊥ resulting from the removal of each input feature in predicting the physical conditions, where a positive change indicates an increase in scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Each of the six panels shows the models for a different ion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the rows within each panel show the models for each of δ, T and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The upper plots show H i (left), Mg ii (middle) and C ii (right) absorber models, while the lower plots show Si iii (left), C iv (middle) and O vi (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In contrast to the feature importance values, the largest increase in scatter when predicting H i absorber overdensity MNRAS 000, 1–14 (2023) 8 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Importance values in predicting H i physical conditions for each remaining input feature, against the input feature removed from the training data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' From left to right, the target predictors are δ, T and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Feature importance values as in Figure 4, for CII absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Feature importance values as in Figure 4, for CIV absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' comes from removing dv and fr200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The loss of accuracy from removing fr200 suggests that the RF model is learning the ra- dial density profile;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' this is also the case for the metal line δ predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' It is less clear why dv is necessary for an accu- rate prediction, since halo absorbers can appear at any ve- locity separation depending on their kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For all ions, removing b causes an increase in scatter in T predictions, confirming that the high feature importance of b reflects the genuine physical relationship with temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For H i, the metallicity predictions are degraded by remov- MNRAS 000, 1–14 (2023) log log T log Z 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 logN- 0.' metadata={'source': 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+page_content=' fr200 log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='M* - sSFR log Removed feature Removed feature Removed featureMachine learning and the CGM 9 Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The change in σ⊥ of the RF models when removing each feature iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Each of the panels shows results for a different ion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the three rows of each panel represent results for each of δ, T and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' ing dv, fr200 or any galaxy property;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' interestingly the model accuracy improves when absorption-related features are re- moved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In other words, gas metallicity in the CGM of a given galaxy can be predicted with reasonable accuracy from only LOS and radial absorber position.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The metal lines broadly show the same changes in scatter with removed features, al- though the changes to the model accuracy are more marginal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 7 PHASE SPACE Having examined the predictive accuracy and inner work- ings of the RF models on individual properties, we now ask whether the RF models can reproduce the two-dimensional (δ, T) phase space structure of the absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Although the RF models can separately predict δ and T, this does not guarantee that they reproduce the relationship between these quantities - particularly since the models are trained sepa- rately for δ and T, so each RF model has no knowledge of the other target quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 8 shows the predicted temperature against predicted overdensity for the 6 species we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The distributions for the truth and predicted data are shown along the top and right hand side of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Note that the plot limits are different for each ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The points are colour coded by the fractional distance from the true value in δ − T phase space: σphase = ��δtrue − δpredict δtrue �2 + �Ttrue − Tpredict Ttrue �2 (5) The colour scale indicates that in general the predicted points lie near their truth values in phase space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the metal line ab- sorbers in particular have a low displacement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The contours show the true distribution in phase space for the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The upper plots show H i (left), Mg ii (middle) and C ii (right) absorbers, while the lower plots show Si iii (left), C iv (mid- dle) and O vi (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The original structure in phase space between overdensity and temperature is reproduced well by the RF models for each of the ions we consider, a success of the ML approach which (since there was no specific tuning of the model to achieve this) arises because temperature in- formation is encoded in the overdensity data and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This is an important test of the RF models which verifies that accurate predictions can be made for multiple physical properties per observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' That said, although the RF models succeed in predicting the phase space structure, the predictions are in general too concentrated near the mean of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' By comparing the predicted distribution with the contours from the original data, it is clear that the predictions ought to be more spread in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This appears to be a generic feature of the RF models, which is also apparent in the 1D distributions of each feature - in general, the RF models produce pre- dicted distributions that are too concentrated towards the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As mentioned in §5, this likely arises from sparse train- ing data at the extremes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thus, the predicted distributions do not capture the important information described by the intrinsic scatter in the original data, which biases the use- fulness of these models for observational analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Therefore some additional step beyond the basic ML model is required in order to capture the full structure in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Em- ploying an oversampling technique (such as Synthetic Minor- ity Over-Sampling Technique for Regression with Gaussian Noise, SMOGN) can boost under-sampled regions of phase space and as such mitigate issues that arise as a result of imbalanced training datasets (de Santi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In order to extend the RF network to also properly capture the full 2-D phase space distribution, we develop a new ap- proach based on a normalising transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' By this, we mean that the predicted and truth data for each feature are mapped onto standard normal distributions, and then the predicted distribution is transformed back onto the shape of the truth data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' To accomplish this we use the quantile transform non- MNRAS 000, 1–14 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04 HI 1215 MgII 2796 CII 1334 log d - log Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' SilII 1206 CIV 1548 OVI 1031 log d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' log Z - logN- EW ap sSFR Krot log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='N- EW ap fr200 sSFR log.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='N- 3oy 6 EW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' ap fr200 log M sSFR log log log Removed feature Removed feature Removed feature10 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Predicted temperature against predicted overdensity for each of the 6 ions we consider, coloured by overall phase space fractional error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The 1D truth (blue curve) and predicted (pink curve) distributions are shown along the top and right of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The contours show the true distribution in phase space for the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The limits of the plots differ for each ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' parametric method implemented in Scikit-Learn (Pe- dregosa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2011), QuantileTransformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The method first maps the cumulative distribution of the data onto a stan- dard Gaussian, and then computes the transformed values using a quantile function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The function also provides the in- verse mapping that transforms a distribution back into the original coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The inverse mapping for the truth data distribution (computed from the training dataset) is used to reconstruct the predicted distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In this way, we can reproduce the larger variance in the truth data without as- suming a shape for the predicted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 9 shows the the predicted temperature against pre- dicted overdensity, using the above normalising transform ap- proach to map the shape of the truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This can be seen in the 1D distributions along the top and right hand side, which in most cases closely follow the truth data distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Crucially, the transformed data also retains the phase space structure of the original predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In addition, trans- forming the predictions onto the shape of the truth data re- sults in no loss of accuracy for the predictions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the MSE, ρ and σ⊥ for the transformed test datasets are very similar to those for the original test dataset predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As such, this is our preferred method for reproducing the scatter in the truth CGM conditions data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We initially explored a simpler approach where we added scatter directly to the predicted data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The results in phase space for the additional scatter approach is shown in Ap- pendix B (Figure B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We found that this approach substan- tially washed out non-Gaussian structure in the predicted distribution, such as the anti-correlation between δ and T at δ > 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' By instead using the normalising transform method, we are preserving these structures in phase space as much as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 10 directly compares the distributions for the truth data (dark purple), the predictions from the RF models (light purple), the transformed predictions (orange), and the pre- dictions with additional scatter (yellow) for the H i absorber overdensities and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The predicted physical con- ditions from the RF models are clearly too closely concen- trated towards the mean.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In contrast, applying the normalis- ing transform approach results in a predicted dataset which closely matches the truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' When additional scatter is instead added to the predictions, the resulting distribution also more closely matches the truth data, although not so precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Quantitatively, a two-sample Kolmogorov–Smirnov test with respect to the truth data gives p-values of > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='95 for the transformed overdensity and temperature distributions, but ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 for the distributions with added scatter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' There are pros and cons to including the normalising trans- form approach to ensure that the phase space scatter is well reproduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' If one wanted to compute distribution functions for physical quantities inferred from absorption line data, not MNRAS 000, 1–14 (2023) logo phase 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 Truth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 Prediction 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 log(T/K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 [MgII 2796 [CII 1334 HI 1215 3.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Predicted temperature against predicted overdensity for each of the 6 ions we consider, mapped to the shape of the truth 1D distributions using a quantile transformer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Points are coloured by overall phase space fractional error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The 1D distributions for the truth data (blue curve) and transformed predictions (pink curve) are shown along the top and right of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The contours show the true distribution in phase space for the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Histograms showing the direct comparison between truth data (dark purple), the predictions (light purple), the transformed predictions (orange) and the predictions with added scatter (yellow) for H i overdensities and temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' MNRAS 000, 1–14 (2023) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 Prediction+Scatter Transformed Prediction 0.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='25 [SilII 1206 CIV 1548 OVI 1031 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 2 3 logd logd logd12 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' including this post-processing step would result in the distri- bution functions improperly capturing the tails, which may be important for some applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, the additional step necessarily degrades the σ⊥ and MSE of the predictions, albeit only marginally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The correlation coefficient does not change since we are only scaling the predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For exam- ple, in the case of H i overdensity, the σ⊥ and MSE increase from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='30 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='31 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='18 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='20 respectively after apply- ing the normalising transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For H i temperature, the σ⊥ increases from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='20 → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='21, while the MSE does not increase (at this level of precision).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Whether or not to employ the above method thus depends on the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 8 CONCLUSIONS We have produced machine-learnt mappings between CGM absorption observables and the underlying gas conditions for H i and selected metal lines using a Random Forest approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' RF models are preferred over other ML techniques for their relative simplicity and interpretability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' These mappings rep- resent a proof of concept for using ML models as part of an analysis pipeline for observational CGM data, which cru- cially does not make simplifying assumptions about the phase or composition of the absorbing gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We identify a general tendency of the RF models to output a narrower predicted distribution than in the input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We demonstrate two methods of reproducing the scatter of the input data: first by adding random Gaussian noise to the predictions, and second by transforming the predictions to the shape of the truth data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Our main results are as follows: The RF models predict reasonable H i overdensities (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='08) and temperatures (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The predictions of overdensity and tem- perature are highly correlated with their truth values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Metal- licity is less well predicted (σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 dex, MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='51);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' metallicity is not directly traced by HI, therefore the learned relationship likely arises from the correlation with density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The RF models also predict reasonable metal absorber conditions and perform to a similar accuracy among all metal lines, with median ρr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='69, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='7, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='68, median σ⊥ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='23, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='15, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='17 and median MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='11, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='04, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='06 for the overdensity, temperature and metallicity predictions, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Lower σ⊥ compared with H i predictions are partly due to the smaller dynamic ranges probed for metals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We report a bimodality in the absorber metallicity dis- tributions for four of the five metal lines (C ii, Si iii, C iv and O vi), suggesting multiple origins for the CGM gas in the Simba model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In terms of feature importances, the RF models learn H i absorber overdensity from column density and equiva- lent width, temperature from the Doppler parameter, and metallicity from the LOS velocity separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Low ion fea- ture importances are similar to H i, except that metallicities are learned from sSFR and κrot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' High ion feature importances are similar to the low ions, except that the overdensities are learned from radial distance and galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In terms of predictive accuracy, the radial distance and LOS separation provide the most useful information for pre- dicting H i overdensity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the radial distance is also most useful for the metal line overdensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The Doppler parameter is again the most important feature for predicting temperature for all lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The LOS separation provides the most useful information for predicting H i metallicity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' the predictions for all lines are degraded by removing galaxy properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The predictions for overdensity and temperature repro- duce the phase space structure seen in the original data for all six ions, despite being trained for separately in the RF models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This verifies that accurate predictions can be made for multiple physical properties per observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' By mapping the predicted data distributions onto the shape of the input distributions using a quantile normalis- ing transformer, we can reproduce the intrinsic scatter in the CGM phase space conditions with no loss of predictive accu- racy or phase space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Although we have considered H i and the metal ions sep- arately, future work on this topic could explore RF models using combinations of absorption lines to assess whether pre- dictions may be improved by using information from multi- ple ion species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' A shortcoming of the ML models presented here is the that the predictions are too concentrated towards the mean of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Further development would be needed on the pipeline in order to reproduce the scatter in the original data without losing information in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The motivation of this project is to develop a useful analy- sis tool for the astronomical community to aid in interpreting absorption observations of the CGM, assuming the galaxy formation model of the Simba simulations and the Faucher- Giguère (2020) UVB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The next phase of this work is to test the method by applying the ML mappings to real observa- tional data and comparing to results derived from ionisation modelling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As such, the trained models produced for this work are available online3 and we encourage others to test the RF models on their own observational data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' A natural extension of this project will be to develop addi- tional ML mappings using absorber data from other simula- tions such as EAGLE and IllustrisTNG to assess the impact of galaxy formation models on the predicted conditions for the CGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Training the RF models on data from one sim- ulation and testing on data from another would provide a robust test of the impact of galaxy formation model on our re- sults.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In addition, developing mappings using absorbers from the CAMELS project (Villaescusa-Navarro et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 2020) would test the dependence of our results on both astrophysical and cosmological models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' ACKNOWLEDGEMENTS We acknowledge helpful discussions with Arif Babul.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' We thank Philip Hopkins for making Gizmo public, Robert Thompson for developing Caesar and Horst Foidl, Thorsten Naab and Bernhard Roettgers for developing Pygad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' SA is supported by a Science & Technology Facilities Council (STFC) studentship through the Scottish Data-Intensive Sci- ence Triangle (ScotDIST).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' RD acknowledges support from the Wolfson Research Merit Award program of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Royal Society.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Throughout this work, DS was supported by the STFC consolidated grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' RA5496 and by the Swiss National Science Foundation (SNSF) Professorship grant no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' 202671.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' CCL acknowledges support from a Dennis Sciama 3 https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='com/sarahappleby/cgm_ml MNRAS 000, 1–14 (2023) Machine learning and the CGM 13 fellowship funded by the University of Portsmouth for the Institute of Cosmology and Gravitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Simba was run on the DiRAC@Durham facility man- aged by the Institute for Computational Cosmology on be- half of the STFC DiRAC HPC Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The equipment was funded by BEIS (Department for Business, Energy & In- dustrial Strategy) capital funding via STFC capital grants ST/P002293/1, ST/R002371/1 and ST/S002502/1, Durham University and STFC operations grant ST/R000832/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' DiRAC is part of the National e-Infrastructure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For the purpose of open access, the author has applied a Creative Commons Attribution (CC BY) licence to any Au- thor Accepted Manuscript version arising from this submis- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' DATA AND SOFTWARE AVAILABILITY The simulation data and galaxy catalogs underlying this ar- ticle are publicly available4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The software used in this work is available on Github5 and the derived data will be shared on request to the corresponding author.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=', van den Bosch F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=', Pakmor R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=', 2019, MNRAS, 482, L85 APPENDIX A: RESULTS FOR OTHER METALS For completeness, here we present figures similar to Figure 1, for Mg ii, Si iii, and O vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The general trends are already cap- tured by the plots in the main text for C ii and C iv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, there are some interesting notable point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' For instance, O vi shows significantly higher temperatures, as expected since it is a higher ionisation line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The metallicity bimodality is still slightly present for O vi, although at a much lower signifi- cance than for the lower ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Si iii has the largest scatter in the recovered T, and also shows some bias such that high-T absorbers are under-predicted while low-T ones are overpre- dicted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This may be because Si iii seems to have absorbers spanning the widest range in temperatures from among the metal ions considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' In terms of the RF performance, how- ever, these ions tell a similar story, which is encouraging since it means the RF methodology is widely applicable with sim- ilar efficacy across a range of commonly observed low-z UV ions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' APPENDIX B: DIRECT GAUSSIAN APPROACH TO ADDING PHASE SPACE SCATTER In §7 we described our normalising transform-based approach to adding scatter to the RF predictions in order to match the 2-D truth distributions in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' A more straightfor- ward approach is to directly add Gaussian scatter to the pre- dicted δ and T distributions to match the truth without first applying a normalising transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' However, the results were less satisfactory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure B1 shows the results, which can be compared to Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' It is clear that the simpler approach causes fea- tures within the true phase space to be more washed out, and substantially degrades the predictive accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Thus we prefer the normalising transform approach presented in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' This paper has been typeset from a TEX/LATEX file prepared by the author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' MNRAS 000, 1–14 (2023) Machine learning and the CGM 15 Figure A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As in Figure 1, showing the predictions and true values for Mg ii absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure A2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As in Figure 1, showing the predictions and true values for Si iii absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure A3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' As in Figure 1, showing the predictions and true values for O vi absorbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' MNRAS 000, 1–14 (2023) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 0.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 101 101 101 n 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 100 10° 100 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='23 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='15 01 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='16 pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='71 Pr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='79 Or = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='4 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='11 MSE = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='1 log OTrue log (T/K)True log (Z/Zo)True16 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Appleby et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Figure B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Predicted temperature against predicted overdensity for each of the 6 ions we consider, with added random Gaussian noise to reproduce the original 1D distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Points are coloured by overall phase space fractional error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The 1D truth and predicted distributions are shown along the top and right of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The contours show the true distribution in phase space for the test dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' The widths of the random noise Gaussians for δ and T are shown in the bottom right of each panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' Compared to the normalising transform results shown in Figure 9, this approach washes out features in phase space substantially more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content=' MNRAS 000, 1–14 (2023) logOphase 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 - 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 0 Truth 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 Prediction+Scatter 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 log(T/K) 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='8 4.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='6 LO LO 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9A0T4oBgHgl3EQfDP-k/content/2301.02001v1.pdf'} +page_content='0 3.' metadata={'source': 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a/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/2301.04275v1.pdf.txt b/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/2301.04275v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..6ff10ec21b8d849524d4e5bf52644b6ec44f7646 --- /dev/null +++ b/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/2301.04275v1.pdf.txt @@ -0,0 +1,888 @@ +LENet: Lightweight And Efficient LiDAR Semantic Segmentation Using +Multi-Scale Convolution Attention +Ben Ding1 and Ji-Chao Jiao2 +Abstract— LiDAR semantic segmentation can provide vehi- +cles with a rich understanding of scene, which is essential to the +perception system in robotics and autonomous driving. In this +paper, we propose LENet, a lightweight and efficient projection- +based LiDAR semantic segmentation network, which has an +encoder-decoder architecture. The encoder consists of a set of +MSCA module, which is a simple convolutional attention mod- +ule to capture multi-scale feature maps. The decoder consists of +IAC module, which uses bilinear interpolation to upsample the +multi-resolution feature maps and a single convolution layer to +integrate the previous and current dimensional features. IAC +is very lightweight and dramatically reduces the complexity +and storage cost. Moreover, we introduce multiple auxiliary +segmentation heads to further refine the network accuracy. We +have conducted detailed quantitative experiments, which shows +how each component contributes to the final performance. +We evaluate our approach on well known public benchmarks +(SemanticKITTI), which demonstrates our proposed LENet is +more lightweight and effective than state-of-the-art semantic +segmentation approaches. Our full implementation will be +available at https://github.com/fengluodb/LENet. +Index Terms— LiDAR point clouds, 3D semantic segmenta- +tion, LiDAR perception, Autonomous Driving +I. INTRODUCTION +Environment perception can help vehicles understand the +surrounding scene, which is essential to autonomous driv- +ing. LiDAR and RGB cameras are common sensors in +the perception system of autonomous driving. Compared to +cameras, LiDAR sensor isn’t affected by lighting and weather +conditions, thus being more robust. Meanwhile, compared to +2D image, LiDAR point cloud can accurately describe the +structure of an object, and provide vehicles with a geometry- +accurate representation of the surroundings. Therefore 3D +point cloud analysis has drawn more and more attention. +Especially, point cloud semantic segmentation aims to assign +the labels for each point, which helps vehicles gain a rich +understanding of the scene. Therefore, point cloud semantic +segmentation is becoming an interesting research topic in +both academic and industrial communities. +Due to the disorder and irregularity of 3d point cloud, we +can’t directly perform standard Convolution Neural Networks +on it. To tackle the problem, extensive research efforts +have been devoted to point cloud semantic segmentation +during the past year. Point-based method [2]–[5] directly +extract features from the raw point cloud, which can reduce +the impact of computational complexity and noise errors +in the pre-processing process. However, they usually have +high computational complexity and limited processing speed. +Voxel-based methods convert the irregular pint clouds into +regular grid representation so that the 3D convolutional neu- +0 +100 +200 +300 +400 +500 +600 +700 +800 +Inference time(ms) +0 +10 +20 +30 +40 +50 +60 +70 +mIoU on SemanticKITTI test(%) +SqueezeSegV3 +RangeNet++ +KPConv +FIDNet +Meta-RangeSeg +RandLA-Net +Lite-HDSeg +PointNet +LatticeNet +Ours +Real-time +Single Scan Semantic Segmentation +point-wise +projection-based +0 +100 +200 +300 +400 +500 +Inference time(ms) +0 +10 +20 +30 +40 +50 +mIoU on SemanticKITTI test(%) +SpSequenceNet +Temporal-LidarSeg +Temporal-LatticeSeg +Meta-RangeSeg +Ours +Real-time +Multiple Scans Semantic Segmentation +point-wise +projection-based +Fig. 1. +Accuracy(mIoU) versus inference time. Our presented LENet +obtains the promising results on both single scan and multiple scans +semantic segmentation in SemanticKITTI test dataset [1]. +ral networks can be employed. Although voxel-base methods +can achieve the state-of-the-art accuracy, they mainly suffer +from heavy computations especially for large-scale LiDAR +point clouds in outdoor scenes of autonomous driving. +Projection-based methods transform the raw point cloud +into an 2D range image by spherical projection strategy. +Compared to point-based methods and voxel-based meth- +ods, projection-based have a superior inference speed and +nice accuracy performance. Meanwhile, they have recently +received increasing attention since the great success of fully +convolutional networks on image semantic segmentation. +In this work, we presents a lightweight and efficient +projection-based LiDAR semantic segmentation network. +Experimental results on well known public benchmarks +(SemanticKITTI) demonstrates that our network have higher +accuracy performance and can run in real time (as shown in +Fig. 2) with less parameters than prior works (as shown in +table III). In summary, the main contribution of this paper +as follows: leftmargin=* +• We present a simple muti-scale convolutional attention +arXiv:2301.04275v1 [cs.CV] 11 Jan 2023 + +module (MSCA), which can capture global and local +context information in the full 360 degrees LiDAR scan. +• A novel IAC module, which uses bilinear interpolation +to upsample the multi-resolution feature maps and a +single convolution layer to integrate the previous and +current dimensional feature. IAC is very lightweight and +dramatically reduces the complexity and storage cost. +• By introducing multiple auxiliary segmentation heads, +we further refine the network accuracy without intro- +ducing additional inference parameters. +• We conduct extensive experiments on the publicly avail- +able datasets, SemanticKITTI [1]. The results show that +our method achieves state-of-the-art performance. +II. RELATED WORK +With the prevalence a large-scale dataset [1], [6], [7] for +the task of point cloud segmentation of autonomous driving +scenes and the rapid development of deep learning, a wide +range of 3D LiDAR point clouds semantic segmentation +methods using deep learning have been proposed over the +past years. Generally, they can be broadly categorized into +four groups according to the representations of input data, +include point, range map. +Point-based methods directly process the raw 3D point +cloud without applying any additional transformation or +pre-processing, which are able to preserve the 3D spatial +structure information. The pioneering methods of this group +are PointNet [2] and PointNet++ [3], which use shared MLPs +to learn the properties of each point. In subsequent series of +works, KPConv [8] develops deformable convolutions that +can use arbitrary number of kernel points to learn local +geometry. Howerver, these approaches have the disadvan- +tages of high computational complexity and large memory +consumption, which hinders them from the large-scale point +cloud. RandLA-Net [5] adopts a random sampling strategy +and uses local feature aggregation to reduce the information +loss caused by random operations, which considerably im- +prove the efficiency of point cloud processing and decrease +the use of memory consumption. +Voxel-based Methods Voxel-based approach to convert +point clouds into voxels for processing, which can effec- +tively solve the irregularity problem. The early voxel-based +methods firstly transform a point cloud into 3D voxel repre- +sentations, then use the standard 3D CNN to predict semantic +labels. However, the regular 3D convolution requires the +huge memory and heavy computational power. Minkowski +[9] CNN chose to use sparse convolution instead of standard +3D convolution and other standard neural network to reduce +the computational cost. Cylinder3D [10] adopts 3D space +partition and designs an asymmetrical residual block to +reduce computation. AF2S3Net [11] achieves state-of-the- +art of voxel-methods, which proposes two novel attention +blocks name Attentive Feature Fustion Module(AF2M) and +Adaptive Feature Fustion(ASFM) to effectively learn local +features and global contexts. +Projection-based Methods project 3D point clouds into +2D image space, which can take advantage of a large amount +of advanced layers for image feature extraction. SqueezeSeg +[12] proposes spherical projection which maps the scatter 3D +laser points into 2D Range-Image, then uses the lightweight +model SqueezeNet and CRF for segmentation. Subsequently, +SqueezeSegV2 [13] proposes context aggregation module +(CAM) to aggregate contextual information from a larger +perceptual field. RangeNet++ [14] integrates Darknet into +SqueezeSeg and proposes an efficient KNN post-processing +method to predict labels for point. SqueezeSegV3 [15] pro- +poses Spatially-Adaptive Convolution (SAC) with different +filters depending on the location of the input image. Sal- +saNext [16] inherits the encoder-decoder architecture from +SalsaNet [17] and presents an uncertainty-aware mechanism +for point feature leaning. Lite-HDSeg [18] achieves state-of- +the-art performance by introducing three different modules, +Inception-like Context Module, Multi-class Spatial Propaga- +tion Network, and a boundary loss. +III. METHOD +A. Range Image Representation. +Using the spherical projection approach, we can transform +the unstructured point cloud into an ordered range image +representation. The advantages of range representation are +that it can use the effective 2D convolutional operation for +fast training and inference, and it can facilitate the mature +deep learning technologies that have been well studied in +image-based tasks. +In the range image representation, each LiDAR point +p = (x, y, z) with Cartesian coordinates, a spherical mapping +R3 → R2 is used to transform it to image coordinates, as +below: +� +u +v +� += +� +1 +2 +� +1 − arctan(y, x) π−1� +w +� +1 − +� +arcsin(z r−1) + fup +� +f−1� +h +� +, +(1) +where (u, v) are image coordinates, (h, w) are the height +and width of the desired range image representation, +f = fup + fdown is the vertical field-of-view of the sensor, +and r = +� +x2 + y2 + z2 is the range of each point. +B. Convolution Attention Encoder +Multi-scale features play a significant role in semantic seg- +mentation since semantic segmentation tasks usually needs +to process objects of different size in a singe image. A +common approach to extract multi-scale features is to use +a combination of a set of convolutions having different +receptive fields, then fusing these respective fields, such as +[16]. Inspired by SegNext [19], we propose a novel multi- +scale convolution attention module (MSCA). +As depicted in Fig.3 (b), MSCA consists of three parts: a +depth-wise convolution to aggregate local information, multi- +branch depth-wise strip convolutions to capture multi-scale +context, and an 1 × 1 convolution to model relationship +between different channels. Finally, the output of 1 × 1 +convolution is used as attention weights directly to re-weight +the input of MSCA. In addition, we adopt the pyramid +structure for our encoder following +[20], [21] and the + +Projection +2D Range Image +64 x 2048 x 5, (x,y,z,d,r) +3x3, channel = 64 +3x3, channel = 128 +3x3, channel = 128 +Input Module +Encoder1 +Encoder2 +Encoder3 +Encoder4 +Backbone Network +IAC +IAC +IAC +IAC +Classification Head +KNN +The Raw Point Cloud +Semantic Prediction +Decoder +Fig. 2. +Illustration of our proposed LENet framework. The backbone network is consisted of MSCA and has the pyramid structure similar to ResNet34. +The ICA module upsamples the low-dimensional feature maps to original size and aggregate it with the output of the previous IAC module. Then, the last +classification head receives the feature maps from the last three ICA module and outputs the label of each point. Finally, we use K Nearest Neighbors +(KNN) to do post-processing. +d, 5 x 5 +d, 1 x 3 +d, 1 x 5 +d, 1 x 7 +d, 3 x 1 +d, 5 x 1 +d, 7 x 1 +1x1 +Convolution Attention +Multi-scale Feature +(a) The BasicBlock +(b) MSCA +3x3 +SiLU +BN +SiLU +MSCA +1x1 +1x1 +Fig. 3. +Illustration of the BasicBlock that is used to build the encoder +and the proposed MSCA. Here, d means a depth-wise convolution, k1 × +k2 means the kernel size of the convolution layer. We extract multi-scale +features using the convolutions and then utilize them as attention weights +to reweight the input of MSCA. +building block in the encoder is composed with a 3 × 3 +convolution layer and MSCA as shown in Fig.3 (a). +C. IAC Decoder +To design a simple and effective decoder, We investigate +several different decoder structure. In +[14]–[16], they use +standard transpose convolutions or pixel-shuffle to produce +the upsampled feature maps, then using a set of convo- +lution to decoder the feature maps, which is effective but +computationally heavy. In FIDNet, it’s decoder uses FID +(fully interpolation decoding) to decode the semantics of +different levels, then using a classification head to fuse these +semantics. Although FID is completely parameter-free, it +doesn’t have the ability to learn from feature, which makes +model’s performance excessively depend on the classification +head. Besides, FIDNet’s classification fuse too much low- +level information and hurts the performance. +In this work, we presented a lightweight decoder as +depicted in Fig. 2. The IAC module contains two parts: a +bilinear interpolation to upsample the feature maps which +come from the encoder, a 3 × 3 convolution to fuse the +information from the encoder and the previous IAC. Finally, +we use the point-wise convolution to fuse the features from +the last three IAC modules. In three different decoder, our +decoder has fewest parameters and best performance. +D. Loss Function +In this work, we train the proposed neural network with +three different loss functions, namely weighted cross-entropy +loss Lwce, Lov´asz loss Lls and boundary loss Lbd. Finally, +our total loss is following: +L = w1Lwce + w2Lls + w3Lbd, +(2) +w1, w2 and w3 are the weights with respect to each loss +function. In our implementation, we set w1 = 1, w2 = 1.5 +and w3 = 1s. +Three loss functions account for three different problem. +To cope with the imbalanced classes problem, the weighted +cross-entropy loss Lwce [32] is employed to maximize the +prediction accuracy for point labels, which is able to balance +the distributions among different classes. It’s defined as +Lwce(y, ˆy) = − +� +i +1 +√fi +p (yi) log (p (ˆyi)) , +(3) +where yi represents the ground truth, and ˆyi is prediction +and fi is the frequency of the ith class. +To solve the problem of optimizing the intersection-over- +union(IoU), the Lov´asz loss Lls [33] is used to maximize the +intersection-over-union (IoU) score that is commonly used +to in performance evaluation on semantic segmentation. It’s +defined as: +Lls = +1 +|C| +� +c∈C +∆Jc(m(c)), mi(c)= +� 1−xi(c) +if c=yi(c) +xi(c) +otherwise +(4) +where |C| is the class number, ∆Jc represents the Lov´asz +extension of the Jaccard index, xi(c) ∈ [0, 1] and yi(c) ∈ +{−1, 1} hold the predicted probability and ground truth label +of pixel i for class c, respectively. +To account for the blurred segmentation boundaries prob- +lem as suggested in [18], the boundary loss function Lbd [34] +is used for LiDAR semantic segmentation, which can be +formulated defined as follows: +Lbd(y, ˆy) = 1 − 2P c +b Rc +b +P c +b + Rc +b +, +(5) + +TABLE I +THE PERFORMANCE COMPARISON ON SEMANTICKITTI MULTIPLE SINGLE SCAN BENCHMARK. +Methods +Size +mean-IoU +FPS (Hz) +car +bicycle +motorcycle +truck +other-vehicle +person +bicyclist +motorcyclist +road +parking +sidewalk +other-ground +building +fence +vegetation +trunk +terrain +pole +traffic-sign +PointNet [2] +50K pts +14.6 +2 +46.3 1.3 +0.3 +0.1 +0.8 +0.2 +0.2 +0.0 61.6 15.8 35.7 1.4 41.4 12.9 31.0 4.6 17.6 2.4 +3.7 +PointNet++ [3] +50K pts +20.1 0.1 53.7 1.9 +0.2 +0.9 +0.2 +0.9 +1.0 +0.0 72.0 18.7 41.8 5.6 62.3 16.9 46.5 13.8 30.0 6.0 +8.9 +SPLATNet [22] +50K pts +22.8 +1 +66.6 0.0 +0.0 +0.0 +0.0 +0.0 +0.0 +0.0 70.4 0.8 41.5 0.0 68.7 27.8 72.3 35.9 35.8 13.8 0.0 +TangentConv [23] +50K pts +35.9 0.3 86.8 1.3 12.7 11.6 10.2 17.1 20.2 0.5 82.9 15.2 61.7 9.0 82.8 44.2 75.5 42.5 55.5 30.2 22.2 +LatticeNet [24] +50K pts +52.9 +7 +92.9 16.6 22.2 26.6 21.4 35.6 43.0 46.0 90.0 59.4 74.1 22.0 88.2 58.8 81.7 63.6 63.1 51.9 48.4 +RandLA-Net [5] +50K pts +53.9 1.3 94.2 26.0 25.8 40.1 38.9 49.2 48.2 7.2 90.7 60.3 73.7 20.4 86.9 56.3 81.4 61.3 66.8 49.2 47.7 +KPConv [8] +50K pts +58.8 3.8 96.0 30.2 42.5 33.4 44.3 61.5 61.6 11.8 88.8 61.3 72.7 31.6 90.5 64.2 84.8 69.2 69.1 56.4 47.4 +BAAF-Net [25] +50K pts +59.9 4.8 95.4 31.8 35.5 48.7 46.7 49.5 55.7 33.0 90.9 62.2 74.4 23.6 89.8 60.8 82.7 63.4 67.9 53.7 52.0 +RangeNet53++ [14] 64 × 2048 52.2 12 91.4 25.7 34.4 25.7 23.0 38.3 38.8 4.8 91.8 65.0 75.2 27.8 87.4 58.6 80.5 55.1 64.6 47.9 55.9 +MINet [26] +64 × 2048 55.2 24 90.1 41.8 34.0 29.9 23.6 51.4 52.4 25.0 90.5 59.0 72.6 25.8 85.6 52.3 81.1 58.1 66.1 49.0 59.9 +3D-MiniNet [27] +64 × 2048 55.8 28 90.5 42.3 42.1 28.5 29.4 47.8 44.1 14.5 91.6 64.2 74.5 25.4 89.4 60.8 82.8 60.8 66.7 48.0 56.6 +SqueezeSegV3 [15] 64 × 2048 55.9 +6 +92.5 38.7 36.5 29.6 33.0 45.6 46.2 20.1 91.7 63.4 74.8 26.4 89.0 59.4 82.0 58.7 65.4 49.6 58.9 +SalsaNext [16] +64 × 2048 59.5 24 91.9 48.3 38.6 38.9 31.9 60.2 59.0 19.4 91.7 63.7 75.8 29.1 90.2 64.2 81.8 63.6 66.5 54.3 62.1 +FIDNet [21] +64 × 2048 59.5 29 93.9 54.7 48.9 27.6 23.9 62.3 59.8 23.7 90.6 59.1 75.8 26.7 88.9 60.5 84.5 64.4 69.0 53.3 62.8 +Meta-RangeSeg [28] 64 × 2048 61.0 26 93.9 50.1 43.8 43.9 43.2 63.7 53.1 18.7 90.6 64.3 74.6 29.2 91.1 64.7 82.6 65.5 65.5 56.3 64.2 +Lite-HDSeg [18] +64 × 2048 63.8 20 92.3 40.0 55.4 37.7 39.6 59.2 71.6 54.1 93.0 68.2 78.3 29.3 91.5 65.0 78.2 65.8 65.1 59.5 67.7 +LENet(Ours) +64 × 2048 64.2 26 93.9 55.7 48.6 40.1 44.0 67.4 64.4 27.7 92.4 68.8 78.2 32.3 92.1 68.8 84.5 69.4 69.1 59.7 62.4 +TABLE II +THE PERFORMANCE COMPARISON ON SEMANTICKITTI MULTIPLE SCANS BENCHMARK. THE ITEM WITH ARROW INDICATES THE MOVING CLASS +Methods +mean-IoU +FPS (Hz) +car +bicycle +motorcycle +truck +other-vehicle +person +bicyclist +motorcyclist +road +parking +sidewalk +other-ground +building +fence +vegetation +trunk +terrain +pole +traffic sign +car +−→ +bicyclist +−−−−→ +person +−−−→ +motorcyclist +−−−−−−−→ +other-vehicle +−−−−−−−−→ +truck +−−→ +TangentConv [23] +34.1 +- +84.9 2.0 18.2 21.1 18.5 1.6 0.0 0.0 83.9 38.3 64.0 15.3 85.8 49.1 79.5 43.2 56.7 36.4 31.2 40.3 1.1 6.4 +1.9 30.1 42.2 +DarkNet53Seg [1] +41.6 +- +84.1 30.4 32.9 20.2 20.7 7.5 0.0 0.0 91.6 64.9 75.3 27.5 85.2 56.5 78.4 50.7 64.8 38.1 53.3 61.5 14.1 15.2 0.2 28.9 37.8 +SpSequenceNet [29] +43.1 +3 +88.5 24.0 26.2 29.2 22.7 6.3 0.0 0.0 90.1 57.6 73.9 27.1 91.2 66.8 84.0 66.0 65.7 50.8 48.7 53.2 41.2 26.2 36.2 2.3 +0.1 +TemporalLidarSeg [30] 47.0 30 92.1 47.7 40.9 39.2 35.0 14.4 0.0 0.0 91.8 59.6 75.8 23.2 89.8 63.8 82.3 62.5 64.7 52.6 60.4 68.2 42.8 40.4 12.9 12.4 2.1 +TemporalLatticeNet [31] 47.1 6.5 91.6 35.4 36.1 26.9 23.0 9.4 0.0 0.0 91.5 59.3 75.3 27.5 89.6 65.3 84.6 66.7 70.4 57.2 60.4 59.7 41.7 9.4 48.8 5.9 +0.0 +Meta-RangeSeg [28] +49.7 22 90.8 50.0 49.5 29.5 34.8 16.6 0.0 0.0 90.8 62.9 74.8 26.5 89.8 62.1 82.8 65.7 66.5 56.2 64.5 69.0 60.4 57.9 22.0 16.6 2.6 +LENet(Ours) +53.0 25.3 92.4 57.0 52.1 38.5 47.0 19.0 0.0 4.3 92.0 68.6 77.3 29.9 90.2 63.4 83.4 68.1 67.6 58.0 62.9 75.2 65.2 62.6 22.5 25.7 2.0 +where P c +b and Rc +b define the precision and recall of predicted +boundary image ˆyb to real one yb for class c. The boundary +image is computed as follows: +yb = pool (1 − y, θ0) − (1 − y) +ˆyb = pool (1 − ˆy, θ0) − (1 − ˆy) +(6) +where pool(·, ·) employs a pixel-wise max-pooling on a +sliding window of size θ0. In addition, we set θ0 = 3. +In our proposed network, the decoder fusion the last three +different dimensional feature map to conduct the output, +which cause it relies heavily on the last three different dimen- +sional feature. Therefore, we use the auxiliary segmentation +head to further refine our proposed network accuracy. These +auxiliary segmentation head compute the weighted loss to- +gether with the main loss. Meanwhile, each loss has the +corresponding weight since the different dimensional feature +maps have different expressive power, which is different from +[15], [35]. The final loss function can be defined as, +Ltotal = Lmain + +3 +� +i=1 +λiL(yi, ˆyi) +(7) +where Lmain is the main loss, yi is the semantic output +obtained from stage i, and ˆyi represents the corresponding +semantic label. L(·) is computed according to Equation 2. In +our implementation, we set λ1 = 1, λ2 = 1 and λ3 = 0.5, +empirically. +E. Implementation details. +We use PyTorch [36] to implement our method and do +all experiments on a PC with 4 NVIDIA RTX 3090 GPUs. +We train network for 50 epochs with initial learning rate +2e−3, which is dynamically adjusted by a cosine annealing +scheduler [37]. The batch size is set to 8, the height and +width of the range image are set to H = 64 and W = 2048, +respectively. The optimizer is AdamW [38] with the default +configuration in Pytorch. During training process, we adopt +random rotation, random point dropout, and flipping the 3D +point cloud to perform data augmentation. +IV. EXPERIMENT +A. Experiment Setups +Datasets. We train and evaluate our network on Se- +manticKITTI dataset, which is a large-scale dataset for the +task of point cloud segmentation of autonomous driving +scenes. It provides the dnese point-wise annotation for 22 se- +quences (43,551 scans) in KITTI Odometry [39] Benchmark. +Sequence 00 to 10 (19,130 scans) are used for training, 11 +to 21 (20,351 scans) for testing. We follow the setting in [1], +and use sequence 08 (4,071 scans) for validation. To evaluate + +(a) Prediction(FIDNet) +(b) Prediction(ours) +(c) Ground Truth +(d) Error Point(FIDNet) +(e) Error Point(ours) +Fig. 4. +Qualitative analysis on the SemanticKITTI validation set (sequence 08). Where (a) and b are the predictions of FIDNet and our method respectively, +c is the semantic segmentation ground truth, d and e are segmentation error maps of FIDNet and our methods, where red indicates wrong prediction. +the effectiveness of our proposed approach, we submit the +output the online evaluation website to obtain the results on +the testing set. +Evaluation Metrics. To faciliate the fair comparison, we +evaluate the performance of different methods with respect +to the mean intersection over union metric (mIoU), which is +defined as below: +mIoU = 1 +n +n +� +c=1 +TPc +TPc + FPc + FNc +, +(8) +where TPc, FPc, and FNc represent true positive, false +positive, and false negative predictions for the class c. +B. Evaluation Results and Comparisons +Table I and Table II show the quantitative results of recent +available and published methods on SemanticKITTI single +scan benchmark and multiple scans benchmark, respectively. +It can be seen from tables that our presented method +achieves the state-of-the-art performance compared to point- +based and image-based on both the single scan benchmark +(64.2%mIoU) and multiple scan benchmark (53.0%mIOU). +Meanwhile, it’s worthy of mentioning that our proposed +network is very lightweight with around 4.7 M parameters +and fast with 24FPS while maintaining high accuracy. +To better visualize the improvements of our proposed +model over the baseline, we provide the qualitative com- +parison exaples in Fig. 4 We compare FIDNet vs LENet in +terms of the generated error map in the same data frame. +It can be seen that our presented method has a significant +improvement over FIDNet. +C. Ablation Studies +In this section, we conducted several ablation experiments +on the SemanticKITTI validation set (sequence 08) to ex- +amine the improvements of each individual module in our +proposed network. For the validation of each setup, the +size of input range image is 64 × 2048, the channel of +input range image is 5. Table III shows the total number +TABLE III +ABLATION STUDY EVALUATED ON SEMANTICKITTI VALIDATION SET. +Baseline +Row +MSCA +IAC +Boundary +Loss +Auxiliary +Loss +mIou +Params(M) +Baseline +1 + + + + +58.3 +6.05 +Ours +2 + + + + +59.3 +5.28 +3 + + + + +59.9 +4.74 +4 + + + + +61.1 +4.74 +5 + + + + +63.1 +4.74 +of model parameters with the corresponding mIoU scores +on the SemanticKITTI validation set. +For fair comparison, we firstly treat FIDNet the baseline +method, which has the similar network structure as ours. +Next, we replace the baseline with proposed MSCA and +ICA in Section III. Finally, we add the boundary loss and +auxiliary loss one by one, to examine the their effectiveness +for our network. As shown in Table III, our network obtains +over 1.0% improvement in accuracy when replacing the basic +block of baseline, which demonstrates that the multi-scale +convolution attention is effective. Next, IAC performs better +than the network around 1.0% after replacing the original +decoder. Additionally, the boundary loss achieves over 1.2% +performance gain and the auxiliary loss gain over 2.0% +performance jump. Finally, our presented LENet approach +has over 4.8% improvement and decreases around 25% +parameters comparing to the baseline, which demonstrate the +efficacy of each module. +V. CONCLUSIONS +In this paper, We presented LENet, a lightweight and +efficient real-time CNN model for LiDAR point cloud seg- +mentation task. 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Urtasun, “Are we ready for autonomous +driving? the kitti vision benchmark suite,” in 2012 IEEE conference +on computer vision and pattern recognition. +IEEE, 2012, pp. 3354– +3361. + diff --git a/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/load_file.txt b/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc78396a76427ee5c0b4bbc9091484858a650c48 --- /dev/null +++ b/PNE3T4oBgHgl3EQfCQnQ/content/tmp_files/load_file.txt @@ -0,0 +1,1094 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf,len=1093 +page_content='LENet: Lightweight And Efficient LiDAR Semantic Segmentation Using Multi-Scale Convolution Attention Ben Ding1 and Ji-Chao Jiao2 Abstract— LiDAR semantic segmentation can provide vehi- cles with a rich understanding of scene, which is essential to the perception system in robotics and autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In this paper, we propose LENet, a lightweight and efficient projection- based LiDAR semantic segmentation network, which has an encoder-decoder architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The encoder consists of a set of MSCA module, which is a simple convolutional attention mod- ule to capture multi-scale feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The decoder consists of IAC module, which uses bilinear interpolation to upsample the multi-resolution feature maps and a single convolution layer to integrate the previous and current dimensional features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' IAC is very lightweight and dramatically reduces the complexity and storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Moreover, we introduce multiple auxiliary segmentation heads to further refine the network accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We have conducted detailed quantitative experiments, which shows how each component contributes to the final performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We evaluate our approach on well known public benchmarks (SemanticKITTI), which demonstrates our proposed LENet is more lightweight and effective than state-of-the-art semantic segmentation approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Our full implementation will be available at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='com/fengluodb/LENet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Index Terms— LiDAR point clouds, 3D semantic segmenta- tion, LiDAR perception, Autonomous Driving I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' INTRODUCTION Environment perception can help vehicles understand the surrounding scene, which is essential to autonomous driv- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' LiDAR and RGB cameras are common sensors in the perception system of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Compared to cameras, LiDAR sensor isn’t affected by lighting and weather conditions, thus being more robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Meanwhile, compared to 2D image, LiDAR point cloud can accurately describe the structure of an object, and provide vehicles with a geometry- accurate representation of the surroundings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Therefore 3D point cloud analysis has drawn more and more attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Especially, point cloud semantic segmentation aims to assign the labels for each point, which helps vehicles gain a rich understanding of the scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Therefore, point cloud semantic segmentation is becoming an interesting research topic in both academic and industrial communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Due to the disorder and irregularity of 3d point cloud, we can’t directly perform standard Convolution Neural Networks on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To tackle the problem, extensive research efforts have been devoted to point cloud semantic segmentation during the past year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Point-based method [2]–[5] directly extract features from the raw point cloud, which can reduce the impact of computational complexity and noise errors in the pre-processing process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' However, they usually have high computational complexity and limited processing speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Voxel-based methods convert the irregular pint clouds into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='regular grid representation so that the 3D convolutional neu- ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='LatticeNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Ours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Real-time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Single Scan Semantic Segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='point-wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='projection-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Inference time(ms) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='mIoU on SemanticKITTI test(%) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='SpSequenceNet ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Temporal-LidarSeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Temporal-LatticeSeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Meta-RangeSeg ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Ours ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Real-time ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Multiple Scans Semantic Segmentation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='point-wise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='projection-based ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Accuracy(mIoU) versus inference time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Our presented LENet obtains the promising results on both single scan and multiple scans semantic segmentation in SemanticKITTI test dataset [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' ral networks can be employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Although voxel-base methods can achieve the state-of-the-art accuracy, they mainly suffer from heavy computations especially for large-scale LiDAR point clouds in outdoor scenes of autonomous driving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Projection-based methods transform the raw point cloud into an 2D range image by spherical projection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Compared to point-based methods and voxel-based meth- ods, projection-based have a superior inference speed and nice accuracy performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Meanwhile, they have recently received increasing attention since the great success of fully convolutional networks on image semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In this work, we presents a lightweight and efficient projection-based LiDAR semantic segmentation network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Experimental results on well known public benchmarks (SemanticKITTI) demonstrates that our network have higher accuracy performance and can run in real time (as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 2) with less parameters than prior works (as shown in table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In summary, the main contribution of this paper as follows: leftmargin=* We present a simple muti-scale convolutional attention arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='04275v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='CV] 11 Jan 2023 module (MSCA), which can capture global and local context information in the full 360 degrees LiDAR scan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' A novel IAC module, which uses bilinear interpolation to upsample the multi-resolution feature maps and a single convolution layer to integrate the previous and current dimensional feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' IAC is very lightweight and dramatically reduces the complexity and storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' By introducing multiple auxiliary segmentation heads, we further refine the network accuracy without intro- ducing additional inference parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We conduct extensive experiments on the publicly avail- able datasets, SemanticKITTI [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The results show that our method achieves state-of-the-art performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' RELATED WORK With the prevalence a large-scale dataset [1], [6], [7] for the task of point cloud segmentation of autonomous driving scenes and the rapid development of deep learning, a wide range of 3D LiDAR point clouds semantic segmentation methods using deep learning have been proposed over the past years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Generally, they can be broadly categorized into four groups according to the representations of input data, include point, range map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Point-based methods directly process the raw 3D point cloud without applying any additional transformation or pre-processing, which are able to preserve the 3D spatial structure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The pioneering methods of this group are PointNet [2] and PointNet++ [3], which use shared MLPs to learn the properties of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In subsequent series of works, KPConv [8] develops deformable convolutions that can use arbitrary number of kernel points to learn local geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Howerver, these approaches have the disadvan- tages of high computational complexity and large memory consumption, which hinders them from the large-scale point cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' RandLA-Net [5] adopts a random sampling strategy and uses local feature aggregation to reduce the information loss caused by random operations, which considerably im- prove the efficiency of point cloud processing and decrease the use of memory consumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Voxel-based Methods Voxel-based approach to convert point clouds into voxels for processing, which can effec- tively solve the irregularity problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The early voxel-based methods firstly transform a point cloud into 3D voxel repre- sentations, then use the standard 3D CNN to predict semantic labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' However, the regular 3D convolution requires the huge memory and heavy computational power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Minkowski [9] CNN chose to use sparse convolution instead of standard 3D convolution and other standard neural network to reduce the computational cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Cylinder3D [10] adopts 3D space partition and designs an asymmetrical residual block to reduce computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' AF2S3Net [11] achieves state-of-the- art of voxel-methods, which proposes two novel attention blocks name Attentive Feature Fustion Module(AF2M) and Adaptive Feature Fustion(ASFM) to effectively learn local features and global contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Projection-based Methods project 3D point clouds into 2D image space, which can take advantage of a large amount of advanced layers for image feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' SqueezeSeg [12] proposes spherical projection which maps the scatter 3D laser points into 2D Range-Image, then uses the lightweight model SqueezeNet and CRF for segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Subsequently, SqueezeSegV2 [13] proposes context aggregation module (CAM) to aggregate contextual information from a larger perceptual field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' RangeNet++ [14] integrates Darknet into SqueezeSeg and proposes an efficient KNN post-processing method to predict labels for point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' SqueezeSegV3 [15] pro- poses Spatially-Adaptive Convolution (SAC) with different filters depending on the location of the input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Sal- saNext [16] inherits the encoder-decoder architecture from SalsaNet [17] and presents an uncertainty-aware mechanism for point feature leaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Lite-HDSeg [18] achieves state-of- the-art performance by introducing three different modules, Inception-like Context Module, Multi-class Spatial Propaga- tion Network, and a boundary loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' METHOD A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Range Image Representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Using the spherical projection approach, we can transform the unstructured point cloud into an ordered range image representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The advantages of range representation are that it can use the effective 2D convolutional operation for fast training and inference, and it can facilitate the mature deep learning technologies that have been well studied in image-based tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In the range image representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' each LiDAR point p = (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' z) with Cartesian coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' a spherical mapping R3 → R2 is used to transform it to image coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' as below: � u v � = � 1 2 � 1 − arctan(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' x) π−1� w � 1 − � arcsin(z r−1) + fup � f−1� h � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' (1) where (u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' v) are image coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' w) are the height and width of the desired range image representation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' f = fup + fdown is the vertical field-of-view of the sensor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' and r = � x2 + y2 + z2 is the range of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Convolution Attention Encoder Multi-scale features play a significant role in semantic seg- mentation since semantic segmentation tasks usually needs to process objects of different size in a singe image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' A common approach to extract multi-scale features is to use a combination of a set of convolutions having different receptive fields, then fusing these respective fields, such as [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Inspired by SegNext [19], we propose a novel multi- scale convolution attention module (MSCA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='3 (b), MSCA consists of three parts: a depth-wise convolution to aggregate local information, multi- branch depth-wise strip convolutions to capture multi-scale context, and an 1 × 1 convolution to model relationship between different channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, the output of 1 × 1 convolution is used as attention weights directly to re-weight the input of MSCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In addition, we adopt the pyramid structure for our encoder following [20], [21] and the Projection 2D Range Image 64 x 2048 x 5, (x,y,z,d,r) 3x3, channel = 64 3x3, channel = 128 3x3, channel = 128 Input Module Encoder1 Encoder2 Encoder3 Encoder4 Backbone Network IAC IAC IAC IAC Classification Head KNN The Raw Point Cloud Semantic Prediction Decoder Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Illustration of our proposed LENet framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The backbone network is consisted of MSCA and has the pyramid structure similar to ResNet34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The ICA module upsamples the low-dimensional feature maps to original size and aggregate it with the output of the previous IAC module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Then, the last classification head receives the feature maps from the last three ICA module and outputs the label of each point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, we use K Nearest Neighbors (KNN) to do post-processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' d, 5 x 5 d, 1 x 3 d, 1 x 5 d, 1 x 7 d, 3 x 1 d, 5 x 1 d, 7 x 1 1x1 Convolution Attention Multi-scale Feature (a) The BasicBlock (b) MSCA 3x3 SiLU BN SiLU MSCA 1x1 1x1 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Illustration of the BasicBlock that is used to build the encoder and the proposed MSCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Here, d means a depth-wise convolution, k1 × k2 means the kernel size of the convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We extract multi-scale features using the convolutions and then utilize them as attention weights to reweight the input of MSCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' building block in the encoder is composed with a 3 × 3 convolution layer and MSCA as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' IAC Decoder To design a simple and effective decoder, We investigate several different decoder structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In [14]–[16], they use standard transpose convolutions or pixel-shuffle to produce the upsampled feature maps, then using a set of convo- lution to decoder the feature maps, which is effective but computationally heavy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In FIDNet, it’s decoder uses FID (fully interpolation decoding) to decode the semantics of different levels, then using a classification head to fuse these semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Although FID is completely parameter-free, it doesn’t have the ability to learn from feature, which makes model’s performance excessively depend on the classification head.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Besides, FIDNet’s classification fuse too much low- level information and hurts the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In this work, we presented a lightweight decoder as depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The IAC module contains two parts: a bilinear interpolation to upsample the feature maps which come from the encoder, a 3 × 3 convolution to fuse the information from the encoder and the previous IAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, we use the point-wise convolution to fuse the features from the last three IAC modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In three different decoder, our decoder has fewest parameters and best performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Loss Function In this work, we train the proposed neural network with three different loss functions, namely weighted cross-entropy loss Lwce, Lov´asz loss Lls and boundary loss Lbd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, our total loss is following: L = w1Lwce + w2Lls + w3Lbd, (2) w1, w2 and w3 are the weights with respect to each loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In our implementation, we set w1 = 1, w2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='5 and w3 = 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Three loss functions account for three different problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To cope with the imbalanced classes problem, the weighted cross-entropy loss Lwce [32] is employed to maximize the prediction accuracy for point labels, which is able to balance the distributions among different classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' It’s defined as Lwce(y, ˆy) = − � i 1 √fi p (yi) log (p (ˆyi)) , (3) where yi represents the ground truth, and ˆyi is prediction and fi is the frequency of the ith class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To solve the problem of optimizing the intersection-over- union(IoU), the Lov´asz loss Lls [33] is used to maximize the intersection-over-union (IoU) score that is commonly used to in performance evaluation on semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' It’s defined as: Lls = 1 |C| � c∈C ∆Jc(m(c)), mi(c)= � 1−xi(c) if c=yi(c) xi(c) otherwise (4) where |C| is the class number, ∆Jc represents the Lov´asz extension of the Jaccard index, xi(c) ∈ [0, 1] and yi(c) ∈ {−1, 1} hold the predicted probability and ground truth label of pixel i for class c, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To account for the blurred segmentation boundaries prob- lem as suggested in [18], the boundary loss function Lbd [34] is used for LiDAR semantic segmentation, which can be formulated defined as follows: Lbd(y, ˆy) = 1 − 2P c b Rc b P c b + Rc b , (5) TABLE I THE PERFORMANCE COMPARISON ON SEMANTICKITTI MULTIPLE SINGLE SCAN BENCHMARK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Methods Size mean-IoU FPS (Hz) car bicycle motorcycle truck other-vehicle person bicyclist motorcyclist road parking sidewalk other-ground building fence vegetation trunk terrain pole traffic-sign PointNet [2] 50K pts 14.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0 where P c b and Rc b define the precision and recall of predicted boundary image ˆyb to real one yb for class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The boundary image is computed as follows: yb = pool (1 − y, θ0) − (1 − y) ˆyb = pool (1 − ˆy, θ0) − (1 − ˆy) (6) where pool(·, ·) employs a pixel-wise max-pooling on a sliding window of size θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In addition, we set θ0 = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In our proposed network, the decoder fusion the last three different dimensional feature map to conduct the output, which cause it relies heavily on the last three different dimen- sional feature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Therefore, we use the auxiliary segmentation head to further refine our proposed network accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' These auxiliary segmentation head compute the weighted loss to- gether with the main loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Meanwhile, each loss has the corresponding weight since the different dimensional feature maps have different expressive power, which is different from [15], [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The final loss function can be defined as, Ltotal = Lmain + 3 � i=1 λiL(yi, ˆyi) (7) where Lmain is the main loss, yi is the semantic output obtained from stage i, and ˆyi represents the corresponding semantic label.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' L(·) is computed according to Equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' In our implementation, we set λ1 = 1, λ2 = 1 and λ3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='5, empirically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Implementation details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We use PyTorch [36] to implement our method and do all experiments on a PC with 4 NVIDIA RTX 3090 GPUs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We train network for 50 epochs with initial learning rate 2e−3, which is dynamically adjusted by a cosine annealing scheduler [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The batch size is set to 8, the height and width of the range image are set to H = 64 and W = 2048, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The optimizer is AdamW [38] with the default configuration in Pytorch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' During training process, we adopt random rotation, random point dropout, and flipping the 3D point cloud to perform data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' EXPERIMENT A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Experiment Setups Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We train and evaluate our network on Se- manticKITTI dataset, which is a large-scale dataset for the task of point cloud segmentation of autonomous driving scenes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' It provides the dnese point-wise annotation for 22 se- quences (43,551 scans) in KITTI Odometry [39] Benchmark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Sequence 00 to 10 (19,130 scans) are used for training, 11 to 21 (20,351 scans) for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' We follow the setting in [1], and use sequence 08 (4,071 scans) for validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To evaluate (a) Prediction(FIDNet) (b) Prediction(ours) (c) Ground Truth (d) Error Point(FIDNet) (e) Error Point(ours) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Qualitative analysis on the SemanticKITTI validation set (sequence 08).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Where (a) and b are the predictions of FIDNet and our method respectively, c is the semantic segmentation ground truth, d and e are segmentation error maps of FIDNet and our methods, where red indicates wrong prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' the effectiveness of our proposed approach, we submit the output the online evaluation website to obtain the results on the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Evaluation Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To faciliate the fair comparison, we evaluate the performance of different methods with respect to the mean intersection over union metric (mIoU), which is defined as below: mIoU = 1 n n � c=1 TPc TPc + FPc + FNc , (8) where TPc, FPc, and FNc represent true positive, false positive, and false negative predictions for the class c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Evaluation Results and Comparisons Table I and Table II show the quantitative results of recent available and published methods on SemanticKITTI single scan benchmark and multiple scans benchmark, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' It can be seen from tables that our presented method achieves the state-of-the-art performance compared to point- based and image-based on both the single scan benchmark (64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='2%mIoU) and multiple scan benchmark (53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0%mIOU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Meanwhile, it’s worthy of mentioning that our proposed network is very lightweight with around 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='7 M parameters and fast with 24FPS while maintaining high accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' To better visualize the improvements of our proposed model over the baseline, we provide the qualitative com- parison exaples in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' 4 We compare FIDNet vs LENet in terms of the generated error map in the same data frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' It can be seen that our presented method has a significant improvement over FIDNet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Ablation Studies In this section, we conducted several ablation experiments on the SemanticKITTI validation set (sequence 08) to ex- amine the improvements of each individual module in our proposed network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' For the validation of each setup, the size of input range image is 64 × 2048, the channel of input range image is 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Table III shows the total number TABLE III ABLATION STUDY EVALUATED ON SEMANTICKITTI VALIDATION SET.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Baseline Row MSCA IAC Boundary Loss Auxiliary Loss mIou Params(M) Baseline 1 \x17 \x17 \x17 \x17 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='05 Ours 2 \x13 \x17 \x17 \x17 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='28 3 \x13 \x13 \x17 \x17 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='9 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='74 4 \x13 \x13 \x13 \x17 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='74 5 \x13 \x13 \x13 \x13 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='1 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='74 of model parameters with the corresponding mIoU scores on the SemanticKITTI validation set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' For fair comparison, we firstly treat FIDNet the baseline method, which has the similar network structure as ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Next, we replace the baseline with proposed MSCA and ICA in Section III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, we add the boundary loss and auxiliary loss one by one, to examine the their effectiveness for our network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' As shown in Table III, our network obtains over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0% improvement in accuracy when replacing the basic block of baseline, which demonstrates that the multi-scale convolution attention is effective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Next, IAC performs better than the network around 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0% after replacing the original decoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Additionally, the boundary loss achieves over 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='2% performance gain and the auxiliary loss gain over 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='0% performance jump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, our presented LENet approach has over 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content='8% improvement and decreases around 25% parameters comparing to the baseline, which demonstrate the efficacy of each module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' CONCLUSIONS In this paper, We presented LENet, a lightweight and efficient real-time CNN model for LiDAR point cloud seg- mentation task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Firstly, We uses encoder based on multi- scale convolution attention module to better capture the features of objects with varying size in LiDAR data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Then, We propose a simple and effective decoder for upsam- pling, which significantly reduces the complexity and storage cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' Finally, We trained our network with embed multiple auxiliary segmentation heads to further improve the power of learned feature without introduction of parameters and efficiency cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PNE3T4oBgHgl3EQfCQnQ/content/2301.04275v1.pdf'} +page_content=' The evaluations on the SemanticKITTI test dataset demonstrate that our proposed method achieves the state-of-the-art performance on both single scan and multiple scans semantic segmentation.' metadata={'source': 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+Ahmad Jamal + + + +Supervisor +Dr. Musa Al-Yaman + +Date +26/05/2021 + +THE:UNIVERSITYOF +JORDAN1Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +i +ABSTRACT +Deaf people are using sign language for communication, and it is a combination of gestures, +movements, postures, and facial expressions that correspond to alphabets and words in spoken +languages. The proposed Arabic sign language recognition model helps deaf and hard hearing people +communicate effectively with ordinary people. +The recognition has four stages of converting the alphabet into letters as follows: Image Loading stage, +which loads the images of Arabic sign language alphabets that were used later to train and test the +model, a pre-processing stage which applies image processing techniques such as normalization, +Image augmentation, resizing, and filtering to extract the features which are necessary to accomplish +the recognition perfectly, a training stage which is achieved by deep learning techniques like CNN, a +testing stage which demonstrates how effectively the model performs for images did not see it before, +and the model was built and tested mainly using PyTorch library. +The model is tested on ArASL2018, consisting of 54,000 images for 32 alphabet signs gathered from +40 signers, and the dataset has two sets: training dataset and testing dataset. We had to ensure that +the system is reliable in terms of accuracy, time, and flexibility of use explained in detail in this report. +Finally, the future work will be a model that converts Arabic sign language into Arabic text. +. + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +ii +TABLE OF CONTENTS + +ABSTRACT ........................................................................................................................................ i +LIST OF FIGURES ............................................................................................................................iv +LIST OF TABLES ..............................................................................................................................vi +GLOSSARY ..................................................................................................................................... vii +Chapter 1 Introduction ....................................................................................................................... 1 +1.1 Background ............................................................................................................................. 1 +1.2 Problem Definition ................................................................................................................... 2 +1.3 Literature Review..................................................................................................................... 2 +1.4 Aims and Objectives ................................................................................................................ 6 +1.5 Report Organization................................................................................................................. 7 +Chapter 2 Design Considerations ...................................................................................................... 8 +2.1 Design Options ........................................................................................................................ 8 +2.1.1 Computer Vision Techniques ............................................................................................ 8 +2.1.2 Software ......................................................................................................................... 16 +2.1.3 Python Frameworks ........................................................................................................ 19 +2.1.4 Hardware ....................................................................................................................... 19 +2.2 Design Constraints and Standards ......................................................................................... 21 +Chapter 3 Model architecture, Training and testing .......................................................................... 23 +3.1 Dataset Examination.............................................................................................................. 26 +3.1.1 The Nature of Images ..................................................................................................... 26 +3.1.2 ArASL2018: Arabic Alphabets Sign Language Dataset ................................................... 27 +3.2 Data Splitting ......................................................................................................................... 29 +3.3 Define the Model ................................................................................................................... 31 +3.3.1 Multilayer Model (ANN) ................................................................................................... 31 +3.3.2 CNN Model..................................................................................................................... 34 +3.3.3 ResNet-18 ...................................................................................................................... 39 + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +iii +3.4 Define A Loss Function and Optimizer ................................................................................... 41 +Chapter 4 Design Testing and Results ............................................................................................. 47 +4.1 Model Training and Validation ................................................................................................ 47 +4.1.1 Model Training................................................................................................................ 47 +4.1.2 Model Validation ............................................................................................................. 47 +4.1.3 Graphs: Training, Validation Accuracy and Loss ............................................................. 48 +4.2 Testing and Results ............................................................................................................... 52 +4.3 Model Inferencing .................................................................................................................. 56 +4.4 New Collected Dataset: ArSLA-2021 Dataset......................................................................... 58 +4.4.1 Overview ........................................................................................................................ 58 +4.4.2 General Notes for ArSLA-2021 Dataset: ......................................................................... 59 +4.5 System Limitations and Compliance with Design Constraints ................................................. 60 +4.5.1 System Limitations: ........................................................................................................ 60 +4.5.2 Design Constrains Compliance ....................................................................................... 60 +4.6 Solution Impact ...................................................................................................................... 61 +4.6.1 Societal Impact ............................................................................................................... 61 +4.6.2 Economic Impact ............................................................................................................ 62 +4.6.3 Environmental Impact ..................................................................................................... 62 +4.6.4 Global Impact ................................................................................................................. 62 +Chapter 5 Conclusion And Future Work ........................................................................................... 63 +5.1 Conclusion ............................................................................................................................ 63 +5.2 Problems Faced .................................................................................................................... 63 +5.3 Recommendations for Future Work........................................................................................ 63 +References ...................................................................................................................................... 64 + + + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +iv +LIST OF FIGURES + +Figure 1-1: Signs of Arabic alphabets ................................................................................................ 1 +Figure 1-2: Sign language recognition system .................................................................................... 3 +Figure 2-1: KNN Classification ........................................................................................................... 8 +Figure 2-2: Sigmoid Function (Logistic Function) ................................................................................ 9 +Figure 2-3: Mapping a Cat Image to Class Scores .......................................................................... 11 +Figure 2-4: Example for Softmax and SVM ...................................................................................... 13 +Figure 2-5: Simple Neuron ............................................................................................................... 13 +Figure 2-6: ANN Architecture, Including ReLU and Softmax ............................................................. 14 +Figure 2-7: An Example of A Convolutional Operation ...................................................................... 15 +Figure 2-8: Machine Learning Processes Which LabVIEW Can Provide ........................................... 17 +Figure 2-9: CPU Operations............................................................................................................. 20 +Figure 2-10: Performance Vs Amount of Data .................................................................................. 21 +Figure 3-1: Machine Learning System Cycle .................................................................................... 23 +Figure 3-2: Flowchart of Recognition Model ..................................................................................... 24 +Figure 3-3: (a) Grayscale Colour Space, (b) RGB Colour Space ...................................................... 26 +Figure 3-4: Samples For ArASL2018 Dataset .................................................................................. 27 +Figure 3-5: A Brief Description of Arabic Sign Classes ..................................................................... 27 +Figure 3-6: General processes on the dataset .................................................................................. 30 +Figure 3-7: ArASL2018 Dataset Histogram ...................................................................................... 31 +Figure 3-8: Multilayer Algorithm ....................................................................................................... 32 +Figure 3-9: Activation Functions and Their Derivatives ..................................................................... 33 +Figure 3-10: CNN Layers with Rectangular Local Receptive Fields .................................................. 35 +Figure 3-11: Connections Between Layers, Adapted from ................................................................ 35 +Figure 3-12: Convolutional Layers with Multiple Feature Maps ......................................................... 36 +Figure 3-13: Max Pooling and Mean Pooling with (2×2 Pooling Kernel, Stride 2, Zero Padding) ....... 37 +Figure 3-14: (a) Network Without Dropout, (b) Network With Dropout. .............................................. 38 + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +v +Figure 3-15: (a) Neuron at training, (b) Neuron at testing. ................................................................ 38 +Figure 3-16: CNN Architecture ......................................................................................................... 39 +Figure 3-17: Residual Learning ........................................................................................................ 40 +Figure 3-18: ResNet-18 Architecture, Adapted From ........................................................................ 40 +Figure 3-19: Mean Square Error ...................................................................................................... 42 +Figure 3-20: Mean Absolute Error .................................................................................................... 43 +Figure 3-21: Loss (BCE) vs. Predicted Probability ............................................................................ 44 +Figure 4-1: Progress of Average Training Loss of ANN .................................................................... 48 +Figure 4-2: Progress of Average Validation Loss of ANN ................................................................. 48 +Figure 4-3: Progress of Average Training Loss of CNN .................................................................... 49 +Figure 4-4: Progress of Average Validation Loss of CNN ................................................................. 50 +Figure 4-5: Progress of Average Training Loss of ResNet-18 ........................................................... 51 +Figure 4-6: Progress of Average Validation Loss of ResNet-18 ........................................................ 51 +Figure 4-7: Confusion Matrix of ResNet-18 ...................................................................................... 53 +Figure 4-8: Confusion Matrix of CNN ............................................................................................... 54 +Figure 4-9: Confusion Matrix of ANN ................................................................................................ 55 +Figure 4-10: Inferencing Flowchart ................................................................................................... 56 +Figure 4-11: (a) Input Image (Y𝒂), (b) Pre-processed Input Image.................................................... 57 +Figure 4-12: ArSL Alphabet Prediction. ............................................................................................ 57 +Figure 4-13: Arabic Sign Language Alphabets Samples ................................................................... 58 + + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +vi +LIST OF TABLES + +Table 1-1: Proposed system with different classifiers ......................................................................... 5 +Table 3-1: Comparison between SGD and ADAM ............................................................................ 46 +Table 4-1: Comparison Between Different Models’ Accuracies ......................................................... 52 +Table 4-2: Arabic Sign Language Alphabets, their numbers, and labels ........................................... 59 +Table 4-3: Inference Time Results ................................................................................................... 61 + + + +Design of Arabic Sign Language Recognition Model + + +Department of Mechatronics Engineering +Senior Design Graduation Project Report +vii +GLOSSARY +ABBREVIATION DESCRIPTION +ArSL +Arabic Sign Language +ANN +Artificial Neural Network +CNN +Convolution Neural Network +RNN +Recurrent Neural Network +ANFIS +Adaptive Neuro-Fuzzy Inference System +KNN +K Nearest Neighbour +SVM +Support Vector Machine +MLP +Multilayer Perceptron +GMM +Gaussian Mixture Model +LDA +Linear Discriminant Analysis +HOG +Histograms of Oriented Gradients +EHD +Edge Histogram Descriptor +DWT +Discrete Wavelet Texture +LBP +Local Binary Pattern +GLCM +Grey-Level Co-occurrence Matrix +BLEU +Bilingual Evaluation Understudy +TER +Translation Error Rate +ReLU +Rectified Linear Unit +SeLU +Scaled Exponential Linear Unit +AFOD +Arab Federation of the Deaf +ANFIS +Adaptive Neuro Fuzzy Inference +ArSLAT +Arabic Sign Language Automatic Translator +TPU +Tensor Processing Unit +ASICs +Application-Specific Integrated Circuits +CUDA +Compute Unified Device Architecture +SM +Streaming Multiprocessor +SGD +Stochastic Gradient Descent + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +1 +Chapter 1 INTRODUCTION +1.1 Background +According to the World Health Organization (WHO), around 466 million people with hearing loss issues, +and 34 million of them are children. It is claimed that by 2050 over 900 million people will have suffering +hearing loss [1]. Hard hearing people can hear up to a specific limited degree and unobvious by a +hearing aid. In contrast, deaf people cannot listen entirely due to head trauma, noise exposure, disease, +or genetic condition [2]. +Sign language is the means of communication between the deaf themselves and with ordinary people, +and every country has its own language. One of these languages is ArSL, used in the Arabic regions; +it was formally introduced in 2001 by the Arab Federation of the Deaf (AFOD) [3]. Sign Language +depends on hand movements and gestures to accomplish what you want. There are various dialects of +ArSL that differ from one country to another; it comprises 28 characters that the different dialects agree +on them. Signs of Arabic alphabets is shown in Figure 1-1. + +Figure 1-1: Signs of Arabic alphabets [4]. + + +2 +2 +2 +L +3 +3 +3 +E +e +JDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +2 +1.2 Problem Definition +The communication gap between ordinary people and deaf people is enormous, and we want to make +it tiny, but it is a long road so, the best way is to study the subject from scratch and absorb the basis of +it. As an initial point, we should learn in detail the signs of Arabic Alphabets to reduce sign language +learners' obstacles, but the matter is not simple for all learners. Many of them will be confused when +they learn a new field and may find this problematic. For this reason, we intend to build a model that +recognizes the alphabet sign from Arabic Sign Language Speakers and then interprets it into text. +1.3 Literature Review +In [5], Omar Al-Jarrah and Alaa Halawani used a collection of ANFIS networks. Each network is trained +to recognize one gesture. The system used images of bare hands, which allows the user to make the +interaction more natural. The subtractive clustering algorithm and the least-squares estimator are used +to identify the fuzzy inference system, and the training is achieved using the hybrid learning algorithm. +The achieved accuracy is 93.55% that resulted from recognizing the 30 Arabic Manual alphabets. +In [6], Khaled Assaleh and M. Al-Rousan used Polynomial Classifiers to recognize the Arabic Sign +Language Alphabet. It had seen that the Polynomial classifier has several advantages compared with +ANFIS-based classification when they work on the same data. The data had been collected from deaf +people and using the same corresponding feature set. The data collected by coloured Marked Glove- +based systems. Polynomial Classifiers showed Significant results over ANFIS based on misclassified +data patterns. Specifically, it has a 36% reduction when the methods were evaluated on the training +data and a 57% reduction when the systems were assessed on the test data. +In [7], the author split the process into three stages; a data collection stage, a feature extraction stage, +and a recognition stage using Hidden Markov Model (HMM). A vision-based methodology is used to +collect the data, and then we need to prepare the data to absorb the necessary features to classify it +using HMM. The collected data were 4500 signs from 15 samples with 300 signs for the single signer, +11 samples were taken for the training set; the accuracy obtained from the experiments is 88.73%. +This paper [8] shows an automated method for the translation of Arabic Sign Language alphabets. Its +data had been collected using images of bare hands. The Experiments showed that the ArSLAT, Arabic +Sign Language Automatic Translator, had an accuracy of 91.3% with 30 Arabic Alphabets. +In [9], the authors presented the stages they used to achieve recognition: skin detection, background +exclusion, face and hands extraction, feature extraction, and classification using Hidden Markov Model +(HMM). The dataset consists of 29 alphabet Arabic letters and numbers from 0to nine with different +brightness. They used 253 training images and 104 testing images with 640×480 pixels. The recognition +system is tested when dividing the handshape's rectangle surrounding it into 4, 9, 16, and 25 zones. At +16 zones, the recognition rate with 19 states reaches 100%, while at 4 and 9 zones cannot match 100%. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +3 +In [10], the author explained the nature of the dataset. It is images for the positive samples with the +hand sign in different scales, different illumination in the complex background for each hand posture, +and the negative samples images from the Internet which do not contain hand posture. Figure 1-2 +shows the stages of translation from Natural language into Sign language. + +Figure 1-2: Sign language recognition system [3]. +In [11], the system was created to recognize the Arabic sign words. The system consists of three stages, +which are Pre-processing, Feature extraction, and classification. Moreover, the training and testing +evaluation methods depend on the database of 23 signs that performed three signers. Each character +is repeated 50 times by the singers, and the training set consists of 70%. The testing set consists of +30%. The model was evaluated on three different frequency domains (viz. Fourier, Hartley, and Log- +Gabor transforms) for the feature extraction stage and assessed on three classifiers: KNN, SVM, and +MLP. The results showed that the best Arabic sign language recognition system is Hartley transform +using SVM classifier based on accuracy, 98.8%. Moreover, when sign images were segmented into 3*3 +segments, the accuracy raised to 99%. +In this paper [12], it is focused on feature extraction. The feature extraction techniques are utilized for +training the classifier. Sign language usually is dynamic where the upper part of the body, head, +shoulder, and hands have a movement while other parts are static. The feature must be utilized by +collecting high-contrast locations such as object edges and corners. +In [13], the authors designed a system to translate the ArSL alphabet gestures into text. The used +dataset is captured from different smartphones by 30 volunteers. Each volunteer worked on a subset +that has 30 images, so the dataset consists of 900 images. The authors used five descriptors to +recognize. When using the Histograms of Oriented Gradients (HOG) descriptor, the proposed ArSL +system accuracy is 63.56%. The accuracy of Edge Histogram Descriptor (EHD) is 42%. The accuracy +of Discrete Wavelet Texture Descriptor (DWT) is 8%. The accuracy of the Local Binary Pattern (LBP) +descriptor is 9.78%. The worst accuracy result is obtained using the 5 Gray-Level Co-occurrence Matrix +(GLCM) descriptor; the proposed ArSL system accuracy is 2.89%. +The authors in [14] present a system that translates isolated Arabic word signals to text with automatic +visual SLRs. The system consists of four stages to obtain the results: hand segmentation using the + +Training Data Set +Preprocessing +Feature Extraction +Text/Audio +Classification +ImageorVideo +Preprocessing +Feature Extraction +AcquisitionDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +4 +dynamic skin detector that depends on the face's colour, tracking using segmented skin points used to +recognize and track the hands with the head's help, extracting the geo-metrical features of the spatial +field. Finally, classification is carried out using Euclidean distance. As a result, the authors achieved a +discrimination rate of 97% using a training set of 300 videos and a test set of 150 videos bearing in +mind that 83% of words had different occlusions. These videos only contain 30 isolated words used in +the daily life of hard of hearing children. +The authors in [15] present a new benchmark dataset publicly accessed along with the Sign Language +Recognition algorithm. The SLR algorithm consists of three phases, which are hand segmentation, +hand shape sequence, and body motion description, and sign classification. Also, the sign classification +phase uses canonical correlation analysis and random forest classifiers. However, the dataset used for +the algorithm was 150 different signs collected from 21 signers using the Kinect v2 sensor. The total +sample is 7500 samples (150 signs * 5 signer groups * 10 samples per sign per group). Finally, the +algorithm achieved a state-of-the-art solution when rated on the public data sets. Also, the achieved +recognition accuracy is 55.57% evaluated on 150 ArSL signs. +The paper [16] starts sorting the sign language into two components; manual and non-manual signs. +The manual signs include hand position, orientation, shape, and trajectory. The non-manual signs +represent body motion and facial expressions. Convolutional Neural Network (CNN) is a deep learning +class employed in image classification; it makes the network quick to learn and find the complex pattern +simplicity. CNN still uses the Backpropagation and its derivatives training methods to learn from data. +The author used a dataset of images containing 2030 images of numbers (from 0 to 10) and 5839 +images of 28 letters of Arab sign language, i.e., 7869 RGB colour images with 256×256 pixels. These +images are taken from different signers and different luminosity intensities. +The authors in [17] present an Arabic Sign Recognition system to overcome finger occlusions and +missing data. The system uses two Leap Motion controllers for data acquisition since they detect hands +and fingers moving. After that, data is put together using the Dempster-Shafer (DS) theory of evidence. +A set of geometric features from both LMCs is chosen to feed them for the classification stage. Finally, +the Bayesian approach with a Gaussian Mixture Model (GMM) and a simple Linear Discriminant +Analysis (LDA) approach, used for classification. There are 2000 samples collected from two native +adult signers by repeated 100 isolated Arabic dynamic signs ten times for each singer. Then, 70% of +the dataset was used for training, and 30% used for testing. The submitted system is considered a +state-of-the-art-glove-based system and single-sensor, and it achieved about 92% recognition +accuracy. + + + +The authors in [4] proposed a real-time ArSL alphabets recognition system. It consists of four +convolution layers, four max-pooling layers, and five dropout layers. However, 54,049 images are used + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +5 +as a dataset for this system, consisting of 32 alphabets obtained from more than 40 participants. It is +divided into 64% for training, 16% for validation, and 20% for testing. Finally, the achieved recognition +accuracy was 97.6%. +The paper [18] shows a novel framework used to recognize isolated Arabic Sign Language words for +signer-independent. This framework depends on three stages to classify input videos. These three +stages are the DeepLabv3+ model used for hand semantic segmentation. The single-layer +convolutional self-organizing map is used to extract hand shape features representation. A deep +recurrent neural network is used to recognize the sequence of extracted feature vectors. The dataset +comprises 150 repetitions for each of the 23 words they used, taken from 3 signers. In conclusion, the +framework model achieves state-of-the-art performance with an average accuracy of 89.5%. +In [19], the authors exhibit sign language differently. Most researchers try to obtain the text rather than +the semantic. To recognize the word with its semantic, they combined CNN with a semantic layer, and +it maps the word to the meaning. A mobile camera picks the dataset in a different surrounding. The +model achieved good recognition accuracy of 88.87%. +In [20], M. M. Kamruzzaman creates a model to convert Arabic Sign Language images to letters by +CNN and then convert generated Arabic letters to Arabic Speech by Google Text to Speech. The CNN +model has 2 Convolution layers, and the first layer has 32 Kernels, and the second has 64 kernels. The +model also trained for 100 epochs on 100 images for every 31 letters and tested 25 images for each +letter. It got an accuracy of about 90% for the testing set. +In [21], the authors focused on the recognition of letters. The collected data was 900 coloured images +have been used to represent the 30 different hand gestures and have been used as a training set; +another 900 images have been taken and used as a test set. They developed the recognition system +and calculated its performance using Feed-Forward Neural Networks and various Recurrent Neural +Networks (RNN) types. The performance that they got is concluded in Table 1-1. + +Table 1-1: Proposed system with different classifiers [21]. +Classifier +Accuracy +Feed-Forward Neural Network +79.33% +Elman neural network +89.66% +Jordan neural network +84.56% +Fully recurrent neural network +95.11% + +The authors in [22] proposed the first ArSL recognition system that converts ArSL to Arabic sentences. +The machine translation system is Rule-based, and it has three stages; the input Arabic Sign language +word is processed for Morphological analysis then Syntactic analysis. Finally, transfer to Arabic + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +6 +sentences. However, the system used a corpus that has sentences that are used in health centres. It +has 600 sentences that consist of 3327 sign words with 593 unique sign words. The proposed dataset +is divided into training, validation, and testing datasets, with a percentage of 70%, 15%, and 15%, +respectively. The results of the system are calculated Manually and automatically. However, the manual +evaluation shows that 80% of the sentences are accurately translated, and 2 ArSL experts do the +evaluation. Also, it is evaluated automatically by BLEU and TER metrics and gets 0.39 and 0.45 +repetitively. +1.4 Aims and Objectives +Facilitation of deaf people's lives and making their communication more straightforward is what we aim +to achieve. The objective is to construct a simple link between deaf people and others by creating an +accurate automated model using deep learning to interpret sign language alphabets to text. We will +study the previous results that the others obtain, enhance the model's performance, and make a +prototype to test the model. In the future, we will do continuous feedback to diagnose any error and fix +it. We will expand our work to include the words and deal with full sentences. + + + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +7 + +1.5 Report Organization +In the rest of this work is organized as follows: Chapter 2 (Design Considerations), Chapter 3 (Model +architecture, Training and testing), Chapter 4 (Design Testing and Results), Chapter 5 (Conclusion And +Future Work). + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +8 +Chapter 2 DESIGN CONSIDERATIONS + +This chapter explains the software, frameworks, techniques, and alternatives that can also be needed +in the project. Also, we will show design constraints and standards. +2.1 Design Options +Image recognition requires complex calculations to accomplish it using the computer. So, we need +powerful techniques and appropriate software to achieve it. In general, computer vision can do this +smoothly, but not all computer vision techniques are suitable for image classification. Also, many +software and frameworks can be used in computer vision. +2.1.1 Computer Vision Techniques +1. K-Nearest-Neighbor Classification (KNN) +In [23]. K-Nearest Neighbour is considered supervised learning in which the features and labels are +given in the model. This technique depends on the closest distance between the point and the +predicted labels to classify the object. An unlabelled query point is assigned the label that has the +K-Nearest Neighbour. The classification process is calculated from most of its K Nearest +Neighbours. To classify the images, each image is converted to a fixed vector, then the distance +can be measured by any function; Euclidean distance is the most common function: +𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖ = √(𝑥 − 𝑦) ∙ (𝑥 − 𝑦) = √∑ +(𝑥𝑖 − 𝑦𝑖)2 +𝑚 +𝑖=1 + +(2.1) + + +Figure 2-1: KNN Classification [23]. + + + +- +- +4Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +9 +2. Linear Classifiers +"Linear classifiers classify data into labels based on a linear combination of input features. +Therefore, these classifiers separate data using a line or plane or a hyperplane. They are suitable +to classify the linear separable data." [24] +2.1 Logistic Regression (Binary Classification): +A statistical model can be used to evaluate (guess) the probability of an event depends on input +data. +For example, we have two classes, e.g., "dog" or "not dog" and those can be represented by 0 and +1. +It can be mathematically represented as follows: 𝑦̂ = 𝜎(𝑧) +𝜎(𝑧) = +1 +1 + ⅇ−𝑧 + +(2.2) + is the logistic function and +𝑧 = 𝑤𝑇𝑥 + 𝑏 + +(2.3) +And these parameters are as follows: +𝜔 : weight +𝑏 : bias +𝑥 : flattened feature input vector +The model takes x as an input, and the probability of the outputs 𝑦̂ = 𝜌(𝑦 = 1|𝑥) + + + + + + + + + + +Figure 2-2: Sigmoid Function (Logistic Function) + +1.0 - +0.5 +6 +0.0 +-8 +-6 +-4 +-2 +0 +2 +4 +6 +8 +2Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +10 +For the images x, the feature vector can be just the pixels' values in RGB channels, and it can represent +by a vector with one dimension. It can be resulted by flattening those three dimensions, and the resulted +size is 𝑛𝑥 = 𝑛ℎ𝑒𝑖𝑔ℎ𝑡 × 𝑛𝑤𝑖𝑑𝑡ℎ × 3. +The goal of this algorithm is to classify the images correctly, and this can do by training the model on +training samples that will change the values of 𝑤 and 𝑏. The optimal values of these parameters can +be justified when 𝑦̂(𝑖) most closely predicts 𝑦(𝑖). Where: +𝑦̂(𝑖) : predicted class value. +𝑦(𝑖) : correct class value. + +In practice, this model usually calculates the loss function: +𝐿 (𝑦̂(𝑖), 𝑦(𝑖)) = −[𝑦(𝑖) log(1 − 𝑦̂(𝑖)) + (1 − 𝑦(𝑖)) log(1 − 𝑦̂(𝑖))] +(2.4) + + +For each training example and minimizing the cost function, +𝐽(𝑤, 𝑏) = 1 +𝑚 ∑ 𝐿 (𝑦̂(𝑖), 𝑦(𝑖)) +𝑚 +𝑖=1 + + +(2.5) + + Overall m training examples. +𝜕𝐿 +𝜕𝑊𝑗 = (𝑦̂(𝑖) − 𝑦(𝑖))𝑥(𝑖) +𝑗 and +𝜕𝐿 +𝜕𝑊 = (𝑦̂(𝑖) − 𝑦(𝑖)) + +(2.6) + +Where 𝑗 = 1,2, . .. , 𝑛𝑥 labels the components of the feature vector. + +To get the optimal value of 𝑤 and 𝑏 ; 𝐽 should be minimized. It can be minimized numerically after +choosing initial values by changing them according to descent along the steepest gradient. + +𝑤𝑗 → 𝑤𝑗 − 𝛼 𝜕𝑗 +𝜕𝑤𝑗 + = 𝑤𝑗 − 𝛼 +𝑚 ∑ 𝜕𝐿 +𝜕𝑤𝑗 +𝑚 +𝑖=1 + + +(2.7) + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +11 +𝛼 is the learning rate (step size), which affects how large each step is taken in the direction of greatest +decrease in 𝐽. Choosing a good value for α is a subtle art (where the too-large value will affect the +training to be fast and the training may not converge steadfastly and too small value so the training will +be slow). +2.2 Softmax and SVM classifiers: +The linear classifier uses the below equation to learn the features of images and stores them in W, b: +𝑓(𝑥𝑖, 𝑊, 𝑏) = 𝑊 𝑥𝑖 + 𝑏 +(2.8) + + +𝑊 : Weights. +𝑏 : bias term. +𝑥𝑖 : input image. +𝑓(𝑥𝑖, 𝑊, 𝑏) : Score function. +W, b (module parameters) will be changed depending on the training dataset. The output module will +classify the image depending on its features (pixel value), and space will be divided by linear functions +[25]. + +Figure 2-3: Mapping a Cat Image to Class Scores [26]. +The above image is added as an example to clarify the idea of the linear classifier. For ease of +visualization, the image is assumed to have 4 pixels only. And 𝑊 is considered as a matrix with a size +of 3 × 4, where 3 is the class number, and 4 is the flattened input size to imply the matrix multiplication +between 𝑊 and 𝑥𝑖 . So, by doing the matrix multiplication of 𝑊 and 𝑥𝑖 then adding the bias 𝑏 so the +results will be the scores for each class. The 𝑊 in the image is bad, and the scores at the end claim +that the image is a dog not a cat. However, the 𝑊 will improve by train the model, and it may get better +results. +There are many loss functions that can be used. For example, the linear classifier model usually uses +a loss called the Multiclass Support Vector Machine (SVM) loss. So, the Multiclass loss can be shown +as below: + +stretchpixelsintosinglecolumn +0.2 +-0.5 +0.1 +2.0 +56 +1.1 +-96.8 +cat score +1.5 +1.3 +2.1 +0.0 +231 ++ +3.2 +437.9 +dog score +0 +0.25 +0.2 +-0.3 +24 +-1.2 +61.95 +input image +ship score +M +6 +2 +f(ci; W,b) +CiDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +12 + +𝐿𝑖 = ∑ max (0,𝑆𝑗 − 𝑆𝑦𝑖 + 𝛥) +𝑗≠𝑦𝑖 + +(2.9) +Where: +𝑆𝑗 : is the score for the jth class. +𝑆𝑦𝑖 : is the score for the ith class. +𝛥 : is the fixed margin. + +Another commonly used classifier is Softmax, which used cross-entropy loss. The function mapping +is still used 𝑓(𝑥𝑖;𝑤) = 𝑤𝑥𝑖 . And the cross-entropy loss has the following form: +𝐿𝑖 = − log ( ⅇ𝑓𝑦𝑖 +𝛴𝑗ⅇ𝑓𝑗) + +(2.10) +Where 𝑓𝑦𝑖: is the class score for the ith element. +Where 𝑓𝑗: is the class score for the jth element. + +And +𝑒𝑧𝑗 +𝛴𝑗𝑒𝑧𝑘 It is called Softmax Function, which has an input vector score (in z) squishes it to a vector of +values between zero and one (Probability), which sum to one. +The below images show the difference between SVM and Softmax classifier for the same input. Both +have the same mapping function, which resulted from the matrix multiplication. But there is a difference +in the interpretation of the score function. SVM interprets the class scores, and it encourages the correct +class to be the higher one by a margin than the other classes. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +13 + +Figure 2-4: Example for Softmax and SVM [26]. +3. Artificial Neural Networks +ANNs are simulated based on the brain's architecture. They consist of connected elements known +as an artificial neuron; it has one or multiple inputs and one output with either zero or one. Each +neuron is associated with a weight, and if their sum is more than the threshold, the neuron will +activate. The following equation will explain the mathematical representation: +𝑥 = +{ + 1, +∑ 𝑤𝑖𝑥𝑖 − 𝑏 +𝑖 + > 0 +0, +∑ 𝑤𝑖𝑥𝑖 − 𝑏 +𝑖 + ≤ 0 + +(2.11) +If we want to make the output smoother between zero and one, we will use the sigmoid function as +the following: +σ = +1 +1 + ⅇ∑ 𝑤𝑖𝑥𝑖−𝑏 +𝑖 + +(2.12) +Figure 2-5 shows a simple neuron with three inputs associated with its weights and the bias and +then applying the activation function to show the result. + +Figure 2-5: Simple Neuron [27]. + + +hinge loss (SVM) +-2.85 +matrix multiply + bias offset +max(0, -2.85 - 0.28 + 1) + +0.86 +max(0, 0.86 - 0.28 + 1) +0.01 +-0.05 +0.1 +0.05 +-15 +0.0 += +1.58 +0.28 +0.7 +0.2 +0.05 +0.16 +22 ++ +0.2 +0.0 +-0.45 +-0.2 +0.03 +-44 +-0.3 +cross-entropy loss (Softmax) +-2.85 +0.058 +0.016 +M +56 +b +exp +normalize +0.86 +2.36 +0.631 +- log(0.353) +Ci +(to sum += +to one) +1.04 +0.28 +1.32 +0.353 +Yi +221w1 +T2 +2 +b + output +Gm +T3 Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +14 +ANNs consist of layers as an input layer, one hidden layer or more, and an output layer. Each layer +consists of neurons that compute the weighted sum of their inputs then specify the output using +some of the activation functions like; sigmoid, ReLU, and SeLU. All the neurons in a layer are +considered an input to the followed layer. ANNs can recognize the output by modifying the weights +and biases each one epoch until minimizing the errors. We need to classify the result of each neuron +in the output layer to the predicted class. So, Softmax function can be used at the output layer. +Figure 2-6 shows the architecture of ANN that includes ReLU and Softmax. + + +Figure 2-6: ANN Architecture, Including ReLU and Softmax [28]. + + + + + + + + + + + + + + + +Softmax +Softmax +output layer +Hiddenlayer +(e.g., ReLU) +Input +ayer +X2Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +15 +4. Convolutional Neural Networks +CNN is the most dominant technique in deep learning that can use in computer vision tasks. CNN +is a mathematical model that consists of three types of layers as follows: +a) Convolutional Layer +It is an essential component of the CNN architecture used for feature extraction. Neurons in +one layer are connected to other neurons in their receptive field. The array that combines the +neurons is called the kernel, and it is typically formed as 3 × 3, but maybe choose 5 × 5 or +7 × 7. This architecture allows the low-level features to concentrate on one layer, then +assemble them into higher-level features in the next layer. +The operation above does not guarantee each kernel's centre to overlap the input layer's +outermost element. Padding, precisely Zero Padding, is a solution to avoid adding zeros around +the inputs that can overlap the outer element of the input layer. +Stride is "the distance between two consecutive Kernels.", the standard option of a stride is +one. Figure 2-7 shows an example of a convolutional operation with a kernel size 3 × 3, Zero +Padding, and a stride of one. + +Figure 2-7: An Example of A Convolutional Operation [29]. +b) Pooling Layer +The goal of this layer is to shrink the inputs to decrease the computations. Max pooling and +Mean pooling are common examples of pooling layer. Max pooling is the most popular form, +which takes the maximum value in the higher-level feature layer. Mean pooling takes the +average of all the elements in the higher-level feature layer. + + + +Element-wise product +Sumup +Kernel +Feature map +InputtensorDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +16 +c) Fully Connected Layer +This layer transforms the last convolutional layer into a one-dimensional array and connects to +one or more dense layers. A non-linear activation function follows the final fully connected layer +to classify the inputs according to the output probabilities. +5. Transfer Learning +Many computer vision cases have small datasets, so the training of the model will be invalid. The +popular approach to deal with this case is to use the transfer function. Transfer learning is a network +that comprises extensive data, and it was trained to absorb generally feature extraction of the image +classification task. Convnet, VGG16, ResNet, Inception, and Xception are examples of +architectures trained on ImageNet (1.4 million labelled images with 1,000 different classes). It is +preferred to choose the understood architecture for you, and no need-to-know new ideas. +2.1.2 Software +1. MATLAB +It is a programming platform that offers toolboxes to help engineers and scientists in academia and +industry to perform the solutions for various aspects of problems. The essence of MATLAB is a +matrix-based language that allows for progressing the calculations smoothly. MATLAB can deal +with data by analysing and visualizing it, improving existing algorithms to coincide with your +requirements, and creating models and apps from scratch [30]. MATLAB includes Many +applications and capabilities that can perform several functions as follows: +1- Applications [31]: +a. Image Processing and Computer Vision: Processing of images and videos using several +techniques to build any visual model. +b. Data Science: Use machine learning to predict and label the data. +c. Deep Learning: Apply deep neural networks and prepare the related data. +d. Signal Processing: Convert the signal and prepare it to analyse. + +2- Capabilities [31]: +a. Algorithm Development: Improve or build algorithms for several tasks. +b. Cloud Computing: Run public clouds like; AWS and AZURE on MathWorks cloud. +c. Data Acquisition: Gain the data from an external source like a camera. +d. GPU Computing: Offer using NVIDIA CUDA to accelerate the training. +e. Parallel Computing: Use CPUs, GPUs, and TPUs simultaneously in large systems. +f. +Real-Time Simulation and Testing: Apply the hardware systems in real-time. + + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +17 +MATLAB is a useful software for Machine Learning because of its simplicity of use and offering +toolboxes that support machine learning algorithms. The toolboxes like; image processing and +computer vision, data science, and deep learning include all the tools to train and test the models. +MATLAB offers parallel computing to operate CPUs, GPUs, TPUs, and clouding to achieve high +performance [32]. +2. LabVIEW +LabVIEW is software designed for engineering problems that require the acquisition of the data, +testing it, measuring it, and controlling it. The most feature of LabView is its ability to create mutuality +environment between the hardware and data insight. LabVIEW provides the user with a graphical +programming approach, toolkits, and modules that help the user visualize any application like +working in a real lab, including hardware configuration, instrumentation to measure the data, and +error debugging. This integration between hardware and software can simplify building a complex +diagram and applying it on hardware, improving the data algorithms, and designing special user +interfaces [33]. +LabVIEW contains Analytics and Machine Learning Toolkit that combines predictive analytics and +machine learning. The toolkit is prepared to deal with large data and do some processes like, +classification, clustering, and anomaly detection. And it has good advantages which are to monitor +the conditions and maintain the predictive [34]. Figure 2-8 shows some of the processes can +LabView applying on data to get some results. + +Figure 2-8: Machine Learning Processes Which LabVIEW Can Provide [35]. + +There are some explanations on the processes: +a. Data Collection: DAQs (Data Acquisitions) allows picking up the required data. +b. Feature Extraction: Some tools like; Vision Development Module, NI Sound and Vibration +Measurement Suite can extract the features from the data based on your domain knowledge. +c. Feature Reduction: Use some techniques to simplify the data and reduce its dimension to +prepare it for training. +d. Model Creation: Give the flexibility to build and train the models. +e. Model Validation: Use evaluation metrics to check the validation of the models. +f. +Deployment: Use deployment data to predict new data. + + + +Data +Feature +Feature +Model +Model +Deployment +Collection +Extraction +Reduction +Creation +ValidationDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +18 +3. Julia +"It is a high-level, high-performance dynamic language, focusing on numerical computing and +general programming." Traditional computing languages were either fast or productive, but not both. +Julia achieves fast and productivity [36]. +Julia contains packages supporting computer vision tasks and includes open-source libraries like +Open CV and Tesseract to find optimum computer vision tasks. Julia can deal with simple images +using Julia Images to advanced Images using Julia's APIs [37]. +4. Python +"Python is an interpreted, object-oriented, high-level programming language with dynamic +semantics," Python is easy to learn because it supports code readability and therefore reduces the +bugs. Python consists of dynamic typing and dynamic biding that make the program shorter and +faster. Python supports packages with a wide range of functionality like; data analytics, databases, +graphical user interface, image processing, and scientific computing, which allows the code to be +reused and decreased the effort required to build the code from scratch [38]. +Python is considered the most common programming language for machine learning and data +science because it allows forgetting the complex parts of programming by putting the concepts +directly into the goal. Python provides us with many libraries and frameworks that offer loading data, +prepare data, label data, visualize data, and apply the different algorithms to train and test the +models [39]. + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +19 +2.1.3 Python Frameworks +A framework is an interface that makes machine learning models simpler and speeds up the processing +of models. Frameworks allow connecting the data with the models as APIs and observe your model +and its performance. The famous frameworks that are used in Python: +1. Scikit-Learn +It is an open-source machine learning framework that implements various model fitting functions, +data extraction, and many other advantages. It is straightforward to use so. It is considered an entry +point to enter the machine learning field [40]. +2. TensorFlow +It is an advanced open-source framework that can achieve complex computations. It allows us to +build huge flexibility models because it has a rich library that contains many functions and prepared +models. TensorFlow offers cloudy hundreds of GPU servers [41]. +3. Keras +It is a high-level Deep Learning API (Application Programming Interface) that can simplify building +the model and training it. Reducing cognitive load is considered one of TensorFlow's most feature, +which can load data efficiently [42]. +4. PyTorch +It is an advanced open-source framework that has tools to improve computer vision and +reinforcement learning fields. It provides cloud platforms and the ability to use GPUs to accelerate +the models [43]. +2.1.4 Hardware +Deep learning algorithms like; computer vision or automatic speech recognition require computational +power because the model becomes deeper and has big data to analyse. Many hardware units can deal +with big data and reduce the training time as follows: +1. Central Processing Unit (CPU) +It is an integrated circuit that performs machine instructions using arithmetic, logic, controlling, and +input/output operations stated by the commands. CPU includes an Arithmetic Logic Unit (ALU), +Central Unit (CU), and Memory Unit (MU). ALU is used to execute arithmetic and logical operations. +CU uses the data bus and control bus to organize the control signals. MU includes the various +aspects of memory such as Random-Access Memory (RAM), Read-Only Memory (ROM), and +CACHE. CPU performs the operations, where registers are loading the values and storing them, +CACHE memory retrieving the values, CU organizes the requests and controls the priorities steps +to process the input according to the ALU requests. Figure 2-9 shows the principal components +of the CPU [44]. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +20 +Most modern CPUs are embedded in IC chip that includes the CPU, memory, and peripherals. +Modern CPUs have multi-cores, where each core can run several threads. Most Machine Learning +algorithms are based on matrix multiplications and additions so, CPUs cannot quickly achieve this +arithmetic calculation; for example, training a deep network with a single chip can continue for days +or weeks. The Frameworks can operate CPU and GPU parallel; the heavy computations on the +GPU, and data processing on the CPU [28]. + + Figure 2-9: CPU Operations [44]. +2. Graphical Processing Unit (GPU) +GPUs become an essential integral part of computing's systems because they become more +complex and need to be faster with high-performance in various aspects like, gaming and Machine +Learning applications. The GPUs are the best choice for large computations applications [45]. GPU +is a high-computational performance processor for graphical processing. GPU was designed for +parallel processing and high memory bandwidth to accomplish high computational power and +increased productivity. The GPU's architecture essential component is the Streaming +Multiprocessor (SM), also called CUDA-Cores by NVidia. SMs contain many ALUs, and each SM +can operate one warp (a pack of 32 threads) simultaneously [27]. + + + +CPU +Control +Unit +Instructions +Processor +Registers +Com binational +Input +Logic +Output +Main +MemoryDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +21 +3. Tensor Processing Unit (TPU) +"It is custom-developed application-specific integrated circuits (ASICs) used to accelerate machine +learning algorithms." Cloud TPUs allow us to train the models on TensorFlow with high performance +and less time. Machine learning's essence is the mathematical computations that minimize the error +between inputs and predicted outputs, so cloud TPUs accelerate calculations' performance. It is +advised to use cloud TPUs in these cases; the models are constructed from matrix computations, +the models that require weeks or months for training, and the large models that contain more and +more layers with huge batch size [46]. +2.2 Design Constraints and Standards +Constraints are restrictions that prevent something from being the best. They can be problems that +arise or issues that come up. Some constraints must be considered in our project as follows: + +1. Availability of Data: + +The AI, Machine Learning and Deep Learning models are hungry for data. Especially, Deep +Learning models need more than 1000 images for each class. And those images should agree +with the real world with many backgrounds, noise, illuminations, and the direction of the image. +And there are few resources for Arabic sign language images. So, we need large data with +various aspects to get a good model with high accuracy and cover all the possibility’s images. +Also, some resources are not available for everyone. + + +Figure 2-10: Performance Vs Amount of Data [47]. + + + + +Industry Giants +Deep Learning +Performance +Quality Gap in Al products +Small and Medium size +Older algorithms +companies +Amount of dataDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +22 +2. Computational Resources: +Training of large-scale data usually needs GPUs or TPUs to accomplish the computations and +memory usage. Also, it may need many GPUs to be used at the same time. Therefore, this will +affect the time required to get results and test the model for many cases to check its +performance. However, these resources are expensive to afford for model training. + +3. Response time (Inference Time): +Real-time systems required short inference time to deploy the model for real-life usage with +minimum computation resources. + +4. Hyperparameter Choosing: +These are considered critical for every model. And choosing these parameters is subtle art +rather than standard choosing. However, you can use similar research and models +hyperparameter as a guide for your research. + +5. Knowledge and Experience in ArSL: +This project is a multi-disciplinary project that combines computer vision with ArSL. So, it needs +an expert in ArSL to take care of the Arabic sign language part of the project. + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +23 +Chapter 3 MODEL ARCHITECTURE, TRAINING +AND TESTING + +This project aims to build a model that can convert the alphabetical image in ArSL to the corresponding +written letter in the Arabic language by applying some algorithms that can extract features and +differences in the image. Choosing the right algorithm depends on the nature of data, complexity, and +required resolution of the images. This model uses the PyTorch framework to complete this task by +testing several models and comparing final results based on the ArASL2018 dataset [48]. +Usually, it would be best to try many approaches before getting the best one for a new problem. Even +experienced machine learning researchers need a lot of ideas before getting the desired results. An +experienced AI engineer suggest following a cycle to get satisfactory results, as follows [49]: + +1. Create an idea on how to build the model. +2. Convert the idea to code that can be implemented. +3. Evaluate the idea by an experiment. +4. Terminate or go back from the beginning and generates more ideas, then keep this continues +iterating. + + +Figure 3-1: Machine Learning System Cycle [49]. + +1. Idea +3.Experiment +2.CodeDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +24 +Before constructing any code, we should do a general flowchart or pseudocode to plan the model's +steps to know the path on it to achieve the task. It helps to be systematically through programming the +code. Figure 3-2 shows the general flowchart for the project: + + +Figure 3-2: Flowchart of Recognition Model + + + + + + + +Start +Hand Images +(Importing from +Google Drive) +Splitting of data into +training& testing +Hyperparameters +tuning +Define the +classification model +Loss function & +optimizer +Classification +Do theresults meet +the requirements? +EndDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +25 +The explanation of the flowchart will be in the following steps: + +Importing of Data: The model will be trained using Google Colab, and we need to feed the +data into the model. One approach to importing data is uploading locally from PC, but this way +is not proper because we will need to upload the data again every time opening Colab. Another +approach is uploading data to Google Drive once, and we just recall the data when we need it. + +Splitting Data: We need to split data perfectly into training, validation, and testing datasets. +These datasets assure to generalize the data and examine the model performance. Scikit-learn +supplies functions that can split the data perfectly with various features like; random state +parameter that generates random datasets and gives the same dataset if you select the same +seed. Another feature is splitting the data into subsets with the same indices. The most +important feature is stratified sampling which means splitting the data into stratified datasets, +which mean that datasets have the same proportion of input dataset. + +Hyperparameters Tuning: The models in deep learning have several parameters like; the +number of epochs, the number of batches, learning rate, and dropout. Parameter’s +manipulation can make the model better or worse, and selecting parameters depends on the +experience or testing several trials until satisfying the results. + +Defining the Model: It means constructing the model's architecture, including the type of +layers, number of layers. + +Loss Function and Optimizer: To assist the model, we need a loss function to ensure that +the training is doing well. Also, we need an optimizer to update the weights inside the network. + +Classification: At this stage, the model is trained, validated, and tested. After that, we see the +results if they satisfy our target or not. + + + + + + + + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +26 +3.1 Dataset Examination +3.1.1 The Nature of Images +In terms of colour, it can be in grayscale representation where it has 1 channel or in RGB representation +where it has 3 channels. [50] + +Grayscale: It contains shades of grey, proportional to the pixels' luminance. The luminance +index changes from 0 to 255, examples of these are black, which is 0, and white which is 255. +Figure 3-3 (a) shows an image that is represented in grayscale. + + + RGB: The colour of each pixel in the image can be represented as a combination of RGB +space colour, and this allows the user to specify the colour intensity between 0 and 255 for +each channel, and the combination of this mixture determines the final colour, examples of +which are RGB (255, 0,0) is red, RGB (255,255,0) is yellow, and RGB (128,0,128) is purple. +These pixels can be combined to form the image we see in real life. Figure 3-3 (b) shows an +image that is represented in RGB colour space. + + +(a) + + +(b) +Figure 3-3: (a) Grayscale Colour Space, (b) RGB Colour Space + +Our project has 54,049 images in grayscale, so that the model will be trained and tested based on +grayscale colour space. The computations in grayscale will be simpler than RGB because we deal with +2D arrays, making the model faster. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +27 +3.1.2 ArASL2018: Arabic Alphabets Sign Language Dataset +ArASL2018 is a dataset that contains 54,049 fully labelled images for 32 alphabets in Arabic sign +language. However, these images are performed from 40 participants of different ages. These images +are grayscale with 64 × 64 pixels JPG format that captured using a smart camera. [4] + + +Figure 3-4: Samples For ArASL2018 Dataset [4]. + + +Figure 3-5: A Brief Description of Arabic Sign Classes [4]. + +# +Letter name +Letter name +# of Images +# +Letter name +Letter name +# of images +in English +in Arabic +in English +in Arabic +Script +script +Script +script +1 +Alif +i(ui) +1672 +17 +Za +上(eL) +1723 +2 +Ba +1(5) +1791 +18 +Ayn +(cF) +2114 +3 +Ta +(e) +1838 +19 +Ghayn +E(c) +1977 +4 +Tha +() +1766 +20 +Fa +(b) +1955 +5 +Jim + () +1552 +21 +Qaf + () +1705 +6 +Ha +c (e) +1526 +22 +Kaf +() +1774 +7 +Kha +t(c) +1607 +23 +Lam +()n +1832 +8 +Dal + (J) +1634 +24 +Mim +r(r) +1765 +6 +Dhal +3 (Ji) +1582 +25 +Nun +i(o) +1819 +10 +Ra +() +1659 +26 +Ha +(e) +1592 +11 +Zay +j(sl) +1374 +27 +Waw +(s) +1371 +12 +sin +u (i) +1638 +28 +Ya +s () +1722 +13 +Shin + () +1507 +29 +() +1791 +14 +pes +-(3) +1895 +30 +Al +(n)n +1343 +15 +pea +c(s) +1670 +31 +Laa +Y(}) +1746 +16 +Ta +(c山) +1816 +32 +Yaa +- (-) +1293i-Alif +L-Ba +i-Ta +L-Tha +-Jim +L-Ha +LE-Kha +Jis-Dal +JIS-Dhal +lj-Ra +slj-Zay +-Sin +-Shin +se-Sad +so-Dad +L-Ta +山-Za +Ce-Ayn +Ce - Ghayn +-Fa +Qaf +s-Kaf +sy-Lam +-Mim +o-Nun +Ls-Ha +g-Waw +ts-Ya +-Taa +JI-Al +y-Laa +-YaaDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +28 + +Problems in The ArASL2018 Dataset: +o +There are different sizes of images in the dataset, which are: + +The number of images with size 𝟔𝟒 × 𝟔𝟒 are 53401. + +The number of images with size 𝟏𝟎𝟐𝟒 × 𝟕𝟔𝟖 are 10. + +The number of images with size 𝟐𝟓𝟔 × 𝟐𝟓𝟔 are 638. +o +These images are collected from a video, which means they are similar in the +surrounding conditions and do not represent the actual life datasets, and this does not +guarantee the variation of data to explore the pattern of images. +o +Some of the images in the dataset need the motion to describe the sign entirely so, +training of them will be useless without the movement. + +Our Solution for The Problems: +o +We resized all images to 𝟐𝟐𝟒 × 𝟐𝟐𝟒. +o +In this stage, we used the available images with greyscale, but we collected new +dataset that has new features. + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +29 +3.2 Data Splitting +To build a suitable model in machine learning, we need to split the dataset into three sets: training, +validation, and testing. The model uses training dataset to tune the parameters, weights, and biases. +Also, testing and validation datasets are the best way to check how well the model can deal with new +data and generalize the new cases. +The validation and testing datasets usually are from the same distribution that the model with work with +in the future, but this is not necessary for the training dataset.[49] +There are some explanations about the three sets briefly as follows: + +Training Dataset: Instance data that the system can predict patterns by finding the optimal +parameters to obtain a good prediction on new data. The training process is like humans +learning, where people can learn complicated techniques by analysis of repeated patterns. + +Validation (Development) Dataset: +o +It is used to get the right parameters through tuning the networks. +o +It is used to solve the overfitting issue; we take an instance of data as a validation +dataset to compare models. + +Testing Dataset: +o +It measures the performance of the network but without making any tunning on the model’s +parameters. +A high percentage of data goes to training as 70%, and validation as 15%, and testing as 15%. +These percentages can differ according to the dataset’s size. Usually, the data cannot be +generalized, or we need to enlarge the dataset so, we manipulate and transform the dataset by, +flipping, cropping, and resizing. This process is called data augmentation. Figure 3-6 shows the +processes on the dataset briefly. + + + + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +30 + +Figure 3-6: General processes on the dataset + + + +The data are distributed to 32 classes, as shown in Figure 3-7, and we need to split it into three sets: +70% training, 15% validation, 15% testing. As a result of this, we got: + +The training data length is 37834. + +The validation data length is 8108. + +The testing data length is 8107. + +Dataset +Training set +validation set +Test set +Machine learning +algorithm +Predictive model +Final Performance estimateDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +31 + + +Figure 3-7: ArASL2018 Dataset Histogram + + + + +3.3 Define the Model +We trained various models to classify the letters. Many models can perform better than others, and the +problems that existed in a model can solve in another model. Our case is considered a complex +problem, so the traditional approaches cannot achieve our target. +3.3.1 Multilayer Model (ANN) +This model consists of one input layer, one or more hidden layers, and one final layer called the output +layer. The number of neurons at the input layer equals the number of data input; in our case, it equals +the number of pixels 64 × 64 = 4096. Hidden layers are chosen according to the complexity of data, +and every hidden layer is responsible for the extraction of specific features. The number of neurons at +the output layer equals the number of classifications, in our case, equals 32. + +2000 +1750 +1500 +Frequency +1250 +1000 +750 +500 +250 +ain +aleff +dha +dhad +gaaf. +ghain +haa +jeem +kaaf +khaa +laam +meem +nun +ra +pees +seen +sheen +taa +thaa +thal +toot +waw +yaa +LabelDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +32 + +Figure 3-8: Multilayer Algorithm + +The following details will explain the algorithm: + +By default, the algorithm deals with input data image by image, and we have more than 50 +thousand images, so the training of the dataset will take much time. As a result, that, we divide +the dataset into many batches to faster our model’s training. And we used to stratify the function +available in the Scikit-learn framework, which takes only the number of images from each class. + +Every neuron has a weight and bias that represent some data features. It must be to initialize +them randomly. The manipulation of them increasingly or decreasingly can decide how the +model’s accuracy. The goal is to get better weights and biases, which achieve less error. + +Every batch will pass to the input layer, then pass to the first hidden layer, and calculate the +neurons’ output. After that, the next hidden layer will receive the previous layer’s output and +apply the same previous calculations, and so on until we get the last layer’s output. + +The calculations are responsible for determining the performance of the model. They called +activation functions like; step, sigmoid, Tanh, and ReLU. Some of them are used to easy +models, and the others to complex ones. + +The output that we will get from the last hidden layer is free to be in any range. The softmax +squeezes the output probabilities between 0 and 1. The architecture supposes that the higher +probability is the correct output of the input. + +Next, the architecture measures the error using the loss function by comparing the obtained +output with predictive output to estimate how much output corresponds with the predictive +values, and the manipulation of the weights and biases will be done using optimizers like; SGD +and Adam by propagating the error backwards from the output through the network. + +Input layer +Hiddcn layer +Hidden layer +Ouput laycr +r(W,b) +f(W.b) +f(W.b) +f2 +13 +P1....32 +Crosscntropy +f(W.b) +f(W,b) +f(W.b) +14096Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +33 +Activation Functions [51]: +Activation functions are used to make the complicated mapping between the inputs and +corresponding outputs by applying the function to the summation of input products with weights and +biases and then feeding them to the next layer. The linear regression tasks can be dealt with by +linear activation functions, but complicated tasks like image classification, speech recognition need +non-linear functions to map between inputs and outputs. An activation function must be +differentiable to implement the optimizer like backpropagation, stochastic gradient descent because +it requires computing the errors for the gradient of weights. Figure 3-9 shows some activation +functions and explains how their derivations seem. + +Figure 3-9: Activation Functions and Their Derivatives [28]. +a. Step Function +It is the simplest activation function and considered a hard limiter, activated when it has a value +greater than a threshold. It cannot be used in multiclass classification tasks because it is limited +between two values, −1 or 1 . The equation that represents the function is: +𝑓(𝑥) = { 1, +𝑥 ≥ 0 +−1, +𝑥 < 0 +(3.1) + +It is obvious from the equation that the derivative of the step function equals zero. +b. Sigmoid Function +It is a non-linear function that bounds the input between 0 to 1. The equations that represent the +function and its derivative are: +𝜎(𝑥) = +1 +1 + ⅇ−𝑥 +𝜎́(𝑥) = 𝜎(𝑥)(1 − 𝜎(𝑥)) +(3.2) +The function has a nonzero derivative everywhere, allowing the optimizer to update the weights +and biases. + +Activationfunctions +Derivatives +1.2 +1.0 +1.0 +0.5 +0.8 +Step +Sigmoid +0.6 +0.0 +Tanh +0.4 +ReLU +-0.5 +0.2 +0.0 +-1.0 +0.2 +4 +-2 +0 +2 +4 +-4 +-2 +0 +2 +4Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +34 +c. Hyperbolic Tangent Function +It is like a sigmoid function, and they have S-shaped and continuously differentiable, but its values +bound the input between −1 to 1. The equations that represent the function and its derivative are: +𝑓(𝑥) = 2𝜎(𝑥)(2𝑥) − 1 +𝑓́(𝑥) = +2 +1 + ⅇ−2𝑥 − 1 +(3.3) +Compared to the sigmoid function, the gradient of tanh function is steeper, and its derivative is +centred around zero, which often helps speed up convergence. +d. ReLU Function +ReLU stands for a rectified linear unit and continuously differentiable except at zero, where the +slope changes abruptly, causing a bouncing around the optimal values. However, it works very +well and can be fast to compute because it is linear, and its derivative is constant. Thereby the +calculations will be simple. The equations that represent the function and its derivative are: +𝑓(𝑥) = {𝑥, +𝑥 ≥ 0 +0, +𝑥 < 0 +𝑓́(𝑥) = { +1, +𝑥 > 0 +0, +𝑥 < 0 +𝑛𝑜𝑡 𝑑ⅇ𝑓𝑖𝑛ⅇ𝑑, +𝑥 = 0 + +(3.4) + +3.3.2 CNN Model +Using multilayer architecture is not sufficient for image recognition tasks because if we get a simple +image with 100 × 100 pixels, we will have 10000 neurons, then feeding them in a layer with 1000 +neurons, we will obtain 10 million connections at the first hidden layer, it is considered a huge number +of links and hard to deal with it. It is noticed that the multilayer model can deal appropriately with small +images that have few features. CNN is a proper solution that uses a specific layer to minimize the +connection to achieve the classification. +1. Convolution Layers +It is an essential component of the CNN architecture used for feature extraction. The neurons in +the first convolution layer do not connect to every neuron in the input data, but neurons in one layer +are connected to other neurons in their receptive field (specific patterns in small regions of the visual +field). Each neuron in the second convolution layer links neurons located within a small rectangle +in the first convolution layer. Figure 3-10 shows the connections between the layers. The first layer +in the architecture is responsible for extracting some features, the next layer for other features, etc. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +35 + +Figure 3-10: CNN Layers with Rectangular Local Receptive Fields [28]. +The array that combines the neurons is called the kernel. The operation above does not guarantee each +kernel’s centre to overlap the input layer’s outermost element. Padding, precisely Zero Padding, is a +solution to avoid adding zeros around the inputs that can overlap the outer element of the input layer. +Stride is “the distance between two consecutive Kernels” [29]. + +Figure 3-11: Connections Between Layers, Adapted froms [28]. + + + + + +Convolutional +layer2 +Convolutional +layer1 +Inputlayerfh= 3 +Zero Padding +fw=3 +S.. +WDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +36 +2. Feature Maps +The convolutional layer can be represented in 3D which every layer has multiple feature map. A +feature map is considered a filter that explores features like vertical lines and horizontal lines. Figure +3-12 shows convolutional layers with multiple feature maps. + +Figure 3-12: Convolutional Layers with Multiple Feature Maps [28]. + +The equation that shows how the convolutional layer computes the output of a given neuron is: +𝑍𝑖,𝑗,𝑘 = 𝑏𝑘 + ∑ ∑ ∑ 𝑋𝑖′,𝑗′,𝑘′ +𝑓𝑛′−1 +𝑘′=0 +𝑓𝑤−1 +𝑣=0 +𝑓ℎ−1 +𝑢=0 +∙ 𝑊𝑢,𝑣,𝑘′,𝑘 𝑤𝑖𝑡ℎ { 𝑖′ = 𝑖 × 𝑠ℎ × 𝑢 +𝑗′ = 𝑗 × 𝑠𝑤 × 𝑤 +(3.5) +Where: + +𝒁𝒊,𝒋,𝒌: the neuron’s output in row i, column j in feature map k of the convolutional layer. + +𝑿𝒊′,𝒋′,𝒌′: the output of the neuron in layer L – 1, row i′, column j′. + +𝒃𝒌 : the bias term in layer L. + +𝑾𝒖,𝒗,𝒌′,𝒌 : the connection weights. + + + + + + + + + + + +Convolutional +Feature +layer2 +map 1 +map2 +Filters +Convolutional +Map 1 +layer1 +Map2 +Inputlayer +Channels +Red +Green +BlueDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +37 +3. Pooling Layers +This layer decreases the computations by shrinking the inputs. The pooling layer is like the +convolutional layer, connected partially to the previous layer’s outputs, located with the small +rectangle receptive field. There are two forms of pooling: Max pooling and Mean pooling. Max +pooling is the most popular form, which takes the maximum value in the higher-level feature layer. +Mean pooling takes the average of all the elements in the higher-level feature layer. Max and Mean +pooling are shown in Figure 3-13. +Figure 3-13: Max Pooling and Mean Pooling with (2×2 Pooling Kernel, Stride 2, Zero +Padding) + + +It is noticed from the above figure that the pooling layer with 2×2 pooling kernel and stride 2 +decreases the image to a quarter of the original image. This stage will reduce the computations, +memory usage, and the number of parameters, thereby easing the extraction of features from the +image. +4. Dropout Layer +A huge number of parameters in the network gives it the flexibility to tend to overfit. One solution to +avoid overfitting is using the early stopping technique, where the network stores the parameters at +the best values when the validation set worsens. With unlimited computations, the early stopping +technique will be aggressive and consume more time so, the best way to avoid overfitting is using +the regularizes. Dropout is one of the most regularization techniques. The term dropout refers to +dropping out the neuron and incoming and outcoming connections temporarily from the network, +as shown in Figure 3-14. + +Original +100 +150 +200 +50 +100 +150 +200 +250 +300 +350 +MaxPooling +MeanPooling +20 +20 +40 +40 +09 +60 +80 +80 +100 +100 +120 +F +120 +406080100120140160 +2.75 +(1+5+3+2)/4 = 2.75Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +38 +The neurons that will be dropped out are chosen randomly, or simply, each neuron in the training +set will be multiplied with a factor called dropout rate ρ. The dropout rate can be selected from +between 0.4-0.5 in CNN. In the testing set, the neurons will not be dropped out. For more +explanation, suppose 𝑃=0.5, the neurons during testing will be connected twice more than neurons +in training. So, each neuron in testing has a total input signal twice larger than what each neuron in +training has, and the performance will not be ok in this case. Each neuron input’s signal in testing +will be multiplied by 0.5 to compensate for the difference in neurons before and after training, as +shown in Figure 3-15. + +(a) + + +(b) + +Figure 3-14: (a) Network Without Dropout, (b) Network With Dropout. [52] + +(a) + + +(b) +Figure 3-15: (a) Neuron at training, (b) Neuron at testing. [52] + + + +X +XPresent with +probability ppw +Always +presentDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +39 +5. Fully Connected Layer +This layer transforms the last convolutional layer into a one-dimensional array and connects to one +or more dense layers, in addition to a dropout layer after each dense layer, with a 0.5 dropout rate +will reduce overfitting. A non-linear activation function follows the final fully connected layer to +estimate inputs classification according to the output probabilities. + +Figure 3-16: CNN Architecture [53]. +Figure 3-16 shows the general architecture of CNN. It is noticed that there are convolutional layers +followed by a ReLU function or others, then another pooling layer, and so on. The previous steps are +considered feature layers and make the image smaller and smaller through the architecture until it +reaches the classification layers. Stages do the classification: flattening the last layer to pass it into a +fully connected layer and then passing the fully connected layer into softmax function to classify the +images according to the estimated probabilities. +3.3.3 ResNet-18 +Many challenges face the building of CNN models, like specifying the number of layers and their size, +initialization of weights, and biases. In bad initialization, the model can consume a lot of time to complete +the task. Many architectures have been developed over the years, and they have a good impact on +improving the trained models. On the other hand, these architectures simplify dealing with data without +deep knowledge in this field. One of them is ResNet-18 architecture. +ResNet-18 is using a skip connection signal to train the model. A skip connection is a signal that passes +into the layer in addition added to the output layer. In a usual architecture, the goal is training the 𝑓(𝑥) +but in the residual training the architecture will be forced to train 𝑓(𝑥) + 𝑥. Figure 3-17 shows the residual +training. + +CAR +TRUCK +VAN +-BICYCLE +INPUT +CONVOLUTION+RELU +POOLING +CONVOLUTION+RELU +POOLING +FLATTEN +FULLY +SOFTMAX +CONNECTED +FEATURE LEARNING +CLASSIFICATIONDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +40 + +Figure 3-17: Residual Learning + +The initial weights are usually close to zero so, the outputs follow them with values close to zero. In the +case of adding the skip connection, the outputs will be clone from the inputs. This will accelerate the +training faster than without the skip connection because the network progresses before the layers start +learning. +ResNet-18 has 18 layers where the convolution layers and fully connected layer are just counted. Let +us analyse ResNet’s architecture that is shown in Figure 3-18. The input image is passed to a +convolutional layer with a 7×7 kernel, 64 feature maps stride 2, zero Padding. Then is fed to a max- +pooling layer with a 7×7 kernel, stride 2, zero Padding. After them, there are four identical convolutional +networks. Every ConvNet has two residual units, and each residual unit consists of two convolution +layers with a 3×3 kernel, stride 2, zero Padding. It is noticed that the feature maps are doubled every +identical ConvNets, and the convolution layers’ size is minimized to half in height and width. Next, the +last layer is fed to the average pooling layer, then a fully connected layer and softmax to estimate the +probabilities. + + Figure 3-18: ResNet-18 Architecture, Adapted From [54]. + + +x +weightlayer +SkipConnection +F(x) +Irelu +x +weightlayer +identity +F(x) +x7 × 7 conv, 64, /2 +3 × 3 conv, 64 +3 × 3 conv, 64 +3 × 3 conv, 64 +3 × 3 conv, 64 +3 × 3 conv, 128, /2 +3 × 3 conv, 128 +3 × 3 conv, 128 +3 × 3 conv, 128 +3 × 3 conv, 256, /2 +3 × 3 conv, 256 +3 × 3 conv, 256 +3 ×3conv,256 +3 × 3 conv, 512, /2 +3 × 3 conv, 512 +3 × 3 conv,512 +3 × 3 conv, 512 +fc 32Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +41 +3.4 Define A Loss Function and Optimizer +To assist the model, we need a loss function to ensure that the Training is doing well. And we need an +optimizer to update the weights inside the network. +A loss function is considered one of the pillars when training the model in image classification. However, +this loss function assists the learning for the model over the training dataset using the weights and +biases through the network. For example, it can be calculated by taking the difference between the +predicted and actual classes. +There are two common categories of loss function like regression loss functions and classification +loss functions. However, regression seeks to find a predicted continuous value depending on many +parameters in the model, while classification chooses an output from a set of categories. Here are many +examples of regression and classification losses that are commonly used in these problems. +Regression Loss Functions: + +Mean Square Error: +It is considered a performance technique that determines how much error the model obtains +comparing with its predictions. The equation is formed by taking the difference between +prediction from the model and the actual, then averaging these differences to get the total +magnitude error. +𝑀𝑆𝐸 =   1 +𝑛   ∑( 𝑦𝑖 −  𝑦̂ 𝑖) +𝑛 +𝑖=1 +  +(3.6) +Where the symbols represent the following: +MSE: Mean Squared Error. +n: total number of predictions/actual data. +𝒚𝒊: actual value. +𝒚̂𝒊: predicted value. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +42 + +Figure 3-19: Mean Square Error + + +Mean Absolute Error: +It is almost like MSE. Using one of them depends on the data distribution. When we have +large outlier districts, MSE will neutralize the negative and positive outliers, giving a wrong +prediction. It is preferred to take the absolute values. The equation is formed by taking the +sum of the absolute differences between the prediction and the actual values, then obtain the +average. +𝑀𝐴𝐸 = 1 +𝑛 ∑ |𝑦𝑖 − 𝑦̂ 𝑖| +𝑛 +𝑗=1 + +(3.7) +Where the symbols represent the following: + +MAE: Mean Absolute Error. +n: total number of predictions/actual data. +𝒚𝒊: actual value. +𝒚̂𝒊: predicted value. + +Regression/ +Best-fit Line +xDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +43 + + +Figure 3-20: Mean Absolute Error + +Classification Loss Functions: + +Mean Squared Error: +Taking the difference between prediction from the model and the actual, then averaging these +differences to get the total magnitude error. + +Cross-Entropy Loss (Log Loss): +It is often used in classification problems. The output probability from a sigmoid or a softmax +then enters the cross-entropy function that assesses how much this model classifies well. +It has two forms, binary cross-entropy, and multi-class cross-entropy. The following equations +clarify each of them. + +Binary Cross-Entropy: +𝐵𝐶𝐸 = 𝑦 ∗ log(𝑝) + (1 − 𝑦) ∗ log (1 − 𝑝)  +(3.8) + P = Prop(y=1), output from a sigmoid activation binary class label y. + +The following graph illustrates the loss vs. predicted probability for a binary classifier for each +y=1 and y=0. + + + +Output +MAE +InputsDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +44 + +Figure 3-21: Loss (BCE) vs. Predicted Probability + + +Categorical Cross-Entropy: + 𝐶𝐶𝐸 = − 1 +𝑚 ∑( 𝑦𝑖 ∗ log(𝑦̂𝑖) + +𝑚 +𝑖=1 +(1 − 𝑦𝑖) ∗ log(1 − 𝑦̂𝑖) ) + +(3.9) +m: number of classes that are represented with one-hot encoding. +𝑦𝑖: ith target class. +𝑦̂𝑖: the predicted probability of that input belongs to the ith class, computed with a softmax +activation. + +Optimizer: +It is an algorithm or method used to tune the model's parameter (e.g., weights, biases, etc.) +during the training to reduce the losses. Also, it affects the results from the model. +Many optimizers can be used with the models during the training. However, every optimizer +has its advantages and disadvantages. + + +Gradient Descent: +It is considered one of the most basic optimization algorithms, and it is a first-order optimization +algorithm dependent on the first order of a loss function. It can be used in classification and +linear regression problems. Also, it is used in backpropagation algorithm. The following +equation can express it: + + + +y=1 +4 +y=0 +3 +ssol +2 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +predicted probabilityDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +45 +𝜃 = 𝜃 − 𝛼 ∇𝐽(𝜃) +(3.10) + +𝜽: is the weight of the model. +𝜶: is the learning rate. +𝛁𝑱(𝜽): is the derivative of the objective (loss) function for the weights. + + +Stochastic Gradient Descent (SGD): +The difference between this algorithm and basic gradient descent is that this algorithm updates +the model’s parameter for each training example in the dataset. + +𝜃 = 𝜃 − 𝛼 ∇𝐽(𝜃; 𝑥(𝑖); 𝑦(𝑖)) + +(3.11) +Where: +{ 𝒙(𝒊), 𝒚(𝒊)}: are the training examples. +𝜽: is the weight of the model. +𝜶: is the learning rate. +𝛁𝑱(𝜽; 𝒙(𝒊); 𝒚(𝒊)): is the derivative of the objective (loss) function for the weights for every +training example. +In this algorithm, the loss function has a lot of fluctuations and variance over the training. + + +Mini-Batch Gradient Descent +It combines the advantages of SGD and standard gradient descent. After every batch of the +training dataset the parameters are updated. +𝜽 = 𝜽 − 𝜶 𝛁𝑱(𝜽; 𝑩(𝒊)) +(3.12) + +Where: +{𝑩(𝒊)}: the batches of training examples. +𝜽: the weight of the model. +𝜶: the learning rate. +𝛁𝑱(𝜽; 𝑩(𝒊)): objective (loss) function derivative for the weights for every batch. + +Learning rate: The gradient tells us where the function has the highest rate of change, but it +does not tell us the value of steps to reach the optimal value. Small steps lead the algorithm to +the best solution but slow progress. Large steps are fast but lead the algorithm to bounce +around the optimal values. + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +46 + +Adam Optimizer: +Adam is derived from adaptive moment estimation. It takes the advantages of both AdaGrad +and RMSProp algorithms to provide an optimization that can deal with sparse gradient on noisy +problems. It decays exponentially average of past gradients so, converging to the optimal +solution fast. Table 3-1 shows a brief comparison between SGD and ADAM. + +Table 3-1: Comparison between SGD and ADAM +Optimizer +Advantages +Disadvantages +SGD + +Can best fit and generalize the +dataset after extensive training + +Cannot deal with global +minima. + +It can be affected by choice of +learning rate. +Adam + +Can deal with sparse gradient on +noisy problems + +Can decay the learning rate +through the learning + +Cannot generalize dataset +perfectly + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +47 +Chapter 4 DESIGN TESTING AND RESULTS +4.1 Model Training and Validation +4.1.1 Model Training +Model training means applying an algorithm to update the model parameters that best fit training data +and predict the new data well. Models differ from others in the ability of data fitting. Each of them is +suitable for a specific type of data and performs depending on the complexity of data. In our project, the +data was trained using; ANN, CNN, ResNet-18. +The training performance depends on several factors, and without them, the training will be invalid and +absorb more time. The factors that can improve the training are: + +Model Architecture: Before building the model, you should know the nature of the data. In our +case for image classification, the deep neural network does not guarantee to extract the image +features and achieve high accuracy, so choosing a powerful architecture like CNN or transfer +learning is preferred. + +Data Plenty: Machine learning models need many data to fit them properly. The data that we +ran was adequate to train the model. In case of lack of data, Data augmentation can be used +or using transfer learning to compensate it. + +The Optimizer: The weights will be initiated randomly, so the optimizer will update the weights +to reach the minimized error. We used ADAM and SGD to update the weights in the models. + +The Number of Epochs: The training of data needs several cycles to reach the proper +parameters. There are two approaches to choose the number of epochs; specifying the number +of epochs directly or using the early stopping technique since the training will stop when there +is no progress. We set the number of epochs to 20 for the models. +4.1.2 Model Validation +It means how the model will estimate performance after the training and is used essentially to avoid +overfitting. Overfitting occurs when the model performs well, generalizes the data quite on the training +but performs worse on the validation. The validation is applied using evaluating the validation error +every training stage to find the minimum error, then stopping the training and saving the parameters. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +48 +4.1.3 Graphs: Training, Validation Accuracy and Loss + +Figure 4-1: Progress of Average Training Loss of ANN with SGD, lr=0.1 Through the Epochs. + + +Figure 4-2: Progress of Average Validation Loss of ANN with SGD, lr=0.1 Through the Epochs. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +49 +Figure 4-1 shows that the average loss of accuracy training for an ANN model using SGD optimizer and +lr = 0.1 decreases slowly with increasing number of epochs. Also, Figure 4-2 shows that the progress +of validation loss for an ANN model does not decrease smoothly with increasing number of epochs. +Finally, we can conclude that ANN needs a lot of time to be trained and to extract the features from the +images. + + + + +Figure 4-3: Progress of Average Training Loss of CNN with SGD, lr=0.1 Through the Epochs. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +50 + +Figure 4-4: Progress of Average Validation Loss of CNN with SGD, lr=0.1 Through the Epochs. + + +CNN performs much better than ANN, Figure 4-3 and Figure 4-4 show that training loss and validation +loss decrease rapidly through the first 5 epochs and there is not obvious decreasing until epoch number +24. Also, it is noted that CNN training loss and validation loss approaches to zero. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +51 + +Figure 4-5: Progress of Average Training Loss of ResNet-18 with SGD, lr=0.1 Through the +Epochs. + +Figure 4-6: Progress of Average Validation Loss of ResNet-18 with SGD, lr=0.1 Through the +Epochs. + +0.7 +0.6 +0.5 +LosS +0.4 +0.3 +0.2 +0.1 +0.0 +0 +1 +2 +m +4 +5 +7 +8 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +EpochNumber0.275 +0.250 +0.225 +0.200 +Validation Loss +0.175 +0.150 +Average +0.125 +0.100 +0.075 +0.050 +0.025 +0.000 +1 +2 +4 +5 +6 +7 +00 +9 +10 +11 +12 +13 +14 +15 +16 +17 +18 +19 +20 +EpochNumberDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +52 +Figure 4-5 and Figure 4-6 show that ResNet-18 model performs better than ANN and CNN. It is noted +that it starts with low loss and it decreases through the first 3-4 epochs. However, the training loss +decreases slightly and stays at low average loss, but average validation loss increases slightly after +epoch 4 which means the model begins overfit over the data. However, we save the model at the least +validation loss to avoid overfitting. +4.2 Testing and Results +After training the model, we need to validate it using a testing dataset. The testing stage performs the +model on a dataset that the model did not see it. The image will pass into the model as an input, and +the output will be one of 32 classes. If a matching between the true output and predicted output occurs, +it contributes rising of model accuracy. The models’ performance and their average accuracies are +shown in Table 4-1. +Table 4-1: Comparison Between Different Models’ Accuracies +Model +Optimizer +Learning rate +Average Test Accuracy +ANN +SGD +0.01 +77.78 % +SGD +0.10 +27.50 % +CNN +SGD +0.01 +93.00 % +SGD +0.10 +95.80 % +Transfer Learning (ResNet-18) +SGD +0.01 +99.21 % +SGD +0.10 +99.36 % +ADAM +0.01 +99.00 % +ADAM +0.10 +96.76 % + +It is noticed that the ResNet-18 has higher accuracy than other architectures. The model corresponds +to the predicted values with actual values at the testing stage, and Figure 4-7 shows how much the +testing set corresponds to the actual values at ResNet-18. +In general, each image will pass into the model and distribute to 32 classes in different accuracy so that +the higher accuracy will be the predicted value. The misleading will have occurred in several cases, +like; the similarity between the images, in which the model usually cannot discern images that have +similarity, and the unclear images, in which the model cannot extract the features correctly because of +the noises in the image. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +53 + +Figure 4-7: Confusion Matrix of ResNet-18 Shows the Matching Between the Predicted Values +with True Values at The Testing Stage. + + + + + + + +0 +0 +。 +0 +0 +240 +0 +0 +1 +0 +2 +0 +1 +0 +1 +245 +0 +3 +0 +0 +0 +0 +248 +0 +0 +m +。 +290 +0 +0 +2 +0 +0 +0 +0 +0 +13 +241 +0 +2 +0 +0 +0 +0 +296 +0 +0 +238 +0 +0 +。 +0 +28 +0 +0 +0 +0 +1 +2 +231 +0 +0 +0 +0 +0 +0 +0 +0 +1 +264 +0 +0 +0 +1 +0 +2 +238 +0 +1 +0 +0 +260 +1 +0 +274 +0 +1 +1 +263 +0 +0 +0 +。 +m +266 +0 +0 +0 +1 +1 +1 +2 +0 +0 +1 +0 +0 +。 +243 +0 +0 +0 +0 +0 +2 +0 +2 +1 +1 +0 +280 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +0 +245 +1 +2 +0 +2 +220 +1 +0 +。 +0 +1 +2 +1 +264 +1 +2 +0 +0 +272 +0 +2 +0 +1 +0 +2 +1 +0 +m +258 +0 +。 +2 +0 +0 +2 +0 +0 +0 +231 +0 +0 +0 +0 +2 +0 +。 +1 +0 +265 +2 +0 +0 +206 +0 +0 +1 +。 +258 +m +190 +。 +0 +0 +0 +0 +0 +0 +0 +0 +1 +0 +2 +0 +203 Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +54 + +Figure 4-8: Confusion Matrix of CNN Shows the Matching Between the Predicted Values with +True Values at The Testing Stage. +Figure 4-8 shows how much the testing set corresponds to the actual values at CNN. It is noticed from +the distribution of predicted and true values that CNN performs well and the misleading between +predicted and true values are considered little much. + +0 +0 +2 +m +0 +241 +0 +0 +1 +2 +0 +3 +0 +0 +0 +1 +0 +238 +4 +0 +2 +4 +1 +1 +2 +281 +4 +0 +2 +1 +0 +11 +229 +1 +0 +10 +1 +1 +0 +1 +3 +1 +0 +1 +0 +28 +1 +1 +0 +1 +5 +5 +226 +0 +0 +1 +0 +1 +2 +217 +2 +0 +2 +m +2 +0 +m +215 +1 +1 +1 +1 +m +0 +1 +2 +0 +1 +1 +0 +1 +1 +255 +0 +0 +0 +2 +1 +1 +1 +0 +2 +1 +1 +228 +0 +0 +1 +0 +2 +1 +256 +0 +0 +m +0 +1 +2 +272 +0 +0 +0 +0 +0 +0 +0 +2 +1 +0 +1 +254 +1 +1 +0 +1 +0 +2 +1 +0 +2 +2 +1 +2 +262 +0 +1 +1 +0 +0 +0 +0 +0 +243 +0 +0 +0 +0 +2 +0 +0 +5 +2 +1 +。 +270 +m +0 +1 +0 +5 +238 +0 +0 +0 +1 +2 +m +2 +0 +1 +214 +1 +1 +0 +0 +m +1 +0 +1 +0 +265 +0 +0 +2 +0 +264 +1 +1 +2 +1 +1 +1 +0 +1 +1 +9 +250 +0 +0 +0 +2 +7 +1 +223 +1 +1 +0 +263 +0 +0 +1 +2 +1 +0 +2 +0 +0 +201 +0 +1 +1 +255 +1 +192 +0 +0 +0 +2 +0 +0 +3 +0 +0 +2 +1 +0 +0 +0 +193 +1Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +55 + +Figure 4-9: Confusion Matrix of ANN Shows the Matching Between the Predicted Values with +True Values at The Testing Stage. +Figure 4-9 shows how much the testing set corresponds to the actual values at ANN. It is noticed from +the distribution of predicted and true values that ANN does not perform well, and there is big misleading +in some of the classes, indicating that ANN is not a valid image classification. + + + + + +9 +0 +0 +0 +5 +1 +1 +169 +3 +33 +2 +1 +1 +m +2 +1 +3 +1 +2 +1 +6 +m +1 +220 +16 +虹 +2 +1 +2 +0 +2 +7 +8 +5 +0 +18 +148 +14 + 42 +4 +2 +m +4 +0 +2 +1 +4 +3 +0 +0 +290 +0 +1 +2 +2 +6 +8 +197 +1 +1 +4 +1 +7 +2 +5 +9 +1 +1 +m +0 +187 + 12 +5 +1 +1 +1 +3 +2 +189 +2 +18 +0 +0 +1 +0 +1 +13 +0 +6 +2 +5 +179 +0 +0 +1 +0 +0 +0 +2 +0 +4 +1 +1 +7 +1 +1 +6 +1 +2 +115 +0 +1 +0 +4 +0 +2 +0 +5 +m +2 +0 +1 +1 +m +。 +225 +7 +2 +1 +1 +0 +2 +9 +10 +1 +80 + 209 +2 +2 +0 +0 +0 +1 +1 +。 +233 +1 +4 +0 +3 +0 +1 +1 +m +m +2 +1 + 20 +1 +m +3 +225 +1 +2 +0 +1 +2 +0 +3 +2 +0 +1 +0 +188 +0 +0 +2 +0 +6 +0 +0 +1 +1 +5 +14 +4 +。 +242 +4 +2 +1 +1 +0 +2 +1 +1 +1 +0 +236 +1 +1 +0 +2 +5 +1 +1 +207 +2 +1 +m +1 +0 +2 +3 +1 +2 +230 +m +14 +7 +0 +8 +0 +0 +0 +2 +1 +9 +226 +m +4 +11 +2 +8 +2 +1 +2 +1 +4 +m +0 +5 +11 +220 +0 +1 +1 +3 +2 +2 +。 +200 +1 +0 +0 +5 +0 +7 +2 +2 +1 +。 +1 +2 +5 +6 +2 +209 +1 +3 +0 +0 +1 +1 +2 +0 +0 +0 +0 +148 +0 +5 +5 +1 +6 +4 +m +10 +1 +1 +1 +1 +182 +1 +5 +0 +。 +152 +0 +0 +0 +3 +0 +0 +0 +0 +5 +20 +1 +0 +2 +0 +0 +0 +129 Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +56 +4.3 Model Inferencing +It is a process of feeding a new input image (unseen image) into a trained DNN model. Figure 4-10 +shows the Flowchart of model inferencing. + +Figure 4-10: Inferencing Flowchart + +We tried to examine the model performance by inserting an image shown in Figure 4-11 (a). The input +image will pass into the pre-processing stage, which does some operations, including resizing, centre +crop, converting to a grayscale image, etc. The image after pre-processing is shown in Figure 4-11 (b). +The pre-processed image will pass into the trained model to classify the image to the predicted class. +Figure 4-12 shows the prediction of the input image, and it is obvious that the model predicts the input +correctly. + +input image +pre-processing +trained mode +top k classesDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +57 + + +(a) + + +(b) +Figure 4-11: (a) Input Image (Y𝒂̅), (b) Pre-processed Input Image + + +Figure 4-12: ArSL Alphabet Prediction. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +58 +4.4 New Collected Dataset: ArSLA-2021 Dataset +4.4.1 Overview +We are creating our dataset that is captured from many people of different ages, and it will be the first +dataset of its kind for Arabic Sign Language Alphabets. Until now, we have collected about +10000 +real-life images for 31 Arabic Sign Alphabets. Figure 4-13 shows from our dataset. + +Figure 4-13: Arabic Sign Language Alphabets Samples + +The dataset has been collected and captured by the people. Also, it is considered a real-life dataset +with images captured under different conditions such as different light, background, image orientation, +image size, image quality, etc. + + + + + + + +Lajf- Alif +-Gi - Ta +-G -Tha +As?- Jim +-Ha +sL-Kha +Jis - Dal +JI3- Dhal +slj - Ra +slj-Zay +w - Sin +o- Shin +Sto-Sad +-Dad +slo-Dad +lis - Za +E - Ghayn +- Fa +ose - Ayn +l-Qaf +l - Kaf +y- Lam +Ase-Mim +ogi-Nun +elo-Ha +gl9 - waw +以- +Y - Laa +8-Taa +JI - AIDesign of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +59 +Table 4-2: Arabic Sign Language Alphabets, their numbers, and labels +# +Letter name +in English +Script +Letter name in +Arabic script +# of Images +# +Letter name +in English +Script +Letter name +in Arabic +script +# of images +1 +Alif أ )فِلَأ( 395 17 +Z𝑎̅ ظ )ءاَظ( 329 +2 +B𝑎̅ ب )ءاَب( 387 18 +Ayn ع )نيَع( 340 +3 +T𝑎̅ ت )ءاَتأ( 385 19 +Ghayn غ )نيَغ( 337 +4 +Th𝑎̅ ث )ءاَث( 379 20 +F𝑎̅ ف )ءاَف( 331 +5 +J𝑖̅𝑚 ج )ْميِج( 388 21 +Q𝑎̅f ق )فاَق( 327 +6 +H𝑎̅ ح )ءاَح( 378 22 +K𝑎̅f ك )فاَك( 332 +7 +Kh𝑎̅ خ )ءاَخ( 347 23 +L𝑎̅m ل )ْمَلا( 335 +8 +D𝑎̅l د )ْلاَد( 343 24 +M𝑖̅m م )ْميِم( 328 +9 +Dh𝑎̅l ذ )لاَذ( 335 25 +N𝑢̅n ن )نوُن( 356 +10 +R𝑎̅ ر )ءاَر( 335 26 +H𝑎̅ ه )ءاَه( 354 +11 +Z 𝑎̅y ز )ياَز( 340 27 +W𝑎̅w و )واَو( 351 +12 +S𝑖̅n س )ْنيِس( 340 28 +Y𝑎̅a ءاَي )ءاَي( 346 +13 +Sh𝑖̅n ش )ْنيِش( 353 29 +T𝑎̅a ة )ة( 344 +14 +S𝑎̅d ص )ْداَص( 340 30 +Al لا )لا( 341 +15 +D𝑎̅d ض )داَض( 339 31 +Laa لا )لا( 339 +16 +T𝑎̅ ط )ءاَط( 334 + + + +This project is supported and consulted by Student Counselling Department at the University +of Jordan, which has specialist interpreters for Arabic Sign Language. They help and consult +us to achieve high-quality work in this field. +4.4.2 General Notes for ArSLA-2021 Dataset: + +It could be a benchmark for researchers in this field. + +It can be used for research and production. + +It will be shared as raw images. Everyone has the choice to do any processing on them. + +It is collected from more than 300 participants. + +Different resolutions have been got by different mobile phones. + +The dataset mainly consists of RGB images. + +These images are static. + +It will be annotated manually in the future. + +It will be made publicly available to support the Sign Language field. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +60 +4.5 System Limitations and Compliance with Design Constraints +4.5.1 System Limitations: + +We faced many limitations while working on this project. The limitations are: +1. Lack of Resources +We have just found one public dataset for ArSL alphabets that is publicly available, the +ArASL2018 dataset. The available datasets have a limited set of conditions, including lighting, +unique images, various background, etc. +2. Hardware Limitations [55] + +The training of deep learning models usually requires GPUs, with enough disk memory and +RAM size that can accelerate the model training, but unfortunately, we do not have available +GPUs at The University of Jordan. + +We have used the available GPUs that are offered by google colab. Unfortunately, google colab +has some limitations: +o +The resources are not guaranteed. +o +The usage limits change depending on the availability. +o +Time limitations for continuous model running, which is at max 12 hours. +o +Memory limits to load the dataset into the model. +o +RAM limitation is used to perform the calculations. +o +GPUs’ type is not guaranteed. + +4.5.2 Design Constrains Compliance +The following points discuss the constraints that were put in place in the beginning, what we were able +to solve, and what we could not: +1. Availability of Data: +We looked for any dataset that has multiple conditions that represent the real-life conditions of the data. +We found one public dataset containing enough images, 54,049 images, that was trained on but with +limitations to the number of unique participants, lighting, and complexity. However, we are collecting +our dataset that overcome these issues. +2. Computational Resources +We have faced a challenge to train models using google colab due to restrictions on it. ArASL2018 +dataset was trained using google colab without any problems and we have achieved an excellent +performance. Unfortunately, we cannot using google collab to train models using ArSLA-2021 dataset. + + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +61 + +3. Response time (Inference Time): +We calculated the inference time required to classify one image per model. It is noted from Table 4-3 +that the induction time responds within a very short time. + +Table 4-3: Inference Time Results +Model +Optimizer +Learning Rate +Inference Time +ANN +SGD +0.1 +258 ms +CNN +SGD +0.1 + 145 ms +Transfer Learning (ResNet-18) +SGD +0.1 + 142 ms + +4. Hyperparameter Choosing +We have tested many hyperparameters depend on research papers that are considered relevant +problem solutions. +5. Knowledge and Experience in ArSL +We have studied a lot of resources about alphabets in ArSL and we have met experts interpreters in +ArSL to deliver a high quality of work. + + +4.6 Solution Impact +4.6.1 Societal Impact +Machine learning broadens our outlook on life. It makes a great leap in various fields like, industry, +medicine, and social life. One of the machine learning branches is computer vision. Computer +vision and its applications simplify the complicated tasks where some of these applications require much +experience to extract the image’s features. +Computer vision is not limited to technical issues but reaches toward humanity issues. In our project, +computer vision is used to build a model that can recognize the Arabic sign language alphabets +automatically. This step positively impacts the community, where it will spread awareness of sign +language and ease the communication between deaf and normal people. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +62 +4.6.2 Economic Impact +Technology has a huge impact on the economy. If the Arabic sign language recognition model were +deployed in smart devices, this would reduce the cost of owning special tools such as gloves, pressure +sensors, and jump motion devices. The cost of hiring more interpreters will also decrease. +4.6.3 Environmental Impact +Our work is digital content, so when you use it there will be no waste. On the other hand, the use of +solid materials will be a waste after corrosion, and this will have bad impacts on the environment. +4.6.4 Global Impact +If our model is deployed in smart devices, it will increase the communication between normal people +and deaf people by interpreting the alphabets in ArSL and converting them into written Arabic text. + + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +63 +Chapter 5 CONCLUSION AND FUTURE +WORK + +5.1 Conclusion +This project was designed to be the first step to help the deaf community by building a model that uses +computer vision techniques to convert the ArSL alphabets into Arabic letters. Many machine learning +techniques were used to build the model, and we chose transfer learning (ResNet-18) which achieved +the highest accuracy. + +5.2 Problems Faced + +It is a new field, so we have needed a lot of time to learn and grasp new concepts especially in +deep learning and ArSL. + +We have not enough fund to buy GPUs to train new models using ArSLA-2021 dataset. + +Since we used the GPUs offered by google colab there was time limitation, and we had no +control over the resources. + +COVID-19 restrictions have prevented us from collecting more data. + +5.3 Recommendations for Future Work +We will expand the project to include: + +Publishing our dataset to be a starting point for someone else to continue the work. + +Training the model using our dataset and collecting more data. + +Collecting dataset for dynamic alphabets and collecting dataset to include words and +continuous speech. + +Creating a model that will be used for real-time application. + +Deploying the model on the mobile application. + +Design of Arabic Sign Language Recognition Model + + +Design of Arabic Sign Language Recognition Model + +64 +REFERENCES +[1] +“Deafness and hearing loss.” https://www.who.int/news-room/fact-sheets/detail/deafness-and- +hearing-loss (accessed Oct. 29, 2020). +[2] +“The +types, +causes +and +treatment +of +hearing +loss.” +https://www.starkey.com/blog/articles/2018/01/The-human-hearing-system-and-hearing-loss +(accessed Oct. 29, 2020). +[3] +M. 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+page_content='School of Engineering Department of Mechatronics Engineering Bachelor of Science in Mechatronics Engineering Senior Design Graduation Project Report Design of Arabic Sign Language Recognition Model Report by Muhammad Al Barham Ahmad Jamal Supervisor Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Musa Al Yaman Date 26/05/2021 THE:UNIVERSITYOF JORDAN1Design of Arabic Sign Language Recognition Model Department of Mechatronics Engineering Senior Design Graduation Project Report i ABSTRACT Deaf people are using sign language for communication, and it is a combination of gestures, movements, postures, and facial expressions that correspond to alphabets and words in spoken languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The proposed Arabic sign language recognition model helps deaf and hard hearing people communicate effectively with ordinary people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The recognition has four stages of converting the alphabet into letters as follows: Image Loading stage,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' which loads the images of Arabic sign language alphabets that were used later to train and test the model,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' a pre-processing stage which applies image processing techniques such as normalization,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Image augmentation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' resizing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' and filtering to extract the features which are necessary to accomplish the recognition perfectly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' a training stage which is achieved by deep learning techniques like CNN,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' a testing stage which demonstrates how effectively the model performs for images did not see it before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' and the model was built and tested mainly using PyTorch library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model is tested on ArASL2018, consisting of 54,000 images for 32 alphabet signs gathered from 40 signers, and the dataset has two sets: training dataset and testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We had to ensure that the system is reliable in terms of accuracy, time, and flexibility of use explained in detail in this report.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, the future work will be a model that converts Arabic sign language into Arabic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Department of Mechatronics Engineering Senior Design Graduation Project Report ii TABLE OF CONTENTS ABSTRACT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 9 Figure 2-3: Mapping a Cat Image to Class Scores .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 11 Figure 2-4: Example for Softmax and SVM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 27 Figure 3-6: General processes on the dataset .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 35 Figure 3-11: Connections Between Layers, Adapted from .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 35 Figure 3-12: Convolutional Layers with Multiple Feature Maps .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 43 Figure 3-21: Loss (BCE) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Predicted Probability .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 44 Figure 4-1: Progress of Average Training Loss of ANN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 48 Figure 4-2: Progress of Average Validation Loss of ANN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 48 Figure 4-3: Progress of Average Training Loss of CNN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 49 Figure 4-4: Progress of Average Validation Loss of CNN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 50 Figure 4-5: Progress of Average Training Loss of ResNet-18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 51 Figure 4-6: Progress of Average Validation Loss of ResNet-18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 51 Figure 4-7: Confusion Matrix of ResNet-18 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 54 Figure 4-9: Confusion Matrix of ANN .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 56 Figure 4-11: (a) Input Image (Y𝒂), (b) Pre-processed Input Image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 58 Design of Arabic Sign Language Recognition Model Department of Mechatronics Engineering Senior Design Graduation Project Report vi LIST OF TABLES Table 1-1: Proposed system with different classifiers .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5 Table 3-1: Comparison between SGD and ADAM .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. 46 Table 4-1: Comparison Between Different Models’ Accuracies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 52 Table 4-2: Arabic Sign Language Alphabets, their numbers, and labels .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 59 Table 4-3: Inference Time Results .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': 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+page_content='Gradient ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Descent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Design of Arabic Sign Language Recognition Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Design of Arabic Sign Language Recognition Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Chapter 1 INTRODUCTION 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Background According to the World Health Organization (WHO), around 466 million people with hearing loss issues, and 34 million of them are children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is claimed that by 2050 over 900 million people will have suffering hearing loss [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hard hearing people can hear up to a specific limited degree and unobvious by a hearing aid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In contrast, deaf people cannot listen entirely due to head trauma, noise exposure, disease, or genetic condition [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Sign language is the means of communication between the deaf themselves and with ordinary people, and every country has its own language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' One of these languages is ArSL, used in the Arabic regions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' it was formally introduced in 2001 by the Arab Federation of the Deaf (AFOD) [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Sign Language depends on hand movements and gestures to accomplish what you want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are various dialects of ArSL that differ from one country to another;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' it comprises 28 characters that the different dialects agree on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Signs of Arabic alphabets is shown in Figure 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 1-1: Signs of Arabic alphabets [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2 2 2 L 3 3 3 E e JDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Problem Definition The communication gap between ordinary people and deaf people is enormous, and we want to make it tiny, but it is a long road so, the best way is to study the subject from scratch and absorb the basis of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" As an initial point, we should learn in detail the signs of Arabic Alphabets to reduce sign language learners' obstacles, but the matter is not simple for all learners." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many of them will be confused when they learn a new field and may find this problematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For this reason, we intend to build a model that recognizes the alphabet sign from Arabic Sign Language Speakers and then interprets it into text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Literature Review In [5], Omar Al-Jarrah and Alaa Halawani used a collection of ANFIS networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each network is trained to recognize one gesture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The system used images of bare hands, which allows the user to make the interaction more natural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The subtractive clustering algorithm and the least-squares estimator are used to identify the fuzzy inference system, and the training is achieved using the hybrid learning algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The achieved accuracy is 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='55% that resulted from recognizing the 30 Arabic Manual alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [6], Khaled Assaleh and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Al-Rousan used Polynomial Classifiers to recognize the Arabic Sign Language Alphabet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It had seen that the Polynomial classifier has several advantages compared with ANFIS-based classification when they work on the same data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The data had been collected from deaf people and using the same corresponding feature set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The data collected by coloured Marked Glove- based systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Polynomial Classifiers showed Significant results over ANFIS based on misclassified data patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Specifically, it has a 36% reduction when the methods were evaluated on the training data and a 57% reduction when the systems were assessed on the test data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [7], the author split the process into three stages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' a data collection stage, a feature extraction stage, and a recognition stage using Hidden Markov Model (HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A vision-based methodology is used to collect the data, and then we need to prepare the data to absorb the necessary features to classify it using HMM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The collected data were 4500 signs from 15 samples with 300 signs for the single signer, 11 samples were taken for the training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the accuracy obtained from the experiments is 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='73%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This paper [8] shows an automated method for the translation of Arabic Sign Language alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Its data had been collected using images of bare hands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The Experiments showed that the ArSLAT, Arabic Sign Language Automatic Translator, had an accuracy of 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3% with 30 Arabic Alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [9], the authors presented the stages they used to achieve recognition: skin detection, background exclusion, face and hands extraction, feature extraction, and classification using Hidden Markov Model (HMM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The dataset consists of 29 alphabet Arabic letters and numbers from 0to nine with different brightness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They used 253 training images and 104 testing images with 640×480 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" The recognition system is tested when dividing the handshape's rectangle surrounding it into 4, 9, 16, and 25 zones." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' At 16 zones, the recognition rate with 19 states reaches 100%, while at 4 and 9 zones cannot match 100%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 3 In [10], the author explained the nature of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is images for the positive samples with the hand sign in different scales, different illumination in the complex background for each hand posture, and the negative samples images from the Internet which do not contain hand posture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 1-2 shows the stages of translation from Natural language into Sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 1-2: Sign language recognition system [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [11], the system was created to recognize the Arabic sign words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The system consists of three stages, which are Pre-processing, Feature extraction, and classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Moreover, the training and testing evaluation methods depend on the database of 23 signs that performed three signers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each character is repeated 50 times by the singers, and the training set consists of 70%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The testing set consists of 30%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model was evaluated on three different frequency domains (viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Fourier, Hartley, and Log- Gabor transforms) for the feature extraction stage and assessed on three classifiers: KNN, SVM, and MLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The results showed that the best Arabic sign language recognition system is Hartley transform using SVM classifier based on accuracy, 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Moreover, when sign images were segmented into 3*3 segments, the accuracy raised to 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In this paper [12], it is focused on feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The feature extraction techniques are utilized for training the classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Sign language usually is dynamic where the upper part of the body, head, shoulder, and hands have a movement while other parts are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The feature must be utilized by collecting high-contrast locations such as object edges and corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [13], the authors designed a system to translate the ArSL alphabet gestures into text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The used dataset is captured from different smartphones by 30 volunteers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each volunteer worked on a subset that has 30 images, so the dataset consists of 900 images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The authors used five descriptors to recognize.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' When using the Histograms of Oriented Gradients (HOG) descriptor, the proposed ArSL system accuracy is 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='56%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The accuracy of Edge Histogram Descriptor (EHD) is 42%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The accuracy of Discrete Wavelet Texture Descriptor (DWT) is 8%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The accuracy of the Local Binary Pattern (LBP) descriptor is 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='78%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The worst accuracy result is obtained using the 5 Gray-Level Co-occurrence Matrix (GLCM) descriptor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the proposed ArSL system accuracy is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='89%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The authors in [14] present a system that translates isolated Arabic word signals to text with automatic visual SLRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" The system consists of four stages to obtain the results: hand segmentation using the Training Data Set Preprocessing Feature Extraction Text/Audio Classification ImageorVideo Preprocessing Feature Extraction AcquisitionDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 4 dynamic skin detector that depends on the face's colour," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" tracking using segmented skin points used to recognize and track the hands with the head's help," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' extracting the geo-metrical features of the spatial field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, classification is carried out using Euclidean distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' As a result, the authors achieved a discrimination rate of 97% using a training set of 300 videos and a test set of 150 videos bearing in mind that 83% of words had different occlusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These videos only contain 30 isolated words used in the daily life of hard of hearing children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The authors in [15] present a new benchmark dataset publicly accessed along with the Sign Language Recognition algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The SLR algorithm consists of three phases, which are hand segmentation, hand shape sequence, and body motion description, and sign classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, the sign classification phase uses canonical correlation analysis and random forest classifiers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, the dataset used for the algorithm was 150 different signs collected from 21 signers using the Kinect v2 sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The total sample is 7500 samples (150 signs * 5 signer groups * 10 samples per sign per group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, the algorithm achieved a state-of-the-art solution when rated on the public data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, the achieved recognition accuracy is 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='57% evaluated on 150 ArSL signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The paper [16] starts sorting the sign language into two components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' manual and non-manual signs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The manual signs include hand position, orientation, shape, and trajectory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The non-manual signs represent body motion and facial expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convolutional Neural Network (CNN) is a deep learning class employed in image classification;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' it makes the network quick to learn and find the complex pattern simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CNN still uses the Backpropagation and its derivatives training methods to learn from data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The author used a dataset of images containing 2030 images of numbers (from 0 to 10) and 5839 images of 28 letters of Arab sign language, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=', 7869 RGB colour images with 256×256 pixels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These images are taken from different signers and different luminosity intensities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The authors in [17] present an Arabic Sign Recognition system to overcome finger occlusions and missing data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The system uses two Leap Motion controllers for data acquisition since they detect hands and fingers moving.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' After that, data is put together using the Dempster-Shafer (DS) theory of evidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A set of geometric features from both LMCs is chosen to feed them for the classification stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, the Bayesian approach with a Gaussian Mixture Model (GMM) and a simple Linear Discriminant Analysis (LDA) approach, used for classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are 2000 samples collected from two native adult signers by repeated 100 isolated Arabic dynamic signs ten times for each singer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Then, 70% of the dataset was used for training, and 30% used for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The submitted system is considered a state-of-the-art-glove-based system and single-sensor, and it achieved about 92% recognition accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The authors in [4] proposed a real-time ArSL alphabets recognition system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It consists of four convolution layers, four max-pooling layers, and five dropout layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, 54,049 images are used Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 5 as a dataset for this system, consisting of 32 alphabets obtained from more than 40 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is divided into 64% for training, 16% for validation, and 20% for testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, the achieved recognition accuracy was 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The paper [18] shows a novel framework used to recognize isolated Arabic Sign Language words for signer-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This framework depends on three stages to classify input videos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These three stages are the DeepLabv3+ model used for hand semantic segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The single-layer convolutional self-organizing map is used to extract hand shape features representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A deep recurrent neural network is used to recognize the sequence of extracted feature vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The dataset comprises 150 repetitions for each of the 23 words they used, taken from 3 signers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In conclusion, the framework model achieves state-of-the-art performance with an average accuracy of 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [19], the authors exhibit sign language differently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Most researchers try to obtain the text rather than the semantic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' To recognize the word with its semantic, they combined CNN with a semantic layer, and it maps the word to the meaning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A mobile camera picks the dataset in a different surrounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model achieved good recognition accuracy of 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='87%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [20], M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Kamruzzaman creates a model to convert Arabic Sign Language images to letters by CNN and then convert generated Arabic letters to Arabic Speech by Google Text to Speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The CNN model has 2 Convolution layers, and the first layer has 32 Kernels, and the second has 64 kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model also trained for 100 epochs on 100 images for every 31 letters and tested 25 images for each letter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It got an accuracy of about 90% for the testing set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In [21], the authors focused on the recognition of letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The collected data was 900 coloured images have been used to represent the 30 different hand gestures and have been used as a training set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' another 900 images have been taken and used as a test set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They developed the recognition system and calculated its performance using Feed-Forward Neural Networks and various Recurrent Neural Networks (RNN) types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The performance that they got is concluded in Table 1-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Table 1-1: Proposed system with different classifiers [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Classifier Accuracy Feed-Forward Neural Network 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='33% Elman neural network 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='66% Jordan neural network 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='56% Fully recurrent neural network 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='11% The authors in [22] proposed the first ArSL recognition system that converts ArSL to Arabic sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The machine translation system is Rule-based, and it has three stages;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the input Arabic Sign language word is processed for Morphological analysis then Syntactic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, transfer to Arabic Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 6 sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, the system used a corpus that has sentences that are used in health centres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It has 600 sentences that consist of 3327 sign words with 593 unique sign words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The proposed dataset is divided into training, validation, and testing datasets, with a percentage of 70%, 15%, and 15%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The results of the system are calculated Manually and automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, the manual evaluation shows that 80% of the sentences are accurately translated, and 2 ArSL experts do the evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it is evaluated automatically by BLEU and TER metrics and gets 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='39 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='45 repetitively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content="4 Aims and Objectives Facilitation of deaf people's lives and making their communication more straightforward is what we aim to achieve." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The objective is to construct a simple link between deaf people and others by creating an accurate automated model using deep learning to interpret sign language alphabets to text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" We will study the previous results that the others obtain, enhance the model's performance, and make a prototype to test the model." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In the future, we will do continuous feedback to diagnose any error and fix it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We will expand our work to include the words and deal with full sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 Report Organization In the rest of this work is organized as follows: Chapter 2 (Design Considerations), Chapter 3 (Model architecture, Training and testing), Chapter 4 (Design Testing and Results), Chapter 5 (Conclusion And Future Work).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 8 Chapter 2 DESIGN CONSIDERATIONS This chapter explains the software, frameworks, techniques, and alternatives that can also be needed in the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, we will show design constraints and standards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Design Options Image recognition requires complex calculations to accomplish it using the computer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, we need powerful techniques and appropriate software to achieve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In general, computer vision can do this smoothly, but not all computer vision techniques are suitable for image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, many software and frameworks can be used in computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Computer Vision Techniques 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' K-Nearest-Neighbor Classification (KNN) In [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' K-Nearest Neighbour is considered supervised learning in which the features and labels are given in the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This technique depends on the closest distance between the point and the predicted labels to classify the object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' An unlabelled query point is assigned the label that has the K-Nearest Neighbour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The classification process is calculated from most of its K Nearest Neighbours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' To classify the images, each image is converted to a fixed vector, then the distance can be measured by any function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Euclidean distance is the most common function: 𝑑(𝑥, 𝑦) = ‖𝑥 − 𝑦‖ = √(𝑥 − 𝑦) ∙ (𝑥 − 𝑦) = √∑ (𝑥𝑖 − 𝑦𝑖)2 𝑚 𝑖=1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1) Figure 2 1: KNN Classification [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Linear Classifiers "Linear classifiers classify data into labels based on a linear combination of input features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Therefore, these classifiers separate data using a line or plane or a hyperplane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They are suitable to classify the linear separable data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='" [24] 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Logistic Regression (Binary Classification): A statistical model can be used to evaluate (guess) the probability of an event depends on input data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For example, we have two classes, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=', "dog" or "not dog" and those can be represented by 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It can be mathematically represented as follows: 𝑦̂ = 𝜎(𝑧) 𝜎(𝑧) = 1 1 + ⅇ−𝑧 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2) is the logistic function and 𝑧 = 𝑤𝑇𝑥 + 𝑏 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3) And these parameters are as follows: 𝜔 : weight 𝑏 : bias 𝑥 : flattened feature input vector The model takes x as an input, and the probability of the outputs 𝑦̂ = 𝜌(𝑦 = 1|𝑥) Figure 2-2: Sigmoid Function (Logistic Function) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content="0 -8 -6 -4 -2 0 2 4 6 8 2Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 10 For the images x, the feature vector can be just the pixels' values in RGB channels, and it can represent by a vector with one dimension." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It can be resulted by flattening those three dimensions, and the resulted size is 𝑛𝑥 = 𝑛ℎ𝑒𝑖𝑔ℎ𝑡 × 𝑛𝑤𝑖𝑑𝑡ℎ × 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The goal of this algorithm is to classify the images correctly, and this can do by training the model on training samples that will change the values of 𝑤 and 𝑏.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The optimal values of these parameters can be justified when 𝑦̂(𝑖) most closely predicts 𝑦(𝑖).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Where: 𝑦̂(𝑖) : predicted class value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑦(𝑖) : correct class value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In practice, this model usually calculates the loss function: 𝐿 (𝑦̂(𝑖), 𝑦(𝑖)) = −[𝑦(𝑖) log(1 − 𝑦̂(𝑖)) + (1 − 𝑦(𝑖)) log(1 − 𝑦̂(𝑖))] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4) For each training example and minimizing the cost function, 𝐽(𝑤, 𝑏) = 1 𝑚 ∑ 𝐿 (𝑦̂(𝑖), 𝑦(𝑖)) 𝑚 𝑖=1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5) Overall m training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜕𝐿 𝜕𝑊𝑗 = (𝑦̂(𝑖) − 𝑦(𝑖))𝑥(𝑖) 𝑗 and 𝜕𝐿 𝜕𝑊 = (𝑦̂(𝑖) − 𝑦(𝑖)) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6) Where 𝑗 = 1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. , 𝑛𝑥 labels the components of the feature vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' To get the optimal value of 𝑤 and 𝑏 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝐽 should be minimized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It can be minimized numerically after choosing initial values by changing them according to descent along the steepest gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑤𝑗 → 𝑤𝑗 − 𝛼 𝜕𝑗 𝜕𝑤𝑗 = 𝑤𝑗 − 𝛼 𝑚 ∑ 𝜕𝐿 𝜕𝑤𝑗 𝑚 𝑖=1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='7) Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 11 𝛼 is the learning rate (step size), which affects how large each step is taken in the direction of greatest decrease in 𝐽.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Choosing a good value for α is a subtle art (where the too-large value will affect the training to be fast and the training may not converge steadfastly and too small value so the training will be slow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Softmax and SVM classifiers: The linear classifier uses the below equation to learn the features of images and stores them in W, b: 𝑓(𝑥𝑖, 𝑊, 𝑏) = 𝑊 𝑥𝑖 + 𝑏 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8) 𝑊 : Weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑏 : bias term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑥𝑖 : input image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑓(𝑥𝑖, 𝑊, 𝑏) : Score function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' W, b (module parameters) will be changed depending on the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The output module will classify the image depending on its features (pixel value), and space will be divided by linear functions [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-3: Mapping a Cat Image to Class Scores [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The above image is added as an example to clarify the idea of the linear classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For ease of visualization, the image is assumed to have 4 pixels only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And 𝑊 is considered as a matrix with a size of 3 × 4, where 3 is the class number, and 4 is the flattened input size to imply the matrix multiplication between 𝑊 and 𝑥𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, by doing the matrix multiplication of 𝑊 and 𝑥𝑖 then adding the bias 𝑏 so the results will be the scores for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The 𝑊 in the image is bad, and the scores at the end claim that the image is a dog not a cat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, the 𝑊 will improve by train the model, and it may get better results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are many loss functions that can be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For example, the linear classifier model usually uses a loss called the Multiclass Support Vector Machine (SVM) loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, the Multiclass loss can be shown as below: stretchpixelsintosinglecolumn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 56 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 -96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8 cat score 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 231 + 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 437.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='9 dog score 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 24 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='95 input image ship score M 6 2 f(ci;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' W,b) CiDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 12 𝐿𝑖 = ∑ max (0,𝑆𝑗 − 𝑆𝑦𝑖 + 𝛥) 𝑗≠𝑦𝑖 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='9) Where: 𝑆𝑗 : is the score for the jth class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑆𝑦𝑖 : is the score for the ith class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝛥 : is the fixed margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Another commonly used classifier is Softmax, which used cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The function mapping is still used 𝑓(𝑥𝑖;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='𝑤) = 𝑤𝑥𝑖 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And the cross-entropy loss has the following form: 𝐿𝑖 = − log ( ⅇ𝑓𝑦𝑖 𝛴𝑗ⅇ𝑓𝑗) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10) Where 𝑓𝑦𝑖: is the class score for the ith element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Where 𝑓𝑗: is the class score for the jth element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And 𝑒𝑧𝑗 𝛴𝑗𝑒𝑧𝑘 It is called Softmax Function, which has an input vector score (in z) squishes it to a vector of values between zero and one (Probability), which sum to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The below images show the difference between SVM and Softmax classifier for the same input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Both have the same mapping function, which resulted from the matrix multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' But there is a difference in the interpretation of the score function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' SVM interprets the class scores, and it encourages the correct class to be the higher one by a margin than the other classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 13 Figure 2-4: Example for Softmax and SVM [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Artificial Neural Networks ANNs are simulated based on the brain's architecture." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They consist of connected elements known as an artificial neuron;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' it has one or multiple inputs and one output with either zero or one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each neuron is associated with a weight, and if their sum is more than the threshold, the neuron will activate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The following equation will explain the mathematical representation: 𝑥 = { 1, ∑ 𝑤𝑖𝑥𝑖 − 𝑏 𝑖 > 0 0, ∑ 𝑤𝑖𝑥𝑖 − 𝑏 𝑖 ≤ 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='11) If we want to make the output smoother between zero and one, we will use the sigmoid function as the following: σ = 1 1 + ⅇ∑ 𝑤𝑖𝑥𝑖−𝑏 𝑖 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='12) Figure 2-5 shows a simple neuron with three inputs associated with its weights and the bias and then applying the activation function to show the result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2 5: Simple Neuron [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' hinge loss (SVM) -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='85 matrix multiply + bias offset max(0, -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='85 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 + 1) + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='86 max(0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='86 - 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 + 1) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='01 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='05 -15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='58 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='16 22 + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='45 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='03 -44 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 cross-entropy loss (Softmax) -2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='85 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='058 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='016 M 56 b exp normalize 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='86 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='631 - log(0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='353) Ci (to sum = to one) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='353 Yi 221w1 T2 2 b output Gm T3 Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 14 ANNs consist of layers as an input layer, one hidden layer or more, and an output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each layer consists of neurons that compute the weighted sum of their inputs then specify the output using some of the activation functions like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' sigmoid, ReLU, and SeLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' All the neurons in a layer are considered an input to the followed layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ANNs can recognize the output by modifying the weights and biases each one epoch until minimizing the errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We need to classify the result of each neuron in the output layer to the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, Softmax function can be used at the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-6 shows the architecture of ANN that includes ReLU and Softmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-6: ANN Architecture, Including ReLU and Softmax [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Softmax Softmax output layer Hiddenlayer (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=', ReLU) Input ayer X2Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 15 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convolutional Neural Networks CNN is the most dominant technique in deep learning that can use in computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CNN is a mathematical model that consists of three types of layers as follows: a) Convolutional Layer It is an essential component of the CNN architecture used for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Neurons in one layer are connected to other neurons in their receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The array that combines the neurons is called the kernel, and it is typically formed as 3 × 3, but maybe choose 5 × 5 or 7 × 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This architecture allows the low-level features to concentrate on one layer, then assemble them into higher-level features in the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" The operation above does not guarantee each kernel's centre to overlap the input layer's outermost element." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Padding, precisely Zero Padding, is a solution to avoid adding zeros around the inputs that can overlap the outer element of the input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Stride is "the distance between two consecutive Kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ", the standard option of a stride is one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-7 shows an example of a convolutional operation with a kernel size 3 × 3, Zero Padding, and a stride of one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-7: An Example of A Convolutional Operation [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' b) Pooling Layer The goal of this layer is to shrink the inputs to decrease the computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Max pooling and Mean pooling are common examples of pooling layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Max pooling is the most popular form, which takes the maximum value in the higher-level feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Mean pooling takes the average of all the elements in the higher-level feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Element-wise product Sumup Kernel Feature map InputtensorDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 16 c) Fully Connected Layer This layer transforms the last convolutional layer into a one-dimensional array and connects to one or more dense layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A non-linear activation function follows the final fully connected layer to classify the inputs according to the output probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Transfer Learning Many computer vision cases have small datasets, so the training of the model will be invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The popular approach to deal with this case is to use the transfer function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Transfer learning is a network that comprises extensive data, and it was trained to absorb generally feature extraction of the image classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convnet, VGG16, ResNet, Inception, and Xception are examples of architectures trained on ImageNet (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 million labelled images with 1,000 different classes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is preferred to choose the understood architecture for you, and no need-to-know new ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Software 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' MATLAB It is a programming platform that offers toolboxes to help engineers and scientists in academia and industry to perform the solutions for various aspects of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The essence of MATLAB is a matrix-based language that allows for progressing the calculations smoothly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' MATLAB can deal with data by analysing and visualizing it, improving existing algorithms to coincide with your requirements, and creating models and apps from scratch [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' MATLAB includes Many applications and capabilities that can perform several functions as follows: 1- Applications [31]: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Image Processing and Computer Vision: Processing of images and videos using several techniques to build any visual model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Data Science: Use machine learning to predict and label the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Deep Learning: Apply deep neural networks and prepare the related data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Signal Processing: Convert the signal and prepare it to analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2- Capabilities [31]: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Algorithm Development: Improve or build algorithms for several tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Cloud Computing: Run public clouds like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' AWS and AZURE on MathWorks cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Data Acquisition: Gain the data from an external source like a camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' GPU Computing: Offer using NVIDIA CUDA to accelerate the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Parallel Computing: Use CPUs, GPUs, and TPUs simultaneously in large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Real-Time Simulation and Testing: Apply the hardware systems in real-time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 17 MATLAB is a useful software for Machine Learning because of its simplicity of use and offering toolboxes that support machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The toolboxes like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' image processing and computer vision, data science, and deep learning include all the tools to train and test the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' MATLAB offers parallel computing to operate CPUs, GPUs, TPUs, and clouding to achieve high performance [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' LabVIEW LabVIEW is software designed for engineering problems that require the acquisition of the data, testing it, measuring it, and controlling it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The most feature of LabView is its ability to create mutuality environment between the hardware and data insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' LabVIEW provides the user with a graphical programming approach, toolkits, and modules that help the user visualize any application like working in a real lab, including hardware configuration, instrumentation to measure the data, and error debugging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This integration between hardware and software can simplify building a complex diagram and applying it on hardware, improving the data algorithms, and designing special user interfaces [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' LabVIEW contains Analytics and Machine Learning Toolkit that combines predictive analytics and machine learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The toolkit is prepared to deal with large data and do some processes like, classification, clustering, and anomaly detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And it has good advantages which are to monitor the conditions and maintain the predictive [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-8 shows some of the processes can LabView applying on data to get some results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-8: Machine Learning Processes Which LabVIEW Can Provide [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are some explanations on the processes: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Data Collection: DAQs (Data Acquisitions) allows picking up the required data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Feature Extraction: Some tools like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Vision Development Module, NI Sound and Vibration Measurement Suite can extract the features from the data based on your domain knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Feature Reduction: Use some techniques to simplify the data and reduce its dimension to prepare it for training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Model Creation: Give the flexibility to build and train the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Model Validation: Use evaluation metrics to check the validation of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Deployment: Use deployment data to predict new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Data Feature Feature Model Model Deployment Collection Extraction Reduction Creation ValidationDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 18 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Julia "It is a high-level, high-performance dynamic language, focusing on numerical computing and general programming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='" Traditional computing languages were either fast or productive, but not both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Julia achieves fast and productivity [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Julia contains packages supporting computer vision tasks and includes open-source libraries like Open CV and Tesseract to find optimum computer vision tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Julia can deal with simple images using Julia Images to advanced Images using Julia's APIs [37]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Python "Python is an interpreted, object-oriented, high-level programming language with dynamic semantics," Python is easy to learn because it supports code readability and therefore reduces the bugs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Python consists of dynamic typing and dynamic biding that make the program shorter and faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Python supports packages with a wide range of functionality like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' data analytics, databases, graphical user interface, image processing, and scientific computing, which allows the code to be reused and decreased the effort required to build the code from scratch [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Python is considered the most common programming language for machine learning and data science because it allows forgetting the complex parts of programming by putting the concepts directly into the goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Python provides us with many libraries and frameworks that offer loading data, prepare data, label data, visualize data, and apply the different algorithms to train and test the models [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Python Frameworks A framework is an interface that makes machine learning models simpler and speeds up the processing of models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Frameworks allow connecting the data with the models as APIs and observe your model and its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The famous frameworks that are used in Python: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Scikit-Learn It is an open-source machine learning framework that implements various model fitting functions, data extraction, and many other advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is straightforward to use so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is considered an entry point to enter the machine learning field [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' TensorFlow It is an advanced open-source framework that can achieve complex computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It allows us to build huge flexibility models because it has a rich library that contains many functions and prepared models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' TensorFlow offers cloudy hundreds of GPU servers [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Keras It is a high-level Deep Learning API (Application Programming Interface) that can simplify building the model and training it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Reducing cognitive load is considered one of TensorFlow's most feature, which can load data efficiently [42]." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' PyTorch It is an advanced open-source framework that has tools to improve computer vision and reinforcement learning fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It provides cloud platforms and the ability to use GPUs to accelerate the models [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 Hardware Deep learning algorithms like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' computer vision or automatic speech recognition require computational power because the model becomes deeper and has big data to analyse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many hardware units can deal with big data and reduce the training time as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Central Processing Unit (CPU) It is an integrated circuit that performs machine instructions using arithmetic, logic, controlling, and input/output operations stated by the commands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CPU includes an Arithmetic Logic Unit (ALU), Central Unit (CU), and Memory Unit (MU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ALU is used to execute arithmetic and logical operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CU uses the data bus and control bus to organize the control signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' MU includes the various aspects of memory such as Random-Access Memory (RAM), Read-Only Memory (ROM), and CACHE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CPU performs the operations, where registers are loading the values and storing them, CACHE memory retrieving the values, CU organizes the requests and controls the priorities steps to process the input according to the ALU requests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-9 shows the principal components of the CPU [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 20 Most modern CPUs are embedded in IC chip that includes the CPU, memory, and peripherals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Modern CPUs have multi-cores, where each core can run several threads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Most Machine Learning algorithms are based on matrix multiplications and additions so, CPUs cannot quickly achieve this arithmetic calculation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' for example, training a deep network with a single chip can continue for days or weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The Frameworks can operate CPU and GPU parallel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the heavy computations on the GPU, and data processing on the CPU [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-9: CPU Operations [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Graphical Processing Unit (GPU) GPUs become an essential integral part of computing's systems because they become more complex and need to be faster with high-performance in various aspects like, gaming and Machine Learning applications." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The GPUs are the best choice for large computations applications [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' GPU is a high-computational performance processor for graphical processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' GPU was designed for parallel processing and high memory bandwidth to accomplish high computational power and increased productivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" The GPU's architecture essential component is the Streaming Multiprocessor (SM), also called CUDA-Cores by NVidia." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' SMs contain many ALUs, and each SM can operate one warp (a pack of 32 threads) simultaneously [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CPU Control Unit Instructions Processor Registers Com binational Input Logic Output Main MemoryDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 21 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Tensor Processing Unit (TPU) "It is custom-developed application-specific integrated circuits (ASICs) used to accelerate machine learning algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='" Cloud TPUs allow us to train the models on TensorFlow with high performance and less time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Machine learning's essence is the mathematical computations that minimize the error between inputs and predicted outputs, so cloud TPUs accelerate calculations' performance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is advised to use cloud TPUs in these cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the models are constructed from matrix computations, the models that require weeks or months for training, and the large models that contain more and more layers with huge batch size [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Design Constraints and Standards Constraints are restrictions that prevent something from being the best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They can be problems that arise or issues that come up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Some constraints must be considered in our project as follows: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Availability of Data: The AI, Machine Learning and Deep Learning models are hungry for data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Especially, Deep Learning models need more than 1000 images for each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And those images should agree with the real world with many backgrounds, noise, illuminations, and the direction of the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And there are few resources for Arabic sign language images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, we need large data with various aspects to get a good model with high accuracy and cover all the possibility’s images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, some resources are not available for everyone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 2-10: Performance Vs Amount of Data [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Industry Giants Deep Learning Performance Quality Gap in Al products Small and Medium size Older algorithms companies Amount of dataDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Computational Resources: Training of large-scale data usually needs GPUs or TPUs to accomplish the computations and memory usage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it may need many GPUs to be used at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Therefore, this will affect the time required to get results and test the model for many cases to check its performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, these resources are expensive to afford for model training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Response time (Inference Time): Real-time systems required short inference time to deploy the model for real-life usage with minimum computation resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hyperparameter Choosing: These are considered critical for every model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And choosing these parameters is subtle art rather than standard choosing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, you can use similar research and models hyperparameter as a guide for your research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Knowledge and Experience in ArSL: This project is a multi-disciplinary project that combines computer vision with ArSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, it needs an expert in ArSL to take care of the Arabic sign language part of the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 23 Chapter 3 MODEL ARCHITECTURE, TRAINING AND TESTING This project aims to build a model that can convert the alphabetical image in ArSL to the corresponding written letter in the Arabic language by applying some algorithms that can extract features and differences in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Choosing the right algorithm depends on the nature of data, complexity, and required resolution of the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This model uses the PyTorch framework to complete this task by testing several models and comparing final results based on the ArASL2018 dataset [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Usually, it would be best to try many approaches before getting the best one for a new problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Even experienced machine learning researchers need a lot of ideas before getting the desired results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' An experienced AI engineer suggest following a cycle to get satisfactory results, as follows [49]: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Create an idea on how to build the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convert the idea to code that can be implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Evaluate the idea by an experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Terminate or go back from the beginning and generates more ideas, then keep this continues iterating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-1: Machine Learning System Cycle [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Idea 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Experiment 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content="CodeDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 24 Before constructing any code, we should do a general flowchart or pseudocode to plan the model's steps to know the path on it to achieve the task." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It helps to be systematically through programming the code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-2 shows the general flowchart for the project: Figure 3-2: Flowchart of Recognition Model Start Hand Images (Importing from Google Drive) Splitting of data into training& testing Hyperparameters tuning Define the classification model Loss function & optimizer Classification Do theresults meet the requirements?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' EndDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 25 The explanation of the flowchart will be in the following steps: \uf0b7 Importing of Data: The model will be trained using Google Colab, and we need to feed the data into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' One approach to importing data is uploading locally from PC, but this way is not proper because we will need to upload the data again every time opening Colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Another approach is uploading data to Google Drive once, and we just recall the data when we need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Splitting Data: We need to split data perfectly into training, validation, and testing datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These datasets assure to generalize the data and examine the model performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Scikit-learn supplies functions that can split the data perfectly with various features like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' random state parameter that generates random datasets and gives the same dataset if you select the same seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Another feature is splitting the data into subsets with the same indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The most important feature is stratified sampling which means splitting the data into stratified datasets, which mean that datasets have the same proportion of input dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Hyperparameters Tuning: The models in deep learning have several parameters like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the number of epochs, the number of batches, learning rate, and dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Parameter’s manipulation can make the model better or worse, and selecting parameters depends on the experience or testing several trials until satisfying the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" Defining the Model: It means constructing the model's architecture, including the type of layers, number of layers." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Loss Function and Optimizer: To assist the model, we need a loss function to ensure that the training is doing well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, we need an optimizer to update the weights inside the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Classification: At this stage, the model is trained, validated, and tested.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' After that, we see the results if they satisfy our target or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Dataset Examination 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 The Nature of Images In terms of colour, it can be in grayscale representation where it has 1 channel or in RGB representation where it has 3 channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" [50] \uf0b7 Grayscale: It contains shades of grey, proportional to the pixels' luminance." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The luminance index changes from 0 to 255, examples of these are black, which is 0, and white which is 255.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-3 (a) shows an image that is represented in grayscale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' RGB: The colour of each pixel in the image can be represented as a combination of RGB space colour, and this allows the user to specify the colour intensity between 0 and 255 for each channel, and the combination of this mixture determines the final colour, examples of which are RGB (255, 0,0) is red, RGB (255,255,0) is yellow, and RGB (128,0,128) is purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These pixels can be combined to form the image we see in real life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3 3 (b) shows an image that is represented in RGB colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' (a) (b) Figure 3-3: (a) Grayscale Colour Space, (b) RGB Colour Space Our project has 54,049 images in grayscale, so that the model will be trained and tested based on grayscale colour space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The computations in grayscale will be simpler than RGB because we deal with 2D arrays, making the model faster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 27 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 ArASL2018: Arabic Alphabets Sign Language Dataset ArASL2018 is a dataset that contains 54,049 fully labelled images for 32 alphabets in Arabic sign language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, these images are performed from 40 participants of different ages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These images are grayscale with 64 × 64 pixels JPG format that captured using a smart camera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' [4] Figure 3-4: Samples For ArASL2018 Dataset [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-5: A Brief Description of Arabic Sign Classes [4].' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Sign ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Language ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Recognition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Design of Arabic Sign Language Recognition Model ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 \uf0b7 Problems in The ArASL2018 Dataset: o There are different sizes of images in the dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' which are: \uf0a7 The number of images with size 𝟔𝟒 × 𝟔𝟒 are 53401.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0a7 The number of images with size 𝟏𝟎𝟐𝟒 × 𝟕𝟔𝟖 are 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0a7 The number of images with size 𝟐𝟓𝟔 × 𝟐𝟓𝟔 are 638.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o These images are collected from a video, which means they are similar in the surrounding conditions and do not represent the actual life datasets, and this does not guarantee the variation of data to explore the pattern of images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o Some of the images in the dataset need the motion to describe the sign entirely so, training of them will be useless without the movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Our Solution for The Problems: o We resized all images to 𝟐𝟐𝟒 × 𝟐𝟐𝟒.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o In this stage, we used the available images with greyscale, but we collected new dataset that has new features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 29 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Data Splitting To build a suitable model in machine learning, we need to split the dataset into three sets: training, validation, and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model uses training dataset to tune the parameters, weights, and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, testing and validation datasets are the best way to check how well the model can deal with new data and generalize the new cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The validation and testing datasets usually are from the same distribution that the model with work with in the future, but this is not necessary for the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' [49] There are some explanations about the three sets briefly as follows: \uf0b7 Training Dataset: Instance data that the system can predict patterns by finding the optimal parameters to obtain a good prediction on new data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The training process is like humans learning, where people can learn complicated techniques by analysis of repeated patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Validation (Development) Dataset: o It is used to get the right parameters through tuning the networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o It is used to solve the overfitting issue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' we take an instance of data as a validation dataset to compare models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Testing Dataset: o It measures the performance of the network but without making any tunning on the model’s parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A high percentage of data goes to training as 70%, and validation as 15%, and testing as 15%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' These percentages can differ according to the dataset’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Usually, the data cannot be generalized, or we need to enlarge the dataset so, we manipulate and transform the dataset by, flipping, cropping, and resizing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This process is called data augmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-6 shows the processes on the dataset briefly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 30 Figure 3-6: General processes on the dataset The data are distributed to 32 classes, as shown in Figure 3-7, and we need to split it into three sets: 70% training, 15% validation, 15% testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' As a result of this, we got: \uf0b7 The training data length is 37834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The validation data length is 8108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The testing data length is 8107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Dataset Training set validation set Test set Machine learning algorithm Predictive model Final Performance estimateDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 31 Figure 3 7: ArASL2018 Dataset Histogram 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Define the Model We trained various models to classify the letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many models can perform better than others, and the problems that existed in a model can solve in another model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Our case is considered a complex problem, so the traditional approaches cannot achieve our target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Multilayer Model (ANN) This model consists of one input layer, one or more hidden layers, and one final layer called the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The number of neurons at the input layer equals the number of data input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' in our case, it equals the number of pixels 64 × 64 = 4096.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hidden layers are chosen according to the complexity of data, and every hidden layer is responsible for the extraction of specific features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The number of neurons at the output layer equals the number of classifications, in our case, equals 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2000 1750 1500 Frequency 1250 1000 750 500 250 ain aleff dha dhad gaaf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ghain haa jeem kaaf khaa laam meem nun ra pees seen sheen taa thaa thal toot waw yaa LabelDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 32 Figure 3 8: Multilayer Algorithm The following details will explain the algorithm: \uf0b7 By default, the algorithm deals with input data image by image, and we have more than 50 thousand images, so the training of the dataset will take much time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' As a result, that, we divide the dataset into many batches to faster our model’s training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And we used to stratify the function available in the Scikit-learn framework, which takes only the number of images from each class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Every neuron has a weight and bias that represent some data features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It must be to initialize them randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The manipulation of them increasingly or decreasingly can decide how the model’s accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The goal is to get better weights and biases, which achieve less error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Every batch will pass to the input layer, then pass to the first hidden layer, and calculate the neurons’ output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' After that, the next hidden layer will receive the previous layer’s output and apply the same previous calculations, and so on until we get the last layer’s output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The calculations are responsible for determining the performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They called activation functions like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' step, sigmoid, Tanh, and ReLU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Some of them are used to easy models, and the others to complex ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The output that we will get from the last hidden layer is free to be in any range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The softmax squeezes the output probabilities between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The architecture supposes that the higher probability is the correct output of the input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Next, the architecture measures the error using the loss function by comparing the obtained output with predictive output to estimate how much output corresponds with the predictive values, and the manipulation of the weights and biases will be done using optimizers like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' SGD and Adam by propagating the error backwards from the output through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Input layer Hiddcn layer Hidden layer Ouput laycr r(W,b) f(W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='b) f(W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='b) f2 13 P1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='.32 Crosscntropy f(W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='b) f(W,b) f(W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='b) 14096Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 33 Activation Functions [51]: Activation functions are used to make the complicated mapping between the inputs and corresponding outputs by applying the function to the summation of input products with weights and biases and then feeding them to the next layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The linear regression tasks can be dealt with by linear activation functions, but complicated tasks like image classification, speech recognition need non-linear functions to map between inputs and outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' An activation function must be differentiable to implement the optimizer like backpropagation, stochastic gradient descent because it requires computing the errors for the gradient of weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-9 shows some activation functions and explains how their derivations seem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-9: Activation Functions and Their Derivatives [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Step Function It is the simplest activation function and considered a hard limiter, activated when it has a value greater than a threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It cannot be used in multiclass classification tasks because it is limited between two values, −1 or 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equation that represents the function is: 𝑓(𝑥) = { 1, 𝑥 ≥ 0 −1, 𝑥 < 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1) It is obvious from the equation that the derivative of the step function equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Sigmoid Function It is a non-linear function that bounds the input between 0 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equations that represent the function and its derivative are: 𝜎(𝑥) = 1 1 + ⅇ−𝑥 𝜎́(𝑥) = 𝜎(𝑥)(1 − 𝜎(𝑥)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2) The function has a nonzero derivative everywhere, allowing the optimizer to update the weights and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Activationfunctions Derivatives 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8 Step Sigmoid 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 Tanh 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 ReLU -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 4 -2 0 2 4 -4 -2 0 2 4Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 34 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hyperbolic Tangent Function It is like a sigmoid function, and they have S-shaped and continuously differentiable, but its values bound the input between −1 to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equations that represent the function and its derivative are: 𝑓(𝑥) = 2𝜎(𝑥)(2𝑥) − 1 𝑓́(𝑥) = 2 1 + ⅇ−2𝑥 − 1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3) Compared to the sigmoid function, the gradient of tanh function is steeper, and its derivative is centred around zero, which often helps speed up convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ReLU Function ReLU stands for a rectified linear unit and continuously differentiable except at zero, where the slope changes abruptly, causing a bouncing around the optimal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, it works very well and can be fast to compute because it is linear, and its derivative is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Thereby the calculations will be simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equations that represent the function and its derivative are: 𝑓(𝑥) = {𝑥, 𝑥 ≥ 0 0, 𝑥 < 0 𝑓́(𝑥) = { 1, 𝑥 > 0 0, 𝑥 < 0 𝑛𝑜𝑡 𝑑ⅇ𝑓𝑖𝑛ⅇ𝑑, 𝑥 = 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 CNN Model Using multilayer architecture is not sufficient for image recognition tasks because if we get a simple image with 100 × 100 pixels, we will have 10000 neurons, then feeding them in a layer with 1000 neurons, we will obtain 10 million connections at the first hidden layer, it is considered a huge number of links and hard to deal with it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noticed that the multilayer model can deal appropriately with small images that have few features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CNN is a proper solution that uses a specific layer to minimize the connection to achieve the classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convolution Layers It is an essential component of the CNN architecture used for feature extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The neurons in the first convolution layer do not connect to every neuron in the input data, but neurons in one layer are connected to other neurons in their receptive field (specific patterns in small regions of the visual field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each neuron in the second convolution layer links neurons located within a small rectangle in the first convolution layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-10 shows the connections between the layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The first layer in the architecture is responsible for extracting some features, the next layer for other features, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 35 Figure 3-10: CNN Layers with Rectangular Local Receptive Fields [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The array that combines the neurons is called the kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The operation above does not guarantee each kernel’s centre to overlap the input layer’s outermost element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Padding, precisely Zero Padding, is a solution to avoid adding zeros around the inputs that can overlap the outer element of the input layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Stride is “the distance between two consecutive Kernels” [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-11: Connections Between Layers, Adapted froms [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convolutional layer2 Convolutional layer1 Inputlayerfh= 3 Zero Padding fw=3 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='. WDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 36 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Feature Maps The convolutional layer can be represented in 3D which every layer has multiple feature map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A feature map is considered a filter that explores features like vertical lines and horizontal lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-12 shows convolutional layers with multiple feature maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-12: Convolutional Layers with Multiple Feature Maps [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equation that shows how the convolutional layer computes the output of a given neuron is: 𝑍𝑖,𝑗,𝑘 = 𝑏𝑘 + ∑ ∑ ∑ 𝑋𝑖′,𝑗′,𝑘′ 𝑓𝑛′−1 𝑘′=0 𝑓𝑤−1 𝑣=0 𝑓ℎ−1 𝑢=0 ∙ 𝑊𝑢,𝑣,𝑘′,𝑘 𝑤𝑖𝑡ℎ { 𝑖′ = 𝑖 × 𝑠ℎ × 𝑢 𝑗′ = 𝑗 × 𝑠𝑤 × 𝑤 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5) Where: \uf0b7 𝒁𝒊,𝒋,𝒌: the neuron’s output in row i, column j in feature map k of the convolutional layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 𝑿𝒊′,𝒋′,𝒌′: the output of the neuron in layer L – 1, row i′, column j′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 𝒃𝒌 : the bias term in layer L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 𝑾𝒖,𝒗,𝒌′,𝒌 : the connection weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Convolutional Feature layer2 map 1 map2 Filters Convolutional Map 1 layer1 Map2 Inputlayer Channels Red Green BlueDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Pooling Layers This layer decreases the computations by shrinking the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The pooling layer is like the convolutional layer, connected partially to the previous layer’s outputs, located with the small rectangle receptive field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are two forms of pooling: Max pooling and Mean pooling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Max pooling is the most popular form, which takes the maximum value in the higher-level feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Mean pooling takes the average of all the elements in the higher-level feature layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Max and Mean pooling are shown in Figure 3-13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-13: Max Pooling and Mean Pooling with (2×2 Pooling Kernel, Stride 2, Zero Padding) It is noticed from the above figure that the pooling layer with 2×2 pooling kernel and stride 2 decreases the image to a quarter of the original image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This stage will reduce the computations, memory usage, and the number of parameters, thereby easing the extraction of features from the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Dropout Layer A huge number of parameters in the network gives it the flexibility to tend to overfit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' One solution to avoid overfitting is using the early stopping technique, where the network stores the parameters at the best values when the validation set worsens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' With unlimited computations, the early stopping technique will be aggressive and consume more time so, the best way to avoid overfitting is using the regularizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Dropout is one of the most regularization techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The term dropout refers to dropping out the neuron and incoming and outcoming connections temporarily from the network, as shown in Figure 3-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Original 100 150 200 50 100 150 200 250 300 350 MaxPooling MeanPooling 20 20 40 40 09 60 80 80 100 100 120 F 120 406080100120140160 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='75 (1+5+3+2)/4 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='75Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 38 The neurons that will be dropped out are chosen randomly, or simply, each neuron in the training set will be multiplied with a factor called dropout rate ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The dropout rate can be selected from between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 in CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In the testing set, the neurons will not be dropped out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For more explanation, suppose 𝑃=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5, the neurons during testing will be connected twice more than neurons in training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' So, each neuron in testing has a total input signal twice larger than what each neuron in training has, and the performance will not be ok in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each neuron input’s signal in testing will be multiplied by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 to compensate for the difference in neurons before and after training, as shown in Figure 3-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' (a) (b) Figure 3-14: (a) Network Without Dropout, (b) Network With Dropout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' [52] (a) (b) Figure 3-15: (a) Neuron at training, (b) Neuron at testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' [52] X XPresent with probability ppw Always presentDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 39 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Fully Connected Layer This layer transforms the last convolutional layer into a one-dimensional array and connects to one or more dense layers, in addition to a dropout layer after each dense layer, with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 dropout rate will reduce overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A non-linear activation function follows the final fully connected layer to estimate inputs classification according to the output probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-16: CNN Architecture [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-16 shows the general architecture of CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noticed that there are convolutional layers followed by a ReLU function or others, then another pooling layer, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The previous steps are considered feature layers and make the image smaller and smaller through the architecture until it reaches the classification layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Stages do the classification: flattening the last layer to pass it into a fully connected layer and then passing the fully connected layer into softmax function to classify the images according to the estimated probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 ResNet-18 Many challenges face the building of CNN models, like specifying the number of layers and their size, initialization of weights, and biases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In bad initialization, the model can consume a lot of time to complete the task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many architectures have been developed over the years, and they have a good impact on improving the trained models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' On the other hand, these architectures simplify dealing with data without deep knowledge in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' One of them is ResNet-18 architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ResNet-18 is using a skip connection signal to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A skip connection is a signal that passes into the layer in addition added to the output layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In a usual architecture, the goal is training the 𝑓(𝑥) but in the residual training the architecture will be forced to train 𝑓(𝑥) + 𝑥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-17 shows the residual training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CAR TRUCK VAN -BICYCLE INPUT CONVOLUTION+RELU POOLING CONVOLUTION+RELU POOLING FLATTEN FULLY SOFTMAX CONNECTED FEATURE LEARNING CLASSIFICATIONDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 40 Figure 3 17: Residual Learning The initial weights are usually close to zero so, the outputs follow them with values close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In the case of adding the skip connection, the outputs will be clone from the inputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This will accelerate the training faster than without the skip connection because the network progresses before the layers start learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ResNet-18 has 18 layers where the convolution layers and fully connected layer are just counted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Let us analyse ResNet’s architecture that is shown in Figure 3-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The input image is passed to a convolutional layer with a 7×7 kernel, 64 feature maps stride 2, zero Padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Then is fed to a max- pooling layer with a 7×7 kernel, stride 2, zero Padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' After them, there are four identical convolutional networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Every ConvNet has two residual units, and each residual unit consists of two convolution layers with a 3×3 kernel, stride 2, zero Padding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noticed that the feature maps are doubled every identical ConvNets, and the convolution layers’ size is minimized to half in height and width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Next, the last layer is fed to the average pooling layer, then a fully connected layer and softmax to estimate the probabilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 3-18: ResNet-18 Architecture, Adapted From [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' x weightlayer SkipConnection F(x) Irelu x weightlayer identity F(x) +x7 × 7 conv, 64, /2 3 × 3 conv, 64 3 × 3 conv, 64 3 × 3 conv, 64 3 × 3 conv, 64 3 × 3 conv, 128, /2 3 × 3 conv, 128 3 × 3 conv, 128 3 × 3 conv, 128 3 × 3 conv, 256, /2 3 × 3 conv, 256 3 × 3 conv, 256 3 ×3conv,256 3 × 3 conv, 512, /2 3 × 3 conv, 512 3 × 3 conv,512 3 × 3 conv, 512 fc 32Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 41 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 Define A Loss Function and Optimizer To assist the model, we need a loss function to ensure that the Training is doing well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' And we need an optimizer to update the weights inside the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' A loss function is considered one of the pillars when training the model in image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, this loss function assists the learning for the model over the training dataset using the weights and biases through the network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' For example, it can be calculated by taking the difference between the predicted and actual classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are two common categories of loss function like regression loss functions and classification loss functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, regression seeks to find a predicted continuous value depending on many parameters in the model, while classification chooses an output from a set of categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Here are many examples of regression and classification losses that are commonly used in these problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Regression Loss Functions: \uf0b7 Mean Square Error: It is considered a performance technique that determines how much error the model obtains comparing with its predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equation is formed by taking the difference between prediction from the model and the actual, then averaging these differences to get the total magnitude error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑀𝑆𝐸 = 1 𝑛 ∑( 𝑦𝑖 − 𝑦̂ 𝑖) 𝑛 𝑖=1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6) Where the symbols represent the following: MSE: Mean Squared Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' n: total number of predictions/actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒚𝒊: actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒚̂𝒊: predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 42 Figure 3 19: Mean Square Error Mean Absolute Error: It is almost like MSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Using one of them depends on the data distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' When we have large outlier districts, MSE will neutralize the negative and positive outliers, giving a wrong prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is preferred to take the absolute values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The equation is formed by taking the sum of the absolute differences between the prediction and the actual values, then obtain the average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑀𝐴𝐸 = 1 𝑛 ∑ |𝑦𝑖 − 𝑦̂ 𝑖| 𝑛 𝑗=1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='7) Where the symbols represent the following: MAE: Mean Absolute Error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' n: total number of predictions/actual data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒚𝒊: actual value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒚̂𝒊: predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Regression/ Best-fit Line xDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 43 Figure 3 20: Mean Absolute Error Classification Loss Functions: \uf0b7 Mean Squared Error: Taking the difference between prediction from the model and the actual, then averaging these differences to get the total magnitude error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Cross-Entropy Loss (Log Loss): It is often used in classification problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The output probability from a sigmoid or a softmax then enters the cross-entropy function that assesses how much this model classifies well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It has two forms, binary cross-entropy, and multi-class cross-entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The following equations clarify each of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Binary Cross-Entropy: 𝐵𝐶𝐸 = 𝑦 ∗ log(𝑝) + (1 − 𝑦) ∗ log (1 − 𝑝) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8) P = Prop(y=1), output from a sigmoid activation binary class label y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The following graph illustrates the loss vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' predicted probability for a binary classifier for each y=1 and y=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Output MAE InputsDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 44 Figure 3-21: Loss (BCE) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Predicted Probability Categorical Cross Entropy: 𝐶𝐶𝐸 = − 1 𝑚 ∑( 𝑦𝑖 ∗ log(𝑦̂𝑖) + 𝑚 𝑖=1 (1 − 𝑦𝑖) ∗ log(1 − 𝑦̂𝑖) ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='9) m: number of classes that are represented with one-hot encoding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑦𝑖: ith target class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑦̂𝑖: the predicted probability of that input belongs to the ith class, computed with a softmax activation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=" \uf0b7 Optimizer: It is an algorithm or method used to tune the model's parameter (e." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=', weights, biases, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=') during the training to reduce the losses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it affects the results from the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many optimizers can be used with the models during the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, every optimizer has its advantages and disadvantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Gradient Descent: It is considered one of the most basic optimization algorithms, and it is a first order optimization algorithm dependent on the first order of a loss function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It can be used in classification and linear regression problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it is used in backpropagation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The following equation can express it: y=1 4 y=0 3 ssol 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 predicted probabilityDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 45 𝜃 = 𝜃 − 𝛼 ∇𝐽(𝜃) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10) 𝜽: is the weight of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜶: is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝛁𝑱(𝜽): is the derivative of the objective (loss) function for the weights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Stochastic Gradient Descent (SGD): The difference between this algorithm and basic gradient descent is that this algorithm updates the model’s parameter for each training example in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜃 = 𝜃 − 𝛼 ∇𝐽(𝜃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑥(𝑖);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑦(𝑖)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='11) Where: { 𝒙(𝒊), 𝒚(𝒊)}: are the training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜽: is the weight of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜶: is the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝛁𝑱(𝜽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒙(𝒊);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝒚(𝒊)): is the derivative of the objective (loss) function for the weights for every training example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In this algorithm, the loss function has a lot of fluctuations and variance over the training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Mini Batch Gradient Descent It combines the advantages of SGD and standard gradient descent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' After every batch of the training dataset the parameters are updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜽 = 𝜽 − 𝜶 𝛁𝑱(𝜽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑩(𝒊)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='12) Where: {𝑩(𝒊)}: the batches of training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜽: the weight of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝜶: the learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝛁𝑱(𝜽;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 𝑩(𝒊)): objective (loss) function derivative for the weights for every batch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Learning rate: The gradient tells us where the function has the highest rate of change, but it does not tell us the value of steps to reach the optimal value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Small steps lead the algorithm to the best solution but slow progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Large steps are fast but lead the algorithm to bounce around the optimal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 46 \uf0b7 Adam Optimizer: Adam is derived from adaptive moment estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It takes the advantages of both AdaGrad and RMSProp algorithms to provide an optimization that can deal with sparse gradient on noisy problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It decays exponentially average of past gradients so, converging to the optimal solution fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Table 3-1 shows a brief comparison between SGD and ADAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Table 3-1: Comparison between SGD and ADAM Optimizer Advantages Disadvantages SGD \uf0b7 Can best fit and generalize the dataset after extensive training \uf0b7 Cannot deal with global minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It can be affected by choice of learning rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Adam \uf0b7 Can deal with sparse gradient on noisy problems \uf0b7 Can decay the learning rate through the learning \uf0b7 Cannot generalize dataset perfectly Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 47 Chapter 4 DESIGN TESTING AND RESULTS 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Model Training and Validation 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Model Training Model training means applying an algorithm to update the model parameters that best fit training data and predict the new data well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Models differ from others in the ability of data fitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Each of them is suitable for a specific type of data and performs depending on the complexity of data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In our project, the data was trained using;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ANN, CNN, ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The training performance depends on several factors, and without them, the training will be invalid and absorb more time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The factors that can improve the training are: \uf0b7 Model Architecture: Before building the model, you should know the nature of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In our case for image classification, the deep neural network does not guarantee to extract the image features and achieve high accuracy, so choosing a powerful architecture like CNN or transfer learning is preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Data Plenty: Machine learning models need many data to fit them properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The data that we ran was adequate to train the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In case of lack of data, Data augmentation can be used or using transfer learning to compensate it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The Optimizer: The weights will be initiated randomly, so the optimizer will update the weights to reach the minimized error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We used ADAM and SGD to update the weights in the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The Number of Epochs: The training of data needs several cycles to reach the proper parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' There are two approaches to choose the number of epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' specifying the number of epochs directly or using the early stopping technique since the training will stop when there is no progress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We set the number of epochs to 20 for the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Model Validation It means how the model will estimate performance after the training and is used essentially to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Overfitting occurs when the model performs well, generalizes the data quite on the training but performs worse on the validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The validation is applied using evaluating the validation error every training stage to find the minimum error, then stopping the training and saving the parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 48 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Graphs: Training, Validation Accuracy and Loss Figure 4-1: Progress of Average Training Loss of ANN with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-2: Progress of Average Validation Loss of ANN with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 49 Figure 4-1 shows that the average loss of accuracy training for an ANN model using SGD optimizer and lr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 decreases slowly with increasing number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, Figure 4-2 shows that the progress of validation loss for an ANN model does not decrease smoothly with increasing number of epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Finally, we can conclude that ANN needs a lot of time to be trained and to extract the features from the images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-3: Progress of Average Training Loss of CNN with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 50 Figure 4-4: Progress of Average Validation Loss of CNN with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' CNN performs much better than ANN, Figure 4-3 and Figure 4-4 show that training loss and validation loss decrease rapidly through the first 5 epochs and there is not obvious decreasing until epoch number 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it is noted that CNN training loss and validation loss approaches to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 51 Figure 4-5: Progress of Average Training Loss of ResNet-18 with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-6: Progress of Average Validation Loss of ResNet-18 with SGD, lr=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Through the Epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 LosS 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='0 0 1 2 m 4 5 7 8 9 10 11 12 13 14 15 16 17 18 19 20 EpochNumber0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='275 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='250 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='225 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='200 Validation Loss 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='175 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='150 Average 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='125 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='075 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='000 1 2 4 5 6 7 00 9 10 11 12 13 14 15 16 17 18 19 20 EpochNumberDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 52 Figure 4-5 and Figure 4-6 show that ResNet-18 model performs better than ANN and CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noted that it starts with low loss and it decreases through the first 3-4 epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, the training loss decreases slightly and stays at low average loss, but average validation loss increases slightly after epoch 4 which means the model begins overfit over the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, we save the model at the least validation loss to avoid overfitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Testing and Results After training the model, we need to validate it using a testing dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The testing stage performs the model on a dataset that the model did not see it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The image will pass into the model as an input, and the output will be one of 32 classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' If a matching between the true output and predicted output occurs, it contributes rising of model accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The models’ performance and their average accuracies are shown in Table 4-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Table 4-1: Comparison Between Different Models’ Accuracies Model Optimizer Learning rate Average Test Accuracy ANN SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='01 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='78 % SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='50 % CNN SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='01 93.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='00 % SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='80 % Transfer Learning (ResNet-18) SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='01 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='21 % SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='36 % ADAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='01 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='00 % ADAM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='76 % It is noticed that the ResNet-18 has higher accuracy than other architectures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The model corresponds to the predicted values with actual values at the testing stage, and Figure 4-7 shows how much the testing set corresponds to the actual values at ResNet-18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In general, each image will pass into the model and distribute to 32 classes in different accuracy so that the higher accuracy will be the predicted value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The misleading will have occurred in several cases, like;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' the similarity between the images, in which the model usually cannot discern images that have similarity, and the unclear images, in which the model cannot extract the features correctly because of the noises in the image.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 53 Figure 4-7: Confusion Matrix of ResNet-18 Shows the Matching Between the Predicted Values with True Values at The Testing Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0 0 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0 0 240 0 0 1 0 2 0 1 0 1 245 0 3 0 0 0 0 248 0 0 m 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 290 0 0 2 0 0 0 0 0 13 241 0 2 0 0 0 0 296 0 0 238 0 0 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0 28 0 0 0 0 1 2 231 0 0 0 0 0 0 0 0 1 264 0 0 0 1 0 2 238 0 1 0 0 260 1 0 274 0 1 1 263 0 0 0 。' metadata={'source': 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m 190 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0 0 0 0 0 0 0 0 1 0 2 0 203 Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 54 Figure 4-8: Confusion Matrix of CNN Shows the Matching Between the Predicted Values with True Values at The Testing Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-8 shows how much the testing set corresponds to the actual values at CNN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noticed from the distribution of predicted and true values that CNN performs well and the misleading between predicted and true values are considered little much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 0 0 2 m 0 241 0 0 1 2 0 3 0 0 0 1 0 238 4 0 2 4 1 1 2 281 4 0 2 1 0 11 229 1 0 10 1 1 0 1 3 1 0 1 0 28 1 1 0 1 5 5 226 0 0 1 0 1 2 217 2 0 2 m 2 0 m 215 1 1 1 1 m 0 1 2 0 1 1 0 1 1 255 0 0 0 2 1 1 1 0 2 1 1 228 0 0 1 0 2 1 256 0 0 m 0 1 2 272 0 0 0 0 0 0 0 2 1 0 1 254 1 1 0 1 0 2 1 0 2 2 1 2 262 0 1 1 0 0 0 0 0 243 0 0 0 0 2 0 0 5 2 1 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 270 m 0 1 0 5 238 0 0 0 1 2 m 2 0 1 214 1 1 0 0 m 1 0 1 0 265 0 0 2 0 264 1 1 2 1 1 1 0 1 1 9 250 0 0 0 2 7 1 223 1 1 0 263 0 0 1 2 1 0 2 0 0 201 0 1 1 255 1 192 0 0 0 2 0 0 3 0 0 2 1 0 0 0 193 1Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 55 Figure 4-9: Confusion Matrix of ANN Shows the Matching Between the Predicted Values with True Values at The Testing Stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-9 shows how much the testing set corresponds to the actual values at ANN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noticed from the distribution of predicted and true values that ANN does not perform well, and there is big misleading in some of the classes, indicating that ANN is not a valid image classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 9 0 0 0 5 1 1 169 3 33 2 1 1 m 2 1 3 1 2 1 6 m 1 220 16 虹 2 1 2 0 2 7 8 5 0 18 148 14 42 4 2 m 4 0 2 1 4 3 0 0 290 0 1 2 2 6 8 197 1 1 4 1 7 2 5 9 1 1 m 0 187 12 5 1 1 1 3 2 189 2 18 0 0 1 0 1 13 0 6 2 5 179 0 0 1 0 0 0 2 0 4 1 1 7 1 1 6 1 2 115 0 1 0 4 0 2 0 5 m 2 0 1 1 m 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 225 7 2 1 1 0 2 9 10 1 80 209 2 2 0 0 0 1 1 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 233 1 4 0 3 0 1 1 m m 2 1 20 1 m 3 225 1 2 0 1 2 0 3 2 0 1 0 188 0 0 2 0 6 0 0 1 1 5 14 4 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 242 4 2 1 1 0 2 1 1 1 0 236 1 1 0 2 5 1 1 207 2 1 m 1 0 2 3 1 2 230 m 14 7 0 8 0 0 0 2 1 9 226 m 4 11 2 8 2 1 2 1 4 m 0 5 11 220 0 1 1 3 2 2 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 200 1 0 0 5 0 7 2 2 1 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 1 2 5 6 2 209 1 3 0 0 1 1 2 0 0 0 0 148 0 5 5 1 6 4 m 10 1 1 1 1 182 1 5 0 。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 152 0 0 0 3 0 0 0 0 5 20 1 0 2 0 0 0 129 Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 56 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Model Inferencing It is a process of feeding a new input image (unseen image) into a trained DNN model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-10 shows the Flowchart of model inferencing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4 10: Inferencing Flowchart We tried to examine the model performance by inserting an image shown in Figure 4-11 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The input image will pass into the pre-processing stage, which does some operations, including resizing, centre crop, converting to a grayscale image, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The image after pre-processing is shown in Figure 4-11 (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The pre-processed image will pass into the trained model to classify the image to the predicted class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-12 shows the prediction of the input image, and it is obvious that the model predicts the input correctly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' input image pre-processing trained mode top k classesDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 57 (a) (b) Figure 4-11: (a) Input Image (Y𝒂̅), (b) Pre-processed Input Image Figure 4 12: ArSL Alphabet Prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 58 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 New Collected Dataset: ArSLA-2021 Dataset 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Overview We are creating our dataset that is captured from many people of different ages, and it will be the first dataset of its kind for Arabic Sign Language Alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Until now, we have collected about +10000 real-life images for 31 Arabic Sign Alphabets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-13 shows from our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Figure 4-13: Arabic Sign Language Alphabets Samples The dataset has been collected and captured by the people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Also, it is considered a real-life dataset with images captured under different conditions such as different light, background, image orientation, image size, image quality, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Lajf- Alif -Gi - Ta -G -Tha As?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='- Jim -Ha sL-Kha Jis - Dal JI3- Dhal slj - Ra slj-Zay w - Sin o- Shin Sto-Sad -Dad slo-Dad lis - Za E - Ghayn - Fa ose - Ayn l-Qaf l - Kaf y- Lam Ase-Mim ogi-Nun elo-Ha gl9 - waw 以- Y - Laa 8-Taa JI - AIDesign of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 59 Table 4-2: Arabic Sign Language Alphabets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' their numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='labels ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Letter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='English ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Letter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Arabic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Letter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='English ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Letter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='name ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='in ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Arabic ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='script ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='# ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='images ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Alif ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='أ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')فِلَأ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='395 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='17 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Z𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ظ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَظ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='329 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='B𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ب ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَب( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='387 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Ayn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ع ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')نيَع( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='T𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ت ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَتأ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='385 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='19 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Ghayn ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='غ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')نيَغ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='337 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Th𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ث ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَث( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='379 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='F𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ف ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَف( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='331 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='J𝑖̅𝑚 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ج ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْميِج( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='388 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Q𝑎̅f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ق ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')فاَق( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='327 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='H𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ح ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَح( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='378 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='K𝑎̅f ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ك ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')فاَك( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='332 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='7 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Kh𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='خ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَخ( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='347 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='L𝑎̅m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ل ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْمَلا( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='335 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='D𝑎̅l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='د ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْلاَد( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='343 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='24 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='M𝑖̅m ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='م ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْميِم( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='328 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ن ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')نوُن( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='356 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='R𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ر ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَر( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='335 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='H𝑎̅ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ه ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَه( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='354 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='11 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Z ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='𝑎̅y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ز ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ياَز( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='W𝑎̅w ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='و ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')واَو( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='351 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='S𝑖̅n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='س ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْنيِس( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='340 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Y𝑎̅a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ءاَي ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ءاَي( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='346 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='13 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='Sh𝑖̅n ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ش ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ْنيِش( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='353 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='29 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='T𝑎̅a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='ة ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=')ة( ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='344 14 S𝑎̅d ص )ْداَص( 340 30 Al لا )لا( 341 15 D𝑎̅d ض )داَض( 339 31 Laa لا )لا( 339 16 T𝑎̅ ط )ءاَط( 334 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='This project is supported and consulted by Student Counselling Department at the University of Jordan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' which has specialist interpreters for Arabic Sign Language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' They help and consult us to achieve high-quality work in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 General Notes for ArSLA-2021 Dataset: \uf0b7 It could be a benchmark for researchers in this field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It can be used for research and production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It will be shared as raw images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Everyone has the choice to do any processing on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It is collected from more than 300 participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Different resolutions have been got by different mobile phones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 The dataset mainly consists of RGB images.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 These images are static.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It will be annotated manually in the future.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 It will be made publicly available to support the Sign Language field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5 System Limitations and Compliance with Design Constraints 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 System Limitations: We faced many limitations while working on this project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The limitations are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Lack of Resources We have just found one public dataset for ArSL alphabets that is publicly available, the ArASL2018 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The available datasets have a limited set of conditions, including lighting, unique images, various background, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hardware Limitations [55] \uf0b7 The training of deep learning models usually requires GPUs, with enough disk memory and RAM size that can accelerate the model training, but unfortunately, we do not have available GPUs at The University of Jordan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 We have used the available GPUs that are offered by google colab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Unfortunately, google colab has some limitations: o The resources are not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o The usage limits change depending on the availability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o Time limitations for continuous model running, which is at max 12 hours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o Memory limits to load the dataset into the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o RAM limitation is used to perform the calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' o GPUs’ type is not guaranteed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Design Constrains Compliance The following points discuss the constraints that were put in place in the beginning, what we were able to solve, and what we could not: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Availability of Data: We looked for any dataset that has multiple conditions that represent the real-life conditions of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' We found one public dataset containing enough images, 54,049 images, that was trained on but with limitations to the number of unique participants, lighting, and complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' However, we are collecting our dataset that overcome these issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Computational Resources We have faced a challenge to train models using google colab due to restrictions on it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' ArASL2018 dataset was trained using google colab without any problems and we have achieved an excellent performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Unfortunately, we cannot using google collab to train models using ArSLA-2021 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 61 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Response time (Inference Time): We calculated the inference time required to classify one image per model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It is noted from Table 4-3 that the induction time responds within a very short time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Table 4 3: Inference Time Results Model Optimizer Learning Rate Inference Time ANN SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 258 ms CNN SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 145 ms Transfer Learning (ResNet 18) SGD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 142 ms 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Hyperparameter Choosing We have tested many hyperparameters depend on research papers that are considered relevant problem solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Knowledge and Experience in ArSL We have studied a lot of resources about alphabets in ArSL and we have met experts interpreters in ArSL to deliver a high quality of work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6 Solution Impact 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Societal Impact Machine learning broadens our outlook on life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' It makes a great leap in various fields like, industry, medicine, and social life.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' One of the machine learning branches is computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Computer vision and its applications simplify the complicated tasks where some of these applications require much experience to extract the image’s features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Computer vision is not limited to technical issues but reaches toward humanity issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' In our project, computer vision is used to build a model that can recognize the Arabic sign language alphabets automatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' This step positively impacts the community, where it will spread awareness of sign language and ease the communication between deaf and normal people.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 62 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Economic Impact Technology has a huge impact on the economy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' If the Arabic sign language recognition model were deployed in smart devices, this would reduce the cost of owning special tools such as gloves, pressure sensors, and jump motion devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' The cost of hiring more interpreters will also decrease.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Environmental Impact Our work is digital content, so when you use it there will be no waste.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' On the other hand, the use of solid materials will be a waste after corrosion, and this will have bad impacts on the environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='4 Global Impact If our model is deployed in smart devices, it will increase the communication between normal people and deaf people by interpreting the alphabets in ArSL and converting them into written Arabic text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Design of Arabic Sign Language Recognition Model Design of Arabic Sign Language Recognition Model 63 Chapter 5 CONCLUSION AND FUTURE WORK 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='1 Conclusion This project was designed to be the first step to help the deaf community by building a model that uses computer vision techniques to convert the ArSL alphabets into Arabic letters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' Many machine learning techniques were used to build the model, and we chose transfer learning (ResNet-18) which achieved the highest accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='2 Problems Faced \uf0b7 It is a new field, so we have needed a lot of time to learn and grasp new concepts especially in deep learning and ArSL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 We have not enough fund to buy GPUs to train new models using ArSLA-2021 dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Since we used the GPUs offered by google colab there was time limitation, and we had no control over the resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 COVID-19 restrictions have prevented us from collecting more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content='3 Recommendations for Future Work We will expand the project to include: \uf0b7 Publishing our dataset to be a starting point for someone else to continue the work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Training the model using our dataset and collecting more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Collecting dataset for dynamic alphabets and collecting dataset to include words and continuous speech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Creating a model that will be used for real-time application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/PdE0T4oBgHgl3EQf1AJg/content/2301.02693v1.pdf'} +page_content=' \uf0b7 Deploying the model on the mobile application.' metadata={'source': 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Jiabei Fan1, Zhengyang +Bai2,3,∗ Weibin Li4, Jianming Zhao1,3,† and Suotang Jia1,3 +1State Key Laboratory of Quantum Optics and Quantum Optics Devices, +Institute of Laser Spectroscopy, Shanxi University, Taiyuan 030006, China +2State Key Laboratory of Precision Spectroscopy, +East China Normal University, Shanghai 200062, China +3Collaborative Innovation Center of Extreme Optics, Shanxi University, Taiyuan 030006, China +4School of Physics and Astronomy, and Centre for the Mathematics +and Theoretical Physics of Quantum Non-equilibrium Systems, +University of Nottingham, Nottingham, NG7 2RD, UK +We study Rydberg electromagnetically induced transparency (EIT) of a cascade three-level atom +involving 80D5/2 state in a strong interaction regime employing a cesium ultracold cloud. In our +experiment, a strong coupling laser couples 6P3/2 to 80D5/2 transition, while a weak probe, driving +6S1/2 to 6P3/2 transition, probes the coupling induced EIT signal. At the two-photon resonance, +we observe that the EIT transmission decreases slowly with time, which is a signature of interaction +induced metastability. The dephasing rate γOD is extracted with optical depth OD = γODt. We +find that the optical depth linearly increases with time at onset for a fixed probe incident photon +number Rin before saturation. +The dephasing rate shows a nonlinear dependence on Rin. +The +dephasing mechanism is mainly attributed to the strong dipole-dipole interactions, which leads to +state transfer from nD5/2 to other Rydberg states. We demonstrate that the typical transfer time +τ0(80D) obtained by the state selective field ionization technique is comparable with the decay time +of EIT transmission τ0(EIT). The presented experiment provides a useful tool for investigating the +strong nonlinear optical effects and metastable state in Rydberg many-body systems. +I. +INTRODUCTION +Due to the strong interaction (∝ n11 with n principal +quantum number) [1], Rydberg atoms provides an ideal +platform to implement quantum information and quan- +tum simulation [2–5] and investigate interaction induced +cooperative optical nonlinearities [6]. The optical nonlin- +ear effects are induced by Rydberg atom interactions [7, +8], i.e., van der Waals (vdW) interactions [9–11] and +dipole-dipole interactions [12, 13]. Strong atomic interac- +tions can be effectively mapped onto photon-photon in- +teractions via electromagnetically induced transparency +(EIT) +[6]. Due to the cooperative effects, the optical +nonlinearity can be greatly enhanced [6, 14–16]. Based +on this, Rydberg EIT experiments are employed to mea- +sure a radio-frequency electric field [17] with a room- +temperature cell, and to realize few-photon optical non- +linearities [18–21] with an ultracold sample, such as the +efficient single photon generation [18], entanglement gen- +eration between light and atomic excitations [20], single- +photon switches [16, 19, 22] and transistors [21, 23, 24]. +The resonant dipole-dipole interaction between two indi- +vidual Rydberg atoms [25] is angular dependent. +Re- +cently, the anisotropic Rydberg interaction have been +adopted to investigate Rydberg polaritons by using Ry- +dberg nD-state [26]. +∗ zhybai@lps.ecnu.edu.cn +† zhaojm@sxu.edu.cn +In this work, we present a Rydberg EIT spectrum of a +cascade three-level cesium atom involving 80D5/2 state in +a dipole trap. Under the two-photon resonance condition, +we observe that EIT transmission displays a slow de- +crease with time. This could be a signature of interaction +induced metastability [27, 28]. Due to the large dipole +matrix elements, the cesium nD-state atom has strong +dipole interactions with energetically close (n+1)P state. +We find that, strong dipole interactions between the nD +and (n+1)P state leads to a fast decay of nD to (n+1)P +state, and the dephasing of the transmission spectrum. +A theoretical model is built to understand EIT dephas- +ing mechanism. The nonlinear dependence on dephasing +rate has also been investigated. +The remainder of the article is arranged as follows. In +Sec. II, we introduce our experimental setup. In Sec. III, +we reveal Rydberg EIT spectrum and its dephasing ex- +perimentally. In Sec. IV, we present a simple theoretical +model to reveal EIT dephasing mechanism. In Sec. V, we +measure the fast decay process on 80D5/2 state. Finally, +we summarize the main results obtained in this work. +II. +EXPERIMENTAL SETUP +Our experiment is performed in a cesium magneto- +optical trap (MOT) with optical dipole trap (ODT).The +beam waist of dipole trap is 45 µm. +A schematic of +relevant levels and the experimental setting are shown +in Fig. 1 (a) and (b). +A three-level system, shown +in Fig. 1(a), consists of a ground state |6S1/2, F = 4⟩ +arXiv:2301.05377v1 [physics.atom-ph] 13 Jan 2023 + +2 +(b) +(a) +(c) +g3 +g2 +Probe +(Coupling) +MOT +ODT +20ms +46ms +46ms +46ms + + +Wc +Wp +|6S1/2> +|6P3/2> +|80D5/2> +p +|g> +|e> +|r> +0 +5 +10 +15 +20 +0.0 +0.1 +0.2 + + +Transmission +Time (ms) + absorption + EIT +Figure 1. +(color online) (a) Atomic level scheme. +A weak +probe laser field with Rabi frequency Ωp drives the lower +transition, |g⟩ = |6S1/2, F = 4⟩ → |e⟩ = |6P3/2, F ′ = 5⟩. +The strong coupling laser (Rabi frequency Ωc) couples the +transition |e⟩ → |r⟩ = |80D5/2⟩. (b) Experiment setup. The +coupling and probe beams are counter-propagated through +the MOT center and overlap with the dipole trap beam. The +transmission of probe beam is detected with a single photon +counting module (SPCM). The inset shows EIT dephasing +behaviors (red) and probe absorption (black; without cou- +pling beam). The data are taken by the sum of 3000 exper- +imental cycles. (c) Experimental timing. After switching off +the MOT and dipole trap beams, Rydberg-EIT coupling and +probe lasers are turned on for 20 µs, during which the probe +laser frequency is ramped through the |g⟩ → |e⟩ transition +over ±15 MHz by a double-passed AOM. +(|g⟩), intermediate state |6P3/2, F ′ = 5⟩ (|e⟩) and Ry- +dberg state |80D5/2⟩ (|r⟩). A weak probe beam (Rabi +frequency Ωp, 852-nm laser with a 100-kHz linewidth), +provided by an external cavity diode laser (Toptica, DL- +pro), drives a lower transition and the frequency is sta- +bilized to the |6S1/2, F = 4⟩ → |6P3/2, F ′ = 5⟩ transition +using the polarization spectroscopy method. The cou- +pling beam (Rabi frequency Ωc) provided by a commer- +cial laser (Toptica, TA-SHG110) with linewidth 1 MHz +drives Rydberg transition, |6P3/2, F ′ = 5⟩ → |80D5/2⟩. +The coupling laser frequency is stabilized to the Ryd- +berg transition using a Rydberg EIT reference signal ob- +tained from a cesium room-temperature vapor cell [29]. +The weak probe laser and strong coupling laser, with +respective Gaussian radius of 9 µm and 30 µm, are over- +lapped and counter-propagated through the MOT center, +see Fig. 1(b). The probe laser is scanned using a double- +passed acousto-optic modulator (AOM) that covers the +lower transition. +The transmission of the probe laser, +Rydberg EIT spectrum, is detected with a single photon +counting module (SPCM) and processed with Labview +program. The glass MOT is surrounded by three pairs +of field-compensation Helmholtz coils, which allow us to +reduce stray magnetic fields via EIT Zeeman splitting, +corresponding stray field less than 5 mG. In our experi- +ment, the peak density of atomic cloud about 1011 cm−3 +is measured by shadow imaging and the temperature of +atomic cloud is about 100 µK. The estimated Rydberg +density is 2.4 × 108 cm−3. +The experimental timing is shown in Fig. 1(c) with the +whole time 200 ms, corresponding to a repetition rate of +5 Hz. In each cycle, after turning off the trap beams, we +switch on the coupling and probe lasers for 20 µs, dur- +ing which the probe-laser frequency is swept across the +|6S1/2, F = 4⟩ → |6P3/2, F ′ = 5⟩ transition, meanwhile +the power is fixed using a proportional-integral-derivative +controller (PID) feedback loop that controls the radio- +frequency power supplied to the 852-nm AOM. The data +shown in Fig. 1(b) is taken by the sum of 3000 cycles. +III. +EXPERIMENTAL OBSERVATION OF +RYDBERG EIT AND TRANSMISSION +DEPHASING +-14 +-7 +0 +7 +14 +0.0 +0.4 +0.8 +0 +4 +8 +12 +16 +20 +0.1 +0.2 +0.3 +0.4 +(a) Rin = 21, 170 Photons/µs + + +Transmission +∆p/2π (ΜΗz) +t0 (EIT)= 4.04 ± 0.10 ms +t0 (EIT)= 4.49 ± 0.05 ms +t0 (EIT)= 4.50 ± 0.06 ms +(b) Rin = 21, 170, 340, 680 Photons/µs + + +Time (µs) +Figure 2. (color online) (a) Rydberg-EIT spectra with a cou- +pling laser, Ωc/2π =10.6 MHz, resonant with the |6P3/2, F ′ = +5⟩ → |80D5/2⟩ transition, and a probe frequency scanning +across the lower transition, |6S1/2, F = 4⟩ → |6P3/2, F ′ = +5⟩, at a probe-photon rate Rin = 21 photons/µs and 170 +photons/µs, respectively. The solid lines are the fittings using +the density matrix equation of a three-level atom. The EIT +transmission displays a strong suppression and blue shift with +increasing probe Rin. (b) EIT transmissions for the indicated +probe photon input rates at two-photon resonance condition. +The transmission remains same for Rin = 21 photons/µs, +whereas decrease with Rin at the beginning and then decay +with the EIT maintain time for larger Rin cases. The solid +lines denote the exponential fittings. For higher Rin the char- +acteristic time are τ0(EIT) = 4.04 ± 0.10 µs, 4.50 ± 0.06 µs +and 4.49 ± 0.05 µs, respectively. +Due to the quantum interference effect [30], the probe +transmission T increases when the coupling and probe +laser frequencies satisfy the two-photon resonance [see +Fig. 1(a)]. Using the rotating wave approximation and in +the interaction picture [31], the eigenstate of three-level +Hamiltonian can be obtained with |D⟩ ∝ Ω∗ +c|g⟩ − Ωp|r⟩ +at two-photon resonance. For conventional EIT, the sys- +tem works in dark state and the probe pulse suffers little +optical absorption. For the three-level scheme, all atoms +are initially prepared in state |g⟩. The EIT system can + +3 +evolve to the dark state |D⟩ with time 1/γ2. However, as +shown in Fig. 1(b), in nD5/2-Rydberg EIT system, the +EIT transmission slowly decreases with time. Interest- +ing, the time scale of the dephasing is much longer than +the lifetime of intermediate state |e⟩. The dipolar inter- +action can lead to many-body dephasing [26, 32] in case +of Rydberg nD state. +We present EIT spectra with 80D5/2 Rydberg state in +Fig. 2(a). The probe field is adopted as incident photon +rates Rin = 21 photons/µs (black dashed line) and 170 +photons/µs (red dashed line) with Ωc = 2π × 10.6 MHz. +It is found that the transmission decreases with in- +creasing Rin and accompanies with the EIT-peak shift. +The EIT transmission rate is around 40% for Rin = 21 +photons/µs, and decreases to 20% when Rin increases +to 170 photons/µs. +We attribute the reduction of op- +tical transmission to atomic interactions between Ryd- +berg states. Besides, when Rin = 170 photons/µs, the +EIT peak has a blue shift about 2.5 MHz compared with +the case for Rin = 21 photons/µs. +For Rydberg EIT, +dark-state polariton is very sensitive to other Rydberg +excitation in the strong interaction regime. This is be- +cause when two Rydberg polaritons propagate inside the +medium with a distance r, they experience an interaction +induced energy shift U(r). It gives rise to a non-vanishing +Im(χ) of probe beam where χ optical susceptibility of sys- +tem and therefore leads to strong absorption and a shift +on EIT spectra [6, 14, 15]. +To further investigate the dephasing feature for nD +state, we vary the probe-photon rate Rin with fixed cou- +pling field Ωc/2π =10.6 MHz. +Both the coupling and +probe lasers frequencies are on resonance. +As shown +in Fig. 2(b), the time dependence of the transmission +is plotted with different Rin. +For low probe photon +rates (i.e., Rin = 21 photons/µs), the transmission is +almost a constant, but when increasing Rin, transmis- +sion exhibits a slow decrease with time t. +By fitting +the data in panel (b) with the exponential function +T = A exp(−t/τ0(EIT)) + T0, the decay time τ0(EIT) = +4.04 ± 0.10 µs, 4.50 ± 0.06 µs and 4.49 ± 0.05 µs can +be extracted for different Rin. The decay time is much +longer than the lifetime in state |e⟩ (i.e., 1/γ2 ∼ 0.03µs). +In order to reveal the dephasing mechanism, we de- +fine the effective optical depth (OD) of the medium as +the logarithm of transmission [i.e., OD = -ln(T)] [26]. +The time evolution of OD is shown in Fig. 3(a) [cor- +responding to the results in Fig. 2(b)]. +One sees that +OD approximately linearly increases with time before t += 5 µs. By neglecting saturation effects, we can redefine +OD = γODt where γOD reflects creation rate of opti- +cal density by decoupled impurities [26]. At small probe +photons (i.e., Rin ≲ 340 µs−1), the extracted rate γOD +displays a linear increase with Rin and then saturates +for large Rin [see Fig. 3(b)]. This is because the system +is not fully in a blockade regime at small probe pho- +tons. Therefore, with increasing photon number, more +Rydberg atoms are excited, which increase the dipole- +dipole interaction, leading to increased γOD. However, +0 +4 +8 +0.8 +1.2 +1.6 +2.0 +2.4 +0 +250 +500 +750 +0.02 +0.04 +0.06 +0.08 +(a) + + + +Time (µs) +Rin=21, 170, 340, 680 Photons/µs + OD +(b) Ωc/2π = 10.6, 8.2, 5.2 ΜΗz + + +γOD (µs-1) +Rin (µs-1) +Figure 3. (color online) (a) The optical depth of the 80D5/2 +EIT transmission taken the logarithm of spectra in Fig. 2(b) +for indicated photon incidence rates Rin at fixed Ωc = +2π×10.6 MHz. The solid lines are linear fittings before t = 5 +µs to the data to extract the dephasing rates γOD. (b) The +dephasing rates γOD as a function of Rin for coupling Rabi +frequency Ωc/2π = 10.6, 8.2 and 5.2 MHz, respectively. +after the system enter fully blockade regime, we can’t +excite more Rydberg atoms so that the dephasing rate +γOD shows saturation. This is a signature of many-body +dephasing. +We calculate the group velocity of probe +photon is around 3920 m/s under our experiment con- +dition with Ωc = 10.6 MHz. By considering the length +of our atomic cloud 1 mm, the propagation time of pho- +ton through the cloud is around 0.26 µs. The Rydberg +blockade radius is about 10 µm for 80D5/2. Under this +condition, almost 100 atoms can be excited to Rydberg +state. Thus we can calculate that in one microsecond, +the maximum number of Rydberg excitation is around +385, where 100 atom/0.26 µs ≃ 385 /µs. Thus, the criti- +cal value for Rin is around 385 /µs. When the photon +incidence rates Rin is smaller than 385 /µs, the system +is not in the blockade regime. When increasing the in- +tensity of the probe laser, the system can works in the +full blockade regime. This estimation is agreed with the +dephasing rates γOD in Fig. 3(b). +We also change Ωc +and measure EIT dephasing rates γOD versus Rin. The +results show a similar nonlinear dependence on Rin [see +Fig. 3(b)]. +IV. +ANALYSIS OF EIT DEPHASING +MECHANISM +The vdW interaction between nD5/2 pair leads to en- +ergy level shifts and dipole interaction of nD5/2 and near- +est Rydberg states yield state transfer of nD5/2 to other +Rydberg states. As no microwave field is present in the +experiment, the other states are excited due to the spon- +taneous decay from nD5/2 state. We have found [33] that +nD-(n+1)P transition is the strongest in our experiment +conditions (see Sec. 5 for details). Hence this leads to +a two stage processes. Rydberg nD5/2 state will decay +to (n + 1)P3/2 through spontaneous decay. The dephas- +ing can then be induced in that regime where dipole- +dipoles interactions couple nearly degenerate Rydberg +pair states [26, 32]. +A full model to describe the de- + +4 +phasing is rather complicated. In this section, we will +focus on the dephasing effects with a simplified model, +which nonetheless captures the main effects. +Considering the three-level scheme in Fig. 1(a), the +dephasing rate of Rydberg state γ3 is around γr + Γre +≈ γr. +Rydberg atom has long lifetimes (1/Γre ∼ n3) +on the order of 100 µs. γr represents the dephasing of +the atomic coherence (originated from atomic collisions, +residue Doppler effect, dipole-dipole interaction between +the Rydberg atoms, finite laser linewidth). The dephas- +ing rate of the intermediate state |e⟩ denotes γ2 = γe ++ Γeg with spontaneous decay rate Γeg and interaction +induced decay γe. +In our physical system, γe is much +smaller than Γeg ≃ 2π × 5.2 MHz, thus γ2 ≈ Γeg . The +collective dissipation can emerge in dense atomic gases, +typically through two-body dipolar couplings [32]. Here +we adopt the effective dephasing γeff +3 +(i.e., γr = γeff +3 ), and +seek the relation between the transmission and dipolar +interaction induced dephasing. +The dynamics of the effective three-level system can be +modeled by the quantum master equation for the many- +atom density operator ρ: +˙ρ = −i[ ˆHeff, ρ] + D1(ρ) + Deff(ρ), +(1) +The effective Hamiltonian in the equation is given by +ˆHeff = +N +� +j=1 +� +−∆pˆσj +ee(r, t) − (∆p + ∆c)ˆσj +rr(r, t) + Ωp +2 ˆσj +eg(r, t) ++Ωc +2 ˆσj +re(r, t) + +N +� +k̸=j +Vjk +2 ˆσj +rrˆσk +rr + H.c. +� +� , +(2) +with ˆσab(zj) ≡ |aj⟩⟨bj| (zj is the position of jth atom +in the respective ensemble) and H.c. representing Her- +mitian conjugate of the preceding terms. Vjk = C6/|rj − +rk|6 is the vdW potential with the dispersive coefficient +C6 ∝ n11. The dissipative effects are described by the +Lindblad form D1(ρ), +D1(ρ) = +N +� +j=1 +Γeg +� +ˆσj +geρˆσj +eg − 1 +2{ˆσj +ee, ρ} +� +, +(3) +where D1(ρ) denotes the decay from state |e⟩ to |g⟩. The +effective dephasing term Deff(ρ) is introduced, +Deff(ρ) = +N +� +j=1 +γeff +3 +� +ˆσj +33ρˆσj +33 − 1 +2{ˆσj +33, ρ} +� +, +(4) +Due to the dephasing and spectral shift of the trans- +parency resonance [see Fig.2(a)], we employ the theo- +retical description of individual atoms coupled to a mean +field (MF) to analyze the EIT spectrum with strong Ry- +dberg interactions [34–36]. +In the MF approximation, +the many-body density matrix ρ is decoupled into indi- +vidual ones through ˆρ ≈ Πi ˆρi. In the thermodynamic +limit, the optical Bloch equations for the three-level sys- +tem can be obtained, where elements of the density ma- +trix are represented with ρab = N −1 � +j⟨ˆσj +ab⟩. +As in- +dicated in our experiment, over a long time evolution, +the interactions between Rydberg state leads to a MF +shift where ∆c → ∆c +V ρrr with MF interaction energy +V = N −1 � +k̸=j Vjk. +Figure 4. (color online) (a) Theoretical results of EIT trans- +mission varies with ∆p for V = 0 and V ̸= 0. (b) EIT trans- +mission and corresponding OD versus effective dephasing rate +γeff +3 +of Rydberg state that accounts for the interaction between +Rydberg atoms with Ωp/2π =1.04 MHz, Ωc/2π =10.6 MHz +and atomic density 1×1011cm−3. +0.0 +0.5 +1.0 +1.5 +2.0 +0 +1 +2 +3 +81P3/2 +tINT = 6 µs +tINT = 3 µs + + +Normalied signal (Arb.units) +Time (µs) +80D5/2 +(a) +0 +4 +8 +12 +0.0 +0.2 +0.4 +0.6 +(b) + +tINT (µs) +80D5/2 Signal (Arb.units) +t0 (80D) = 3.15 ± 0.38 ms +tINT = 0 µs +Figure 5. (color online) (a) Normalized time of flight (TOF) +signals for laser excitation to 80D5/2 state with indicated in- +teraction times, tINT. The three traces are set vertically offset +for clarity, with the respective zero levels shown as horizon- +tal dashed lines. The gates for the 80D5/2 and 81P3/2 states +signals are shown as a light pink and gray shaded regions, +respectively. (b) Measurements of the 80D5/2 state as a func- +tion of the interaction time tINT. +Initial prepared 80D5/2 +Rydberg atoms transfer to nearby Rydberg states |r′⟩ during +tINT due to the strong resonant dipole interaction. The solid +line shows the exponential fitting with the characteristic time +τ0(80D) = 3.15 ± 0.38 µs. +By numerically solving the MF Bloch equation, one +can obtain EIT transmission varies with ∆p. As shown +in Fig. 4(a), there is an increasing blue shift of trans- +parency away from the non-interaction EIT resonance. +This is because that the shifted Rydberg state detunes +the EIT windows. The similar EIT signal is also observed +experimentally [see Fig.2(a)]. To obtain the dependence +of the EIT transmission on the effective dephasing in- +duced by Rydberg interactions, we calculate EIT spectra +for a series of effective γeff +3 , accounting for the many- +body dephasing by Rydberg atoms. +In Fig. 4(b), the +EIT transmission as a function of γeff +3 +is plotted. The +system works at two-photon detuning δ = 0 with the +probe Ωp/2π =1.04 MHz, Ωc/2π =10.6 MHz and atomic +density Na = 1×1011 cm−3. It shows that the EIT trans- + +0.8 +0.8 +3 +(a) +(b) +- -Non-Interaction +Transmission +0.6 +0.6 +Interaction +2 +OD +0.4 +0.4 +1 +0.2 +0.2 +0 +0 +0 +-20 +-10 +0 +10 +20 +0 +5 +10 +15 +20 +f /2(MHz) +Ap/2π(MHz)5 +mission decrease with γeff +3 . +The EIT transmission de- +creases to 50% when γeff +3 += 2π×1.5 MHz. It is consistent +with the trend of our experimental data [see Fig. 2(a)]. +For comparing, we also plot the corresponding OD in +Fig. 4(b), as expected, calculated OD of the probe beam +increase as γeff +3 . We should note that the presented theo- +retical model is based on MF equation by simply varying +the effective decay rate γeff +3 . Beyond the present model, +many-body quantum model need to be developed to gain +better understanding of the metastable dynamics [27, 28]. +We will discuss it somewhere else. +V. +TEST OF THE DECAY IN 80D5/2 STATE +The dephasing effects arise from many physical rea- +sons, e.g., atomic collisions, residue Doppler effect, de- +population between the Rydberg atoms, or finite laser +linewidth. To test the collective dephasing process, we +also conduct the experiment to measure the fast pop- +ulation transfer from 80D5/2 to 81P3/2. +For 80D5/2 +state used in this work, the space to nearest 81P3/2 +state is 1.3124 GHz, corresponding dipole matrix element +5649.1 ea0 with e electron charge and a0 Bohr radius. +Therefore, 80D5/2 Rydberg state displays strong dipole +interactions with 81P3/2, resulting to the state transfer of +80D5/2 → 81P3/2 [33]. This transformation leads to the +decay of 80D5/2 Rydberg atoms and further decreases of +EIT transmission. In order to verify this conjecture, we +carry out more test that is performed in an additional +MOT [not shown in Fig. 1(b)], in which Rydberg atoms +is detected with a state select field ionization detection. +The temperature of the atomic cloud is almost same with +the main setup, and the peak density of atomic cloud is +8.0×1010cm−3, which is comparable with the main setup +of 1.0 × 1011cm−3. In addition, we make the probe and +coupling Rabi frequency on the almost same to that of the +main setup so that the Rydberg population is comparable +with that of the EIT in the main apparatus. Therefore, +we obtain the similar EIT spectra and field ionization sig- +nal simultaneously in the test setup. The details of the +setup can be seen in our previous work [33, 37]. In the +test experiments, after switching off the MOT beams, we +apply a two-photon excitation pulse with duration 4 µs +for preparing 80D5/2 Rydberg atoms, an optional inter- +action time tINT before the ionization detection allows us +to study the decay of 80D5/2 Rydberg and state trans- +formation. +In Fig. 5(a), we present the time of flight (TOF) sig- +nals for laser excitation to 80D5/2 state. Initially, atoms +are populated in 80D5/2 state (see the black curve) in +the TOF signal. Due to the resonant dipole interaction, +atoms in 80D5/2 state in light pink shaded region partly +transfer to nearby 81P3/2 state in the gray shaded region +[see the red curve for tINT = 3 µs]. With further increas- +ing tINT, most of 80D5/2 state Rydberg atoms transfer +to 81P3/2 state (see the blue curve for tINT = 6 µs). The +population in state 81P3/2 reaches maximal at tINT = +6 µs, and then decays to other states. +For better un- +derstanding of the transfer process, we make a series of +measurements for different tINT. Figure 5(b) displays the +measured 80D5/2 as a function of tINT. It is clear to ob- +serve the fast decay process on 80D5/2 state within 8µs. +To obtain the decay characteristic of 80D5/2 state, we fit +the experimental data using exponential function [see the +solid line of Fig. 5(b)]. The fitted results show decay time +τ0(80D) = 3.15 ± 0.38 µs is close to the decay of the EIT +transmission rate τ0(EIT ). Therefore, we conclude that +the dipole interaction induced the state transfer may be +the reason that leads to the decay of 80D5/2 Rydberg +three-level EIT transmission. +VI. +CONCLUSION +We have presented Rydberg EIT spectra in a cascade +three-level scheme involving 80D5/2 state of Cs atoms. +Rydberg EIT spectrum shows strong dependence on the +probe incident photon number. 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A 97 +(2018), 10.1103/PhysRevA.97.031403, arXiv:1708.02768. + diff --git a/TNE5T4oBgHgl3EQfAQ5w/content/tmp_files/load_file.txt b/TNE5T4oBgHgl3EQfAQ5w/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6025b48bfff91b856c7587e3aae0dfe13ac8e02f --- /dev/null +++ b/TNE5T4oBgHgl3EQfAQ5w/content/tmp_files/load_file.txt @@ -0,0 +1,605 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf,len=604 +page_content='Dephasing of ultracold cesium 80D5/2-Rydberg Electromagnetically Induced Transparency Yuechun Jiao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Liping Hao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Jingxu Bai1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Jiabei Fan1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Zhengyang Bai2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='∗ Weibin Li4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Jianming Zhao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='† and Suotang Jia1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3 1State Key Laboratory of Quantum Optics and Quantum Optics Devices,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Institute of Laser Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Shanxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Taiyuan 030006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' China 2State Key Laboratory of Precision Spectroscopy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' East China Normal University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Shanghai 200062,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' China 3Collaborative Innovation Center of Extreme Optics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Shanxi University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Taiyuan 030006,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' China 4School of Physics and Astronomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' and Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' University of Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Nottingham,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' NG7 2RD,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' UK We study Rydberg electromagnetically induced transparency (EIT) of a cascade three-level atom involving 80D5/2 state in a strong interaction regime employing a cesium ultracold cloud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In our experiment, a strong coupling laser couples 6P3/2 to 80D5/2 transition, while a weak probe, driving 6S1/2 to 6P3/2 transition, probes the coupling induced EIT signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' At the two-photon resonance, we observe that the EIT transmission decreases slowly with time, which is a signature of interaction induced metastability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephasing rate γOD is extracted with optical depth OD = γODt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We find that the optical depth linearly increases with time at onset for a fixed probe incident photon number Rin before saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephasing rate shows a nonlinear dependence on Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephasing mechanism is mainly attributed to the strong dipole-dipole interactions, which leads to state transfer from nD5/2 to other Rydberg states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We demonstrate that the typical transfer time τ0(80D) obtained by the state selective field ionization technique is comparable with the decay time of EIT transmission τ0(EIT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The presented experiment provides a useful tool for investigating the strong nonlinear optical effects and metastable state in Rydberg many-body systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' INTRODUCTION Due to the strong interaction (∝ n11 with n principal quantum number) [1], Rydberg atoms provides an ideal platform to implement quantum information and quan- tum simulation [2–5] and investigate interaction induced cooperative optical nonlinearities [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The optical nonlin- ear effects are induced by Rydberg atom interactions [7, 8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', van der Waals (vdW) interactions [9–11] and dipole-dipole interactions [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Strong atomic interac- tions can be effectively mapped onto photon-photon in- teractions via electromagnetically induced transparency (EIT) [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Due to the cooperative effects, the optical nonlinearity can be greatly enhanced [6, 14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Based on this, Rydberg EIT experiments are employed to mea- sure a radio-frequency electric field [17] with a room- temperature cell, and to realize few-photon optical non- linearities [18–21] with an ultracold sample, such as the efficient single photon generation [18], entanglement gen- eration between light and atomic excitations [20], single- photon switches [16, 19, 22] and transistors [21, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The resonant dipole-dipole interaction between two indi- vidual Rydberg atoms [25] is angular dependent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Re- cently, the anisotropic Rydberg interaction have been adopted to investigate Rydberg polaritons by using Ry- dberg nD-state [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' ∗ zhybai@lps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='ecnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='cn † zhaojm@sxu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='cn In this work, we present a Rydberg EIT spectrum of a cascade three-level cesium atom involving 80D5/2 state in a dipole trap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Under the two-photon resonance condition, we observe that EIT transmission displays a slow de- crease with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This could be a signature of interaction induced metastability [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Due to the large dipole matrix elements, the cesium nD-state atom has strong dipole interactions with energetically close (n+1)P state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We find that, strong dipole interactions between the nD and (n+1)P state leads to a fast decay of nD to (n+1)P state, and the dephasing of the transmission spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A theoretical model is built to understand EIT dephas- ing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The nonlinear dependence on dephasing rate has also been investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The remainder of the article is arranged as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' II, we introduce our experimental setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' III, we reveal Rydberg EIT spectrum and its dephasing ex- perimentally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' IV, we present a simple theoretical model to reveal EIT dephasing mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' V, we measure the fast decay process on 80D5/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Finally, we summarize the main results obtained in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' EXPERIMENTAL SETUP Our experiment is performed in a cesium magneto- optical trap (MOT) with optical dipole trap (ODT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='The beam waist of dipole trap is 45 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A schematic of relevant levels and the experimental setting are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1 (a) and (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A three-level system, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(a), consists of a ground state |6S1/2, F = 4⟩ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='05377v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='atom-ph] 13 Jan 2023 2 (b) (a) (c) g3 g2 Probe (Coupling) MOT ODT 20ms 46ms 46ms 46ms \uf044 \uf044 Wc Wp |6S1/2> |6P3/2> |80D5/2> \uf044p |g> |e> |r> 0 5 10 15 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 Transmission Time (ms) absorption EIT Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (color online) (a) Atomic level scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A weak probe laser field with Rabi frequency Ωp drives the lower transition, |g⟩ = |6S1/2, F = 4⟩ → |e⟩ = |6P3/2, F ′ = 5⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The strong coupling laser (Rabi frequency Ωc) couples the transition |e⟩ → |r⟩ = |80D5/2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (b) Experiment setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The coupling and probe beams are counter-propagated through the MOT center and overlap with the dipole trap beam.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The transmission of probe beam is detected with a single photon counting module (SPCM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The inset shows EIT dephasing behaviors (red) and probe absorption (black;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' without cou- pling beam).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The data are taken by the sum of 3000 exper- imental cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (c) Experimental timing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' After switching off the MOT and dipole trap beams, Rydberg-EIT coupling and probe lasers are turned on for 20 µs, during which the probe laser frequency is ramped through the |g⟩ → |e⟩ transition over ±15 MHz by a double-passed AOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (|g⟩), intermediate state |6P3/2, F ′ = 5⟩ (|e⟩) and Ry- dberg state |80D5/2⟩ (|r⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A weak probe beam (Rabi frequency Ωp, 852-nm laser with a 100-kHz linewidth), provided by an external cavity diode laser (Toptica, DL- pro), drives a lower transition and the frequency is sta- bilized to the |6S1/2, F = 4⟩ → |6P3/2, F ′ = 5⟩ transition using the polarization spectroscopy method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The cou- pling beam (Rabi frequency Ωc) provided by a commer- cial laser (Toptica, TA-SHG110) with linewidth 1 MHz drives Rydberg transition, |6P3/2, F ′ = 5⟩ → |80D5/2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The coupling laser frequency is stabilized to the Ryd- berg transition using a Rydberg EIT reference signal ob- tained from a cesium room-temperature vapor cell [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The weak probe laser and strong coupling laser, with respective Gaussian radius of 9 µm and 30 µm, are over- lapped and counter-propagated through the MOT center, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The probe laser is scanned using a double- passed acousto-optic modulator (AOM) that covers the lower transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The transmission of the probe laser, Rydberg EIT spectrum, is detected with a single photon counting module (SPCM) and processed with Labview program.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The glass MOT is surrounded by three pairs of field-compensation Helmholtz coils, which allow us to reduce stray magnetic fields via EIT Zeeman splitting, corresponding stray field less than 5 mG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In our experi- ment, the peak density of atomic cloud about 1011 cm−3 is measured by shadow imaging and the temperature of atomic cloud is about 100 µK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The estimated Rydberg density is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 × 108 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The experimental timing is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(c) with the whole time 200 ms, corresponding to a repetition rate of 5 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In each cycle, after turning off the trap beams, we switch on the coupling and probe lasers for 20 µs, dur- ing which the probe-laser frequency is swept across the |6S1/2, F = 4⟩ → |6P3/2, F ′ = 5⟩ transition, meanwhile the power is fixed using a proportional-integral-derivative controller (PID) feedback loop that controls the radio- frequency power supplied to the 852-nm AOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(b) is taken by the sum of 3000 cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' EXPERIMENTAL OBSERVATION OF RYDBERG EIT AND TRANSMISSION DEPHASING 14 7 0 7 14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='8 0 4 8 12 16 20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 (a) Rin = 21, 170 Photons/µs Transmission ∆p/2π (ΜΗz) t0 (EIT)= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='10 ms t0 (EIT)= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='05 ms t0 (EIT)= 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='06 ms (b) Rin = 21, 170, 340, 680 Photons/µs Time (µs) Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (color online) (a) Rydberg-EIT spectra with a cou- pling laser, Ωc/2π =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz, resonant with the |6P3/2, F ′ = 5⟩ → |80D5/2⟩ transition, and a probe frequency scanning across the lower transition, |6S1/2, F = 4⟩ → |6P3/2, F ′ = 5⟩, at a probe-photon rate Rin = 21 photons/µs and 170 photons/µs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The solid lines are the fittings using the density matrix equation of a three-level atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The EIT transmission displays a strong suppression and blue shift with increasing probe Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (b) EIT transmissions for the indicated probe photon input rates at two-photon resonance condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The transmission remains same for Rin = 21 photons/µs, whereas decrease with Rin at the beginning and then decay with the EIT maintain time for larger Rin cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The solid lines denote the exponential fittings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For higher Rin the char- acteristic time are τ0(EIT) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='10 µs, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='06 µs and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='05 µs, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Due to the quantum interference effect [30], the probe transmission T increases when the coupling and probe laser frequencies satisfy the two-photon resonance [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Using the rotating wave approximation and in the interaction picture [31], the eigenstate of three-level Hamiltonian can be obtained with |D⟩ ∝ Ω∗ c|g⟩ − Ωp|r⟩ at two-photon resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For conventional EIT, the sys- tem works in dark state and the probe pulse suffers little optical absorption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For the three-level scheme, all atoms are initially prepared in state |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The EIT system can 3 evolve to the dark state |D⟩ with time 1/γ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' However, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(b), in nD5/2-Rydberg EIT system, the EIT transmission slowly decreases with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Interest- ing, the time scale of the dephasing is much longer than the lifetime of intermediate state |e⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dipolar inter- action can lead to many-body dephasing [26, 32] in case of Rydberg nD state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We present EIT spectra with 80D5/2 Rydberg state in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The probe field is adopted as incident photon rates Rin = 21 photons/µs (black dashed line) and 170 photons/µs (red dashed line) with Ωc = 2π × 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' It is found that the transmission decreases with in- creasing Rin and accompanies with the EIT-peak shift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The EIT transmission rate is around 40% for Rin = 21 photons/µs, and decreases to 20% when Rin increases to 170 photons/µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We attribute the reduction of op- tical transmission to atomic interactions between Ryd- berg states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Besides, when Rin = 170 photons/µs, the EIT peak has a blue shift about 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='5 MHz compared with the case for Rin = 21 photons/µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For Rydberg EIT, dark-state polariton is very sensitive to other Rydberg excitation in the strong interaction regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This is be- cause when two Rydberg polaritons propagate inside the medium with a distance r, they experience an interaction induced energy shift U(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' It gives rise to a non-vanishing Im(χ) of probe beam where χ optical susceptibility of sys- tem and therefore leads to strong absorption and a shift on EIT spectra [6, 14, 15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' To further investigate the dephasing feature for nD state, we vary the probe-photon rate Rin with fixed cou- pling field Ωc/2π =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Both the coupling and probe lasers frequencies are on resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 2(b), the time dependence of the transmission is plotted with different Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For low probe photon rates (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', Rin = 21 photons/µs), the transmission is almost a constant, but when increasing Rin, transmis- sion exhibits a slow decrease with time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' By fitting the data in panel (b) with the exponential function T = A exp(−t/τ0(EIT)) + T0, the decay time τ0(EIT) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='10 µs, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='06 µs and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='05 µs can be extracted for different Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The decay time is much longer than the lifetime in state |e⟩ (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', 1/γ2 ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='03µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In order to reveal the dephasing mechanism, we de- fine the effective optical depth (OD) of the medium as the logarithm of transmission [i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', OD = -ln(T)] [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The time evolution of OD is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 3(a) [cor- responding to the results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 2(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' One sees that OD approximately linearly increases with time before t = 5 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' By neglecting saturation effects, we can redefine OD = γODt where γOD reflects creation rate of opti- cal density by decoupled impurities [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' At small probe photons (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', Rin ≲ 340 µs−1), the extracted rate γOD displays a linear increase with Rin and then saturates for large Rin [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This is because the system is not fully in a blockade regime at small probe pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Therefore, with increasing photon number, more Rydberg atoms are excited, which increase the dipole- dipole interaction, leading to increased γOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' However, 0 4 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 0 250 500 750 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='08 (a) Time (µs) Rin=21, 170, 340, 680 Photons/µs OD (b) Ωc/2π = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 ΜΗz γOD (µs-1) Rin (µs-1) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (color online) (a) The optical depth of the 80D5/2 EIT transmission taken the logarithm of spectra in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 2(b) for indicated photon incidence rates Rin at fixed Ωc = 2π×10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The solid lines are linear fittings before t = 5 µs to the data to extract the dephasing rates γOD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (b) The dephasing rates γOD as a function of Rin for coupling Rabi frequency Ωc/2π = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 MHz, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' after the system enter fully blockade regime, we can’t excite more Rydberg atoms so that the dephasing rate γOD shows saturation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This is a signature of many-body dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We calculate the group velocity of probe photon is around 3920 m/s under our experiment con- dition with Ωc = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' By considering the length of our atomic cloud 1 mm, the propagation time of pho- ton through the cloud is around 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='26 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The Rydberg blockade radius is about 10 µm for 80D5/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Under this condition, almost 100 atoms can be excited to Rydberg state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Thus we can calculate that in one microsecond, the maximum number of Rydberg excitation is around 385, where 100 atom/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='26 µs ≃ 385 /µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Thus, the criti- cal value for Rin is around 385 /µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' When the photon incidence rates Rin is smaller than 385 /µs, the system is not in the blockade regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' When increasing the in- tensity of the probe laser, the system can works in the full blockade regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This estimation is agreed with the dephasing rates γOD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 3(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We also change Ωc and measure EIT dephasing rates γOD versus Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The results show a similar nonlinear dependence on Rin [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 3(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' ANALYSIS OF EIT DEPHASING MECHANISM The vdW interaction between nD5/2 pair leads to en- ergy level shifts and dipole interaction of nD5/2 and near- est Rydberg states yield state transfer of nD5/2 to other Rydberg states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' As no microwave field is present in the experiment, the other states are excited due to the spon- taneous decay from nD5/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We have found [33] that nD-(n+1)P transition is the strongest in our experiment conditions (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 5 for details).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Hence this leads to a two stage processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Rydberg nD5/2 state will decay to (n + 1)P3/2 through spontaneous decay.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephas- ing can then be induced in that regime where dipole- dipoles interactions couple nearly degenerate Rydberg pair states [26, 32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' A full model to describe the de- 4 phasing is rather complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In this section, we will focus on the dephasing effects with a simplified model, which nonetheless captures the main effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Considering the three-level scheme in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(a), the dephasing rate of Rydberg state γ3 is around γr + Γre ≈ γr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Rydberg atom has long lifetimes (1/Γre ∼ n3) on the order of 100 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' γr represents the dephasing of the atomic coherence (originated from atomic collisions, residue Doppler effect, dipole-dipole interaction between the Rydberg atoms, finite laser linewidth).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephas- ing rate of the intermediate state |e⟩ denotes γ2 = γe + Γeg with spontaneous decay rate Γeg and interaction induced decay γe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In our physical system, γe is much smaller than Γeg ≃ 2π × 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 MHz, thus γ2 ≈ Γeg .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The collective dissipation can emerge in dense atomic gases, typically through two-body dipolar couplings [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Here we adopt the effective dephasing γeff 3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', γr = γeff 3 ), and seek the relation between the transmission and dipolar interaction induced dephasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dynamics of the effective three-level system can be modeled by the quantum master equation for the many- atom density operator ρ: ˙ρ = −i[ ˆHeff, ρ] + D1(ρ) + Deff(ρ), (1) The effective Hamiltonian in the equation is given by ˆHeff = N � j=1 � −∆pˆσj ee(r, t) − (∆p + ∆c)ˆσj rr(r, t) + Ωp 2 ˆσj eg(r, t) +Ωc 2 ˆσj re(r, t) + N � k̸=j Vjk 2 ˆσj rrˆσk rr + H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' � � , (2) with ˆσab(zj) ≡ |aj⟩⟨bj| (zj is the position of jth atom in the respective ensemble) and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' representing Her- mitian conjugate of the preceding terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Vjk = C6/|rj − rk|6 is the vdW potential with the dispersive coefficient C6 ∝ n11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dissipative effects are described by the Lindblad form D1(ρ), D1(ρ) = N � j=1 Γeg � ˆσj geρˆσj eg − 1 2{ˆσj ee, ρ} � , (3) where D1(ρ) denotes the decay from state |e⟩ to |g⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The effective dephasing term Deff(ρ) is introduced, Deff(ρ) = N � j=1 γeff 3 � ˆσj 33ρˆσj 33 − 1 2{ˆσj 33, ρ} � , (4) Due to the dephasing and spectral shift of the trans- parency resonance [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2(a)], we employ the theo- retical description of individual atoms coupled to a mean field (MF) to analyze the EIT spectrum with strong Ry- dberg interactions [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In the MF approximation, the many-body density matrix ρ is decoupled into indi- vidual ones through ˆρ ≈ Πi ˆρi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In the thermodynamic limit, the optical Bloch equations for the three-level sys- tem can be obtained, where elements of the density ma- trix are represented with ρab = N −1 � j⟨ˆσj ab⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' As in- dicated in our experiment, over a long time evolution, the interactions between Rydberg state leads to a MF shift where ∆c → ∆c +V ρrr with MF interaction energy V = N −1 � k̸=j Vjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (color online) (a) Theoretical results of EIT trans- mission varies with ∆p for V = 0 and V ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (b) EIT trans- mission and corresponding OD versus effective dephasing rate γeff 3 of Rydberg state that accounts for the interaction between Rydberg atoms with Ωp/2π =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 MHz, Ωc/2π =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz and atomic density 1×1011cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 0 1 2 3 81P3/2 tINT = 6 µs tINT = 3 µs Normalied signal (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='units) Time (µs) 80D5/2 (a) 0 4 8 12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 (b) tINT (µs) 80D5/2 Signal (Arb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='units) t0 (80D) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='38 ms tINT = 0 µs Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (color online) (a) Normalized time of flight (TOF) signals for laser excitation to 80D5/2 state with indicated in- teraction times, tINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The three traces are set vertically offset for clarity, with the respective zero levels shown as horizon- tal dashed lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The gates for the 80D5/2 and 81P3/2 states signals are shown as a light pink and gray shaded regions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' (b) Measurements of the 80D5/2 state as a func- tion of the interaction time tINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Initial prepared 80D5/2 Rydberg atoms transfer to nearby Rydberg states |r′⟩ during tINT due to the strong resonant dipole interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The solid line shows the exponential fitting with the characteristic time τ0(80D) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='38 µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' By numerically solving the MF Bloch equation, one can obtain EIT transmission varies with ∆p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 4(a), there is an increasing blue shift of trans- parency away from the non-interaction EIT resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This is because that the shifted Rydberg state detunes the EIT windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The similar EIT signal is also observed experimentally [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' To obtain the dependence of the EIT transmission on the effective dephasing in- duced by Rydberg interactions, we calculate EIT spectra for a series of effective γeff 3 , accounting for the many- body dephasing by Rydberg atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 4(b), the EIT transmission as a function of γeff 3 is plotted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The system works at two-photon detuning δ = 0 with the probe Ωp/2π =1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='04 MHz, Ωc/2π =10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 MHz and atomic density Na = 1×1011 cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' It shows that the EIT trans- 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='8 3 (a) (b) -Non-Interaction Transmission 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='6 Interaction 2 OD 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='4 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='2 0 0 0 20 10 0 10 20 0 5 10 15 20 f /2(MHz) Ap/2π(MHz)5 mission decrease with γeff 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The EIT transmission de- creases to 50% when γeff 3 = 2π×1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='5 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' It is consistent with the trend of our experimental data [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 2(a)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For comparing, we also plot the corresponding OD in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 4(b), as expected, calculated OD of the probe beam increase as γeff 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We should note that the presented theo- retical model is based on MF equation by simply varying the effective decay rate γeff 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Beyond the present model, many-body quantum model need to be developed to gain better understanding of the metastable dynamics [27, 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We will discuss it somewhere else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' TEST OF THE DECAY IN 80D5/2 STATE The dephasing effects arise from many physical rea- sons, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=', atomic collisions, residue Doppler effect, de- population between the Rydberg atoms, or finite laser linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' To test the collective dephasing process, we also conduct the experiment to measure the fast pop- ulation transfer from 80D5/2 to 81P3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For 80D5/2 state used in this work, the space to nearest 81P3/2 state is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='3124 GHz, corresponding dipole matrix element 5649.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='1 ea0 with e electron charge and a0 Bohr radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Therefore, 80D5/2 Rydberg state displays strong dipole interactions with 81P3/2, resulting to the state transfer of 80D5/2 → 81P3/2 [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' This transformation leads to the decay of 80D5/2 Rydberg atoms and further decreases of EIT transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In order to verify this conjecture, we carry out more test that is performed in an additional MOT [not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 1(b)], in which Rydberg atoms is detected with a state select field ionization detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The temperature of the atomic cloud is almost same with the main setup, and the peak density of atomic cloud is 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0×1010cm−3, which is comparable with the main setup of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='0 × 1011cm−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In addition, we make the probe and coupling Rabi frequency on the almost same to that of the main setup so that the Rydberg population is comparable with that of the EIT in the main apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Therefore, we obtain the similar EIT spectra and field ionization sig- nal simultaneously in the test setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The details of the setup can be seen in our previous work [33, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In the test experiments, after switching off the MOT beams, we apply a two-photon excitation pulse with duration 4 µs for preparing 80D5/2 Rydberg atoms, an optional inter- action time tINT before the ionization detection allows us to study the decay of 80D5/2 Rydberg and state trans- formation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 5(a), we present the time of flight (TOF) sig- nals for laser excitation to 80D5/2 state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Initially, atoms are populated in 80D5/2 state (see the black curve) in the TOF signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Due to the resonant dipole interaction, atoms in 80D5/2 state in light pink shaded region partly transfer to nearby 81P3/2 state in the gray shaded region [see the red curve for tINT = 3 µs].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' With further increas- ing tINT, most of 80D5/2 state Rydberg atoms transfer to 81P3/2 state (see the blue curve for tINT = 6 µs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The population in state 81P3/2 reaches maximal at tINT = 6 µs, and then decays to other states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' For better un- derstanding of the transfer process, we make a series of measurements for different tINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Figure 5(b) displays the measured 80D5/2 as a function of tINT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' It is clear to ob- serve the fast decay process on 80D5/2 state within 8µs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' To obtain the decay characteristic of 80D5/2 state, we fit the experimental data using exponential function [see the solid line of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' 5(b)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The fitted results show decay time τ0(80D) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content='38 µs is close to the decay of the EIT transmission rate τ0(EIT ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Therefore, we conclude that the dipole interaction induced the state transfer may be the reason that leads to the decay of 80D5/2 Rydberg three-level EIT transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' CONCLUSION We have presented Rydberg EIT spectra in a cascade three-level scheme involving 80D5/2 state of Cs atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' Rydberg EIT spectrum shows strong dependence on the probe incident photon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The optical transmission displays decay behavior with time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' An optical depth of the medium is defined to characterize the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We have shown that the optical depth displays linear in- crease with the time at onset for fixed probe Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We have further obtained the dephasing rate γOD by redefining OD=γODt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The dephasing rate linearly increases with weak probe field and then saturates for large Rin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' We have shown that the dephasing mechanism is mainly at- tributed to the strong dipole-dipole interactions, which lead to strong population decay on |nD5/2⟩ state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/TNE5T4oBgHgl3EQfAQ5w/content/2301.05377v1.pdf'} +page_content=' The experimental setting provides a platform to explore quan- tum nonlinear optics and quantum information process- ing, and creates metastable state in 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Espino,1, 2 Aviral Prakash,1, 3 David Radice,1, 3, 4, ∗ and Domenico Logoteta5, 6 +1Institute for Gravitation & the Cosmos, The Pennsylvania State University, University Park, PA 16802 +2Department of Physics, University of California, Berkeley, CA 94720, USA +3Department of Physics, The Pennsylvania State University, University Park, PA 16802 +4Department of Astronomy & Astrophysics, The Pennsylvania State University,University Park, PA 16802 +5Dipartimento di Fisica, Università di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy +6INFN, Sezione di Pisa, Largo B. Pontecorvo, 3 I-56127 Pisa, Italy +ABSTRACT +We use numerical relativity simulations of binary neutron star mergers to show that high density +deconfinement phase transitions (PTs) to quark matter can be probed using multimodal postmerger +gravitational wave (GW) spectroscopy. Hadron-quark PTs suppress the one-armed spiral instability in +the remnant. This is manifested in an anti-correlation between the energy carried in the l = 2, m = 1 +GW mode and energy density gap which separates the two phases. Consequently, a single measurement +of the signal-to-noise ratios of the l = 2, m = 1 and l = 2, m = 2 GW modes could constrain the energy +density gap of the PT. +Keywords: Neutron stars — Equation of state — Gravitational waves — Hydrodynamics — Instabilities +1. INTRODUCTION +Binary neutron star (BNS) mergers produce some +of the most extreme conditions in nature, compressing +matter to several times the nuclear density and to tem- +peratures of tens of MeV (Perego et al. 2019). +More +extreme conditions are only found in the early Universe +and in the interior of black holes (BHs). Multimessen- +ger observations of binary neutron star (BNS) mergers +can be used to probe the properties of matter in these +conditions, providing a unique avenue to study the non- +perturbative regime of QCD (Shibata 2005; Hinderer +et al. 2010; Damour et al. 2012; Sekiguchi et al. 2011; +Hotokezaka et al. 2011; Bauswein et al. 2013; Radice +et al. 2017; Abbott et al. 2017a; Margalit & Metzger +2017; Bauswein et al. 2017; Radice et al. 2018b; Most +et al. 2019, 2020; Bauswein et al. 2019; Coughlin et al. +2019; De et al. 2018; Abbott et al. 2019, 2018; Radice & +Dai 2019; Dietrich et al. 2020; Breschi et al. 2021, 2022; +Kashyap et al. 2022; Perego et al. 2022; Fujimoto et al. +2022; Prakash et al. 2021). +Presently, there are large uncertainties in the funda- +mental physics of strongly-interacting matter at densi- +∗ Alfred P. Sloan fellow +ties of a few times nuclear saturation (Capano et al. +2020; Pang et al. 2021; Annala et al. 2022). It is not +even clear what the relevant degrees of freedom are for +the densities and temperatures reached in the core of +remnant massive neutron stars (RMNS) of BNS merg- +ers. It is possible that matter remains composed of nu- +cleons, together with leptons (electrons, positrons, and +muons) and photons (Perego et al. 2019; Loffredo et al. +2022). The appearance of more exotic baryons, such as +hyperons, is not excluded (Sekiguchi et al. 2011; Radice +et al. 2017; Logoteta 2021). +It is also possible for a +transition to the deconfined quark-gluon plasma phase +to take place in BNS mergers (Most et al. 2019, 2020; +Bauswein et al. 2019; Prakash et al. 2021). The deter- +mination of the state of matter formed in BNS mergers +is one of the most pressing scientific objectives of mul- +timessenger astronomy (Evans et al. 2021; Lovato et al. +2022). +Previous work has shown that the presence of phase +transitions to deconfined quarks can be revealed by a +shift of the postmerger gravitational wave (GW) peak +frequency f2 from the value expected for hadronic equa- +tions of state (EOSs) (Bauswein et al. 2019; Weih et al. +2020; Blacker et al. 2020; Kedia et al. 2022). However, +such frequency shifts can be degenerate with deviations +arXiv:2301.03619v1 [astro-ph.HE] 9 Jan 2023 + +2 +from universal relations due to hadronic physics or other +effects (Most et al. 2019; Weih et al. 2020; Liebling et al. +2021; Prakash et al. 2021; Fujimoto et al. 2022; Tootle +et al. 2022). It has also been suggested that the presence +of a phase transition could be inferred from a measure- +ment of the threshold mass for prompt collapse of BNS +systems (Bauswein et al. 2020, 2021; Perego et al. 2022; +Kashyap et al. 2022). +In this Letter, we use 8 state- +of-the-art numerical relativity simulations to show, for +the first time, that the presence and strength of a QCD +phase transition could be unambiguously determined +through multimodal GW spectroscopy of RMNS. Such +measurements will be possible with the next-generation +of GW experiments like Cosmic Explorer (Reitze et al. +2019), Einstein Telescope (Punturo et al. 2010), and +NEMO (Ackley et al. 2020). +2. METHODS +We consider binaries in quasi-circular orbits and ec- +centric encounters on nearly parabolic orbits. Although +BNS mergers with highly eccentric orbits are expected to +be significantly more rare than those with quasi-circular +inspirals, these events may still have appreciable rates +of as high as 50 Gpc−3 yr−1 (Lee et al. 2010; Paschalidis +et al. 2015); we include results from both types of merg- +ers to consider as wide a variety of scenarios as possible. +Initial data for the quasi-circular binaries is created us- +ing the conformal thin sandwich formalism (York 1999) +and assuming a helical Killing vector and irrotational +flows. +The resulting elliptic equations are solved us- +ing the pseudo-spectral code LORENE (Gourgoulhon et al. +2001; Taniguchi et al. 2001; Taniguchi & Gourgoulhon +2002). Initial data for the eccentric encounters is con- +structed by superimposing two isolated, boosted, neu- +tron stars, following Radice et al. (2016b). The initial +separation of the stellar barycenters for parabolic en- +counters is set to 100 km, which is sufficiently large so +that the level of constraint violation in the initial data is +comparable to that of the quasi-circular binaries (Radice +et al. 2016b). +We +perform +BNS +merger +simulations +using +the +WhiskyTHC code (Radice & Rezzolla 2012; Radice et al. +2014a,b). WhiskyTHC makes use of the CTGamma space- +time solver (Pollney et al. 2011), which is a part of the +Einstein Toolkit (Zlochower et al. 2022). The adap- +tive mesh refinement driver Carpet (Schnetter et al. +2004) is used to generate the dynamical grid structure +employed in the simulations. All simulations considered +in the present work have been performed using at least +two grid resolutions. +Although there are quantitative +differences in the GW waveforms computed at different +resolutions, the qualitative features discussed here are +robust across all simulations. +Unless otherwise speci- +fied, we discuss results from simulations using the fidu- +cial grid resolution (with grid spacing ∆x ≃ 184.6 m in +the finest refinement level). The grid structure for the +simulations is described in detail in Radice et al. (2018a) +and Radice et al. (2016b) for the quasi-circular and ec- +centric simulations, respectively. +For a clear understanding of the role that high-density +deconfinement phase transitions could play in the de- +velopment of the one-armed spiral instability, we con- +sider a total of 7 EOS models and run a total of 8 +simulations with varying phase transition features. In +particular, the size of the energy density gap which +separates the hadronic and quark phases is a useful +way to classify hybrid hadron-quark EOS models and +provides a qualitative measure of the ‘strength’ of the +phase transition (Alford & Han 2016). As such, we con- +sider EOS models that cover several sizes of the energy +density gap, ranging from non-existent (i.e., a purely +hadronic EOS) to large, while maintaining consistency +with current astrophysical constraints on the dense mat- +ter EOS. We consider both phenomenological EOS mod- +els (Paschalidis et al. 2018; Alvarez-Castillo & Blaschke +2017; Alford & Sedrakian 2017; Bozzola et al. 2019; Es- +pino & Paschalidis 2022) (in the form of piecewise poly- +tropic approximations using the prescription of Read +et al. (2009)) and microphysical, finite temperature EOS +models (Bastian 2021; Bombaci & Logoteta 2018; Lo- +goteta et al. 2021; Prakash et al. 2021). We only con- +sider equal-mass ratio binary configurations, with the +total binary mass ranging from 2.6 M⊙ − 2.7 M⊙. The +lack of π-rotational symmetry in BNS configurations +with unequal-masses may be a suitable way of effec- +tively seeding non-axisymmetric fluid instabilities that +can take hold in the post-merger environment. +Neu- +trino emission and reabsorption are not included for bi- +naries in eccentric orbits, while all quasi-circular bina- +ries include a neutrino treatment via the moment based +M0 scheme (Radice et al. 2018a). However, neutrinos +are not expected to influence the dynamics on the time +scales considered in our study (Radice et al. 2020, 2022). +Additionally, magnetic fields are not accounted for in +any of our simulations, but these are also expected to +be subdominant (Palenzuela et al. 2022). We find that, +despite the diversity in binary properties and differences +in the evolution, the effects presented in this work are +robust. +3. RESULTS +The +one-armed +spiral +instability +is +a +non- +axisymmetric mode in a rapidly rotating fluid which, +when saturated, leads to the dominance of a single + +3 +0 +10 +20 +30 +40 +100 +101 +102 +103 +104 +t − tmerger (ms) +E2,1 +GW/E2,1 +GW(tnorm) +hadron (DD2F) +hadron-quark (DD2F-SF5) +0 +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +−3 +−2 +−1 +0 +1 +∆e (1015g/cm3) +log10(⟨E2,1 +GW/E2,1 +GW(tnorm)⟩)hadron−quark +log10(⟨E2,1 +GW/E2,1 +GW(tnorm)⟩)hadron +Quasi-circular +Eccentric +Figure 1. Left panel: Energy carried by GWs in the l = 2, m = 1 mode as a function of time, scaled by the value at a fixed +normalization time tnorm = tmer + 0.5 ms. The development of the one-armed spiral instability can be observed in the purely +hadronic simulation as the energy in the l = 2, m = 1 GW mode continues to grow, but is suppressed in the hadron-quark +simulation. Right panel: The same quantity depicted in the left panel, but time-averaged over a fixed time window and shown +as a function of the energy density gap ∆e for each simulation. We normalize by the same quantity for the complementary +hadronic EOS. We include approximate error bars, obtained using lower resolution simulations, to represent the uncertainty +introduced by our numerical methods. The black solid line represents a linear fit to the data from our simulations. We find that +the normalized energy emitted by the l = 2, m = 1 GW mode decreases for simulations that employ EOS models with larger +values of ∆e. +high-density mode in the fluid density which is dis- +placed from the fluid barycenter (Pickett et al. 1996; +Centrella et al. 2001; Saijo et al. 2002; Ou & Tohline +2006). +The one-armed spiral instability has been ob- +served to develop commonly in BNS merger simulations +that produce long-lived, massive post-merger remnants +on timescales of O(10ms) (Paschalidis et al. 2015; East +et al. 2016; East et al. 2016; Radice et al. 2016a; Lehner +et al. 2016) and in simulations of many other astro- +physical systems including supernovae (Ott et al. 2005; +Kuroda et al. 2014), white dwarfs (Kashyap et al. 2015, +2017) and accretion disks (Kashyap et al. 2017; Wessel +et al. 2021). Each fluid density mode that arises during +the evolution of a massive NS remnant is associated +with GW emission at characteristic frequencies stem- +ming from its pattern speed. As such, the development +of the one-armed spiral instability in astrophysical sys- +tems may be observed by considering multimodal GW +spectroscopy (Radice et al. 2016a). For the simulations +considered in this work we extract multimodal GW in- +formation within the Newman-Penrose formalism. We +compute the coefficients of s = −2 spin-weighted spher- +ical harmonic decompositions of the Newman-Penrose +scalar Ψ4 which we label as Ψl,m +4 +. The one-armed spiral +instability can therefore be observed in the GW spec- +trum extracted from our simulations as a growth in the +power and amplitude of the l = 2, m = 1 GW mode +(i.e., Ψ2,1 +4 ) and simultaneous decay of the dominant +l = 2, m = 2 GW mode (i.e., Ψ2,2 +4 ). +Our simulations show that high-density deconfinement +phase transitions act to suppress the one-armed spiral +instability. Depending on the features of the phase tran- +sition, the one-armed spiral mode may either arise on a +significantly longer timescale when compared to simula- +tions which employ purely hadronic EOS models, or it +may be suppressed altogether on the timescales probed +by our simulations. There are several potential mecha- +nisms via which the instability may be suppressed. For +example, it has been shown that the physical extent of +the remnant plays an important role in the development +of the instability, with larger remnants being more con- +ducive to the development of the instability on shorter +timescales (Radice et al. 2016a; Lehner et al. 2016; Saijo +& Yoshida 2016; Saijo 2018). The significant softening +at high densities introduced by the phase transition re- +sults in more compact post-merger remnants (relative +to scenarios that consider only hadronic degrees of free- +dom). As such, the more compact hybrid star remnants +may see a weaker development of the one-armed spiral +instability when compared to neutron star remnants. +In the left panel of Fig. 1 we show the energy car- +ried by the l = 2, m = 1 GW mode (scaled by the en- +ergy emitted at a time shortly after the merger tnorm = +tmer + 0.5 ms) as a function of time for simulations em- +ploying the DD2F (hadronic) and DD2F-SF5 (hybrid + +4 +102 +103 +104 +f [Hz] +10−27 +10−26 +10−25 +10−24 +10−23 +10−22 +10−21 +10−20 +2√f|h+(f)|, +√ +S [Hz−1/2], 40Mpc +hadron (BLh) +l = 2, m = 2 +l = 2, m = 1 +aLIGO +ET +CE20 +CE40 +102 +103 +104 +f [Hz] +10−27 +10−26 +10−25 +10−24 +10−23 +10−22 +10−21 +10−20 +2√f|h+(f)|, +√ +S [Hz−1/2], 40Mpc +hadron − quark (BLQ) +l = 2, m = 2 +l = 2, m = 1 +aLIGO +ET +CE20 +CE40 +Figure 2. +Multimodal GW amplitude spectrum computed for symmetric binaries of total mass M = 2.6 M⊙ in an edge-on +configuration. Also shown are the noise sensitivity curves for advanced LIGO (aLIGO), Einstein Telescope (ET), the 20 km +postmerger-optimized configuration for the Cosmic Explorer (CE20) and the 40 km configuration for Cosmic Explorer (CE40). +A suppression in the amplitude spectral density (ASD) as a result of the deconfinement phase transition may be detectable with +the third generation detectors and most cleanly with CE40. +hadron-quark) EOSs. We find that the energy carried +in the l = 2, m = 1 mode of the GWs is significantly +smaller in the simulation employing a hybrid hadron- +quark EOS, indicating that the one-armed spiral in- +stability is suppressed in scenarios with deconfinement +phase transitions at densities relevant for BNS mergers. +We emphasize that in the left panel of Fig. 1 we show- +case results for a set of EOS models which are identical +below the threshold for a phase transition, and as such +the simulations have identical initial conditions. In the +right panel of Fig. 1, we show the time-averaged energy +emitted by the l = 2, m = 1 GW mode ⟨E2,1 +GW⟩ (again +scaled by the energy emitted at a time shortly after the +merger tnorm = tmer+0.5 ms) as a function of the energy +density gap ∆e, where we define the energy density gap +as the difference between the energy density e at the end +of the hadronic phase and beginning of the quark phase +for cold matter in β-equilibrium (Alford & Han 2016), +∆e ≡ equark,initial − ehadron,final, +(1) +where we assume units where the speed of light c = 1. +We identify the end and beginning of each phase by +considering the change in the approximate adiabatic in- +dex Γ = d log p/d log(ρ), where p is the fluid pressure +of the cold, beta-equilibrium, barotropic EOS for each +EOS model considered. +The region corresponding to +the phase transition is always unambiguously marked +by discontinuities in, or sudden changes in the slope of, +the adiabatic index for the EOS models we consider. +For the results depicted in the right panel of Fig. 1, we +time-average over a window of ∆t ≈ 40 ms after the +merger except for cases that lead to a remnant collapse +on shorter timescales (in such cases, we time-average +until the collapse of the NS remnant). Additionally, for +each simulation, we normalize by a complementary sim- +ulation that uses identical initial data but employs a +hadronic EOS having the same low-density behavior be- +low the phase transition threshold as the hybrid hadron- +quark EOS. As such, we depict the point corresponding +to all hadronic EOS simulations with a black square at +∆e = 0. Each simulation is time-averaged to the same +extent as its complementary hadronic simulation. We +find an anti-correlation between the energy carried in the +l = 2, m = 1 GW mode and the size of the energy density +gap. In other words, as the size of the energy density +gap (and thereby the qualitative ‘strength’ of the phase +transition) increases, GW emission in the l = 2, m = 1 +mode decreases, which signifies that the one-armed spi- +ral instability is further suppressed for EOS models with +‘stronger’ deconfinement phase transitions. We elabo- +rate on the choice of quantities depicted in Fig. 1 in +App. A. +4. DISCUSSION +The characteristic frequency associated with peak +emission in the l = 2, m = 1 GW mode has half the +value of that associated with the l = 2, m = 2 mode +(i.e., f 2,1 +peak = f 2,2 +peak/2). +Observationally, a GW signal +would contain information at all contributing frequen- +cies. However, the dominant GW emission associated +with binary coalescence is always expected to be from +the l = 2, m = 2 contribution, such that fpeak = f 2,2 +peak. + +5 +Therefore, a potential observational signature of the +one-armed spiral instability is the growth in power of an +incoming GW signal at a frequency that is half of the +dominant frequency; if it develops in the post-merger +environment, the one-armed spiral instability will con- +tinuously power the emission of GWs at fpeak/2, while +emission in the dominant fpeak decays in time (Bernuzzi +et al. 2015). +In Fig. 2 we show the post-merger GW amplitude +spectrum density (ASD) for a symmetric, edge-on binary +situated at a distance of 40 Mpc, which is consistent with +the luminosity distance observed for GW170817 (Abbott +et al. 2017b). The edge-on configuration is the most op- +timal for the detection of an m = 1 mode. As expected, +we see a relative suppression of power in the m = 1 +mode (with respect to the complementing hadronic sim- +ulation) with the onset of a deconfinement phase tran- +sition. In this realistic configuration, coupled with the +40 km Cosmic Explorer detector (Reitze et al. 2019), the +appearance of quarks in the post-merger remnant results +in a suppression of the postmerger signal-to-noise ratio +(SNR) of the (l = 2, m = 1) mode by a factor of 2, from +2.14 in the hadronic case to 1.08 in the hadron-quark +case. The GW ASD peak of the l = 2, m = 1 mode +(between 1-2 kHz) and the postmerger ASD peak of the +l = 2, m = 2 mode (between 2-4 kHz), lie respectively +in the most sensitive regions of the 40 km and the 20 km +postmerger optimized Cosmic Explorer configurations. +Our analysis recommends an increase in detector sen- +sitivities in the high-frequency regimes (see also Zhang +et al. (2022)) for best possible constraints on deconfine- +ment phase transitions in BNS mergers. +In this letter we have highlighted, for the first time, +that high-density deconfinement phase transitions act +to suppress the one-armed spiral instability. +We find +an anti-correlation between the energy carried in the +l = 2, m = 1 GW mode and the size of the energy den- +sity gap which qualitatively separates the hadronic and +quark phases. +Our findings reveal a deep connection +between observable multimodal GW emission and the +microphysical description of matter in the post-merger +environment. We expect the one-armed spiral instability +to be detectable at distances of 40 Mpc using future gen- +eration detectors (Radice et al. 2016a). If evidence of a +strong one-armed spiral mode can be inferred from GW +observations of the post-BNS merger environment, our +findings suggest that a strong high-density deconfine- +ment phase transition at the densities relevant to BNS +mergers would be disfavored. On the other hand, if ev- +idence for the one-armed spiral instability is not found +for close-by BNS mergers, this could also point to the +possibility of a deconfinement phase transition taking +place at densities relevant to BNS mergers. +We point out that other effects relevant in the post- +merger environment - such as the presence of strong +magnetic fields (Franci et al. 2013) and additional de- +grees of freedom that can cause a sudden softening of +the EOS - may affect the development of the one-armed +spiral instability. However, the relevant timescales and +extent to which the aforementioned phenomena can af- +fect the development of non-axisymmetric instabilities +or the GW spectrum remains uncertain (Radice et al. +2016a; Muhlberger et al. 2014), and may not impact +our conclusions (Palenzuela et al. 2022; Zappa et al. +2022). The effects discussed in the present work arise +on dynamical timescales ∼ O(10 ms), and may be the +dominant mechanism for suppression of the one-armed +spiral instability. Additionally, although we find a trend +in the decrease of energy carried by the l = 2, m = 1 +GW mode for larger values of ∆e, additional studies +will help establish a more robust trend and provide +an understanding of the potential spread in the trend. +In particular, future lines of investigation will include: +(1) considering the combined effects of the mass ratio +and high-density phase transitions on the development +of the one-armed spiral instability; (2) considering the +effects of accurate neutrino transport on high-density +deconfinement phase transitions, as neutrinos may mod- +ify the composition of matter and thereby potentially +affect the onset of the phase transition; (3) employing +EOS models at systematically increasing values of ∆e +while holding the hadronic region of the EOS fixed, as +a limitation of the present work is the assumption that +the l = 2, m = 1 GW mode is perfectly known in the +case of hadronic EOSs; and (4) investigating the effects +discussed in this work in scenarios with a crossover to +quark matter, as our present work only considers EOS +models with phase transitions. We leave such studies to +future work. +PE acknowledges funding from the National Science +Foundation under Grant No. +PHY-2020275. +DR ac- +knowledges funding from the U.S. Department of En- +ergy, Office of Science, Division of Nuclear Physics under +Award Number(s) DE-SC0021177 and from the National +Science Foundation under Grants No. +PHY-2011725, +PHY-2116686, and AST-2108467. Simulations were per- +formed on Bridges2 and Expanse (NSF XSEDE allo- +cation TG-PHY160025). This research used resources +of the National Energy Research Scientific Computing +Center, a DOE Office of Science User Facility supported +by the Office of Science of the U.S. Department of En- +ergy under Contract No. DE-AC02-05CH11231. + +6 +REFERENCES +Abbott, B. P., et al. 2017a, Astrophys. J. Lett., 851, L16, +doi: 10.3847/2041-8213/aa9a35 +—. 2017b, Phys. Rev. Lett., 119, 161101 +—. 2018, Phys. Rev. Lett., 121, 161101, +doi: 10.1103/PhysRevLett.121.161101 +—. 2019, Phys. Rev. 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Left panel: Energy in the l = 2, m = 1 GW mode as a function of time for simulations employing a hadronic +(DD2F) and hadron-quark (BBKF1.5) EOS; the simulations use identical initial conditions and are run with a grid resolution of +∆x = 369.2 m in the finest grid. These results showcase that the one-armed spiral instability may be seeded at different levels in +the postmerger environment for different simulations. Right panel: Same quantity as the left panel, but normalized to the value +at a time shortly after merger, tnorm = tmerger + 0.5 ms. Normalizing at this time accounts for the one-armed spiral instability +being seeded at disparate levels across simulations. +In Fig. 1 we depict an example of GW quantities that exhibit the suppression of the one-armed spiral instability for +simulations that employ hadron-quark EOSs. In particular, we calculate the GW energy carried in the l = 2, m = 1 +mode. In Fig. 1 we show E2,1 +GW normalized to its value at a time shortly after the merger; we depict normalized quantities +because of the variable nature in which the one-armed spiral instability is seeded in the immediate post-merger +environment in the context of numerical studies. Unless it is explicitly excited as a non-axisymmetric perturbation of +a known amplitude (e.g., as a fixed-amplitude perturbation in the rest mass density), the one-armed spiral instability +arises numerically from error at the level of floating-point precision (Espino et al. 2019). As such, small differences in +the early post-merger evolution of the fluid can result in the instability being seeded at different strengths. We do not +explicitly seed the one-armed spiral instability using fluid perturbations in this work and, as a result, simulations that +either run on different machines, use different grid resolutions, or use different numerical libraries result in different +strengths for the initial instability seed. In Fig. 3 we show the energy in the l = 2, m = 1 GW mode E2,1 +GW as a +function of time for a set of low resolution simulations used to produce the error bars of Fig. 1. The left panel of Fig. 3 +shows E2,1 +GW as extracted from our simulations and appears to show that the simulation employing a hadron-quark +EOS produces a larger energy in the l = 2, m = 1 GW mode. However, it is clear the energy at a time shortly after the +merger E2,1 +GW(tmerger + ϵ) (where ϵ is a small additive time) is larger for the hadron-quark simulation, suggesting that +the one-armed spiral instability was seeded at a larger amplitude in that case. In order to account for the different +levels at which the one-armed spiral instability is seeded in the immediate post-merger environment, we normalize +the quantities depicted in Fig. 1 at a time shortly after the merger tnorm = tmer + ϵ. We find that setting ϵ = 0.5 ms +results in all simulations in our work having roughly equal values of E2,1 +GW in the few ms immediately following merger. +Normalizing at a time shortly after merger ensures that all simulations have approximate parity in the level at which +the one-armed spiral instability is seeded and leads to the robust trend established in the right panel of Fig. 1, regardless +of grid resolution used. + diff --git a/U9E2T4oBgHgl3EQfDAYv/content/tmp_files/load_file.txt b/U9E2T4oBgHgl3EQfDAYv/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..6bc1b99f998c02f70478671877648f4265edf002 --- /dev/null +++ b/U9E2T4oBgHgl3EQfDAYv/content/tmp_files/load_file.txt @@ -0,0 +1,1162 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf,len=1161 +page_content='Draft version January 11, 2023 Typeset using LATEX twocolumn style in AASTeX631 Revealing Phase Transition in Dense Matter with Gravitational Wave Spectroscopy of Binary Neutron Star Mergers Pedro L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Espino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2 Aviral Prakash,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 3 David Radice,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='1,' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PA 16802 2Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' University of California,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Berkeley,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' CA 94720,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' USA 3Department of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PA 16802 4Department of Astronomy & Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The Pennsylvania State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='University Park,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PA 16802 5Dipartimento di Fisica,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Università di Pisa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Pontecorvo, 3 I-56127 Pisa, Italy 6INFN, Sezione di Pisa, Largo B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Pontecorvo, 3 I-56127 Pisa, Italy ABSTRACT We use numerical relativity simulations of binary neutron star mergers to show that high density deconfinement phase transitions (PTs) to quark matter can be probed using multimodal postmerger gravitational wave (GW) spectroscopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Hadron-quark PTs suppress the one-armed spiral instability in the remnant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' This is manifested in an anti-correlation between the energy carried in the l = 2, m = 1 GW mode and energy density gap which separates the two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Consequently, a single measurement of the signal-to-noise ratios of the l = 2, m = 1 and l = 2, m = 2 GW modes could constrain the energy density gap of the PT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Keywords: Neutron stars — Equation of state — Gravitational waves — Hydrodynamics — Instabilities 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' INTRODUCTION Binary neutron star (BNS) mergers produce some of the most extreme conditions in nature, compressing matter to several times the nuclear density and to tem- peratures of tens of MeV (Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' More extreme conditions are only found in the early Universe and in the interior of black holes (BHs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Multimessen- ger observations of binary neutron star (BNS) mergers can be used to probe the properties of matter in these conditions, providing a unique avenue to study the non- perturbative regime of QCD (Shibata 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Hinderer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Damour et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Sekiguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Hotokezaka et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Margalit & Metzger 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Most et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Coughlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' De et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice & Dai 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Dietrich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Breschi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Kashyap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Prakash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Presently, there are large uncertainties in the funda- mental physics of strongly-interacting matter at densi- ∗ Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Sloan fellow ties of a few times nuclear saturation (Capano et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Pang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Annala et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' It is not even clear what the relevant degrees of freedom are for the densities and temperatures reached in the core of remnant massive neutron stars (RMNS) of BNS merg- ers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' It is possible that matter remains composed of nu- cleons, together with leptons (electrons, positrons, and muons) and photons (Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Loffredo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The appearance of more exotic baryons, such as hyperons, is not excluded (Sekiguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Logoteta 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' It is also possible for a transition to the deconfined quark-gluon plasma phase to take place in BNS mergers (Most et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Prakash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The deter- mination of the state of matter formed in BNS mergers is one of the most pressing scientific objectives of mul- timessenger astronomy (Evans et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Lovato et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Previous work has shown that the presence of phase transitions to deconfined quarks can be revealed by a shift of the postmerger gravitational wave (GW) peak frequency f2 from the value expected for hadronic equa- tions of state (EOSs) (Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Weih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Blacker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Kedia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' However, such frequency shifts can be degenerate with deviations arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='03619v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='HE] 9 Jan 2023 2 from universal relations due to hadronic physics or other effects (Most et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Weih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Liebling et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Prakash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Tootle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' It has also been suggested that the presence of a phase transition could be inferred from a measure- ment of the threshold mass for prompt collapse of BNS systems (Bauswein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Perego et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Kashyap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In this Letter, we use 8 state- of-the-art numerical relativity simulations to show, for the first time, that the presence and strength of a QCD phase transition could be unambiguously determined through multimodal GW spectroscopy of RMNS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Such measurements will be possible with the next-generation of GW experiments like Cosmic Explorer (Reitze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019), Einstein Telescope (Punturo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2010), and NEMO (Ackley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' METHODS We consider binaries in quasi-circular orbits and ec- centric encounters on nearly parabolic orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Although BNS mergers with highly eccentric orbits are expected to be significantly more rare than those with quasi-circular inspirals, these events may still have appreciable rates of as high as 50 Gpc−3 yr−1 (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Paschalidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2015);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' we include results from both types of merg- ers to consider as wide a variety of scenarios as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Initial data for the quasi-circular binaries is created us- ing the conformal thin sandwich formalism (York 1999) and assuming a helical Killing vector and irrotational flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The resulting elliptic equations are solved us- ing the pseudo-spectral code LORENE (Gourgoulhon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Taniguchi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Taniguchi & Gourgoulhon 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Initial data for the eccentric encounters is con- structed by superimposing two isolated, boosted, neu- tron stars, following Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The initial separation of the stellar barycenters for parabolic en- counters is set to 100 km, which is sufficiently large so that the level of constraint violation in the initial data is comparable to that of the quasi-circular binaries (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We perform BNS merger simulations using the WhiskyTHC code (Radice & Rezzolla 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2014a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' WhiskyTHC makes use of the CTGamma space- time solver (Pollney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2011), which is a part of the Einstein Toolkit (Zlochower et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The adap- tive mesh refinement driver Carpet (Schnetter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2004) is used to generate the dynamical grid structure employed in the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' All simulations considered in the present work have been performed using at least two grid resolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Although there are quantitative differences in the GW waveforms computed at different resolutions, the qualitative features discussed here are robust across all simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Unless otherwise speci- fied, we discuss results from simulations using the fidu- cial grid resolution (with grid spacing ∆x ≃ 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='6 m in the finest refinement level).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The grid structure for the simulations is described in detail in Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2018a) and Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2016b) for the quasi-circular and ec- centric simulations, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' For a clear understanding of the role that high-density deconfinement phase transitions could play in the de- velopment of the one-armed spiral instability, we con- sider a total of 7 EOS models and run a total of 8 simulations with varying phase transition features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In particular, the size of the energy density gap which separates the hadronic and quark phases is a useful way to classify hybrid hadron-quark EOS models and provides a qualitative measure of the ‘strength’ of the phase transition (Alford & Han 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As such, we con- sider EOS models that cover several sizes of the energy density gap, ranging from non-existent (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', a purely hadronic EOS) to large, while maintaining consistency with current astrophysical constraints on the dense mat- ter EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We consider both phenomenological EOS mod- els (Paschalidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Alvarez-Castillo & Blaschke 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Alford & Sedrakian 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bozzola et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Es- pino & Paschalidis 2022) (in the form of piecewise poly- tropic approximations using the prescription of Read et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2009)) and microphysical, finite temperature EOS models (Bastian 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Bombaci & Logoteta 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Lo- goteta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Prakash et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We only con- sider equal-mass ratio binary configurations, with the total binary mass ranging from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='6 M⊙ − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='7 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The lack of π-rotational symmetry in BNS configurations with unequal-masses may be a suitable way of effec- tively seeding non-axisymmetric fluid instabilities that can take hold in the post-merger environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Neu- trino emission and reabsorption are not included for bi- naries in eccentric orbits, while all quasi-circular bina- ries include a neutrino treatment via the moment based M0 scheme (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2018a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' However, neutrinos are not expected to influence the dynamics on the time scales considered in our study (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2020, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Additionally, magnetic fields are not accounted for in any of our simulations, but these are also expected to be subdominant (Palenzuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find that, despite the diversity in binary properties and differences in the evolution, the effects presented in this work are robust.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' RESULTS The one-armed spiral instability is a non- axisymmetric mode in a rapidly rotating fluid which, when saturated, leads to the dominance of a single 3 0 10 20 30 40 100 101 102 103 104 t − tmerger (ms) E2,1 GW/E2,1 GW(tnorm) hadron (DD2F) hadron-quark (DD2F-SF5) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='6 −3 −2 −1 0 1 ∆e (1015g/cm3) log10(⟨E2,1 GW/E2,1 GW(tnorm)⟩)hadron−quark log10(⟨E2,1 GW/E2,1 GW(tnorm)⟩)hadron Quasi-circular Eccentric Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Left panel: Energy carried by GWs in the l = 2, m = 1 mode as a function of time, scaled by the value at a fixed normalization time tnorm = tmer + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The development of the one-armed spiral instability can be observed in the purely hadronic simulation as the energy in the l = 2, m = 1 GW mode continues to grow, but is suppressed in the hadron-quark simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Right panel: The same quantity depicted in the left panel, but time-averaged over a fixed time window and shown as a function of the energy density gap ∆e for each simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We normalize by the same quantity for the complementary hadronic EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We include approximate error bars, obtained using lower resolution simulations, to represent the uncertainty introduced by our numerical methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The black solid line represents a linear fit to the data from our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find that the normalized energy emitted by the l = 2, m = 1 GW mode decreases for simulations that employ EOS models with larger values of ∆e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' high-density mode in the fluid density which is dis- placed from the fluid barycenter (Pickett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Centrella et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Saijo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Ou & Tohline 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The one-armed spiral instability has been ob- served to develop commonly in BNS merger simulations that produce long-lived, massive post-merger remnants on timescales of O(10ms) (Paschalidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' East et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' East et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016) and in simulations of many other astro- physical systems including supernovae (Ott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Kuroda et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2014), white dwarfs (Kashyap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2015, 2017) and accretion disks (Kashyap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Wessel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Each fluid density mode that arises during the evolution of a massive NS remnant is associated with GW emission at characteristic frequencies stem- ming from its pattern speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As such, the development of the one-armed spiral instability in astrophysical sys- tems may be observed by considering multimodal GW spectroscopy (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' For the simulations considered in this work we extract multimodal GW in- formation within the Newman-Penrose formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We compute the coefficients of s = −2 spin-weighted spher- ical harmonic decompositions of the Newman-Penrose scalar Ψ4 which we label as Ψl,m 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The one-armed spiral instability can therefore be observed in the GW spec- trum extracted from our simulations as a growth in the power and amplitude of the l = 2, m = 1 GW mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Ψ2,1 4 ) and simultaneous decay of the dominant l = 2, m = 2 GW mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Ψ2,2 4 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Our simulations show that high-density deconfinement phase transitions act to suppress the one-armed spiral instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Depending on the features of the phase tran- sition, the one-armed spiral mode may either arise on a significantly longer timescale when compared to simula- tions which employ purely hadronic EOS models, or it may be suppressed altogether on the timescales probed by our simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' There are several potential mecha- nisms via which the instability may be suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' For example, it has been shown that the physical extent of the remnant plays an important role in the development of the instability, with larger remnants being more con- ducive to the development of the instability on shorter timescales (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Lehner et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Saijo & Yoshida 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Saijo 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The significant softening at high densities introduced by the phase transition re- sults in more compact post-merger remnants (relative to scenarios that consider only hadronic degrees of free- dom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As such, the more compact hybrid star remnants may see a weaker development of the one-armed spiral instability when compared to neutron star remnants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 we show the energy car- ried by the l = 2, m = 1 GW mode (scaled by the en- ergy emitted at a time shortly after the merger tnorm = tmer + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 ms) as a function of time for simulations em- ploying the DD2F (hadronic) and DD2F-SF5 (hybrid 4 102 103 104 f [Hz] 10−27 10−26 10−25 10−24 10−23 10−22 10−21 10−20 2√f|h+(f)|, √ S [Hz−1/2], 40Mpc hadron (BLh) l = 2, m = 2 l = 2, m = 1 aLIGO ET CE20 CE40 102 103 104 f [Hz] 10−27 10−26 10−25 10−24 10−23 10−22 10−21 10−20 2√f|h+(f)|, √ S [Hz−1/2], 40Mpc hadron − quark (BLQ) l = 2, m = 2 l = 2, m = 1 aLIGO ET CE20 CE40 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Multimodal GW amplitude spectrum computed for symmetric binaries of total mass M = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='6 M⊙ in an edge-on configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Also shown are the noise sensitivity curves for advanced LIGO (aLIGO), Einstein Telescope (ET), the 20 km postmerger-optimized configuration for the Cosmic Explorer (CE20) and the 40 km configuration for Cosmic Explorer (CE40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' A suppression in the amplitude spectral density (ASD) as a result of the deconfinement phase transition may be detectable with the third generation detectors and most cleanly with CE40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' hadron-quark) EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find that the energy carried in the l = 2, m = 1 mode of the GWs is significantly smaller in the simulation employing a hybrid hadron- quark EOS, indicating that the one-armed spiral in- stability is suppressed in scenarios with deconfinement phase transitions at densities relevant for BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We emphasize that in the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 we show- case results for a set of EOS models which are identical below the threshold for a phase transition, and as such the simulations have identical initial conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1, we show the time-averaged energy emitted by the l = 2, m = 1 GW mode ⟨E2,1 GW⟩ (again scaled by the energy emitted at a time shortly after the merger tnorm = tmer+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 ms) as a function of the energy density gap ∆e, where we define the energy density gap as the difference between the energy density e at the end of the hadronic phase and beginning of the quark phase for cold matter in β-equilibrium (Alford & Han 2016), ∆e ≡ equark,initial − ehadron,final, (1) where we assume units where the speed of light c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We identify the end and beginning of each phase by considering the change in the approximate adiabatic in- dex Γ = d log p/d log(ρ), where p is the fluid pressure of the cold, beta-equilibrium, barotropic EOS for each EOS model considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The region corresponding to the phase transition is always unambiguously marked by discontinuities in, or sudden changes in the slope of, the adiabatic index for the EOS models we consider.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' For the results depicted in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1, we time-average over a window of ∆t ≈ 40 ms after the merger except for cases that lead to a remnant collapse on shorter timescales (in such cases, we time-average until the collapse of the NS remnant).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Additionally, for each simulation, we normalize by a complementary sim- ulation that uses identical initial data but employs a hadronic EOS having the same low-density behavior be- low the phase transition threshold as the hybrid hadron- quark EOS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As such, we depict the point corresponding to all hadronic EOS simulations with a black square at ∆e = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Each simulation is time-averaged to the same extent as its complementary hadronic simulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find an anti-correlation between the energy carried in the l = 2, m = 1 GW mode and the size of the energy density gap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In other words, as the size of the energy density gap (and thereby the qualitative ‘strength’ of the phase transition) increases, GW emission in the l = 2, m = 1 mode decreases, which signifies that the one-armed spi- ral instability is further suppressed for EOS models with ‘stronger’ deconfinement phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We elabo- rate on the choice of quantities depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 in App.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' DISCUSSION The characteristic frequency associated with peak emission in the l = 2, m = 1 GW mode has half the value of that associated with the l = 2, m = 2 mode (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', f 2,1 peak = f 2,2 peak/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Observationally, a GW signal would contain information at all contributing frequen- cies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' However, the dominant GW emission associated with binary coalescence is always expected to be from the l = 2, m = 2 contribution, such that fpeak = f 2,2 peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 5 Therefore, a potential observational signature of the one-armed spiral instability is the growth in power of an incoming GW signal at a frequency that is half of the dominant frequency;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' if it develops in the post-merger environment, the one-armed spiral instability will con- tinuously power the emission of GWs at fpeak/2, while emission in the dominant fpeak decays in time (Bernuzzi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2 we show the post-merger GW amplitude spectrum density (ASD) for a symmetric, edge-on binary situated at a distance of 40 Mpc, which is consistent with the luminosity distance observed for GW170817 (Abbott et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2017b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The edge-on configuration is the most op- timal for the detection of an m = 1 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As expected, we see a relative suppression of power in the m = 1 mode (with respect to the complementing hadronic sim- ulation) with the onset of a deconfinement phase tran- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In this realistic configuration, coupled with the 40 km Cosmic Explorer detector (Reitze et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019), the appearance of quarks in the post-merger remnant results in a suppression of the postmerger signal-to-noise ratio (SNR) of the (l = 2, m = 1) mode by a factor of 2, from 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='14 in the hadronic case to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='08 in the hadron-quark case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The GW ASD peak of the l = 2, m = 1 mode (between 1-2 kHz) and the postmerger ASD peak of the l = 2, m = 2 mode (between 2-4 kHz), lie respectively in the most sensitive regions of the 40 km and the 20 km postmerger optimized Cosmic Explorer configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Our analysis recommends an increase in detector sen- sitivities in the high-frequency regimes (see also Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2022)) for best possible constraints on deconfine- ment phase transitions in BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In this letter we have highlighted, for the first time, that high-density deconfinement phase transitions act to suppress the one-armed spiral instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find an anti-correlation between the energy carried in the l = 2, m = 1 GW mode and the size of the energy den- sity gap which qualitatively separates the hadronic and quark phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Our findings reveal a deep connection between observable multimodal GW emission and the microphysical description of matter in the post-merger environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We expect the one-armed spiral instability to be detectable at distances of 40 Mpc using future gen- eration detectors (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' If evidence of a strong one-armed spiral mode can be inferred from GW observations of the post-BNS merger environment, our findings suggest that a strong high-density deconfine- ment phase transition at the densities relevant to BNS mergers would be disfavored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' On the other hand, if ev- idence for the one-armed spiral instability is not found for close-by BNS mergers, this could also point to the possibility of a deconfinement phase transition taking place at densities relevant to BNS mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We point out that other effects relevant in the post- merger environment - such as the presence of strong magnetic fields (Franci et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2013) and additional de- grees of freedom that can cause a sudden softening of the EOS - may affect the development of the one-armed spiral instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' However, the relevant timescales and extent to which the aforementioned phenomena can af- fect the development of non-axisymmetric instabilities or the GW spectrum remains uncertain (Radice et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2016a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Muhlberger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2014), and may not impact our conclusions (Palenzuela et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Zappa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The effects discussed in the present work arise on dynamical timescales ∼ O(10 ms), and may be the dominant mechanism for suppression of the one-armed spiral instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Additionally, although we find a trend in the decrease of energy carried by the l = 2, m = 1 GW mode for larger values of ∆e, additional studies will help establish a more robust trend and provide an understanding of the potential spread in the trend.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In particular, future lines of investigation will include: (1) considering the combined effects of the mass ratio and high-density phase transitions on the development of the one-armed spiral instability;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (2) considering the effects of accurate neutrino transport on high-density deconfinement phase transitions, as neutrinos may mod- ify the composition of matter and thereby potentially affect the onset of the phase transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' (3) employing EOS models at systematically increasing values of ∆e while holding the hadronic region of the EOS fixed, as a limitation of the present work is the assumption that the l = 2, m = 1 GW mode is perfectly known in the case of hadronic EOSs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' and (4) investigating the effects discussed in this work in scenarios with a crossover to quark matter, as our present work only considers EOS models with phase transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We leave such studies to future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PE acknowledges funding from the National Science Foundation under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PHY-2020275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' DR ac- knowledges funding from the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Department of En- ergy, Office of Science, Division of Nuclear Physics under Award Number(s) DE-SC0021177 and from the National Science Foundation under Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' PHY-2011725, PHY-2116686, and AST-2108467.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Simulations were per- formed on Bridges2 and Expanse (NSF XSEDE allo- cation TG-PHY160025).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' This research used resources of the National Energy Research Scientific Computing Center, a DOE Office of Science User Facility supported by the Office of Science of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='S.' metadata={'source': 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+page_content='org/abs/2210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='11491 Zhang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Yang, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Martynov, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Schmidt, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', & Miao, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='org/abs/2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='12144 Zlochower, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Brandt, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', Diener, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2022, The Einstein Toolkit, The "Berhard Riemann" release, ET_2022_05, Zenodo, doi: 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5281/zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='6588641 9 APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' GW PROBE OF THE ONE-ARMED SPIRAL INSTABILITY 10−1 100 101 10−10 10−9 10−8 10−7 10−6 10−5 10−4 10−3 t − tmerger (ms) E2,1 GW hadron (DD2F) hadron-quark (BBKF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5) 10−1 100 101 100 101 102 103 104 105 106 107 t − tmerger (ms) E2,1 GW/E2,1 GW(tnorm) hadron (DD2F) hadron-quark (BBKF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5) Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Left panel: Energy in the l = 2, m = 1 GW mode as a function of time for simulations employing a hadronic (DD2F) and hadron-quark (BBKF1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5) EOS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' the simulations use identical initial conditions and are run with a grid resolution of ∆x = 369.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='2 m in the finest grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' These results showcase that the one-armed spiral instability may be seeded at different levels in the postmerger environment for different simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Right panel: Same quantity as the left panel, but normalized to the value at a time shortly after merger, tnorm = tmerger + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Normalizing at this time accounts for the one-armed spiral instability being seeded at disparate levels across simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 we depict an example of GW quantities that exhibit the suppression of the one-armed spiral instability for simulations that employ hadron-quark EOSs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In particular, we calculate the GW energy carried in the l = 2, m = 1 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 we show E2,1 GW normalized to its value at a time shortly after the merger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' we depict normalized quantities because of the variable nature in which the one-armed spiral instability is seeded in the immediate post-merger environment in the context of numerical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Unless it is explicitly excited as a non-axisymmetric perturbation of a known amplitude (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=', as a fixed-amplitude perturbation in the rest mass density), the one-armed spiral instability arises numerically from error at the level of floating-point precision (Espino et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' As such, small differences in the early post-merger evolution of the fluid can result in the instability being seeded at different strengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We do not explicitly seed the one-armed spiral instability using fluid perturbations in this work and, as a result, simulations that either run on different machines, use different grid resolutions, or use different numerical libraries result in different strengths for the initial instability seed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 3 we show the energy in the l = 2, m = 1 GW mode E2,1 GW as a function of time for a set of low resolution simulations used to produce the error bars of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' The left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 3 shows E2,1 GW as extracted from our simulations and appears to show that the simulation employing a hadron-quark EOS produces a larger energy in the l = 2, m = 1 GW mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' However, it is clear the energy at a time shortly after the merger E2,1 GW(tmerger + ϵ) (where ϵ is a small additive time) is larger for the hadron-quark simulation, suggesting that the one-armed spiral instability was seeded at a larger amplitude in that case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' In order to account for the different levels at which the one-armed spiral instability is seeded in the immediate post-merger environment, we normalize the quantities depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1 at a time shortly after the merger tnorm = tmer + ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' We find that setting ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content='5 ms results in all simulations in our work having roughly equal values of E2,1 GW in the few ms immediately following merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' Normalizing at a time shortly after merger ensures that all simulations have approximate parity in the level at which the one-armed spiral instability is seeded and leads to the robust trend established in the right panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} +page_content=' 1, regardless of grid resolution used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9E2T4oBgHgl3EQfDAYv/content/2301.03619v1.pdf'} diff --git 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100644 index 0000000000000000000000000000000000000000..a6d275b9b6d571ab2ac5175cc4378d29e8b39b95 --- /dev/null +++ b/U9FLT4oBgHgl3EQfRi9C/content/tmp_files/2301.12037v1.pdf.txt @@ -0,0 +1,766 @@ +Possible depth-resolved reconstruction of +shear moduli in the cornea following collagen +crosslinking (CXL) with optical coherence +tomography and elastography +GABRIEL REGNAULT,1,* MITCHELL A. KIRBY,1 RUIKANG K. WANG1,2, TUENG +T. SHEN2,3, MATTHEW O’DONNELL1 AND IVAN PELIVANOV1 +1Department of Bioengineering, University of Washington, Seattle, Washington 98105, USA +2Department of Ophthalmology, University of Washington, Seattle, Washington 98104, USA +3School of Medicine, University of Washington, Seattle, Washington 98195, USA +gregnaul@uw.edu +Abstract: Collagen crosslinking of the cornea (CXL) is commonly employed to prevent or +treat keratoconus. Although the global change of stiffness before and after treatment can be +monitored with OCE, the depth dependence of this change is still unclear. Here we propose to +combine phase-decorrelation measurement applied to OCT structural images and acoustic +micro-tapping optical coherence elastography (AµT-OCE) to explore possible reconstruction +of stiffness with depth within crosslinked corneas in an ex vivo human cornea sample. The +analysis of experimental OCT images can help define the penetration depth of CXL into the +cornea, which varies from ∼100μm in the periphery to ∼150μm in the central area and +demonstrates a sharp transition between areas. This information was used in numerical +simulations of the mechanical guided wave propagation in the two-layer medium with different +elastic moduli corresponding to those for CXL and normal corneas. The simulation supports +our hypothesis that ‘averaged’ elastic moduli can be reconstructed from guided wave +propagation in the cornea with partial penetration of CXL, which may then be used to decouple +the moduli in each layer. However, additional studies are required to define and characterize +the reconstruction algorithm for the more general case of arbitrary CXL penetration depth and +its possible variation over the cornea. + +1. Introduction +At the interface between air and the inner eye, the cornea provides both protection and the +primary optical element focusing light onto the retina. It contains multiple layers, including the +epithelium and the stroma. The first acts as a barrier against the external environment and the +latter maintains its stiffness, transparency and focusing power [1,2]. The microstructure of the +stroma is composed of collagen fibrils, arranged in lamellae, lying within a protein rich, +hydrated proteoglycan mesh [3,4]. +Corneal diseases (such as keratoconus (KC)) and surgical complications from refractive +surgeries (such as LASIK) may deform the cornea (ectasia) and reduce vision. The prevalence +of KC in the general population is estimated to be 1.38 per 1000 [5], and nearly 1 million +refractive surgeries are performed each year in the USA. Despite their overall success, however, +suboptimal visual outcomes and post-refractive corneal decompensation cannot always be +predicted for an individual patient. Corneal collagen crosslinking (CXL) is a minimally +invasive procedure that can potentially slow the progression of corneal ectasia [6-9]. UV light +modifies the microstructure of cornea soaked in riboflavin and forms additional chemical bonds +between collagen fibers in the stroma [10]. Post-treatment corneas become stiffer and more +resistant to enzymatic digestion [11-13]. Although corneal topography (curvature map) and + +thickness map can be obtained prior to surgery, and needed refractive corrections can be +estimated, there is an unmet need to predict corneal decompensation from interventions such +as LASIK and CXL therapies. Unfortunately, surgical planning cannot be customized and +outcomes (e.g. post-surgery corneal ectasia risks) cannot be accurately predicted without +quantitatively mapping corneal elasticity. Thus, methods that can quantitatively reconstruct +corneal elastic moduli are in high demand. +Ocular response analyzer (ORA - Reichert Technologies) and Dynamic Scheimpflug Analyzer +(DSA - Corvis ST – Oculus Opitkgerate GmbH) are the state-of-the-art in clinical +measurements of corneal mechanics. They estimate stiffness as the pressure at inward +applanation divided by corneal displacement [14-16]. Over 100 papers are published annually +on the topic, and the Journal of Cataract and Refractive Surgery devoted an entire issue to the +topic [15]. However, measurements induce large corneal deformations that are often clinically +unacceptable, require a non-trivial IOP correction in simulations [17] and assume a simple +isotropic mechanical model leading to high variability with experimental conditions. Results +obtained with the Corvis ST on KC may be contradictory, and some even show no significant +change in corneal stiffness pre- and post-CXL surgery [18, 19]. In addition, the result is +averaged over the entire cornea with no spatial resolution, and the reconstruction is questionable +if corneal thickness varies. +There is no consensus in the literature on the elastic model for cornea and the stiffness range +even for healthy subjects. The most common model assumes an incompressible, isotropic, and +linear elastic solid, where a single parameter, the Young’s modulus 𝐸 (or equivalently the shear +modulus 𝜇 = 𝐸/3), defines elasticity. Destructive mechanical tests can determine 𝐸 ex vivo, +with reported values for human cornea (at low-strain) of 800 kPa to 4.7 MPa for tensile loading +[20-24], and 100 kPa to 3 MPa for inflation loading [25-26]. Note that the destructive nature of +mechanical tests precludes their clinical translation. +Optical coherence elastography (OCE) is a promising tool to probe corneal biomechanics. +Dynamic OCE can potentially deliver a non-contact and non-invasive measurement in a clinical +environment [27-31] as it excites mechanical waves in the cornea (for example, using an air- +puff or acoustic micro-tapping (AT [27]) and tracks them using phase-sensitive optical +coherence tomography (OCT). Analyzing the shear wave propagation speed or dispersion leads +to an estimate of the shear modulus. OCE studies reported corneal shear moduli in the range of +1.8 – 52.3 kPa [32, 33], in close agreement with values obtained from torsional and rheometry +testing of ex vivo cornea (2.5 – 47.3 kPa) [34-36]. However, all shear-based methods produce +moduli differing by 1-2 orders of magnitude from those reported by tensile and inflation tests. +For years the Rayleigh wave group velocity was used to reconstruct corneal elasticity [31, 38, +27, 28-32, 39, 40], resulting in a 2 orders of magnitude mismatch in Young’s modulus +compared to tensile and inflation tests. We clearly showed that this approach is inappropriate +[41] since the cornea is a layer bounded by air at the top and aqueous humor on the bottom. +Thus, guided waves propagate with strong geometric dispersion [41, 42]. +Recently, we hypothesized that corneal anisotropy is the primary cause of these discrepancies +[42]. Corneal microstructure supports this hypothesis. The stroma contains collagen lamellae +running in-plane across its width. Lamellae make up approximately 90% of tissue thickness +and account for the majority of the cornea’s mechanical structure. They are stacked vertically +in approximately 200-500 separate planes [43, 44], suggesting an anisotropic mechanical +behavior with very different responses to in-plane versus out-of-plane loads. We introduced a +model of a nearly-incompressible transverse isotropic (NITI) medium [42] to explain the +discrepancy. It is defined by two (in-, 𝜇, and out-of-plane, 𝐺 ) shear moduli, decoupling +tensile/inflation properties from shear responses. Based on this model, an algorithm utilizing + +guided mechanical waves in a bounded NITI medium to reconstruct both moduli from AT- +OCE measurements has been developed. +In [45] we showed that both in- and out-of-plane post-CXL corneal shear moduli increase +compared to their pre-surgery values. It is interesting that at the end of the CXL procedure +corneal Young’s modulus changes on average by about 2 times whereas the out-of-plane shear +modulus 𝐺 changes by almost 4 times. Indeed, the CXL procedure is intended to better +crosslink corneal lamellae, which it does. However, deformational properties are defined by the +Young’s modulus, which does not change nearly as much and has implications for potential +refractive changes. +Note, however, that the effects of crosslinking are always homogeneous in depth as the +penetration of riboflavin is more pronounced in the anterior region of the cornea, where the +solution is applied during soaking [46]. Riboflavin penetration is purposely limited to minimize +damage to the endothelium and ensure the biological integrity of the cornea [47]. +As noted above, mechanical waves generated in the cornea are usually not bulk shear or surface +Rayleigh waves; they are guided waves. Reconstructing corneal moduli using local estimates +of the propagation speed at different depths leads to large errors. In fact, guided waves occupy +the entire depth of the cornea and, therefore, average information over depth. Thus, +reconstructing the depth dependence of corneal moduli is practically impossible without a good +estimate of CXL penetration within the cornea. +Blackburn et al. [48] have recently introduced a novel metric to track CXL penetration within +the cornea using time-resolved OCT. They demonstrated that the phase decorrelation decay rate +of the complex OCT signal is reduced in CXL areas and can be used to distinguish treated and +untreated areas after a CXL procedure. +In this letter, we combine the methods described in [48] with AT-OCE measurements to +explore possible reconstruction of both in- and out-of-plane corneal elastic moduli over depth. +In addition, we use numerical simulations of guided wave propagation in a two-layer medium +consisting of a CXL cornea layer on the top of the untreated cornea to demonstrate how guided- +wave characteristics can be used to reconstruct corneal elasticity in both parts of the medium. +Although our results show promise in monitoring the effects of CXL using a two-layer model, +a more detailed study on model-based reconstruction of the in-depth distribution of elastic +properties in CXL cornea is required to generalize our findings. + +2. Method + +2.1 Cornea preparation + +One corneal-scleral ring, stored in Optisol (Chiron Ophtalmics) was obtained from CorneaGen. +This sample from a 26 years-old donor was stored for less than 30 days. The corneal-scleral +button is mounted on an artificial anterior chamber (Barron, CorzaMedical; see figure 1), +connected through the inlet port to an elevated bath filled with balanced saline solution (BSS) +to apply a controlled pressure on the anterior segment of the cornea. The outlet port remained +closed to allow the pressure to settle at 15 mmHg within the chamber, corresponding to human +physiological conditions [49]. Crosslinking was performed following the Dresden protocol [6]. +First, the epithelial membrane was removed from the sample. Then, the cornea was soaked in +riboflavin for 30 min, by applying a 50µL drop of 0.1% riboflavin in 20% dextran solution +every two minutes. The cornea was then exposed to a 3mW/cm2 of 370 nm ultra-violet (UV) +light for 30 minutes, while a drop was re-applied every 5 minutes. + + + +Figure 1. Picture of the experimental set up during UV-CXL. a) Acoustic micro-tapping transducer. b) +Artificial anterior chamber with c) the inlet port connected to the elevated bath and outlet port closed for +controlling IOP. + +2.2 AµT-OCE imaging system + +A spectral domain OCT system with a 46.5 kHz frame rate was used to track wave propagation +and structural changes within the cornea. As described in previous studies [27,31,42,45], a +cylindrically focused air-coupled transducer, operating at 1MHz, generates a spatio-temporal +sharp displacement at the surface of the cornea, using reflection based radiation force. Because +of the cylindrical geometry of the transducer, the push is line-shaped and generates planar +guided waves within the bounded tissue. The OCT system operated in M-B mode. A single +push is triggered by the system while 512 consecutive A-scans are taken at a fixed location (M- +scan). The M-scan sequence and push excitation are repeated for 256 locations, creating a three- +dimensional volume with 256 𝑥-samples, 1024 𝑧-samples and 512 𝑡-samples (see figure 2.a)), +with an effective imaging range of 6mm × 1.2mm × 11ms. The particle axial vibration is +obtained from the optical phase difference between two consecutive A-line scans at each +location [51]. The spatio-temporal (𝑥-𝑡) surface signature of the guided wave is computed from +an exponential weighted-average of the vertical velocity over the first 180µm of the anterior +part of the cornea. As shown in Figure 2.b), the guided wave only propagates during the first +4ms of the scans. This first part will be used to determine the stiffness of the material by fitting +the measured dispersion curve in the frequency-wavenumber domain (𝑓-𝑘), obtained from the +2D Fourier spectrum (see Figure 2.c)). This procedure is detailed in section 2.3. On the other +hand, data from the last 7ms are used to study the structural changes with phase decorrelation +(see section 2.4). + + +a) +D +Figure 2. Stiffness reconstruction and depth penetration measurements using spectral domain time- +resolved OCT. a) 3D (𝑥, 𝑧 and 𝑡), wave field after AµT excitation. The top surface wavefield of the initial +time sequence (black dotted region) is used for stiffness reconstruction and data at the end of the sequence +is used for the phase decorrelation measurement. b) 𝑥-𝑡-plot showing the top surface signature of the +guided mode. c) 𝑓-𝑘-plot obtained by 2D-FFT of the 𝑥-𝑡-plot showing the dispersion signature of the first +anti-symmetric mode 𝐴0. The red curve indicates the best fit obtained with the NITI model. d) Structural +OCT image obtained from averaging the last 7ms of the raw OCT signal. e) Phase decorrelation function +𝑔(𝜏) at the location indicated by the red square on d). +2.3 NITI model and fitting + +The microstructure of the cornea implies a transverse isotropy in the direction normal to the +corneal interface [52]. Like most biological tissue, the cornea is nearly-incompressible. +Consequently, an exact description of its behavior under external loading requires a NITI (for +Nearly Incompressible Transversely Isotropic) model [42]. In Voigt notation, the Hook’s law +of stress and strain for a TI material takes the form: + +a) +OCT +7ms +4ms +AuT-OCE +Verticalvelocity,Vz(a.u.) +PowerSpectrum(dB） +b) +c +6 +0.2 +5 +0.1 +(1/mm) +4 +-5 +(uu) +3 +0 +X +2 +-10 +-0.1 +1 +-0.2 +0 +0 +-15 +0 +2 +4 +0 +1 +2 +3 +4 +t (ms) +f (kHz) +...Linear fit +0.999 +0.998 +0 +0.1 +0.2 +0.3[ + + + + + +𝜎𝑥𝑥 +𝜎𝑦𝑦 +𝜎𝑧𝑧 +𝜏𝑦𝑧 +𝜏𝑥𝑧 +𝜏𝑥𝑦] + + + + + += +[ + + + + + 𝜆 + 2𝜇 +𝜆 +𝜆 +𝜆 +𝜆 + 2𝜇 +𝜆 +𝜆 +𝜆 +𝜆 + 𝛿 +𝐺 +𝐺 +𝜇 ] + + + + + + +[ + + + + + +ϵ𝑥𝑥 +ϵ𝑦𝑦 +ϵ𝑧𝑧 +γ𝑦𝑧 +γ𝑥𝑧 +γ𝑥𝑦] + + + + + + +where 𝜎𝑖𝑗 denotes engineering stress, 𝜖𝑖𝑗 denotes engineering strain, 𝜏𝑖𝑗 denotes shear stress, +𝛾𝑖𝑗 = 2 𝜖𝑖𝑗 denotes shear strains, the subscript 𝑥, 𝑦 and 𝑧 refer to the Cartesian axes and 𝐺, 𝜇, 𝜆 +and 𝛿 are four independent elastic constants. In previous studies [52,53], we have demonstrated +that 𝛿, which accounts for tissue tensile anisotropy, cannot be determined from guided wave +propagation but that the in-plane Young’s modulus can be approximated as 𝐸𝑇 = 𝐸 ≅ 3𝜇 +assuming tensile isotropy (𝛿 = 0). Thus, among the four elastic constants, only 𝐺 and 𝜇, +respectively the out-of-plane and in-plane shear moduli, are needed to predict deformations of +a nearly incompressible tissue under mechanical loading. Accounting for the appropriate +boundary conditions (water below and air above) and the finite thickness of the medium, the +dispersion relationship of guided waves can be determined directly from 𝐺 and 𝜇. Note that in +cornea, only the first anti-symmetric mode, referred to as 𝐴0, propagates in the range of +frequencies that can be recorded in elastography (typically < 5kHz). + +The experimental 𝑓-𝑘 spectrum (see Figure 2.c)) is obtained by computing the 2D-FFT of the +𝑥-𝑡-plot. Because the boundary conditions of the cornea on the anterior chamber highly differ +with physiological ones (the cornea is clamped here), a structural resonance of the cornea is +observed in the low-frequency regime (below 1kHz). To remove this artifact, we applied a +temporal super-Gaussian filter (∼ 0.5ms FWHM) that follows the maximum of vibration +velocity prior to computing the Fourier transform. The shear moduli 𝐺 and 𝜇 are obtained from +fitting the measured 𝑓-𝑘-spectrum with the analytical dispersion relationship of the 𝐴0 mode. +To ensure reliable fitting, we computed a goodness of fit metric 𝜙 = +∑ 𝜒𝑓𝑖𝑡(𝑓) +𝑓 +∑ 𝜒𝑚𝑎𝑥(𝑓) +𝑓 +, where 𝜒𝑓𝑖𝑡(𝑓) +corresponds to the energy of the 2D spectrum covered by the best analytical solution at a given +frequency 𝑓 and 𝜒𝑚𝑎𝑥(𝑓) is the unconstrained maximal energy of the spectrum at frequency 𝑓. +Based on recent results (see supplemental material in [12]), reliable fitting in human ex vivo +corneas are associated with values of 𝜙 > 0.9. More details about the complete fitting +procedure can be found in [45]. An example of a 2D-spectrum and the fitted 𝐴0 mode obtained +with this procedure are shown in Figure 2.c). + +2.4 Phase Decorrelation OCT (PhD-OCT) + +Blackburn et al. [48] have recently introduced a novel metric to track CXL penetration within +the cornea using time-resolved OCT. They demonstrated that the phase decorrelation decay rate +of the complex OCT signal is reduced in CXL areas and can be used to distinguish treated and +untreated areas of the cornea after a procedure. The complex-valued autocorrelation of the +signal 𝑔(𝜏) is computed over 15 consecutive sample at 46,500 Hz, for six consecutive pixels +within a given A-line: + +𝑔(𝜏) = ⟨ +⟨𝐸(𝑡) 𝐸∗(𝑡 + 𝜏)⟩𝑝𝑖𝑥𝑒𝑙𝑠 +√⟨𝐸(𝑡) 𝐸∗(𝑡)⟩𝑝𝑖𝑥𝑒𝑙𝑠 × √⟨𝐸(𝑡 + 𝜏) 𝐸∗(𝑡 + 𝜏)⟩𝑝𝑖𝑥𝑒𝑙𝑠 +⟩, + +which is expected to follow an exponential decay [54]: + +𝑔(𝜏) = 𝑒−𝛤. 𝜏 ≈ 1 − 𝛤. 𝜏, + + +where Γ is the decorrelation coefficient that is inversely proportional to the Brownian diffusion +coefficient [54], meaning that the more coherent the material, the smaller the decorrelation +coefficient. This procedure is performed starting at the 𝑛, 𝑛 + 1, 𝑛 + 2, … A-line, where 𝑛 is +the first time-sample used for the phase-decorrelation ( 𝑡(𝑛) = 4ms). The decorrelation +coefficient Γ is then computed using the averaged 𝑔(𝜏) over the number of starting points by +fitting with a first order polynomial (see Figure 2.e)): < 𝑔(𝜏) > = 𝑏 − Γ. 𝜏, where <> denotes +the average over the number of starting points. In crosslinked parts of the cornea (anterior), the +tissue stiffens, which implies that Γ should be smaller than in the untreated region (posterior). +For post-processing, we rejected all fits were b < 0.95, corresponding in general to peripheral +regions where the SNR (Signal to Noise Ratio) is too low. + +3. Results + +3.1 Corneal stiffness before and after CXL + +A single scan of AµT-OCE, taking approximatively 3s to acquire and save, was taken before +and after CXL. The signature of the vertically polarized velocity was used to compute the 𝑓-𝑘- +spectrum, which was fitted using the procedure detailed in section 2.3. The results for the +reconstructed stiffness before and after CXL are shown in Table 1. As generally observed in +the literature [55,56], the thickness is slightly reduced following crosslinking. The measured +stiffnesses increased for 𝐺 from 61.8kPa to 132.2kPa and for 𝜇 from 5.3MPa to 8.1MPa. Such +increase in both stiffness moduli following CXL is in good agreement with previously reported +results for ex vivo human corneas [45]. The goodness of fit metric is 𝜙 = 0.96 before CXL and +𝜙 = 0.95 after CXL, which confirms the robustness of the fit and the validity of the results. + +Table 1. Measured stiffnesses from the experimental dataset. + +Thickness, +ℎ (µm) +Out-of-plane shear +modulus, G (kPa) +In-plane shear +modulus, 𝜇 (MPa) +Goodness of +fit, 𝜙 +Before CXL +575 +61.8 +5.3 +0.96 +After CXL +520 +132.2 +8.1 +0.95 + +3.2 Depth dependance of CXL + +The results for phase decorrelation are shown in Figure 3. For the untreated case (Figures 3.a, +c), both the OCT intensity and Γ are constant in directions normal to the interface. In the center, +the higher scattering of laser light slightly influences the results, suggesting that Γ depends on +the signal to noise ratio of the system. The CXL area can be seen in both structural OCT and Γ +maps after CXL (Figures 3.b, d)). As shown in Figure 3.e), the drop of OCT intensity is +associated with an increase of Γ, which demonstrates that both metrics can be used to identify +the CXL area. The thickness of the CXL layer varies from ∼100μm in the periphery to ∼150μm +in the central area of the cornea, which agrees with recent observation of lateral changes in the +effect of crosslinking in corneas [46]. + + + +Figure 3. Short time decorrelation before and after CXL. Structural OCT images obtained from the last +7ms of the OCT scan for a) before CXL and b) after CXL. Maps of decorrelation coefficient 𝛤 for c) +before and d) after CXL. e) Profile of OCT intensity and 𝛤 along the red dotted line shown in b) and d) +for the CXL cornea. + +3.3 Numerical simulations + +We designed a finite element (FEM) simulation to study the layering effect of CXL (section +3.2) on the reconstructed stiffness using the NITI model. The geometry is shown in Figure 4.a). +Boundary conditions of the cornea were replicated so that the material is bounded above by air +and below by water. The speed of sound in all layers (material and water) was fixed to avoid +reflection of compressional waves at the different boundaries. It also improved the absorption +of waves at the absorbing boundaries and, thus, avoided divergence of the simulations. We used +transient excitation mimicking AµT experiments to generate broadband elastic waves within +the material. More details about the simulations can be found in [42,50]. Based on the phase- +decorrelation measurements (see Section 3.2), we assumed that after CXL, two layers with +distinct stiffnesses but with identical thicknesses (each 0.25mm) were formed within the +cornea, the top layer being stiffer than the bottom one. + +We fixed 𝐺1 = 200kPa, 𝜇1 = 10MPa, 𝐺2 = 50kPa and 𝜇2 = 4MPa, and a total thickness ℎ = +0.5mm. Similar to OCE experiments, we used the top surface vertical signature of the guided +wave (Figure 4.b) to compute its 2D 𝑓-k spectrum (Figure 4.d). We also performed simulations +in a single-layer medium with moduli corresponding to that averaged over the two-layer +structure, i.e. 𝐺1 = 125kPa, 𝜇1 = 7MPa, 𝐺2 = 125kPa and 𝜇2 = 7MPa. The computed 𝑥-𝑡 +plot (Figure 4.c) for the homogeneous layer with averaged parameters is nearly identical to that +for the two-layer medium. Furthermore, an analytically calculated dispersion curve for the case +of averaged parameters fits well the 2D spectrum computed for the two-layer case with 𝜙 = +0.961 (red line in Figure 4.d). Thus, the simulation supports our hypothesis that ‘averaged’ +moduli can be reconstructed from guided wave propagation in the cornea with partial +penetration of CXL. These preliminary results show that defining the CXL penetration depth +from structural OCT images may help decouple elastic moduli in each cornea layer. However, +additional studies are required to define and characterize the reconstruction algorithm for the +more general case of arbitrary CXL penetration depth and its possible variation over the cornea. + + + +0.01 +0.05 +a +intensity +0.04 +10-2 +rate +0.005 +10°2 +0.03 +00 +10-3 +decorrelation +100 +0.01 +0.02 +d +0.01 +0.005 +10~2 +0 +200 +400 +600 +Distance (μm) +Figure 4. Finite element simulations to study the effects of a layered structure for a CXL cornea. a) +Geometry of the two layered material used in simulations, bounded above by air and below by water. b) +Top surface spatio-temporal signature (𝑥-𝑡-plot) of the guided wave for the two-layer case with 𝐺1 = +200kPa, 𝜇1 = 10MPa, 𝐺2 = 50kPa, 𝜇2 = 4MPa and a total thickness ℎ = 0.5mm. c) 𝑥-𝑡 plot of the +guided wave for one layer averaged over the depth parameters, i.e. 𝐺 = 125kPa, 𝜇 = 7MPa and +thickness ℎ = 0.5mm. d) 2D Fourier spectrum of the wave studied in b), showing the main propagating +𝐴0 mode. An analytical dispersion curve (red line) computed for the case c) is plotted on the top of the +2D spectrum. + +4. Discussion and Conclusions +In this study we combined structural OCT imaging with dynamic OCE to assess the penetration +depth of CXL treatment in the cornea. Analyzing brightness of structural OCT images and +image decorrelation between consecutive B-scans, we conclude that there is a sharp transition +between CXL and untreated cornea. This finding allowed us to suggest a model of a two-part +medium for the treated cornea, where cornea elastic properties remain constant in each part. +This finding can be confirmed with destructive measurements of both layers, but it is outside +the scope of the current study. + +Both experimental results and numerical simulations show that guided waves deliver +‘averaged’ or ‘effective’ elastic properties of the cornea. 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Determination of the modulus of elasticity of the +human cornea. Journal of refractive surgery, 23(8), 808-818 (2007). + +27. Ambroziński, Ł., Song, S., Yoon, S. J., Pelivanov, I., Li, D., Gao, L., ... & O’Donnell, M. +Acoustic micro-tapping for non-contact 4D imaging of tissue elasticity. Scientific +reports, 6(1), 38967 (2016). + +28. Song, S., Huang, Z., Nguyen, T. M., Wong, E. Y., Arnal, B., O’Donnell, M., & Wang, R. +K. Shear modulus imaging by direct visualization of propagating shear waves with +phase-sensitive optical coherence tomography. Journal of biomedical optics, 18(12), +121509-121509 (2013). + +29. Han, Z., Aglyamov, S. R., Li, J., Singh, M., Wang, S., Vantipalli, S., ... & Larin, K. V. +Quantitative assessment of corneal viscoelasticity using optical coherence elastography +and a modified Rayleigh–Lamb equation. Journal of biomedical optics, 20(2), 020501- +020501 (2015). + +30. Larin, K. V., & Sampson, D. D. Optical coherence elastography–OCT at work in tissue +biomechanics. Biomedical optics express, 8(2), 1172-1202 (2017). + +31. Kirby, M. A., Pelivanov, I., Song, S., Ambrozinski, Ł., Yoon, S. J., Gao, L., ... & +O’Donnell, M. Optical coherence elastography in ophthalmology. Journal of biomedical +optics, 22(12), 121720-121720 (2017). + +32. Han, Z., Li, J., Singh, M., Wu, C., Liu, C. H., Raghunathan, R., ... & Larin, K. V. Optical +coherence elastography assessment of corneal viscoelasticity with a modified Rayleigh- +Lamb wave model. Journal of the mechanical behavior of biomedical materials, 66, 87- +94 (2017). + +33. Singh, M., Li, J., Han, Z., Raghunathan, R., Nair, A., Wu, C., ... & Larin, K. V. +Assessing the effects of riboflavin/UV-A crosslinking on porcine corneal mechanical +anisotropy with optical coherence elastography. Biomedical Optics Express, 8(1), 349- +366 (2017). + +34. Petsche, S. J., Chernyak, D., Martiz, J., Levenston, M. E., & Pinsky, P. M. Depth- +dependent transverse shear properties of the human corneal stroma. Investigative +ophthalmology & visual science, 53(2), 873-880 (2012). + +35. Hatami-Marbini, H. Viscoelastic shear properties of the corneal stroma. Journal of +biomechanics, 47(3), 723-728 (2014). + +36. Sloan, S. R., Khalifa, Y. M., & Buckley, M. R. The location-and depth-dependent +mechanical response of the human cornea under shear loading. Investigative +ophthalmology & visual science, 55(12), 7919-7924 (2014). + +37. Kirby, M. A., Pelivanov, I., Song, S., Ambrozinski, Ł., Yoon, S. J., Gao, L., ... & +O’Donnell, M. Optical coherence elastography in ophthalmology. Journal of biomedical + +optics, 22(12), 121720-121720 (2017). + +38. Han, Z., Li, J., Singh, M., Aglyamov, S. R., Wu, C., Liu, C. H., & Larin, K. V. Analysis +of the effects of curvature and thickness on elastic wave velocity in cornea-like structures +by finite element modeling and optical coherence elastography. Applied physics letters, +106(23), 233702 (2015). + +39. Li, C., Guan, G., Cheng, X., Huang, Z., & Wang, R. K. Quantitative elastography +provided by surface acoustic waves measured by phase-sensitive optical coherence +tomography. Optics letters, 37(4), 722-724 (2012). + +40. Zhu, J., Qi, L., Miao, Y., Ma, T., Dai, C., Qu, Y., ... & Chen, Z. 3D mapping of elastic +modulus using shear wave optical micro-elastography. Scientific reports, 6(1), 35499 +(2016). + +41. Pelivanov, I., Gao, L., Pitre, J., Kirby, M. A., Song, S., Li, D., ... & O’Donnell, M. Does +group velocity always reflect elastic modulus in shear wave elastography?. Journal of +biomedical optics, 24(7), 076003-076003 (2019). + +42. Pitre Jr, J. J., Kirby, M. A., Li, D. S., Shen, T. T., Wang, R. K., O’Donnell, M., & +Pelivanov, I. Nearly-incompressible transverse isotropy (NITI) of cornea elasticity: +model and experiments with acoustic micro-tapping OCE. Scientific reports, 10(1), +12983 (2020). + +43. 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Experimental Eye +Research, 219, 109064 (2022). + +47. Wollensak, G., Spoerl, E., & Seiler, T. Riboflavin/ultraviolet-A–induced collagen +crosslinking for the treatment of keratoconus. American journal of ophthalmology, +135(5), 620-627 (2003). + +48. Blackburn, B. J., Gu, S., Ford, M. R., de Stefano, V., Jenkins, M. W., Dupps, W. J., & +Rollins, A. M. Noninvasive assessment of corneal crosslinking with phase-decorrelation + +optical coherence tomography. Investigative ophthalmology & visual science, 60(1), 41- +51 (2019). + +49. Shiose, Y. Intraocular pressure: new perspectives. Survey of ophthalmology, 34(6), 413- +435 (1990). + +50. Regnault, G., Kirby, M. A., Kuriakose, M., Shen, T., Wang, R. K., O’Donnell, M., & +Pelivanov, I. Spatial resolution in optical coherence elastography of bounded media. +Biomedical Optics Express, 13(9), 4851-4869 (2022). + +51. Wang, R. K., Kirkpatrick, S., & Hinds, M. Phase-sensitive optical coherence +elastography for mapping tissue microstrains in real time. Applied Physics Letters, +90(16), 164105 (2007). + +52. Kirby, M. A., Pitre Jr, J. J., Liou, H. C., Li, D. S., Wang, R. K., Pelivanov, I., ... & Shen, +T. T. Delineating corneal elastic anisotropy in a porcine model using noncontact OCT +elastography and ex vivo mechanical tests. Ophthalmology Science, 1(4), 100058 (2021). + +53. Kirby, M. A., Tang, P., Liou, H. C., Kuriakose, M., Pitre Jr, J. J., Pham, T. N., ... & +Pelivanov, I. Probing elastic anisotropy of human skin in vivo with light using non- +contact acoustic micro-tapping OCE and polarization sensitive OCT. Scientific Reports, +12(1), 3963 (2022). + +54. Chu, B. Laser light scattering. Annual review of physical chemistry, 21(1), 145-174 +(1970). + +55. Greenstein, S. A., Shah, V. P., Fry, K. L., & Hersh, P. S. Corneal thickness changes after +corneal collagen crosslinking for keratoconus and corneal ectasia: one-year results. +Journal of Cataract & Refractive Surgery, 37(4), 691-700 (2011). + +56. Barbosa, M. M. C., Barbosa Jr, J. B., Hirai, F. E., & Hofling-Lima, A. L. Effect of cross- +linking on corneal thickness in patients with corneal edema. Cornea, 29(6), 613-617 +(2010). + diff --git a/U9FLT4oBgHgl3EQfRi9C/content/tmp_files/load_file.txt b/U9FLT4oBgHgl3EQfRi9C/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b0547fc68b40f916b62de759280cc559e6295ae1 --- /dev/null +++ b/U9FLT4oBgHgl3EQfRi9C/content/tmp_files/load_file.txt @@ -0,0 +1,842 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf,len=841 +page_content='Possible depth-resolved reconstruction of shear moduli in the cornea following collagen crosslinking (CXL) with optical coherence tomography and elastography GABRIEL REGNAULT,1,* MITCHELL A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' KIRBY,1 RUIKANG K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' WANG1,2, TUENG T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' SHEN2,3, MATTHEW O’DONNELL1 AND IVAN PELIVANOV1 1Department of Bioengineering, University of Washington, Seattle, Washington 98105, USA 2Department of Ophthalmology, University of Washington, Seattle, Washington 98104, USA 3School of Medicine, University of Washington, Seattle, Washington 98195, USA gregnaul@uw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='edu Abstract: Collagen crosslinking of the cornea (CXL) is commonly employed to prevent or treat keratoconus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Although the global change of stiffness before and after treatment can be monitored with OCE, the depth dependence of this change is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Here we propose to combine phase-decorrelation measurement applied to OCT structural images and acoustic micro-tapping optical coherence elastography (AµT-OCE) to explore possible reconstruction of stiffness with depth within crosslinked corneas in an ex vivo human cornea sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The analysis of experimental OCT images can help define the penetration depth of CXL into the cornea, which varies from ∼100μm in the periphery to ∼150μm in the central area and demonstrates a sharp transition between areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This information was used in numerical simulations of the mechanical guided wave propagation in the two-layer medium with different elastic moduli corresponding to those for CXL and normal corneas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The simulation supports our hypothesis that ‘averaged’ elastic moduli can be reconstructed from guided wave propagation in the cornea with partial penetration of CXL, which may then be used to decouple the moduli in each layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' However, additional studies are required to define and characterize the reconstruction algorithm for the more general case of arbitrary CXL penetration depth and its possible variation over the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Introduction At the interface between air and the inner eye, the cornea provides both protection and the primary optical element focusing light onto the retina.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' It contains multiple layers, including the epithelium and the stroma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The first acts as a barrier against the external environment and the latter maintains its stiffness, transparency and focusing power [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The microstructure of the stroma is composed of collagen fibrils, arranged in lamellae, lying within a protein rich, hydrated proteoglycan mesh [3,4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Corneal diseases (such as keratoconus (KC)) and surgical complications from refractive surgeries (such as LASIK) may deform the cornea (ectasia) and reduce vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The prevalence of KC in the general population is estimated to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='38 per 1000 [5], and nearly 1 million refractive surgeries are performed each year in the USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Despite their overall success, however, suboptimal visual outcomes and post-refractive corneal decompensation cannot always be predicted for an individual patient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Corneal collagen crosslinking (CXL) is a minimally invasive procedure that can potentially slow the progression of corneal ectasia [6-9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' UV light modifies the microstructure of cornea soaked in riboflavin and forms additional chemical bonds between collagen fibers in the stroma [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Post-treatment corneas become stiffer and more resistant to enzymatic digestion [11-13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Although corneal topography (curvature map) and thickness map can be obtained prior to surgery, and needed refractive corrections can be estimated, there is an unmet need to predict corneal decompensation from interventions such as LASIK and CXL therapies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Unfortunately, surgical planning cannot be customized and outcomes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' post-surgery corneal ectasia risks) cannot be accurately predicted without quantitatively mapping corneal elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thus, methods that can quantitatively reconstruct corneal elastic moduli are in high demand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Ocular response analyzer (ORA - Reichert Technologies) and Dynamic Scheimpflug Analyzer (DSA - Corvis ST – Oculus Opitkgerate GmbH) are the state-of-the-art in clinical measurements of corneal mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' They estimate stiffness as the pressure at inward applanation divided by corneal displacement [14-16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Over 100 papers are published annually on the topic, and the Journal of Cataract and Refractive Surgery devoted an entire issue to the topic [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' However, measurements induce large corneal deformations that are often clinically unacceptable, require a non-trivial IOP correction in simulations [17] and assume a simple isotropic mechanical model leading to high variability with experimental conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Results obtained with the Corvis ST on KC may be contradictory, and some even show no significant change in corneal stiffness pre- and post-CXL surgery [18, 19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In addition, the result is averaged over the entire cornea with no spatial resolution, and the reconstruction is questionable if corneal thickness varies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' There is no consensus in the literature on the elastic model for cornea and the stiffness range even for healthy subjects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The most common model assumes an incompressible, isotropic, and linear elastic solid, where a single parameter, the Young’s modulus 𝐸 (or equivalently the shear modulus 𝜇 = 𝐸/3), defines elasticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Destructive mechanical tests can determine 𝐸 ex vivo, with reported values for human cornea (at low-strain) of 800 kPa to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='7 MPa for tensile loading [20-24], and 100 kPa to 3 MPa for inflation loading [25-26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Note that the destructive nature of mechanical tests precludes their clinical translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Optical coherence elastography (OCE) is a promising tool to probe corneal biomechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Dynamic OCE can potentially deliver a non-contact and non-invasive measurement in a clinical environment [27-31] as it excites mechanical waves in the cornea (for example, using an air- puff or acoustic micro-tapping (A\uf06dT [27]) and tracks them using phase-sensitive optical coherence tomography (OCT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Analyzing the shear wave propagation speed or dispersion leads to an estimate of the shear modulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' OCE studies reported corneal shear moduli in the range of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='8 – 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3 kPa [32, 33], in close agreement with values obtained from torsional and rheometry testing of ex vivo cornea (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 – 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3 kPa) [34-36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' However, all shear-based methods produce moduli differing by 1-2 orders of magnitude from those reported by tensile and inflation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' For years the Rayleigh wave group velocity was used to reconstruct corneal elasticity [31, 38, 27, 28-32, 39, 40], resulting in a 2 orders of magnitude mismatch in Young’s modulus compared to tensile and inflation tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' We clearly showed that this approach is inappropriate [41] since the cornea is a layer bounded by air at the top and aqueous humor on the bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thus, guided waves propagate with strong geometric dispersion [41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Recently, we hypothesized that corneal anisotropy is the primary cause of these discrepancies [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Corneal microstructure supports this hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The stroma contains collagen lamellae running in-plane across its width.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Lamellae make up approximately 90% of tissue thickness and account for the majority of the cornea’s mechanical structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' They are stacked vertically in approximately 200-500 separate planes [43, 44], suggesting an anisotropic mechanical behavior with very different responses to in-plane versus out-of-plane loads.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' We introduced a model of a nearly-incompressible transverse isotropic (NITI) medium [42] to explain the discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' It is defined by two (in-, 𝜇, and out-of-plane, 𝐺 ) shear moduli, decoupling tensile/inflation properties from shear responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Based on this model, an algorithm utilizing guided mechanical waves in a bounded NITI medium to reconstruct both moduli from A\uf06dT- OCE measurements has been developed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In [45] we showed that both in- and out-of-plane post-CXL corneal shear moduli increase compared to their pre-surgery values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' It is interesting that at the end of the CXL procedure corneal Young’s modulus changes on average by about 2 times whereas the out-of-plane shear modulus 𝐺 changes by almost 4 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Indeed, the CXL procedure is intended to better crosslink corneal lamellae, which it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' However, deformational properties are defined by the Young’s modulus, which does not change nearly as much and has implications for potential refractive changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Note, however, that the effects of crosslinking are always homogeneous in depth as the penetration of riboflavin is more pronounced in the anterior region of the cornea, where the solution is applied during soaking [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Riboflavin penetration is purposely limited to minimize damage to the endothelium and ensure the biological integrity of the cornea [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' As noted above, mechanical waves generated in the cornea are usually not bulk shear or surface Rayleigh waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' they are guided waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Reconstructing corneal moduli using local estimates of the propagation speed at different depths leads to large errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In fact, guided waves occupy the entire depth of the cornea and, therefore, average information over depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thus, reconstructing the depth dependence of corneal moduli is practically impossible without a good estimate of CXL penetration within the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Blackburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' [48] have recently introduced a novel metric to track CXL penetration within the cornea using time-resolved OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' They demonstrated that the phase decorrelation decay rate of the complex OCT signal is reduced in CXL areas and can be used to distinguish treated and untreated areas after a CXL procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In this letter, we combine the methods described in [48] with A\uf06dT-OCE measurements to explore possible reconstruction of both in- and out-of-plane corneal elastic moduli over depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In addition, we use numerical simulations of guided wave propagation in a two-layer medium consisting of a CXL cornea layer on the top of the untreated cornea to demonstrate how guided- wave characteristics can be used to reconstruct corneal elasticity in both parts of the medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Although our results show promise in monitoring the effects of CXL using a two-layer model, a more detailed study on model-based reconstruction of the in-depth distribution of elastic properties in CXL cornea is required to generalize our findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 Cornea preparation One corneal-scleral ring, stored in Optisol (Chiron Ophtalmics) was obtained from CorneaGen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This sample from a 26 years-old donor was stored for less than 30 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The corneal-scleral button is mounted on an artificial anterior chamber (Barron, CorzaMedical;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' see figure 1), connected through the inlet port to an elevated bath filled with balanced saline solution (BSS) to apply a controlled pressure on the anterior segment of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The outlet port remained closed to allow the pressure to settle at 15 mmHg within the chamber, corresponding to human physiological conditions [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Crosslinking was performed following the Dresden protocol [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' First, the epithelial membrane was removed from the sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Then, the cornea was soaked in riboflavin for 30 min, by applying a 50µL drop of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1% riboflavin in 20% dextran solution every two minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The cornea was then exposed to a 3mW/cm2 of 370 nm ultra-violet (UV) light for 30 minutes, while a drop was re-applied every 5 minutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Picture of the experimental set up during UV-CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' a) Acoustic micro-tapping transducer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' b) Artificial anterior chamber with c) the inlet port connected to the elevated bath and outlet port closed for controlling IOP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 AµT OCE imaging system A spectral domain OCT system with a 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 kHz frame rate was used to track wave propagation and structural changes within the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' As described in previous studies [27,31,42,45], a cylindrically focused air-coupled transducer, operating at 1MHz, generates a spatio-temporal sharp displacement at the surface of the cornea, using reflection based radiation force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Because of the cylindrical geometry of the transducer, the push is line-shaped and generates planar guided waves within the bounded tissue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The OCT system operated in M-B mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' A single push is triggered by the system while 512 consecutive A-scans are taken at a fixed location (M- scan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The M-scan sequence and push excitation are repeated for 256 locations, creating a three- dimensional volume with 256 𝑥-samples, 1024 𝑧-samples and 512 𝑡-samples (see figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='a)), with an effective imaging range of 6mm × 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2mm × 11ms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The particle axial vibration is obtained from the optical phase difference between two consecutive A-line scans at each location [51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The spatio-temporal (𝑥-𝑡) surface signature of the guided wave is computed from an exponential weighted-average of the vertical velocity over the first 180µm of the anterior part of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' As shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='b), the guided wave only propagates during the first 4ms of the scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This first part will be used to determine the stiffness of the material by fitting the measured dispersion curve in the frequency-wavenumber domain (𝑓-𝑘), obtained from the 2D Fourier spectrum (see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This procedure is detailed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' On the other hand, data from the last 7ms are used to study the structural changes with phase decorrelation (see section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' a) D Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Stiffness reconstruction and depth penetration measurements using spectral domain time- resolved OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' a) 3D (𝑥, 𝑧 and 𝑡), wave field after AµT excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The top surface wavefield of the initial time sequence (black dotted region) is used for stiffness reconstruction and data at the end of the sequence is used for the phase decorrelation measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' b) 𝑥-𝑡-plot showing the top surface signature of the guided mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' c) 𝑓-𝑘-plot obtained by 2D-FFT of the 𝑥-𝑡-plot showing the dispersion signature of the first anti-symmetric mode 𝐴0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The red curve indicates the best fit obtained with the NITI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' d) Structural OCT image obtained from averaging the last 7ms of the raw OCT signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' e) Phase decorrelation function 𝑔(𝜏) at the location indicated by the red square on d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3 NITI model and fitting The microstructure of the cornea implies a transverse isotropy in the direction normal to the corneal interface [52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Like most biological tissue, the cornea is nearly-incompressible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Consequently, an exact description of its behavior under external loading requires a NITI (for Nearly Incompressible Transversely Isotropic) model [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In Voigt notation, the Hook’s law of stress and strain for a TI material takes the form: a) OCT 7ms 4ms AuT OCE Verticalvelocity,Vz(a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=') PowerSpectrum(dB） b) c 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 (1/mm) 4 5 (uu) 3 0 X 2 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 0 0 15 0 2 4 0 1 2 3 4 t (ms) f (kHz) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='Linear fit 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='999 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='998 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3[ 𝜎𝑥𝑥 𝜎𝑦𝑦 𝜎𝑧𝑧 𝜏𝑦𝑧 𝜏𝑥𝑧 𝜏𝑥𝑦] = [ 𝜆 + 2𝜇 𝜆 𝜆 𝜆 𝜆 + 2𝜇 𝜆 𝜆 𝜆 𝜆 + 𝛿 𝐺 𝐺 𝜇 ] [ ϵ𝑥𝑥 ϵ𝑦𝑦 ϵ𝑧𝑧 γ𝑦𝑧 γ𝑥𝑧 γ𝑥𝑦] where 𝜎𝑖𝑗 denotes engineering stress, 𝜖𝑖𝑗 denotes engineering strain, 𝜏𝑖𝑗 denotes shear stress, 𝛾𝑖𝑗 = 2 𝜖𝑖𝑗 denotes shear strains, the subscript 𝑥, 𝑦 and 𝑧 refer to the Cartesian axes and 𝐺, 𝜇, 𝜆 and 𝛿 are four independent elastic constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In previous studies [52,53], we have demonstrated that 𝛿, which accounts for tissue tensile anisotropy, cannot be determined from guided wave propagation but that the in-plane Young’s modulus can be approximated as 𝐸𝑇 = 𝐸 ≅ 3𝜇 assuming tensile isotropy (𝛿 = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thus, among the four elastic constants, only 𝐺 and 𝜇, respectively the out-of-plane and in-plane shear moduli, are needed to predict deformations of a nearly incompressible tissue under mechanical loading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Accounting for the appropriate boundary conditions (water below and air above) and the finite thickness of the medium, the dispersion relationship of guided waves can be determined directly from 𝐺 and 𝜇.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Note that in cornea, only the first anti-symmetric mode, referred to as 𝐴0, propagates in the range of frequencies that can be recorded in elastography (typically < 5kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The experimental 𝑓-𝑘 spectrum (see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='c)) is obtained by computing the 2D-FFT of the 𝑥-𝑡-plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Because the boundary conditions of the cornea on the anterior chamber highly differ with physiological ones (the cornea is clamped here), a structural resonance of the cornea is observed in the low-frequency regime (below 1kHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' To remove this artifact, we applied a temporal super-Gaussian filter (∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5ms FWHM) that follows the maximum of vibration velocity prior to computing the Fourier transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The shear moduli 𝐺 and 𝜇 are obtained from fitting the measured 𝑓-𝑘-spectrum with the analytical dispersion relationship of the 𝐴0 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' To ensure reliable fitting, we computed a goodness of fit metric 𝜙 = ∑ 𝜒𝑓𝑖𝑡(𝑓) 𝑓 ∑ 𝜒𝑚𝑎𝑥(𝑓) 𝑓 , where 𝜒𝑓𝑖𝑡(𝑓) corresponds to the energy of the 2D spectrum covered by the best analytical solution at a given frequency 𝑓 and 𝜒𝑚𝑎𝑥(𝑓) is the unconstrained maximal energy of the spectrum at frequency 𝑓.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Based on recent results (see supplemental material in [12]), reliable fitting in human ex vivo corneas are associated with values of 𝜙 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' More details about the complete fitting procedure can be found in [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' An example of a 2D-spectrum and the fitted 𝐴0 mode obtained with this procedure are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='4 Phase Decorrelation OCT (PhD-OCT) Blackburn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' [48] have recently introduced a novel metric to track CXL penetration within the cornea using time-resolved OCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' They demonstrated that the phase decorrelation decay rate of the complex OCT signal is reduced in CXL areas and can be used to distinguish treated and untreated areas of the cornea after a procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The complex-valued autocorrelation of the signal 𝑔(𝜏) is computed over 15 consecutive sample at 46,500 Hz, for six consecutive pixels within a given A-line: 𝑔(𝜏) = ⟨ ⟨𝐸(𝑡) 𝐸∗(𝑡 + 𝜏)⟩𝑝𝑖𝑥𝑒𝑙𝑠 √⟨𝐸(𝑡) 𝐸∗(𝑡)⟩𝑝𝑖𝑥𝑒𝑙𝑠 × √⟨𝐸(𝑡 + 𝜏) 𝐸∗(𝑡 + 𝜏)⟩𝑝𝑖𝑥𝑒𝑙𝑠 ⟩, which is expected to follow an exponential decay [54]: 𝑔(𝜏) = 𝑒−𝛤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 𝜏 ≈ 1 − 𝛤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 𝜏, where Γ is the decorrelation coefficient that is inversely proportional to the Brownian diffusion coefficient [54], meaning that the more coherent the material, the smaller the decorrelation coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This procedure is performed starting at the 𝑛, 𝑛 + 1, 𝑛 + 2, … A-line, where 𝑛 is the first time-sample used for the phase-decorrelation ( 𝑡(𝑛) = 4ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The decorrelation coefficient Γ is then computed using the averaged 𝑔(𝜏) over the number of starting points by fitting with a first order polynomial (see Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='e)): < 𝑔(𝜏) > = 𝑏 − Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 𝜏, where <> denotes the average over the number of starting points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In crosslinked parts of the cornea (anterior), the tissue stiffens, which implies that Γ should be smaller than in the untreated region (posterior).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' For post-processing, we rejected all fits were b < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='95, corresponding in general to peripheral regions where the SNR (Signal to Noise Ratio) is too low.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Results 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 Corneal stiffness before and after CXL A single scan of AµT-OCE, taking approximatively 3s to acquire and save, was taken before and after CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The signature of the vertically polarized velocity was used to compute the 𝑓-𝑘- spectrum, which was fitted using the procedure detailed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The results for the reconstructed stiffness before and after CXL are shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' As generally observed in the literature [55,56], the thickness is slightly reduced following crosslinking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The measured stiffnesses increased for 𝐺 from 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='8kPa to 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2kPa and for 𝜇 from 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3MPa to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Such increase in both stiffness moduli following CXL is in good agreement with previously reported results for ex vivo human corneas [45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The goodness of fit metric is 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='96 before CXL and 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='95 after CXL, which confirms the robustness of the fit and the validity of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Measured stiffnesses from the experimental dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thickness, ℎ (µm) Out of plane shear modulus, G (kPa) In plane shear modulus, 𝜇 (MPa) Goodness of fit, 𝜙 Before CXL 575 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='8 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='96 After CXL 520 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='95 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2 Depth dependance of CXL The results for phase decorrelation are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' For the untreated case (Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='a, c), both the OCT intensity and Γ are constant in directions normal to the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' In the center, the higher scattering of laser light slightly influences the results, suggesting that Γ depends on the signal to noise ratio of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The CXL area can be seen in both structural OCT and Γ maps after CXL (Figures 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='b, d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' As shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='e), the drop of OCT intensity is associated with an increase of Γ, which demonstrates that both metrics can be used to identify the CXL area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The thickness of the CXL layer varies from ∼100μm in the periphery to ∼150μm in the central area of the cornea, which agrees with recent observation of lateral changes in the effect of crosslinking in corneas [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Short time decorrelation before and after CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Structural OCT images obtained from the last 7ms of the OCT scan for a) before CXL and b) after CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Maps of decorrelation coefficient 𝛤 for c) before and d) after CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' e) Profile of OCT intensity and 𝛤 along the red dotted line shown in b) and d) for the CXL cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='3 Numerical simulations We designed a finite element (FEM) simulation to study the layering effect of CXL (section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2) on the reconstructed stiffness using the NITI model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The geometry is shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Boundary conditions of the cornea were replicated so that the material is bounded above by air and below by water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The speed of sound in all layers (material and water) was fixed to avoid reflection of compressional waves at the different boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' It also improved the absorption of waves at the absorbing boundaries and, thus, avoided divergence of the simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' We used transient excitation mimicking AµT experiments to generate broadband elastic waves within the material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' More details about the simulations can be found in [42,50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Based on the phase- decorrelation measurements (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='2), we assumed that after CXL, two layers with distinct stiffnesses but with identical thicknesses (each 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='25mm) were formed within the cornea, the top layer being stiffer than the bottom one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' We fixed 𝐺1 = 200kPa, 𝜇1 = 10MPa, 𝐺2 = 50kPa and 𝜇2 = 4MPa, and a total thickness ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Similar to OCE experiments, we used the top surface vertical signature of the guided wave (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='b) to compute its 2D 𝑓-k spectrum (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' We also performed simulations in a single-layer medium with moduli corresponding to that averaged over the two-layer structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 𝐺1 = 125kPa, 𝜇1 = 7MPa, 𝐺2 = 125kPa and 𝜇2 = 7MPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' The computed 𝑥-𝑡 plot (Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='c) for the homogeneous layer with averaged parameters is nearly identical to that for the two-layer medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Furthermore, an analytically calculated dispersion curve for the case of averaged parameters fits well the 2D spectrum computed for the two-layer case with 𝜙 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='961 (red line in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Thus, the simulation supports our hypothesis that ‘averaged’ moduli can be reconstructed from guided wave propagation in the cornea with partial penetration of CXL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' These preliminary results show that defining the CXL penetration depth from structural OCT images may help decouple elastic moduli in each cornea layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' However, additional studies are required to define and characterize the reconstruction algorithm for the more general case of arbitrary CXL penetration depth and its possible variation over the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='05 a intensity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='04 10-2 rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='005 10°2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='03 00 10-3 decorrelation 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='02 d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='005 10~2 0 200 400 600 Distance (μm) Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Finite element simulations to study the effects of a layered structure for a CXL cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' a) Geometry of the two layered material used in simulations, bounded above by air and below by water.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' b) Top surface spatio-temporal signature (𝑥-𝑡-plot) of the guided wave for the two-layer case with 𝐺1 = 200kPa, 𝜇1 = 10MPa, 𝐺2 = 50kPa, 𝜇2 = 4MPa and a total thickness ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' c) 𝑥-𝑡 plot of the guided wave for one layer averaged over the depth parameters, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 𝐺 = 125kPa, 𝜇 = 7MPa and thickness ℎ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5mm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' d) 2D Fourier spectrum of the wave studied in b), showing the main propagating 𝐴0 mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' An analytical dispersion curve (red line) computed for the case c) is plotted on the top of the 2D spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Discussion and Conclusions In this study we combined structural OCT imaging with dynamic OCE to assess the penetration depth of CXL treatment in the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Analyzing brightness of structural OCT images and image decorrelation between consecutive B-scans, we conclude that there is a sharp transition between CXL and untreated cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This finding allowed us to suggest a model of a two-part medium for the treated cornea, where cornea elastic properties remain constant in each part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' This finding can be confirmed with destructive measurements of both layers, but it is outside the scope of the current study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Both experimental results and numerical simulations show that guided waves deliver ‘averaged’ or ‘effective’ elastic properties of the cornea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' When the thickness of the CXL layer can be measured, the two-layer model can be used to reconstruct mechanical moduli in both parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Note, however, that such reconstructions may not be simple and require additional simulations, experimental studies, and depth-resolved mechanical tests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Freo G1, μ1, Ai, Cp, p h G2, μ2, 2, Cp, P cw=CppW=p Absorbing 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='. 10 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='. - lu, averaged moduli 8 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 (1/mm) -5 (uw) 6 mm 0 0 X 4 X 4 -10 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 2 -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='5 2 0 0 0 -15 0 2 0 - 2 0 2 4 t (ms) t (ms) f (kHz)References 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Sridhar, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Anatomy of cornea and ocular surface.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Marcovich, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Barkana, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Mandel, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Hirsh, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Morad, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Ambrozinski, Ł.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Yoon, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Gao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' & O’Donnell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Optical coherence elastography in ophthalmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Journal of biomedical optics, 22(12), 121720-121720 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Li, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Singh, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Han, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Raghunathan, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Nair, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', Wu, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=', .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' & Larin, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Assessing the effects of riboflavin/UV-A crosslinking on porcine corneal mechanical anisotropy with optical coherence elastography.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Biomedical Optics Express, 8(1), 349- 366 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Petsche, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' & O’Donnell, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Optical coherence elastography in ophthalmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Journal of biomedical optics, 22(12), 121720 121720 (2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' 38.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/U9FLT4oBgHgl3EQfRi9C/content/2301.12037v1.pdf'} +page_content=' Han, Z.' metadata={'source': 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0000000000000000000000000000000000000000..2daaf291ddbbb9a963af8f575a53190b63f0348c --- /dev/null +++ b/V9E0T4oBgHgl3EQfVgDO/vector_store/index.pkl @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:45c160ec1c7936b3e3c76af47b2c181260b4e90a8721ed22353bb1e57cd0c17f +size 78620 diff --git a/VtE0T4oBgHgl3EQflwHV/content/tmp_files/2301.02491v1.pdf.txt b/VtE0T4oBgHgl3EQflwHV/content/tmp_files/2301.02491v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..b41f4758264d388e0a149af796919cbf189c1fbc --- /dev/null +++ b/VtE0T4oBgHgl3EQflwHV/content/tmp_files/2301.02491v1.pdf.txt @@ -0,0 +1,23406 @@ +arXiv:2301.02491v1 [math.CT] 6 Jan 2023 +A CATEGORIFICATION OF QUINN’S FINITE TOTAL +HOMOTOPY TQFT WITH APPLICATION TO TQFTS AND +ONCE-EXTENDED TQFTS DERIVED FROM STRICT +OMEGA-GROUPOIDS +JO˜AO FARIA MARTINS AND TIMOTHY PORTER +Abstract. We first revisit the construction of Quinn’s Finite Total Homotopy +TQFT, which depends on the choice of a homotopy finite space, B. We build +our construction directly from homotopy theoretical techniques, and hence, as +in Quinn’s original notes from 1995, the construction works in all dimensions. +Our aim in this is to provide background for giving in detail the construc- +tion of a once-extended TQFT categorifying Quinn’s TQFT, in the form of a +symmetric monoidal bifunctor from the bicategory of manifolds, cobordisms +and extended cobordisms, initially to the symmetric monoidal bicategory of +profunctors (enriched over vector spaces), and then to the Morita bicategory +of algebras, bimodules and bimodule maps. These once-extended versions of +Quinn’s TQFT likewise are defined for all dimensions, and, as with the original +version, depend on the choice of a homotopy finite space B. +To show the utility of this approach, we explicitly compute both Quinn’s +finite total homotopy TQFT, and its extended version, for the case when B is +the classifying space of a homotopically finite omega-groupoid, in this paper +taking the form of a crossed complex, following Brown and Higgins. +The constructions in this paper include, in particular, the description of +once-extended TQFTs derived from the classifying space of a finite strict 2- +group, of relevance for modelling discrete higher gauge theory, but the tech- +niques involved are considerably more general. +Acknowledgement: The greatest part of the research for this paper was financed +by the Leverhulme Trust research project grant RPG-2018-029: Emergent Physics +from Lattice Models of Higher Gauge Theory. +JFM would like to express his gratitude to Alex Bullivant for discussion that +framed some of the ideas supporting the construction of the once-extended version +of Quinn’s finite total homotopy TQFT, to Fiona Torzewska for discussions on +cobordism categories and on cofibrant cospans, to Ben Horton for discussions on +bicategories of extended cobordisms, and also to Paul Martin, Vincent Koppen and +Jack Rom¨o for useful discussions. We both thank Ronnie Brown for discussions on +classifying spaces of crossed complexes. The final stages of the writing of this paper +were partially funded by the project PTDC/MAT-PUR/31089/2017: Higher Struc- +tures and Applications, of FCT (Portugal), and then by EPSRC, via the Programme +Grant EP/W007509/1: Combinatorial Representation Theory: Discovering the In- +terfaces of Algebra with Geometry and Topology. +Date: January 6, 2023. +Key words and phrases. TQFTs, Extended TQFTs, Symmetric Monoidal Bicategory, Sym- +metric Monoidal Bifunctor, Profunctors / Distributeurs, Morita Bicategory (of algebras, bimod- +ules and bimodule maps), Homotopy Finite Spaces, Function Spaces, Crossed Complexes, Strict +Omega-Groupoids, 2-groups, Discrete Higher Gauge Theory +©2022: Jo˜ao Faria Martins and Tim Porter. +1 + +A CATEGORIFICATION OF QUINN’S TQFT +2 +Note: This is a preliminary version for commenting. +Comments, suggestions, +corrections, etc, are very welcome, and can be sent to either (or both) of the authors, +j.fariamartins@leeds.ac.uk and t.porter.maths@gmail.com. +1. Introduction +In Lecture 4 of his lecture notes, [101, 1995], on axiomatic topological quantum +field theory, Quinn described, what he called the finite total homotopy TQFT, +following on from a suggestion of Kontsevich, [73]. This family of TQFTs, whose +construction works in any spatial dimension, had, in special cases, been studied by +Dijkgraaf and Witten, [43], and also by Segal, [107]. In that lecture, Quinn sketches +the construction, starting from a space, B, which has ‘finite total homotopy’ or, +as we will say, ‘is homotopy finite’. This finiteness is to ensure that the resulting +theory takes values in the category of finite dimensional vector spaces. +The basic construction used is quite simple in its main idea. Let d be any non- +negative integer. Let Cob(d,d+1) be the symmetric monoidal category of closed +smooth d-manifolds, and diffeomorphism classes of (d + 1)-cobordisms between +them. Given B, and a (smooth and closed) d-manifold, Σ, then the TQFT, which +we will denote by QB : Cob(d,d+1) → VectQ, assigns Q[Σ, B] to Σ. The vector +space corresponding to Σ is thus based on the set, [Σ, B], of homotopy classes of +maps from Σ to B. Important examples are when B is the classifying space of a +finite group, or of a finite (strict) 2-group, where one retrieves well known examples +of TQFTs, such as Dijkgraff-Witten’s TQFT, [43], and the Yetter-Porter TQFT, +[124, 97, 53], but there are many other possibilities. Given a cobordism, M, from +Σ to another manifold, Σ′, the construction gives a matrix, and hence a linear +transformation from QB(Σ) to QB(Σ′), the matrix being with respect to the given +bases of the two vector spaces. +Our main purpose in this paper is to categorify this construction of Quinn to +get what we call the once-extended Quinn TQFT, which will be formulated in three +different, closely related, ways. +What do we mean by ‘categorification’? In very general terms, when categori- +fying a theory, one wants to try to replace sets by categories or groupoids, cat- +egories themselves by 2-categories, or better bicategories, functions between sets +by functors between categories, etc., and, when all that is done, to add another +layer corresponding to natural transformations. Here, for instance, we want to re- +place the category, Cob(d,d+1), by a bicategory / weak 2-category, 2Cob(d,d+1,d+2), +incorporating a form of 2-cobordism, or cobordism between cobordisms, between +manifolds. We want to replace [Σ, B], which is the same as π0(BΣ), the set of path- +components of the mapping space1, BΣ, by the fundamental groupoid, π1(BΣ, BΣ), +and then do corresponding adaptations of Quinn’s methodology to obtain Vect- +valued profunctors between these groupoids, associated to cobordisms. Finally we +then construct appropriate transformations between profunctors to be associated to +extended cobordisms connecting cobordisms. All this we want to linearise, forming +the free linear categories on the groupoids, etc. +To make all this work, we need to start by taking apart Quinn’s original method, +and, noting that in the published version, [101], a lot is merely sketched, we have in- +cluded a more detailed rendition of his theory, and, in fact, will give a parametrised +family of variants of his theory. Partially because of this, we work not only over +Q, but over a more general subfield, κ, of C, as on occasion we will need the extra +freedom that that gives us. +1which may sometimes be more conveniently written as TOP(Σ, B). + +A CATEGORIFICATION OF QUINN’S TQFT +3 +In developing this theory, we hoped that it would allow calculations that will +generalise known ones, and, to this end, we develop methods of explicit calculation, +of both Quinn’s finite total homotopy TQFT, and its categorified versions, in a +particular family of cases, namely when B is the classifying space of a homotopy +finite strict ∞-groupoid, using the crossed complex model of such algebraic objects, +as developed by Brown–Higgins–Sivera, [27], and Tonks, [116]. In particular, our +framework includes the case when B is the classifying space of a strict 2-group, +which is relevant for understanding TQFTs and extended TQFTs derived from +discrete higher gauge theory, [6, 31]. +The framework for explicit calculations developed here can likely be extended +in order to allow for combinatorial calculation of Quinn’s finite total homotopy +TQFT, and its once-extended versions, whenever the homotopy finite space, B, is +represented combinatorially, for instance when B is the classifying space of a finite +simplicial group. (Note that finite simplicial groups are considerably more general +than crossed complexes, and do model all homotopy fnite spaces [46].) Such a study +will be deferred to a future paper. +We also expect that the categorification constructed here of Quinn’s finite to- +tal homotopy TQFT, to a once-extended TQFT, can be further categorified to +a doubly-extended, perhaps even fully-extended TQFT, [82]. +This analysis will +likewise be deferred to a subsequent paper. +... in a bit more detail +1.1. The ‘classical’ Quinn finite total homotopy TQFT. Let Vectκ = Vect +be the category of κ-vector spaces and linear maps, which will usually be considered +together with its usual symmetric monoidal category structure. Throughout the +paper, given a non-negative integer, d, the symmetric monoidal category of closed +d-manifolds and equivalence classes of cobordisms between them will be denoted +Cob(d,d+1). By a d-dimensional TQFT, we will mean a symmetric monoidal functor +from Cob(d,d+1) to Vect, as in, for instance, [82, 37]. We note that there is no +assumption made, nor needed, that our manifolds or cobordisms be oriented, or +even orientable. +In this paper, we will need to work over the category, CGWH, of compactly +generated, weak Hausdorff spaces, [87, 113]. Such a space, B, is called homotopy +finite if B has only a finite number of path-components, each of which has only a +finite number of non-trivial homotopy groups, all of which are finite. Also recall +that given a homotopy finite space, its homotopy content is defined by the formula +below (cf. also [101, Lecture 4], [4] and [56, §3]): +χπ(B) = +� +[x]∈π0(B) +��π2(B, x) +�� ��π4(B, x) +�� ��π6(B, x) +�� . . . +��π1(B, x) +�� ��π3(B, x) +�� ��π5(B, x) +�� . . . ∈ Q. +Given a fixed homotopy finite space B, Quinn defined what he called the finite +total homotopy TQFT, denoted here by QB : Cob(d,d+1) → VectQ, defined for all +d ≥ 0. In Part 2 of this paper, particularly in Section 4, we provide a thorough +description of the construction of Quinn’s finite total homotopy TQFT, giving full +mathematical details, in particular defining, more generally, a parametrised version, +Q(s) +B : Cob(d,d+1) → VectC, of it, where s is a complex parameter. All TQFTs, +Q(s) +B , for fixed B, are related by natural isomorphisms. The latter parameter, s, was +not present in Quinn’s original construction, and is closely related to the parameter +appearing in the similar groupoidification functor in [5]. +Part 2 of this paper is subdivided into two sections. In Section 4, we give full +details of the construction of Quinn’s finite total homotopy TQFT. Before that, in + +A CATEGORIFICATION OF QUINN’S TQFT +4 +Section 3, which is considerably more technical, we formulate the required homotopy +theoretical setting supporting its construction. In particular, we introduce one of +the main technical tools used in this paper, which is the idea of a fibrant span, +(p, M, p′): B → B′, of homotopy finite spaces, meaning that we have a diagram of +homotopy finite spaces, +M +p +�❥❥❥❥❥❥ +p′ +�❚ +❚ +❚ +❚ +❚ +❚ +B +B′, +and, crucially, the induced map, +� +p, p′� +: M → B×B′, is required to be a fibration2. +These fibrant spans of homotopy finite spaces can be composed by perform- +ing the obvious pullback. Moreover, we have a category, HFspan, whose objects +are homotopy finite spaces, and morphisms are fibred homotopy classes of fibrant +spans connecting them3. The identity on a homotopy finite space, B, in HFspan, is +given by the fibred homotopy class of the fibrant span of homotopy finite spaces, +(sB, B, tB): B → B, obtained from the path-space fibration, that is, +BI +sB +�❥❥❥❥❥❥ +tB �❯ +❯ +❯ +❯ +❯ +❯ +B +B. +Here BI is the space of all maps from the unit interval, I to B, where I = [0, 1], +and sB(γ) = γ(0) and tB(γ) = γ(1). +A crucial step for the construction of Quinn’s finite total homotopy TQFT is +a family of functors, R(s) : HFspan → Vect, (working now over C), sending each +homotopy finite space, B, to the free vector space, C(π0(B)), over the set π0(B), +and, given a fibrant span, (p, M, p′): B → B′, the matrix elements of the linear +map, R(s)� +(p, M, p′) +� +: R(s)(B) → R(s)(B′), are given by the equation +� +PCb(B) +��R(s)� +(p, M, p′) +���PCb′(B′) +� += χπ� +⟨p, p′⟩−1(b, b′) +� � +χπ(PCb(B)) +�s � +χπ(PCb′(B′)) +�1−s. +Here b ∈ B, b′ ∈ B′, and we denote the path component of b in B by PCb(B), and +the same for b′. +The existence of the functor, R(s) : HFspan → Vect, is implicit in the construc- +tion in [101], and it is also addressed by G´alvez-Carrillo, Kock, and Tonks, [56], +albeit in the framework of ∞-groupoids, and using homotopy pullbacks instead of +the usual pullbacks (which we can use here, since we are working with fibrant spans +only). This also generalises the “groupoidification” construction in [5]. +Let B be a fixed homotopy finite space. The final step of the construction of +Quinn’s final total homotopy TQFT, Q(s) +B : Cob(d,d+1) → Vect, made explicit in +Section 4, relies on the existence of a functor, FB : Cob(d,d+1) → HFspan, sending a +d-manifold, Σ, to the space, BΣ, of continuous functions from Σ to B, and sending +the equivalence class of a (d + 1)-cobordism, (i, S, j): Σ → Σ′, from Σ to Σ′, seen +as a cospan, +Σ +i +�❘ +❘ +❘ +❘ +❘ +❘ +Σ′ +j +�❧❧❧❧❧❧ +S +, +to the equivalence class of the following fibrant span of homotopy finite spaces, +BS +i∗ +�❥❥❥❥❥❥ +j∗ +�❚ +❚ +❚ +❚ +❚ +❚ +BΣ +BΣ′. +2In this paper “fibration” means Hurewicz fibration. +3A ‘dual’ category of cofibrant cospans is treated in [117]. + +A CATEGORIFICATION OF QUINN’S TQFT +5 +(Here given f : S → B, then i∗(f) = f ◦i and j∗(f) = f ◦j.) This allows us to define +Quinn’s finite total homotopy TQFT, Q(s) +B : Cob(d,d+1) → Vect, as the composite +functor, +Cob(d,d+1) FB +−−→ HFspan +R(s) +−−−→ Vect. +This functor, Q(s) +B : Cob(d,d+1) → Vect, can be canonically given the structure of +a symmetric monoidal functor, and hence of a TQFT. +In Section 4, we also show some properties of Q(s) +B , as we change B. For instance, +we show that Q(s) +B depends only on the homotopy type of B, up to natural isomor- +phisms, which fact is not immediate. Furthermore, given a homotopy finite space, +B, we have an action of the group, E(B), of homotopy classes of homotopy equiv- +alences of B, on the TQFTs Q(s) +B : Cob(d,d+1) → Vect, by natural isomorphisms. +These are Theorems 90 and 91. +We will show, much later on, in Subsection 8.2, how to calculate Q(s) +B , explicitly, +in the case in which B is the classifying space of a strict ∞-groupoid, [2], noting +that such a structure is often, more classically, called an ω-groupoid as in [27], and +is often considered (as we will do here) in its form as a crossed complex, in the sense +of [27]. The cases for Q(s) +B treated here, when B is the classifying space of a crossed +complex, include that in which B is the classifying space of a finite 2-group, and +are hence relevant for understanding discrete higher gauge theory, [31, 93]. The +explicit formulae that we will give for the TQFTs derived from crossed complexes, +via the special case of 2-groups / crossed modules, complement and generalise those +of [53]. The latter reference considered only invariants of closed manifolds derived +from crossed complexes, however providing a homotopy interpretation of the Yetter +homotopy 2-type TQFT [124, 97] for the case of closed manifolds. +1.2. The once-extended versions of Quinn’s finite total homotopy TQFT. +Part 3 of this paper, which again is subdivided into two sections, is devoted to the +construction of once-extended versions of Quinn’s finite total homotopy TQFT. +Let d be a non-negative integer. +We let 2Cob(d,d+1,d+2) be the symmetric +monoidal bicategory of d-dimensional closed (and smooth) manifolds, (d + 1)- +cobordisms between closed d-manifolds, and diffeomorphism classes of (d + 2)- +extended cobordisms connecting (d + 1)-cobordisms; see [106], for instance, for +precise definitions. +We let Prof κ = vProf be the symmetric monoidal bicategory with objects small +linear categories (meaning categories enriched over Vect). Given two small linear +categories, C and C′, 1-morphisms from C to C′ are Vect-enriched profunctors, +H: C ↛ C′, so, according to our conventions, they are enriched functors, H: Cop × +C′ → Vect. The 2-morphisms are natural transformations of such enriched functors +from Cop × C′ to Vect; see e.g. [10, 63] for complete definitions. We review the +definition of the bicategory, vProf, in Subsection 2.8. +In this paper, a once-extended TQFT, sometimes called here, more briefly and +more vaguely, an extended TQFT, will be, by definition, a symmetric monoidal +bifunctor, +2Cob(d,d+1,d+2) → vProf, +thought of as a categorified version of a classical TQFT. We will also consider +once-extended TQFTs formulated as symmetric monoidal bifunctors, +2Cob(d,d+1,d+2) → Mor, +where Mor is the bicategory of algebras, bimodules and bimodule maps. +The +latter constructions however always originate from bifunctors with target vProf +by ‘linearisation’, in a similar way to [92]. + +A CATEGORIFICATION OF QUINN’S TQFT +6 +We will work with the subbicategory, vProfGrp, of vProf, whose objects are +groupoids, G = (s, t: G1 → G0), each made into a linear category by applying +the free vector space functor, Lin: Set → Vect, to the hom-sets of G. +Given +groupoids, G and G′, 1-morphisms in vProfGrp, from G to G′, are hence, by +definition, Vect-profunctors, H: G ↛ G′, in our conventions these being functors, +H: Gop × G′ → Vect. Given Vect-profunctors, H, H′ : Gop × G′ → Vect, a 2- +morphism, η: H =⇒ H′, in vProfGrp, is a natural transformation of functors, +Gop × G′ → Vect. +There are two subbicategories of vProfGrp that we will consider here, namely +vProfGrphf, the sub-bicategory of vProf, whose objects are the homotopy finite +groupoids, and vProfGrpfin, the sub-bicategory of vProfGrp whose objects are +the finite groupoids. +1.3. The first version of the construction of the once-extended Quinn +TQFT. Our first construction of a once-extended version of Quinn’s finite total +homotopy TQFT is a (symmetric monoidal) bifunctor, +2QB : 2Cob(d,d+1,d+2) −→ vProfGrphf, +constructed in Section 6, particularly Subsection 6.2. +As in the construction of Quinn’s finite total homotopy TQFT, we will factor +2QB by an intermediate homotopy theoretical construction, which we describe in +Section 5, where we develop most of the homotopy-theoretical underpinning for +the once-extended Quinn TQFT. In particular, we consider a bicategory-like object +(however not quite a bicategory), denoted 2span(HF). The objects of 2span(HF) +are homotopy finite spaces, and the 1-cells of 2span(HF), from X to Y , are fibrant +spans, (p, M, p′): X → Y , of homotopy finite spaces. The (not strictly associative) +composition, •, of 1-cells in 2span(HF) is again given by the obvious pullback. +Each homotopy finite space, X, has a ‘horizontal identity’ of X, given by the path- +space fibrant span, (sX, XI, tX). +Given two fibrant spans of homotopy finite spaces, +(p, M, p′), (q, N, q′): X → Y, +2-cells in 2span(HF), connecting them, are given by homotopy finite fibrant re- +solved 2-spans, W: (p, M, p′) =⇒ (q, N, q′), by definition consisting of diagrams +as below, +W = +X +M +p +� +p′ +� Y +XI +sX +� +tX +� +L +lX +� +rY +� +P +� +Q +� +Y I +sY +� +tY +� +X +N +q +� +q′ +� Y. +Here X, Y, M, N, L are homotopy finite spaces, and, crucially for the construction +to work, the induced map below, called the filler of W, is a fibration: +(1) +L +PL +−→ M +× +X×Y (XI × Y I) +× +X×Y N. +(On the right-hand-side, we have the obvious pullback arising from the limit along +the exterior faces of the diagram defining W.) +Again, homotopy finite fibrant +resolved 2-spans compose horizontally and vertically, though not associatively. +In the crucial cases arising in the once-extended Quinn TQFT, some particular 1- +cells, (p, M, p′): X → Y , have ‘vertical units’, id(p,M,p′) : (p, M, p′) =⇒ (p, M, p′), + +A CATEGORIFICATION OF QUINN’S TQFT +7 +and we moreover have ‘unitor’ 2-cells: +ρ(p,M,p′) +X +: (X +(p,M,p′) +−−−−−→ Y ) • (Y +(sY ,Y I,tY ) +−−−−−−−→ Y ) =⇒ (X +(p,M,p′) +−−−−−→ Y ), +and +λ(p,M,p′) +X +: (X +(sX,XI,tX) +−−−−−−−→ X) • (X +(p,M,p′) +−−−−−→ Y ) =⇒ (X +(p,M,p′) +−−−−−→ Y ). +Throughout Section 5, we construct an ‘assignment’, +H = +� +π1(−, −), H, 2H) +� +: 2span(HF) → vProfGrphf, +that gives the following. +(1) Each homotopy finite space, X, is sent to its fundamental groupoid, π1(X, X). +(2) Given a, homotopy finite, fibrant span, X +(p,M,p′) +−−−−−→ Y , we have a Vect- +profunctor, +H +� +X +(p,M,p′) +−−−−−→ Y +� +: π1(X, X)op × π1(Y, Y ) → Vect, +such that: +(a) given x ∈ X and y ∈ Y , the (by construction, finite dimensional) +vector space +H(X +(p,M,p′) +−−−−−→ Y )(x, y) +is the free vector space over the path components of the fibre ⟨p, p′⟩−1(x, y). +(b) Given morphisms in π1(X, X) and π1(Y, Y ), i.e. equivalence classes of +paths, in X and Y , +x +[γX] +−−−→ x′ +and +y +[γY ] +−−−→ y′, +the linear map, +H +� +X +(p,M,p′) +−−−−−→ Y +�� +[γX], [γY ] +� +: H +� +X +(p,M,p′) +−−−−−→ Y +� +(x′, y) → H +� +X +(p,M,p′) +−−−−−→ Y +� +(x, y′), +is induced by any of the homotopy equivalences, between fibres, +⟨p, p′⟩−1(x′, y) → ⟨p, p′⟩−1(x, y′), +arising by applying the homotopy lifting property of ⟨p, p′⟩: M → +X ×Y to γX and γY , together. Here γX is the reverse of the path γX. +(3) Finally, given W: (p, M, p′) +=⇒ +(q, N, q′), as above, we have a natural +transformation, of functors π1(X, X)op × π1(Y, Y ) → Vect, +2HW : H +� +X +(p,M,p′) +−−−−−→ Y +� +=⇒ H +� +X +(q,N,q′) +−−−−−→ Y +� +. +Explicitly, given objects x ∈ X and y ∈ Y , the linear map, +2HW +(x,y) : H +� +X +(p,M,p′) +−−−−−→ Y +� +(x, y) → H +� +X +(q,N,q′) +−−−−−→ Y +� +(x, y), +has the following matrix elements, where m ∈ ⟨p, p′⟩−1(x, y) and n ∈ +⟨q, q′⟩−1(x, y), and PL is defined in (1), +� +PCm +� +⟨p, p′⟩−1(x, y) +� +| 2HW +(x,y) | PCn +� +⟨q, q′⟩−1(x, y) +�� += χπ� +P −1 +L (m, constx, consty, n) +� +χπ� +PCn(⟨q, q′⟩−1(x, y)) +� +. +Here constx and consty are the constant paths at x and y. The proof that +2HW, defined this way, is indeed a natural transformation requires a wealth +of careful verifications. + +A CATEGORIFICATION OF QUINN’S TQFT +8 +Our main result in Section 5 is that the assignment, +H: 2span(HF) → vProfGrphf, +preserves all various compositions, and the horizontal identities, up to applying ap- +propriate natural isomorphisms. Moreover, vertical units and unitors are preserved +by H, whenever they exist. The hardest calculation is that indeed the natural +transformations 2HW are well behaved with respect to the horizontal composition +of fibrant resolved 2-spans of homotopy finite spaces. This is done in §5.5.2. +Having developed the homotopy theoretical context supporting the once-extended +Quinn TQFT, Section 6 is devoted to its explicit construction, in three different +forms. Let B be a homotopy finite space. Similarly to the case of Quinn’s finite total +homotopy TQFT, we have an assignment, B(−) : 2Cob(n,n+1,n+2) → 2span(HF), +sending each manifold, cobordism, or extended cobordism to its space of maps to +B. This preserves all compositions, identities, and unitors, up to natural homeo- +morphisms. Finally, the once-extended Quinn TQFT, +2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, +is defined by the composite functor, +2Cob(n,n+1,n+2) +B(−) +−−−→ 2span(HF) +H +−→ vProfGrphf. +This is treated in Subsection 6.2. We check later, in Subsection 6.6, that indeed this +bifunctor can naturally be given the structure of a symmetric monoidal bifunctor. +1.4. The finitary once-extended Quinn TQFT. If Σ is a d-dimensional closed +smooth manifold, then the groupoid that 2QB associates to Σ is 2QB(Σ) = +π1(BΣ, BΣ). This is a homotopy finite groupoid, however its set of objects is, in +general, uncountable. In Subsection 6.3, we will explain how the size of the image +groupoids under 2QB can be reduced by considering a closely related bifunctor, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +that we call the finitary once-extended Quinn TQFT. +Here the objects of the bicategory, denoted 2Cob +(n,n+1,n+2) +B +, are now B-decorated +manifolds, (Σ, f Σ). +These are, by definition, closed (and smooth) d-manifolds, +Σ, equipped with a B-decoration, fΣ, that is, a finite subset, f Σ, of BΣ, con- +taining at least one function in each path-component of BΣ. The 1-morphisms, +(Σ, f Σ) → (Σ′, f Σ′), are given by cobordisms, Σ → Σ′, with no further struc- +ture, and the same for 2-morphisms. On objects, the finitary once-extended Quinn +TQFT gives +2QB(Σ, f Σ) = π1(BΣ, fΣ), +and on 1-morphisms and 2-morphisms, we make use of the obvious restrictions of +the profunctors and natural transformations given by 2QB. +The groupoids that 2QB associates to a B-decorated manifold, (Σ, fΣ), explic- +itly depend on the B-decoration, fΣ, of Σ. However this dependence is only up +to a canonically defined invertible profunctor, which is functorial (up to natural +isomorphism) with respect to further changes in the B-decoration, and also natural +with respect to the profunctors associated to cobordisms. +1.5. The Morita-valued once-extended Quinn TQFT. In Subsection 6.4, we +change the target bicategory of our categorification of Quinn’s finite total homotopy +TQFT from vProfGrpfin to Mor, the bicategory of (finite dimensional) algebras, +bimodules and bimodule maps. + +A CATEGORIFICATION OF QUINN’S TQFT +9 +Our starting point will be the discussion of a naturally defined linearisation +bifunctor, +Lin(2) : vProfGrpfin → Mor, +essentially defined in [90], as part of a Morita equivalence between a linear category +C and the algebra [C] that is associated to it. On objects, Lin(2) sends a groupoid +G to its groupoid algebra, [123, 34], here denoted Lin(2)(G). At the level of 1- +morphisms, a Vect-profunctor, H: G ↛ G′, of groupoids is then easily converted +into a +� +Lin(2)(G), Lin(2)(G′) +� +-bimodule, whose underlying vector space is +� +x∈ob(G),y∈ob(G′) +H(x, y). +Likewise, natural transformations of profunctors naturally linearise to bimodule +maps. +These simple observations allow us to define yet one more version of the once- +extended Quinn TQFT, the Morita valued once-extended Quinn TQFT, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor, +by considering the following composite of bifunctors: +2Cob +(n,n+1,n+2) +B +2QB +−−−→ vProfGrpfin +Lin(2) +−−−−→ Mor. +This is done in §6.4.5. +The algebra, 2QB(Σ, f Σ), that is associated to a B-decorated d-manifold de- +pends on the decoration, fΣ, of Σ. However, this dependence is up to a canonically +defined Morita equivalence, which is functorial with respect to further changes in +the decoration, and natural with respect to the bimodules associated to cobordisms. +1.6. Explicit calculations for classifying spaces of crossed complexes. Quinn’s +finite total homotopy TQFT, Q(s) +B , and its ‘finitary’ once-extended versions, 2QB +and 2Q +Mor +B +, can, in theory, be combinatorially calculated by passing to one of the +existing combinatorial models for homotopy theory, for instance, simplicial sets, or +simplicial groups, [41, 86]. Explicit formulae in this general setting will be deferred +to a future paper. +In the last part of this paper, Part 4, we will work within a ‘truncation’ of +homotopy theory, obtained by passing to the category of strict infinity groupoids, +or ω-groupoids in the nomenclature of [27], which we will consider in their equivalent +form as crossed complexes, following Brown and Higgins, [24]. (See also the more +recent monograph, [27], by Brown, Higgins and Sivera.) +We will give explicit formula for Quinn’s finite total homotopy TQFT, Q(s) +B , and +the two finitary versions of the once-extended Quinn TQFT, for cases in which B +is the classifying space, BA, of a homotopy finite crossed complex, A. Note that +the spaces of the form BA, where A is a homotopy finite crossed complex, do not +include all possible homotopy classes of homotopy finite spaces, but include, for +instance, those that are 2-types (i.e. whose homotopy groups, πi, vanish for i ≥ 3). +To this end, in Section 7, we review the homotopy theory of crossed complexes, +closely following work of Brown, Higgins, Sivera, [27, 26], and Tonks, [116]. Our +main new results are in §7.5.4, and, given a subsimplicial set, Y , of a simplicial +set, X, and a crossed complex, A, they give a crossed complex model for the fibres +of the induced fibration, TOP(|X|, BA) → TOP(|Y |, BA), obtained by restricting +a function, f : |X| → BA, to |Y |. This has direct application to giving explicit +formulae for Quinn’s finite total homotopy TQFT, and its extended versions. +In Section 8, we finally give explicit formulae for Q(s) +BA, 2QBA, and 2Q +Mor +BA , where +A is a homotopy finite crossed complex. The formulae are given in terms of what we + +A CATEGORIFICATION OF QUINN’S TQFT +10 +call simplicial stratifications, iX : |XΣ| → Σ, of manifolds, Σ, and analogously for +cobordisms between manifolds, and extended cobordisms. Here XΣ is a simplicial +set and iX is a homeomorphism. Simplicial stratifications are more general than +triangulations of manifolds, and typically allow us to decompose a manifold utilising +a smaller number of simplices. +It will not be necessary to prove that the formulae given do not depend on +the chosen simplicial stratifications, since they are instead proved to coincide with +quantities that are, by construction, topological invariants, except when it comes +to what the once-extended TQFTs assign to d-manifolds, where the dependence on +a simplicial stratification, only, up to naturally defined invertible profunctors, or +bimodules, is naturally a feature of the construction. +On that token, when treating the finitary once-extended versions of the Quinn +TQFT, derived from a homotopy finite crossed complex, we will address, in §8.3.4 +and §8.3.5, yet two more versions of the once-extended Quinn TQFT, denoted +2QA : 2Cob +(d,d+1,d+2) +st +−→ vProfGrpfin, +and +2Q +Mor +A +: 2Cob +(d,d+1,d+2) +st +−→ Mor. +Here the bicategory, 2Cob +(d,d+1,d+2) +st +, has objects pairs, (Σ, iXΣ), where Σ is a +closed smooth d-manifold, and iXΣ is a simplicial stratification of Σ. The 1- and +2-morphisms of 2Cob +(d,d+1,d+2) +st +are cobordisms, and diffeomorphism classes of ex- +tended cobordisms, without any choice of simplicial stratification. +This will, in turn, give rise to the construction of (albeit non canonical) once- +extended TQFTs, +� +2QA : 2Cob(d,d+1,d+2) −→ vProfGrpfin, +and +� +2QMor +A +: 2Cob(d,d+1,d+2) −→ Mor, +obtained by picking a simplicial stratification of each path-connected closed d- +manifold. This step in general requires using the choice axiom, but its full force is +not necessary when the domain bicategory of a once-extended TQFTs is restricted +to a ‘finitary’ sub-bicategory of 2Cob(d,d+1,d+2), for instance when considering pre- +sentations of the symmetric monoidal bicategory, 2Cob(1,2,3), as done for example +in [11, 10]. +One useful general theorem proved in this paper is the following (see Theorem +263, in Section 8). +Theorem. Let A be a finite crossed complex, and d a non-negative integer. We +have once-extended TQFTs, +� +2QA : 2Cob(d,d+1,d+2) −→ vProfGrpfin, +and +� +2QMor +A +: 2Cob(d,d+1,d+2) −→ Mor. +They can be ‘normalised’ so that, if Σ is any chosen path-connected closed d- +manifold, and iΣ : |XΣ| → Σ is a simplicial stratification of Σ, then +• � +2QA(Σ) is the groupoid whose objects are the crossed complex maps from +the fundamental crossed complex, Π(XΣ), of the simplicial set, XΣ, to A, + +A CATEGORIFICATION OF QUINN’S TQFT +11 +and whose morphisms are crossed complex homotopies (considered up to +2-fold homotopy) between those crossed complex maps from Π(XΣ) to A, +and +• � +2QMor +A +(Σ) is the groupoid algebra of � +2QA(Σ). +This is quite a general result, of which we will give some representative examples +in Subsection 8.4. The category of crossed complexes includes that of groupoids +and of strict 2-groups, as full subcategories. Taking A to be a finite group, G, or, +more generally, a finite groupoid, the theorem above gives a homotopy theoretical +interpretation, and a proof of existence, of the (0, 1, 2)-extended TQFTs derived, +as in [106, §3.8], from the fact that the groupoid algebra of a finite groupoid is a +“separable symmetric Frobenius algebra”, see [75, Example 5.1.]. Passing to the +(1, 2, 3)-extended TQFTs context, and considering a simplicial stratification of S1, +with single 0- and 1-simplices, and with A a finite group, � +2QMor +A +associates the +quantum double of the group algebra of G to S1. This gives a new proof of, and a +homotopy theoretical interpretation for, the fact that there exists a Morita valued +(1, 2, 3)-extended TQFT, sending S1 to the quantum double of the group algebra +of G, which is essentially proven in [92, 10, 100]. +We note that the overall construction is considerably more general, and it works +in all dimensions, and for all finite crossed complexes. In particular, we also develop, +at the end of the paper, the case when A = G, a crossed module of finite groups, +which is of relevance for higher gauge theory, [6, 3, 52]. Concretely, we write down, +in Subsection 8.4, some explicit formulae for the (1, 2, 3)- and (2, 3, 4)-extended +TQFTs derived from G. Passing to the language of discrete higher gauge theory, +as treated in [31, 93], the algebras that � +2QMor +G +associates to S1 and to the torus +coincide with the ‘tube algebras’ considered in [30, Sections 10 and 13], [32] and [33, +Section 3], in the context of models for excitations of topological phases, derived +from higher gauge theory. These algebras were one of the initial motivations for +the work in this paper. +A general result, directly following from the theorem above, is that if Σ is an +n-manifold, with a simplicial stratification, then there exists an (n, n + 1, n + 2)- +extended TQFT that sends Σ to the groupoid of discrete G-connections in Σ and +gauge transformation (considered up to 2-gauge transformation) between them. +We expect that, if G is a finite simplicial group – so that G can represent any +finite homotopy type by Ellis’ theorem, [46], – then there will similarly exist once- +extended TQFTs, +2QG : 2Cob +(d,d+1,d+2) +st +−→ vProfGrpfin, +and +2Q +Mor +G +: 2Cob +(d,d+1,d+2) +st +−→ Mor. +sending (Σ, iX : |XΣ| → Σ), to the groupoid of simplicial maps from XΣ to W(G), +the simplicial classifying space of G, and homotopy classes of maps between them, +up to 2-fold homotopy. In that case, 2QG(Σ, iΣ) will be the fundamental groupoid +of the simplicial function space, W(G)XΣ. We hope to address this in a future +publication. In particular, we expect that the recent construction of topological +invariants of 4-manifolds derived from 3-groups (2-crossed modules), in [102], is +a particular case of the Quinn finite total homotopy TQFT, using a 3-type, B, +represented by a 2-crossed module of finite groups. + +A CATEGORIFICATION OF QUINN’S TQFT +12 +In a future publication, we also expect to address the construction of homotopy +quantum field theories, including extended ones, derived from crossed complexes. +This should be closely related to the construction in [108]. +We also hope to address whether Quinn finite total homotopy TQFT can be +further categorified, and also explore its twisting by cohomology classes of homotopy +finite spaces. +Contents +1. +Introduction +2 +1.1. +The ‘classical’ Quinn finite total homotopy TQFT +3 +1.2. +The once-extended versions of Quinn’s finite total homotopy TQFT +5 +1.3. +The first version of the construction of the once-extended Quinn +TQFT +6 +1.4. +The finitary once-extended Quinn TQFT +8 +1.5. +The Morita-valued once-extended Quinn TQFT +8 +1.6. +Explicit calculations for classifying spaces of crossed complexes +9 +Part 1. +Preliminaries +13 +2. +Some general conventions and notation +13 +2.1. +General notation +13 +2.2. +Conventions for groupoids +14 +2.3. +Conventions for topological spaces +15 +2.4. +Review of fibrations +17 +2.5. +Review of fibre homotopy equivalence +19 +2.6. +Brief crib-sheet on manifolds, cobordisms, etc. +20 +2.7. +Conventions on bicategories +24 +2.8. +Conventions on profunctors +35 +Part 2. +The homotopy theoretical underpinning of Quinn’s finite +total homotopy TQFT +43 +3. +Homotopically finite spaces and the category HFspan +43 +3.1. +Homotopically finite (HF) spaces +43 +3.2. +Fibrant spans of HF spaces and their composition +47 +3.3. +A family of functors, R(s) : HFspan → Vect, derived from the +homotopy content +54 +4. +A more detailed review of Quinn’s finite total homotopy TQFT +60 +4.1. +Cobordism categories +61 +4.2. +Quinn’s results on HF function spaces +63 +4.3. +Quinn’s finite total homotopy TQFT +64 +4.4. +A discussion of the monoidality of Quinn’s finite total homotopy +TQFT +66 +4.5. +Some examples and properties of Quinn’s finite total homotopy TQFT 67 +4.6. +Changing B +69 +Part 3. +Once-extended versions of Quinn’s finite total homotopy +TQFT +72 +5. +The homotopy-theoretical underpinning of the once-extended Quinn +TQFT +73 +5.1. +Notation and some more basic results about fibrations +73 +5.2. +The profunctor construction +80 +5.3. +HF fibrant resolved 2-spans connecting fibrant spans +85 +5.4. +HF resolved fibrant 2-spans and natural transformations of profunctors 89 + +A CATEGORIFICATION OF QUINN’S TQFT +13 +5.5. +The horizontal composition of HF resolved 2-spans +98 +5.6. +The vertical composition of HF resolved 2-spans +108 +5.7. +Towards horizontal and vertical identities +112 +5.8. +Comment and Summary +115 +6. +Once-extended versions of Quinn’s TQFT +117 +6.1. +Conventions for the bicategory 2Cob(n,n+1,n+2) +117 +6.2. +A once-extended version of Quinn’s TQFT +121 +6.3. +A finitary version of the once-extended Quinn TQFT +125 +6.4. +The Morita valued extended Quinn TQFT +128 +6.5. +The symmetric monoidal structure in 2Cob(n,n+1,n+2) +139 +6.6. +The symmetric monoidal structure of the bifunctor 2QB +157 +Part 4. +Calculations for classifying spaces of ω-groupoids +163 +7. +Crossed complexes: their homotopy theory and their classifying spaces 165 +7.1. +Definition of crossed complexes, and related notions +165 +7.2. +Fundamental crossed complexes of filtered spaces +169 +7.3. +Homotopy of crossed complexes +173 +7.4. +The classifying space of a crossed complex +180 +7.5. +Fibrations of crossed complexes and profunctors from fibrant spans +188 +7.6. +Computing the homotopy content of a finite crossed complex +195 +8. +TQFTs and once-extended TQFTs derived from homotopy finite crossed +complexes +199 +8.1. +Conventions and nomenclature +201 +8.2. +TQFTs from homotopy finite and finite crossed complexes +202 +8.3. +The once-extended TQFTs derived from finite crossed complexes +206 +8.4. +Some explicit calculations for the once-extended TQFTs derived from +finite groupoids and 2-groups +214 +8.5. +Final note: why should we bother with crossed modules? +229 +References +230 +Part 1. Preliminaries +We will review some of the background theory in the various areas that will feed +into this paper. Many readers will not need to read these short sections and need +only refer to them when the ideas and results, mentioned here, are needed in later +sections. As we also set up some of the necessary notation, a quick skim through +to see what is here is probably necessary however. +2. Some general conventions and notation +2.1. General notation. +• Let V and W be vector spaces, over a field κ, with given bases, X and Y , +respectively. +We will specify a linear map, f : V → W, by giving its matrix +elements, denoted ⟨x | f | y⟩ ∈ κ, where x ∈ X and y ∈ Y , hence f(x) = +� +y∈Y ⟨x | f | y⟩y for each x ∈ X. +• If X is a finite set, then its cardinality is denoted |X| 4. +4We note that the same notation is used for the geometric realisation of a simplicial set, +X. +For most of the time, the risk of confusion is slight, and on the few occasions that both +notations occur in the same expression, we expect that the context allows the meaning to be +gleaned unambiguously with little pain. + +A CATEGORIFICATION OF QUINN’S TQFT +14 +• The category of sets is denoted Set. The category of κ-vector spaces will be +denoted Vect, or Vectκ. The category of topological spaces will be denoted +Top. +• If C is a monoidal category, the tensor product functor is denoted by ⊗C : C×C → +C. In the cases where the tensor product arises from a coproduct or a product +in C, we will also use the notation (respectively), +×C : C × C → C +and +⊔C : C × C → C. +2.2. Conventions for groupoids. A groupoid, G, will be denoted +G = (s, t: G1 → G0), +where G1 and G0 are, respectively, the set of morphisms and the set of objects of +G. Morphisms of G are frequently denoted as (s(g) +g−→ t(g)), so s(g) is the source +of g and t(g) its target. The identity of x ∈ G0 is denoted 1x = (x +1x +−→ x), or 1G +x . +Our convention for notation for composition in this context is that the composition +of (x +g−→ y) and (y +h−→ z) is (x +gh +−→ z). The set of arrows from x to y is denoted +G(x, y), or homG(x, y). The vertex group at x ∈ G is G(x) := homG(x, x). +A totally disconnected groupoid is a groupoid for which the source and target +maps coincide. Totally disconnected groupoids are frequently denoted in the style +A = (β : A1 → A0), where β := s = t. +We will identify a groupoid having just a single object with its group of mor- +phisms of that single object. +A groupoid, G, is said to be discrete if it has no non-identity arrows. In this case, +it is more or less indistinguishable from its set, G0, of objects. Such a groupoid +may also be called trivial. +We often identify a set, X, with the corresponding discrete groupoid having X as +its set of objects, and, of course, just the identity arrows as the arrows. This gives +an inclusion of the category of sets into that of groupoids. This inclusion functor +has a left adjoint, sending a groupoid, G, to the set of connected components, +π0(G). For basic information on the theory of groupoids, see [21]. +A simple, but useful, example of a groupoid is the ‘unit interval groupoid’, often +denoted I. In this groupoid, we have objects 0 and 1, and only two non-identity +morphisms, 0 +(0,1) +−−−→ 1 and 1 +(1,0) +−−−→ 0, with the evident compositions and identities. +We will often think of groupoids as modelling very simple homotopy types (1- +types). We will also recall the notion of homotopically finite space; see Subsection +3.1. Combining the two notions, we will have a notion of homotopy finite, or ho- +motopically finite groupoid. This is just one of several related finiteness conditions +on groupoids, so we list some of the main ones that may be used later on. +• A groupoid, G, will be said to be finite if both G0 and G1 are finite sets. +• G may be called locally finite if each ‘hom-set’ G(x, y) is a finite set, (so in +particular its vertex groups are all finite). +• G will be called homotopically finite (or briefly to be a HF-groupoid) if it +has finitely many connected components and each vertex group, G(x), is a +finite group, (so both π0(G) and π1(G, x) ∼= G(x) are finite for each possible +base point, x ∈ G0). +We introduce some notation that may be used in this context. +• Grp will be the category of groupoids, +• finGrp will be that of finite groupoids, +• locfinGrp that of locally finite groupoids, + +A CATEGORIFICATION OF QUINN’S TQFT +15 +and +• hfGrp that of HF-groupoids. +2.3. Conventions for topological spaces. We will require a certain background +of concepts and notation when handling topological spaces, not all of which is +considered in many sources on topology. +(1) Recall that a space X is called weak Hausdorff, see [87, 111] and [114, §7.9], +if given any continuous map, f : K → X, where K is compact Hausdorff, then +f(K) is closed in X. +(2) We will denote by CG the full subcategory of the category Top of topological +spaces with objects the compactly generated spaces; for definitions see, for in- +stance, [111], [66, §2.4], [114, §7.9] or [55, page 242] – note that these are called +k-spaces in [87, 55, 66]. +(3) We have a k-ification functor, denoted k: Top → CG. Definitions are in [55, +page 242] and [114, §7.9]. It is a right adjoint to the inclusion functor CG → Top; +see [55, page 243]. If X is a space, then the map k(X) → X given by the identity +function, which we will sometimes denote ǫX : k(X) → X, is continuous. This +gives the counit of the adjunction. If f : X → Y is a continuous map between +topological spaces, then k(f): k(X) → k(Y ) is f itself, at the level of maps +between sets. +(4) If K is a compact Hausdorff space, then a set map, f : K → k(X), is continuous +if, and only if the same map f : K → X is continuous. +(The same holds if +K is compactly generated.) +In particular, by noting that all disks, Dn, are +compact Hausdorff, it follows that the map ǫX : k(X) → X is a weak homotopy +equivalence. +(5) It follows by the discussion above that if X is weak Hausdorff, then so is k(X). +(6) As in [87, 55, 113], we will work in the category CGWH, the full subcategory +of Top with objects the compactly generated and weak Hausdorff topological +spaces, (which we will refer to as CGWH spaces). These include all compact +Hausdorff spaces and all metric spaces. Recall that CGWH has all small limits +and colimits, [80, 111]. Note further, see [66, Proposition 2.4.22], the limits in +CGWH are computed by computing the limits in Top, and then applying the +k-ification functor, so, for example, given a pair of CGWH spaces, their product +is X × Y = k(X ×0 Y ), where ×0 is the product in Top. +Unless otherwise +specified, all limits and colimits of CGWH spaces will be calculated in CGWH. +(7) We note, moreover, that CGWH is a cartesian closed monoidal category, [80, +111]. Given CGWH spaces, X and Y , the space of maps from X to Y , will +be denoted both by Y X and by TOP(X, Y ). If X is compact Hausdorff, the +topology on Y X is the k-ification of the compact-open topology on the set of +maps from X to Y . +(8) As in [111, 80], a subset, A, of a CGWH space, X, will be always be given the +k-ification of the topology induced by X, called the CGWH induced topology. +Note that +• if F is closed in X, then F with the induced topology from X is already +CGWH, so k will not modify the topology, hence the CGWH induced topol- +ogy on F is the usual induced topology on F as a subspace of X; +• if A ⊂ B ⊂ X, then the k-ification of the topology that X induces on A +coincides with the k-ification of the topology that B, with the k-ification of +the induced topology from X, induces on A; +• the inclusion, A → X, is continuous, +and + +A CATEGORIFICATION OF QUINN’S TQFT +16 +• if A ⊂ X, and f : Y → A, with both X and Y being CGWH spaces, then f +is continuous (where A has the k-ification of the induced topology) if, and +only if, f is continuous when considered as a map from Y to X. +(9) Given a CGWH space, X, then XI denotes the space of maps from I = [0, 1] to +X with the k-ification of the compact-open topology. We have continuous maps, +which we will often denote, s := sX, t := tX : XI → X with s(γ) = γ(0) and +t(γ) = γ(1). The notation emphasises that these pick out the source and target +of a path. Alternative notation will sometimes be used, for instance, e0(X) and +e1(X), standing for evaluation at 0 or 1, as these are more natural if considering +XI as being a ‘cocylinder’ on X, i.e. dual to the usual cylinder X × I, when +dealing with homotopies between maps. In a particular circumstance, in which +such a notation might lead to clashes or confusion, we may substitute other +notation, but will note the substitution nearer the place it is used5 +(10) Given a CGWH space X, and x ∈ X, we write Fx(X) for the space of paths, +γ : I → X, starting at x, and Ωx(X) for the space of all paths in D starting and +ending in x. These are given the CGWH induced topology. +(11) Given a CGWH space X, and j ∈ {0, 1}, also define the inclusions, ιX +j +:= +ιj : X → X × I, by ιj(x) = (x, j). These are continuous. (We may occasionally +use simplified notation for these end inclusions.) +(12) Given a CGWH space X, we define π0(X), as usual, as the coequaliser, +π0(X) := Coeq(XI +s +−→ +−→ +t +X). +We note, however, that π0(X) is a CGWH space, and not just a set, and need +not have the discrete topology here. There is, of course, a natural projection +map, p : X → π0(X), and the fibres of that map (in the category CGWH) are +obtained by pulling back along maps from singletons to the space π0(X). The +safe way to handle this (so as not to end up with spaces that are not CGWH) is +given in the following. +(13) Given a CGWH space, X, and an element, x ∈ X, the path-component that x +belongs to will be denoted by PCx(X). Each path component of X is given the +k-ification of the topology induced by X. This CGWH interpretation of path +component corresponds to the formula PCx(X) = p−1(p(x)). +(14) The category with objects the CGHW spaces and morphisms from X to Y , being +the maps, X → Y , considered up to homotopy will be denoted CGWH/ ≃. +(15) We will consider a functor, �π0 : CGWH/ ≃ → Set. This sends a CGWH space, +X, to the set, �π0(X), of (k-ified) path components in X, in other words to the set +of PCx(X), for x ∈ X. Of course, we note that two or more different elements of +X may correspond to, and hence may label, the same path component. (We do +not want to invoke the axiom of choice and, for instance, choose a representative +point in each path component.) +Let X and Y be CGWH spaces. Given a homotopy class, [f]: X → Y , of +maps from X to Y , we put +�π0(f) +� +PCx(X) +� +:= PCf(x)(Y ). +There is an obvious one-to-one correspondence, π0(X) ↔ �π0(X), if we forget +the topology in π0(X). Throughout the paper, it will be useful to distinguish +between π0(X) and �π0(X). +We will write �π0(X) = {PCx(X) | x ∈ X}, but note again that different x in +X may correspond to the same element PCx(X) ∈ �π0(X). +5An instant of this is when handling simplicial sets, where s is used, almost universally in the +literature, for the degeneracy mappings, so conflicts with s as ‘source’. + +A CATEGORIFICATION OF QUINN’S TQFT +17 +(16) Another very useful way to view �π0(X) is as actually being the set consisting of +the fibres of the continuous function, p: X → π0(X), mentioned above. As fibres +naturally are k-ified, this provides a quick categorical way of handling the index- +ation. The same double interpretation of the set of connnected components will +be useful later on when we look at connected components of crossed complexes, +see page 168. +2.4. Review of fibrations. Let us recall, for instance from [87, Chapter 7] or +[113, Chapter 5] the definition. +Definition 1 (Fibration). Let E and B be CGWH spaces. We say that p: E → B +is a Hurewicz fibration (abbr. fibration) if the following homotopy lifting property +holds: given any CGWH space X, any homotopy, H : X × I → B, and any map, +ˆf : X → E, making the diagram with solid arrows, below, commutative, then there +exists a map, H′ : X × I → E, making the full diagram commutative. +X +ι0 +� +ˆ +f +� E +p +� +X × I +H′ +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +H +� B. +This H′ : X × I → E is called a lifting of H starting at ˆf. +Remark 2. Differently from the conventions in [87, Chapter 7], we do not impose +that fibrations are surjective6, hence, given any space B, we have a fibration, ∅ → B. +We recall that: +Lemma 3. • The composite of fibrations is a fibration. +• If X and Y are CGWH-spaces, then pX : X × Y → X and pY : X × Y → Y are +both fibrations. +• Pullbacks of fibrations are fibrations. This means that if p: E → B is a fibration, +and f : X → B is any map (of CGWH spaces), then the map, q: X ×B E → X, +appearing in the pullback diagram below is a fibration ([87, §6.1 Lemma]), +X ×B E +q +� +� E +p +� +X +f +� B. +If p: E → X is a fibration, then, given x ∈ X, the fibre of p at x is +Ex := p−1(x). +Following our conventions in Subsection 2.3, the fibre Ex is given the induced +CGWH-topology from E, i.e. the k-ification of the topology that E induces on Ex. +Given that p is continuous, p−1(x) is closed in E, so p−1(x) is compactly generated +already, with the induced topology, so the k-ification step will not modify the +topology in Ex. +We also note that we have the following pullback diagram in CGWH, +p−1(x) +� +inc +� E +p +� +{x} +inc +� X, +6This surjectivity condition was dropped in the subsequent [88]; see footnote on page 25. + +A CATEGORIFICATION OF QUINN’S TQFT +18 +where the inc denote the obvious inclusion maps. +We will make extensive use of the fact that if p: E → B is a fibration, and +x, y ∈ B are in the same path-component, then the fibres, p−1(x) and p−1(y), are +homotopy equivalent; see e.g. [87, Chapter 7]. We will also need that if E is path- +connected and x ∈ X, it follows that all path-components of Ex are homotopically +equivalent, [48, Proposition 3]. We will review some of these results in more detail, +later, starting with Lemma 94, page 74. +Let A is a subset of X, being considered with the k-ification of the induced +topology, as above. Consider EA = p−1(A), with the induced CGWH-topology, +and the induced map, pA : EA → A. It is easy to see that we have a pullback +diagram in CGWH: +EA +pA +� +inc +� E +p +� +A +inc � X. +Indeed given a CGWH space, Z, and continuous maps, f : Z → E and g : Z → A, +such that p ◦ f = inc ◦ g, there is a unique set map, f ′ : Z → EA, obtained by +restricting the codomain of f to EA, which arises from the fact that the diagram +above gives a pullback in Set. The map, f ′, is continuous as a map f ′ : Z → E, +hence it is continuous as a map Z → EA, since Z is CGWH and EA has the k- +ification of the induced topology, so the diagram above is a pullback in CGWH +as well. Since pullbacks of fibrations are fibrations, we have that pA : EA → A is a +fibration. +More generally, suppose that p: E → A is a fibration. +Let e ∈ E and set +x = p(e). It is easy to see that p(PCe(E)) = PCx(X). Moreover the induced map +pe : PCe(E) → PCx(X) is a surjective fibration. This is [109, Lemma 2.3.1], (which +can easily be adapted to the CGWH setting). +2.4.1. Cofibrations. Looking at the dual setting, recall that a map, f : A → X, +of CGWH spaces is a cofibration, [87, Chapter 6], or [113, §5.1], if it satisfies the +homotopy extension property: +For any CGWH space, B, any map, g : X → B, and any homotopy, +H : A × I → B, as in the diagram, +A +ιA +0 � +f +� X +ιX +0 � +g +� B +A × I +H +� +f×idI � X × I +H′ +�① +① +① +① +there is a homotopy, H′, making the diagram commute, so extend- +ing the homotopy H. +The following two well known results will be used without further comment. +• Let f : A → X be a cofibration and let B be a CGWH space, then the induced +map on mapping spaces, f ∗ : BX → BA, sending φ: X → B to φ ◦ f : A → B, is +a fibration. (For example, see [87, Section 7.2].) +• If (X, Y ) is a CW-complex pair, meaning that Y is a subcomplex of the CW- +complex, X, then the inclusion map i: Y → X is a cofibration; see, for instance, +[55, Corollary 1.4.7]. + +A CATEGORIFICATION OF QUINN’S TQFT +19 +2.5. Review of fibre homotopy equivalence. Let f, g : X → Y be maps of +CHWH spaces. A homotopy, H : X × I → Y , connecting f to g, will frequently be +denoted by f +H +=⇒ g. +Given two fibrations, p: X → B and q: Y → B, over the same space, a fibre +map, or fibred map, f : X → Y , is a map such that the diagram below commutes, +(2) +X +p +�❅ +❅ +❅ +❅ +❅ +❅ +❅ +❅ +f +� Y +q +�⑦⑦⑦⑦⑦⑦⑦⑦ +B +Two fibre maps, f, g : X → Y , are fibre homotopic if there exists a homotopy, +H : X × I → Y , called a fibre or fibred homotopy, connecting f and g, and such +that for each (x, t) ∈ X × I, p(x) = q(H(x, t)). We write f +H +=⇒ +B +g. +We say that a pair of fibre maps, f : X → Y and f ′ : Y → X, realises a fibre +homotopy equivalence if we have fibre homotopies, +f ◦ f ′ H +=⇒ +B +idX and f ′ ◦ f +H′ +==⇒ +B +idY . +This means that H : X × I → X satisfies p(H(x, t)) = p(x), for each (x, t) ∈ X × I, +and similarly for H′. +The following non-immediate, but well-known, result will be used later. It is the +dual version of Dold’s Theorem; see the discussion and references in [69, Chapter +I, section 6, p. 33]. +Lemma 4 (Fibre homotopy equivalence). Suppose that in (2), f : X → Y is a +homotopy equivalence, then there exists a homotopy inverse, f ′ : Y → X, of f, +which is a fibre map, and such that f and f ′ realise a fibre homotopy equivalence. +For a proof, see [87, Chapter 7.5], [23, Theorem 3.4]. Very thorough discussions +in the dual case of cofibrations appear in [21, 7.4.2: Addendum] and in [69], as +mentioned above. +Almost by definition, we have: +Lemma 5. Suppose that f : X → Y , as in (2), is a fibre homotopy equivalence. +Given any b ∈ B, the map, f, restricts to a homotopy equivalence, p−1(b) → +q−1(b). +□ +We will also need a ‘relative’ version of the result from Lemma 4. This is also +to be found in [87, Chapter 7.5, p. 53]. First we need two definitions. The first +is simply the restriction of the notion of morphism between maps in a category, +applied to CGWH and, in particular, to fibrations there. +Definition 6. Given two fibrations, p : D → A and q : E → B, a map from p to q +is a pair, (g, f), of maps as in the square, +D +g +� +p +� +E +q +� +A +f +� B, +making that square commute. We write (g, f) : p → q. +Definition 7. Given two fibrations, p: D → A and q: E → B, a map, (g, f): p → +q, as above, is a homotopy equivalence of fibrations if there are homotopy inverses, +f ′ of f and g′ of g, such that p ◦ g′ = f ′ ◦ q, and, in addition, there are homotopies, + +A CATEGORIFICATION OF QUINN’S TQFT +20 +H : g′ ◦ g ≃ idD, and K : g ◦ g′ ≃ idE, that cover homotopies, h: f ′ ◦ f ≃ idA and +k: f ◦ f ′ ≃ idB. +The relative version of Lemma 4 is then: +Proposition 8. If (g, f): p → q is a map of fibrations in which both g and f are +homotopy equivalences, then (g, f): p → q is a homotopy equivalence of fibrations. +□ +A proof can be given by pulling q back along g, and then applying Lemma 4. +There is also a relative version of Lemma 5. +Corollary 9. Suppose p: D → A and q: E → B are fibrations, and (g, f): p → q +is a homotopy equivalence between them, then, for any a ∈ A, the induced map on +fibres, f : p−1(a) → q−1(f(a)) is a homotopy equivalence. +□ +We note that, as this assumes that f is a homotopy equivalence, π0(f) is a +bijection and so, if b ∈ B, there is an a ∈ A with f(a) ∈ PCb(B). This means that +q−1(b) is homotopy equivalent to p−1(a). The actual homotopy equivalence will, of +course, depend on the path used to join f(a) and b. +2.6. Brief crib-sheet on manifolds, cobordisms, etc. Let n be a non-negative +integer. A topological manifold of dimension n is a Hausdorff and second countable +topological space, S, such that each point of S has a neighbourhood homeomorphic +to an open subset of the upper half plane of Rn. A smooth manifold, (S, smtS), is +a pair, consisting of a topological manifold, S, and a smooth structure, smtS, on S; +see, for instance, [65], or [89, §1]. We call S the underlying topological manifold of +(S, smtS), and will usually abbreviate (S, smtS) to S, when the context makes this +unambiguous. We note that a topological manifold being smooth is a structure, +not a property, and some topological manifolds do not have a smooth structure at +all. +If M is a compact smooth manifold, then it can be given a finite triangulation, +and, in particular, it can be given the structure of a finite CW-complex, see [94]. +We also note that if M is a smooth manifold with border, then we can find, again +see [94], a triangulation of the pair, (M, ∂M), making ∂M a subcomplex of M. In +particular, the inclusion, ι: ∂M → M, is a cofibration. +We next give a fairly standard definition of cobordism. We will need to shift +the viewpoint slightly before generalising this to ‘extended cobordisms’, but we will +do that in two stages, rewording the definition and notation, before giving that +generalisation. General references include [72], for the standard idea of cobordism, +and, for the higher dimensional extended case, see [106, 91, 92]. +Definition 10. Two n-manifolds, Σ1 and Σ2, are said to be cobordant, if they +‘jointly’ or ‘together’ (‘co’) form the ‘bord’, i.e. +the boundary, of an (n + 1)- +manifold, S. This means that there are embeddings, ij : Σj → ∂S → S, for j = 1, 2, +so that the induced map, Σ1 ⊔Σ2 → ∂S, is a homeomorphism (or a diffeomorphism +if we are working with smooth manifolds). +This gives a cospan in CGWH, +Σ1 +i1 +�❏ +❏ +❏ +❏ +❏ +❏ +Σ2 +i2 +�tttttt +S +, +and we note that +� i1 +i2 +� +: Σ1 ⊔ Σ2 → S is an embedding, and is thus a cofibration, +as it is, essentially, given by the inclusion of the boundary of a smooth manifold + +A CATEGORIFICATION OF QUINN’S TQFT +21 +into the manifold. (Here +� i1 +i2 +� +is the obvious induced morphism from the disjoint +union, Σ1 ⊔ Σ2, to S.) We say that the cospan is cofibrant. Such cofibrant cospans +are treated in [117, 118]. +In the above setting, we say: +Definition 11. The (n + 1)-manifold, S, or, more precisely, the 5-tuple, +(S, Σ1, Σ2, i1, i2), +is called a cobordism from Σ1 to Σ2. +If the dimension is not evident from the context, we may say that S is an (n+1)- +cobordism, or even, occasionally, an (n, n + 1)-cobordism. +The manifold, Σ1, is often referred to as the inward boundary, whilst Σ2 is then +the outward boundary of the cobordism. +Example 12. For any n-manifold, Σ, the trivial or identity cobordism, is given +by S = Σ × I, for I = [0, 1], with the two ends being copies of Σ, via the obvious +embeddings, +ei(Σ) : Σ ∼= Σ × {i} ֒−→ Σ × I, +i = 0, 1. +Cobordisms compose in the well-known way by gluing the outward boundary +of the first to the inward boundary of the second. +This is only defined up to +isomorphism, so it is usual to pass to diffeomorphism classes of cobordisms, relative +to the boundaries. With smooth cobordisms, a collar on the boundaries has to be +chosen so as to ensure that the resulting composite cobordism also has a smooth +structure; see [89, §1 and §3] or [37]. A collar also comes in in the proof that the +trivial cobordism acts as an identity arrow for the composition. We refer the reader +to the various sources mentioned earlier, but will be considering related issues later +on. +We will need to consider certain types of cobordisms between cobordisms, which +are thought of as being certain ‘2-cospans’. To introduce these, we will need to +have the notion of a manifold with corners. The study of manifolds with corners +was originally developed by Cerf, [39], and Douady, [44], in the early 1960s, as a +natural generalisation of the concept of manifolds with boundary. +A (differential) manifold with corners is a generalisation of the standard defini- +tion above obtained by replacing the upper half plane of Rn by Rn ++, that is [0, ∞)n. +We thus have maximal atlases that consist of families of compatible charts, where +the charts, (ϕ, U), are of the form +ϕ : U → [0, ∞)n, +and where two charts, (ϕi, Ui), for i = 1, 2, are said to be compatible if, and only if, +ϕ2 ◦ ϕ−1 +1 +: ϕ1(U1 ∩ U2) → ϕ2(U1 ∩ U2) +is a diffeomorphism; see, for instance, the original sources and Laures, [77], for more +details. +For each x ∈ X, with x ∈ U, the number, c(x), of zeros in the coordinates of +ϕ(x) does not depend on the choice of chart, (U, ϕ). That number is called the +index of x. +Definition 13. A manifold with corners is a topological manifold with boundary +equipped with a maximal atlas of charts, as above. +A face of a manifold with corners is the closure of some connected component of +the set of points, {x ∈ X | c(x) = 1}. + +A CATEGORIFICATION OF QUINN’S TQFT +22 +We often need to know how the faces fit together. For this, J¨anich, [67], in- +troduced the notion of an ⟨n⟩-manifold, which is reviewed by Laures, [77], in the +context that we need. +Definition 14. A manifold with corners is called a manifold with faces if each +x ∈ X belongs to c(x) different connected faces. +There is an obvious set of simple examples of such objects. +Example 15. Take X = Rm ++. This is a manifold with faces in an obvious way. +The faces are the m-different coordinate hyperplanes, say Hi = {x ∈ Rm ++ | xi+1 = +0}, i = 0, . . . , m − 1, and adopting the obvious order (H0, . . . , Hm−1), and taking +∂i = Hi we note that each Hk is a manifold with faces using the faces Hi ∩ Hk, for +i ̸= k. +This gives us the basic examples of the following: +Definition 16. An ⟨n⟩-manifold is a manifold, X, with faces together with a spe- +cific ordered n-tuple, (∂0X, ∂1X, . . . , ∂n−1X), of faces of X which satisfy the con- +ditions: +1) ∂0X ∪ ∂1X ∪ . . . ∪ ∂n−1X = ∂X, the boundary of X; +2) ∂iX ∩ ∂jX is a (possibly empty) face of both ∂iX and ∂jX if i ̸= j. +Any ⟨n⟩-manifold gives rise to an n-dimensional cubical diagram of topological +spaces. More precisely let 2 be the ordered set, {0 < 1}. This gives a category with +two objects plus a single non-identity arrow, 0 ← 1. We set ⟨n⟩ to be the n-fold +product category, 2n. Given an ⟨n⟩-manifold, X, we can form a functor, +X : ⟨n⟩ → CGWH, +as follows. If a = (a0, . . . , an−1) is an object of ⟨n⟩, set X(a) = � +ai=1 ∂iX, with +X(0, . . . , 0) = X, the morphisms in ⟨n⟩ are then sent to the evident inclusions. +We note that a ⟨0⟩-manifold is just a manifold without boundary. A ⟨1⟩-manifold +is a manifold with boundary and a ⟨2⟩-manifold is a manifold with corners with +∂X = ∂0X ∪ ∂1X, and ∂0X ∩ ∂1X consists of ‘corners’ of index 2. +For us, the key case will be that with n = 2 and a ⟨2⟩-manifold is a manifold +with corners, with the specified faces being (∂0X, ∂1X) and it gives +∂0X ∩ ∂1X +� +� +∂0X +� +∂1X +� X. +Laures, in [77], introduced cobordisms of ⟨n⟩-manifolds. We will adopt and adapt +the form of his definition that is given in [106]. +We will shift our viewpoint, and notation, on cobordisms, ready for the extended +higher dimensional case. Let Y1 and Y2 be smooth closed n-manifolds. If W is +a cobordism from Y1 to Y2, then it is an (n + 1)-manifold with boundary, and +hence a ⟨1⟩-manifold, but, in addition, there is a specified decomposition of ∂W, +traditionally written something like ∂inW ⊔ ∂outW, together with embeddings, as +before, i1 : Y1 → ∂W, i2 : Y2 → ∂W, so that i1(Y1) = ∂inW and i2(Y2) = +∂outW, making i := +� i1 +i2 +� +: Y1 ⊔ Y2 → ∂inW ⊔ ∂outW = ∂W an isomorphism / +diffeomorphism. +For later use, we note a cobordism, as above, corresponds to a cospan, +Y1 +i1 +−→ W +i2 +←− Y2, + +A CATEGORIFICATION OF QUINN’S TQFT +23 +but also note that although this process preserves the composition (up to isomor- +phism), it does not preserve identities. The identity cobordism on a manifold, Y, +gives the cospan +Y +e0(Y ) +−−−−→ Y × I +e1(Y ) +←−−−− Y, +whilst the identity cospan on Y is +Y +idY +−−→ Y +idY +←−− Y. +Let now W0, W1 be two (n, n + 1)-cobordisms from Y1 to Y2. We will write +∂Wi = ∂inWi ⊔ ∂outWi ∼= Y0 ⊔ Y1, for i = 0, 1. +Definition 17. An (extended) (d − 2, d − 1, d)-cobordism from W0 to W1 will be a +⟨2⟩-manifold, S, equipped with +• a decomposition and isomorphism, +W0 ⊔ W1 +g−→ ∂0,inS ⊔ ∂0,outS = ∂0S; +• a decomposition and isomorphism, +(Y1 × I) ⊔ (Y2 × I) +f−→ ∂1,inS ⊔ ∂1,outS = ∂1S; +so that +g−1 ◦ f : Y1 × {0} ⊔ Y2 × {0} → ∂inW0 ⊔ ∂outW0, +and +g−1 ◦ f : Y1 × {1} ⊔ Y2 × {1} → ∂inW1 ⊔ ∂outW1, +which are to be compatible with the structural isomorphisms of W0 and W1 and the +decompositions. +A good way of getting a picture of such a structure is via a double cospan diagram +of a special kind. (It is in fact a cofibrantly resolved double cospan, but to justify +that name here would prolong the discussion too much for a mere ‘crib-sheet’.) +This looks like: +Y1 +i1,0 +� +e0(Y1) +� +W0 +gin +� +Y2 +i2,0 +� +e0(Y2) +� +Y1 × I +fin +� S +Y2 × I +fout +� +Y1 +i1,1 +� +e1(Y1) +� +W1 +gout +� +Y2 +i2,1 +� +e1(Y2) +� +where the top and bottom rows give the two cobordisms, W0 and W1, but the left +and right ‘vertical’ cobordisms are trivial / identity ones. Such a double cobordism +is not the most general form one could imagine, but corresponds to the fact that +such ‘2-cospans’ form something more like a bicategory with these as the 2-cells, +rather than a double bicategory of 2-cospans in some sense. This point is explored +in Jeff Morton’s paper, [91]. That being said, a second point to note is that the left +and right cospans are not trivial cospans, as such would have form Y +idY +−−→ Y +idY +←−− Y , +but that could not correspond to a cobordism as the middle term would need to be +of higher dimension than the ends. +We will be applying mapping space constructions, BY , etc., to these various +types of cobordisms and cospans, and will then take up some of these ideas again. + +A CATEGORIFICATION OF QUINN’S TQFT +24 +2.7. Conventions on bicategories. We will have frequent need to use the termi- +nology of the theory of bicategories7. +2.7.1. The basics of bicategories. A basic introduction to this theory can be found +in Leinster’s [78], with a more thorough and complete description given in Borceux’s +[17]. We may also use the summary to be found in [106], and the relevant parts +of the draft book, [68], by Johnson and Yau. We will also use that last reference +as one of the sources for definitions relating to monoidal categories and monoidal +bicategories. +Definition 18. A bicategory, B, is specified by the following: +• a collection of objects, denoted Ob(B), or sometimes B0; +• for each pair of objects, a, b in B, a locally small category, B(a, b), whose ob- +jects are 1-morphisms from a to b, whose morphisms are called 2-morphisms +and whose composition is sometimes referred to as vertical composition; +• for objects, a, b, c in B, there are composition functors, +ca,b,c : B(a, b) × B(b, c) → B(a, c), +and, for each object a in B, a functor, Ia : [0] → B(a, a), (where [0] is the +‘singleton’ category). The functors, c, are called horizontal compositions; +and +• natural isomorphisms, +α : ca,b,d ◦ (cb,c,d × id) ⇒ ca,c,d ◦ (id × ca,b,c), +λ : ca,b,b ◦ (Ib × id) ⇒ id, +and +ρ : ca,a,b ◦ (id × Ia) ⇒ id, +called, respectively, the associator and the left and right unitors. +These are required to satisfy the pentagon and triangle identities, which we omit +here, referring the reader to Borceux, [17], and the many other sources in the liter- +ature. +Notation 19. Although when discussing specific bicategories, we will more often +than not use generic composition symbols such as • or ◦, but when it is clear whether +we intend horizontal or vertical composition, it can be useful to have available some +specific notation that distinguishes them. In such cases, we may use #0 for hori- +zontal composition and #1 for the vertical one. The motivation for the symbolism +is that in horizontal composites the ‘intersection’ of the two 2-morphisms is the +object which is the codomain of the first 2-morphism and the domain of the second. +Objects live in dimension 0 of the data specifying the bicategory, hence #0 would +be used. For vertical composition of suitable 2-morphisms, the ‘intersection’ is a +1-morphism, so lives in dimension 1. +The two classes of examples of bicategories usually given are the following. +(1) Any (strict) 2-category is a bicategory in which the natural isomorphisms, +α, λ and ρ, are all identity arrows, (in particular, the composition is associa- +tive, and the identities are identities ‘on the nose’). We thus have that any +category gives a ‘locally discrete’ bicategory in which each hom-category, +B(a, b), is ‘just a set’, or more exactly is a discrete category. +(2) Any monoidal category, (M, ⊗, I), gives a bicategory, M[1], in which there +is just one object, which we will denote by ∗, and where M[1](∗, ∗) = M. +The composition, c∗∗, is the tensor, ⊗, and the identity, I∗, is the monoidal +identity, I. +7We recall that these are weak 2-categories. + +A CATEGORIFICATION OF QUINN’S TQFT +25 +We will see several other examples of bicategories later on coming from spans, +profunctors, both set and vector space valued ones, from bimodules, and, of course +given the context, from cobordisms. Much of this paper will involve checking that +various constructions are compatible with the bicategorical properties of their do- +main and codomain. +Definition 20. Let A and B be bicategories. A bifunctor, sometimes also called +a homomorphism, or a pseudo-functor8, F : A → B, or more completely (F, ϕ), +consists of +(1) a function, F : A0 → B0, mapping objects to objects; +(2) for each pair of objects, a, a′ in A, a functor, +Fa,a′ : A(a, a′) → B(F(a), F(a′)); +(3) natural isomorphisms, ϕa0,a1,a2, for each triple, a0, a1, a2, of objects in A, +as shown in the diagrams, +A(a0, a1) × A(a1, a2) +cA +� +Fa0,a1 ×Fa1,a2 +� +✖✖✖✖� +ϕa0,a1,a2 +A(a0, a2) +Fa0,a2 +� +B(F(a0), F(a1)) × B(F(a1), F(a2)) +cB � B(F(a0), F(a2)) +and, for each object, a in A, +[0] +IA +a +� +IB +F (a) +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +☞☞☞☞� +ϕa +A(a, a) +Fa,a +� +B(F(a), F(a)) +such that certain diagrams, expressing compatibility with the corresponding +associators and unitors, commute, and, again, we will not give them here +as they can easily be found in the literature and we will give them in a +simplified case slightly later. +If the ϕa0,a1,a2 and ϕa are all identities, F is said to be a strict homo- +morphism. +It may sometimes be useful to weaken the conditions on the ϕ requiring that +they just be natural transformations. In this case, the term ‘morphism’ may be +used, as in Leinster’s notes, [78]. +As any 2-functor, F : A → B, between two 2-categories, yields a homomorphism +between the corresponding bicategories, the above should be seen as a natural gen- +eralisation of a 2-functor. It is, thus, natural to consider transformations between +homomorphisms between bicategories. +Notation: Given a 1-morphism, b1 +h−→ b2, in a bicategory B, we denote by h∗ +and h∗, the natural transformations / induced morphisms, +h∗ : B(b, b1) → B(b, b2), +and +h∗ : B(b2, b) → B(b1, b). +8We will tend to use this last term in a particular case. + +A CATEGORIFICATION OF QUINN’S TQFT +26 +Definition 21. Let (F, ϕ), (G, ψ) : A → B be two homomorphisms between bicate- +gories. A transformation, also called a pseudo-natural transformation, σ : F → G, +is given by +• 1-morphisms, σa : F(a) → G(a), for each object a in A; +• natural isomorphism, as in the diagram, +A(a, a′) +Fa,a′ � +Ga,a′ +� +✒✒✒✒� +σa.a′ +B(F(a), F(a′)) +(σa)∗ +� +B(G(a), G(a′)) +(σa′ )∗ +� B(F(a), G(a′), +and, thus, for each f : a → a′, a 2-morphism, σf : G(f)σa ⇒ σa′F(f). +As before we omit the conditions for compatibility with the other struc- +ture, referring to the literature. +Given the fact that bicategories have an extra layer of structure than do cate- +gories, it is not surprising that not only does one consider bifunctors / homomor- +phisms, and transformations between them, but also some sort of 2-transformation +between transformations. These are called modifications. +Definition 22. Given F, G : A → B, as before, and σ, θ : F → G, two transforma- +tions, a modification, Γ : σ → θ, consists of a 2-morphism, Γ1 : σa ⇒ θa, for every +object a in A. These are required to make the following square commute, +G(f)σa +id#Γa � +σf +� +G(f)θa +θf +� +σa′F(f) +� θa′F(f), +for every 1-morphism f : a → a′ in A. +Remark 23. We refer the reader to [106] for how to compose transformations, +etc., so that one gets a bicategory Bicat(A, B) with the resulting structure, provided +the bicategories are small. We will not be using this that much, so will refer to the +sources if we need to use it, rather than introduce more conventions here +It will be useful, from time to time, to be able to use the following analogues of +some basic categorical notions adapted to work internally within a bicategory. +We suppose that A is a bicategory. +Definition 24. (i) Given a pair of 1-morphisms, f : A → B and u : B → A, in +A, we say f is left adjoint to u9, and written f ⊣ u, if there are two 2-morphisms, +η : 1A ⇒ uf, +and +ε : fu ⇒ 1B, +such that the following equations hold: +(u +∼ += +←→ 1Au +η·u +−−→ uf · u +∼ += +←→ u · fu +u·ε +−−→ u · 1B +∼ += +←→ u) = idu, +and +(f +∼ += +←→ f1A +fη +−→ f · uf +∼ += +←→ fu · f +εf +−→ 1B · f +∼ += +←→ f) = idf, +where we have written ∼= to label the evident unitors and associators, or their in- +verses. +9and u is right adjoint to f, + +A CATEGORIFICATION OF QUINN’S TQFT +27 +(ii) A 1-morphism, f : A → B, in A is an equivalence if there is a 1-morphism, +g : B → A, and two 2-isomorphisms, gf +∼ += +=⇒ 1A and fg +∼ += +=⇒ 1B. +It is said to be an adjoint equivalence if f ⊣ g and both η and ε are isomorphisms. +In Section 6, we will consider several examples of bifunctors / pseudo-functors, +F : A → B, in which +• the domain, A, is ‘locally discrete’ in the sense mentioned on page 24, so +each A(x, y) is a discrete category, i.e., a set. We will often just say that +A is a category. In fact, in most of the cases that we will need, A will be a +groupoid as all its 1-morphisms are invertible; +but +• the codomain, B, is a bicategory, usually that of 2Cob(n,n+1,n+2), any of +the span or cospan bicategories, vProf, one of its variants, or Mor. +In this case, F allows one to specify structure very easily in B. We repeat the +specification of a pseudo-functor, but in the simplified form that this context allows. +We have +• for each object, a in A, an object, F(a), in B; +• for each pair, a0, a1, of objects in A, a functor, +Fa0,a1 : A(a0, a1) → B(F(a0), F(a1)), +and, as A(a0, a1) is discrete, this just means a family of 1-morphisms, F(f) : +F(a0) → F(a1), where f : a0 → a1 ∈ A; +• for each composable pair of morphisms, a0 +f−→ a1, a1 +g−→ a2, in A, an +invertible 2-morphism, ϕg,f : F(g)F(f) ⇒ F(gf) in B; +• for each object, a, in A, an invertible 2-morphism, ϕa : idF (a) ⇒ F(ida), +in B. +These must satisfy the following conditions: +• compatibility with the associator in B, so given, in addition, h : a2 → a3, +the following diagram commutes: +(F(h)F(g))F(f) +� +aB +� +F(hg)F(f) +�❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +F((hg)f) = F(h(gf)), +F(h)(F(g)F(f)) +� F(h)F(gf) +�❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +❧ +where the unlabelled arrow are derived from the various ϕ 2-cells10, +• compatibility with the right and left unitors, (which are ‘equalities’ in A), +so, for each f : a0 → a1 in A, the diagrams: +F(f) · idF (a0) +� +ρB +F (a0) +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +F(f) · F(ida0) +� +F(f · ida0) = F(f) +10Note that as A is a category (hg)f = h(gf). + +A CATEGORIFICATION OF QUINN’S TQFT +28 +and +idF (a1) · F(f) +� +λB +F (a1) +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +F(ida0) · F(f) +� +F(ida1 · f) = F(f), +where the unlabelled arrows are the evident ones. +When formalising or analysing a structure in a category or bicategory, the struc- +ture is often expressed in terms of the commutativity of certain diagrams. Suppose +we have a commutative diagram in A. We can think of this as a functor D : I → A, +where I is some ‘template’ for the commutative diagram. We can then think of D +as a (trivially structured) pseudo-functor and compose it with our given F : A → B. +The result will be a pseudo-functor from I to B, so a ‘pseudo-commutative’ diagram +in B. We note all the 2-cells in this diagram will be invertible. +As a more-or-less trivial example, we can take I = [2], the small category corre- +sponding to the ordered set, 0 → 1 → 2, and the D will correspond to a commuta- +tive diagram of form +a1 +a12 +�❈ +❈ +❈ +❈ +❈ +❈ +❈ +❈ +a0 +a01 +�⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +⑤ +a02 +� a2. +The corresponding 2-diagram, FD, in B will be the pseudo-commutative one having +an invertible 2-arrow from F(a12)F(a01) to F(a02), together with three invertible +2-arrows, idF (ai) ⇒ F(ida1), for i = 0, 1, 2. +(In some contexts, these latter 2- +arrows may, themselves, be identity 2-arrows, i.e., the pseudo-functor, F, may be +‘normalised’, but the general case is important to us.) +We will need such constructions especially when looking at ‘inducing’ a descrip- +tion of structure on certain bicategories, D, in the context of monoidal bicategories. +2.7.2. Monoidal bicategories. We will also need the bicategorical analogues of monoidal +and symmetric monoidal categories. +One of the motivating examples for the notion of a symmetric monoidal bicate- +gory is the following. +Example 25. Let R be a commutative ring, and let Alg(R) be the bicategory such +that +• the objects are R-algebras, denoted A, B, etc.; +• the morphisms from A to B are the left-right (A, B)-bimodules; +• the 2-morphisms are the bimodule homomorphisms. +The monoidal product is the tensor product over R, so the unit is R itself, considered +as an (R, R)-algebra. This bicategory is also often denoted Alg2(R) or, as later in +this paper, by MorR, or simply by Mor if the commutative ring, R, is ‘understood’ +or ‘fixed’ for the relevant section; see Definition 164. +Later on, around that definition, we will introduce this more formally as we need +some more explicit detail as to its construction and its structure. It will be one of +the main codomain bicategories for the once-extended Quinn theory. +A monoidal bicategory is a bicategory that also has a monoidal structure, up +to the equivalence inherent in the bicategorical context. They can be defined in +various ways, for instance, as a tricategory having just one object, [58, 61]. Other +definitions mention Gray categories11. Each of these is fairly complex to give, and +11For which see, for example, [36]. + +A CATEGORIFICATION OF QUINN’S TQFT +29 +needs a few more definitions. The following is one of the simpler ones in as much +as it seems fairly clearly motivated by the definition of monoidal category suitably +weakened with equality replaced by equivalence. It does use some bicategorical +language that we have not given earlier, but is, perhaps, fairly self explanatory12. +Definition 26. A monoidal bicategory, A, consists of +• a bicategory, A; +• a pseudofunctor/homomorphism, ⊗ : A × A → A, so ⊗ is ‘functorial up to +isomorphism’; +• a pseudofunctor/homomorphism, I : 1 → A, where 1 is the unit bicategory; +• an adjoint equivalence, sometimes called the monoidal associator, diagram- +matically denoted +A3 +⊗×A � +A×⊗ +� +✂✂✂✂� α +A2 +⊗ +� +A2 +⊗ +� A, +in Bicat(A3, A), corresponding to associativity of ⊗ in a monoidal category. +This adjoint equivalence consists of α, its adjoint, α∗, with unit, ηα, and +counit, εα. To see what these do, we take a triple, A, B, C, of objects in A, +so (A, B, C) is in A3, and then we have that +αCBA : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A), +whilst +α∗ +CBA : C ⊗ (B ⊗ A) → (C ⊗ B) ⊗ A, +with unit and counit, +ηα +CBA : Id ⇒ α∗ +CBA ◦ αCBA +and +εα +CBA: αCBA ◦ α∗ +CBA ⇒ Id, +being isomorphisms. Furthermore, given 1-morphisms, f : C → C′, g : B → +B′ and h: A → A′, we have natural 2-morphisms, for instance, +(C ⊗ B) ⊗ A +αCBA +� +✖✖✖✖� α(f,g,h) +(f⊗g)⊗h +� (C′ ⊗ B′) ⊗ A′ +αC′B′A′ +� +C ⊗ (B ⊗ A) +f⊗(g⊗h) +� C′ ⊗ (B′ ⊗ A′); +• adjoint equivalences, sometimes called the monoidal unitors, +A2 +⊗ +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +A +I×A +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +A +� +✤✤ ✤✤� ℓ +A +12The structure and laws are well illustrated in A. S. Corner’s thesis, [40] in §1.6, in the +draft book by Johnson and Yau, [68] and in Mike Stay’s article, [110], which has some excellent +diagrams. + +A CATEGORIFICATION OF QUINN’S TQFT +30 +and +A2 +⊗ +�❆ +❆ +❆ +❆ +❆ +❆ +❆ +❆ +A +A×I +�⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +⑥ +A +� +✤✤ ✤✤� r +A +in Bicat(A2, A), corresponding to left and right unitors13; +and +• an invertible modification giving the analogue of the pentagon rule for monoidal +product in a monoidal category. This is called the pentagonator. Its com- +ponent 2-morphisms, for objects A, B, C, D in A looks like, +(D ⊗ (C ⊗ B)) ⊗ A +� D ⊗ ((C ⊗ B) ⊗ A) +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +((D ⊗ C) ⊗ B) ⊗ A +�♦ +♦ +♦ +♦ +♦ +♦ +♦ +♦ +♦ +♦ +♦ +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +⇓πDCBA +D ⊗ (C ⊗ (B ⊗ A)), +(D ⊗ C) ⊗ (B ⊗ A) +�❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +where each of the unlabelled arrows corresponds to a use of an associator, +possibly combined with an identity on an object, as in the usual pentagon +rule for monoidal categories14; +and +• invertible modifications, µ +� +, λ +� +and ρ +� +, called the middle, left and right 2- +unitors, respectively, with component 2-morphisms, for objects A and B in +A, +(B ⊗ I) ⊗ A +� +✏✏✏✏� µB,A +B ⊗ (I ⊗ A) +B⊗ℓA +� +B ⊗ A +r∗ +B⊗A +� += +� B ⊗ A, +(I ⊗ B) ⊗ A +ℓB⊗A +� +α +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +⇓λ +� +BA +B ⊗ A, +I ⊗ (B ⊗ A) +ℓB⊗A +�✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +B ⊗ A +B⊗r∗ +A +� +r∗ +B⊗A +�● +● +● +● +● +● +● +● +● +⇓ρ +� +BA +B ⊗ (A ⊗ I). +(B ⊗ A) ⊗ I +α +�r +r +r +r +r +r +r +r +r +r +This data is required to satisfy three pasting diagrams, which we omit15, but which +are well presented in Johnson and Yau, [68], and in [58, 61], from the point of view +of the more general tricategories. In a string diagram form, they are also to be +found in Corner, [40]. These are easier to draw, but still quite complex to read16. +To ease our way towards a sketch of the definition of symmetric monoidal bi- +category, we will briefly recall the corresponding definition of symmetric monoidal +category. Although originally introduced directly by specifying that there was a +natural isomorphism, X ⊗Y ∼= Y ⊗X, satisfying certain axioms, for our purposes it +is slightly better to go via the definition of braided monoidal category, so we briefly +recall that first. +13Leaving the reader to expand that condition on the lines above. +14see Johnson and Yau, [68], for a fully labelled version. +15as being far too large and complicated to include, and to type set, for the limited use we will +have of them, +16and we have not used this string diagram notation elsewhere in this paper. + +A CATEGORIFICATION OF QUINN’S TQFT +31 +(We have adapted the definition given in Etingof, Gelaki, Nikshych and Ostrik, +[47].) +Definition 27. A braided monoidal category is a monoidal category, (C, ⊗, I), +equipped with a natural isomorphism, +RX,Y : X ⊗ Y ∼= Y ⊗ X, +called the braiding, such that the diagrams, +X ⊗ (Y ⊗ Z) +� (Y ⊗ Z) ⊗ X +�P +P +P +P +P +P +P +P +P +P +P +P +(X ⊗ Y ) ⊗ Z +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +�P +P +P +P +P +P +P +P +P +P +P +P +Y ⊗ (Z ⊗ X) +(Y ⊗ X) ⊗ Z +� Y ⊗ (X ⊗ Z) +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +and +(X ⊗ Y ) ⊗ Z +� Z ⊗ (X ⊗ Y ) +�P +P +P +P +P +P +P +P +P +P +P +P +X ⊗ (Y ⊗ Z) +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +�P +P +P +P +P +P +P +P +P +P +P +P +(Z ⊗ X) ⊗ Y +X ⊗ (Z ⊗ Y ) +� (X ⊗ Z) ⊗ Y +�♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +♥ +commute for all choices of objects, X, Y, Z, in C, and where each arrow is an evident +application of the associator, its inverse or of the braiding. +Definition 28. A braided monoidal category, C, is said to be symmetric if, for all +X, Y in C, +RY,X ◦ RX,Y = idX⊗Y . +A symmetric monoidal bicategory categorifies the above, so replacing equalities +by structural morphisms. A complete description of symmetric monoidal bicate- +gories can be found in [60, 62]. Loosely speaking, we have a bicategory, B, which +is monoidal as above (Definition 26). The monoidal structure is assumed to be +braided, so, we have a pseudo-natural transformation, R: ⊗ → ⊗ ◦ τ, of bifunctors +from B × B to B, where τ arises from swapping coordinates, [60, page 4234], so, in +particular, for every X, Y in B, there is an equivalence (within B), +RX,Y : X ⊗ Y +≃ +−→ Y ⊗ X, +and also invertible 2-cells between the two obvious composites from (X ⊗Y )⊗Z to +Y ⊗(Z ⊗X), and similarly from X ⊗(Y ⊗Z) to (Z ⊗X)⊗Y , so replacing equality, +in the diagrams of Definition 27, by invertible 2-cells. (We will denote these 2-cells +by RX|Y Z and RXY |Z as seems to be the fairly standard notation currently in use; +see [68] and [110].) A full definition of braided monoidal bicategories can be found +in [60, Subsection 2.4]. +There is another intermediate step before getting to the final form for ‘sym- +metric’, rather than merely ‘braided’. Following [62, 1.1. Definitions], a sylleptic +monoidal bicategory is a braided monoidal bicategory with a syllepsis. +Such a + +A CATEGORIFICATION OF QUINN’S TQFT +32 +structure is a natural isomorphism, given by, for each pair X, Y of objects, an +isomorphism, +νXY : RY XRXY +∼ += +−→ IdX⊗Y . +The structure of a symmetric monoidal bicategory is, then, to satisfy one additional +axiom which says that the two ways of rewriting RXY RY XRXY to RXY , one using +νXY , the other using νY X, agree. +Example 29. An excellent list of examples of symmetric monoidal bicategories +can be found on page 2 of Mike Stay’s paper, [110]. We will select a few of most +relevance to this work, adapting some to fit the context here. We will also add a +few others. Yet others will be included later on, once the necessary terminology has +been introduced, and, for those here, we will simply mention them briefly, with a +reference to where they are discussed later. +• V −Cat: If V is a symmetric monoidal category, then the 2-category, V − +Cat, of V-categories forms a symmetric monoidal 2-category, and thus a +symmetric monoidal bicategory; see Kelly, [70], page 12. In particular, this +applies when V is the symmetric monoidal category of small categories. +• Bicategories with finite products: One has that any bicategory, A, +with binary product, − × −, and terminal object, 1, underlies a symmetric +monoidal bicategory with −×− as its tensor product and 1 as its unit object; +see Theorem 2.15 of Carboni, Kelly, Walters and Wood, [35]. Essentially +the same arguments work for bicategories with finite coproducts. +• Span(C): As is well known, a span from A to B in a category, C, is a +diagram +C +f +�❥❥❥❥❥❥ +g +�❯ +❯ +❯ +❯ +❯ +❯ +A +B. +For each pair A, B, there is a category, Span(C)(A, B), of spans, as above, +and, if C has pullbacks, then we can compose spans in a well known way. +This gives a bicategory, Span(C); see Borceux, [17], Examples 7.7.3. +If C is a category with finite products, then the bicategory, Span(C), is +a symmetric monoidal bicategory17, in which the tensor product on both +objects and spans is given by the product. +This needs taking apart a little as there are subtleties that are important +later on. Of course, the objects of Span(C) are just the objects of C. Given +any two objects, A1 and A2, in Span(C), and thus in C, their tensor product +will be A1 ⊗ A2 := A1 × A2, whilst the tensor product of two spans is +(A1 ← C1 → B1) ⊗ (A2 ← C2 → B2) := (A1 × A2 ← C1 × C2 → B1 × B2), +where the maps are as one would expect. +To define the associator on objects, we suppose that we have three objects, +A1, A2 and A3, and we need a ‘morphism’ (in Span(C)), +αA1A2A3 : (A1 ⊗ A2) ⊗ A3 → A1 ⊗ (A2 ⊗ A3). +Within the base category, C, we have an associator (iso)morphism, (the +usual one coming from the universal property of products), +aA1A2A3 : (A1 × A2) × A3 → A1 × (A2 × A3), +which satisfies the requirements that the pentagon diagrams commute. (Re- +member C is a monoidal category with the product as tensor, so it is in a +simpler setting than Span(C).) +17In fact, Span(C) is a compact closed bicategory in the sense of Stay’s paper. + +A CATEGORIFICATION OF QUINN’S TQFT +33 +In Span(C), as we said, the associator transformation is to be made of +spans, and the one that works is +(A1 × A2) × A3 +id +←− (A1 × A2) × A3 +aA1A2A3 +−−−−−−→ A1 × (A2 × A3), +in other words, using the way that C can be thought if as being ‘embedded’ +in Span(C), using the second legs of the spans. +The associator is not just αA1A2A3, but has to be part of an adjoint +equivalence, so we need a α∗ +A1A2A3 going the other way, which is given by +the reverse span, i.e. using the ‘first leg’, and we also need η and ε as in +Definition 26. That ‘second leg’ then quickly shows how to specify η and ε +for Span(C) in terms of the corresponding ones in C. +The unitors and the braidings are similarly handled giving them first in +C before transferring them to Span(C) using the second leg process. More +general results are described in much more detail in [110]. +• A special case of this is when C = Set and this gives, after a bit of adap- +tation and verification, a symmetric monoidal bicategory structure to the +category of relations. +• Prof: The bicategory of profunctors, or distributeurs. This will be revisited +after we have recalled the basic theory in the next section. +• vProf: The bicategory of Vect-enriched categories, enriched profunctors +and enriched natural transformations. Again discussion of this is postponed +to the next section. +• Cospan(C): If we replace C by the opposite category then we have that, if +C has finite colimits, Cospan(C) will be a symmetric monoidal bicategory, +having coproduct, ⊔, as its tensor product. +• 2Cob(d,d+1,d+2): Let d be a non-negative integer. It is well known that +Cob(d,d+1), the category of closed smooth d-manifolds and diffeomorphism +classes of cobordisms between them, [89, 37], forms a symmetric monoidal +category with coproduct / disjoint union, ⊔, as the tensor product. +As +proved in [106], 2Cob(d,d+1,d+2), the bicategory of closed smooth d-manifolds, +their cobordisms, and diffeomorphism classes of extended cobordisms be- +tween cobordisms, is a symmetric monoidal bicategory, again having ⊔ as its +tensor product. We will sketch the construction of the symmetric monoidal +structure of 2Cob(d,d+1,d+2) in §6.5.2. +• Alg(R), also denoted Mor: We mentioned this important example earlier +in Example 25. Here R is a commutative ring, the objects are R-algebras, +the 1-morphisms are bimodules and the 2-morphisms are homomorphisms +of bimodules. The tensor product is the tensor product over R. +This example is sometimes denoted Alg2(R), to indicate that it is the 2- +categorical version of the category of algebras and bimodules between them. +We, in fact, will use yet another name namely the Morita bicategory of R +and consequently denote it MorR, as it is a good context for emphasising +Morita equivalences. We will introduce it in more detail in Definition 164, +where R will be a subfield, κ, of C, as that will be the only case that we will +be needing later on. +• BimodR: In the previous example, there is no reason to restrict the ‘al- +gebras’ to having a single object18. We can form a bicategory, which we +will denote by BimodR for the moment, having R-linear categories as ob- +jects, with bimodules over them as its 1-morphisms and (R-linear) natural +transformations as its 2-morphisms. +18in the sense of Mitchell’s paper, [90]. + +A CATEGORIFICATION OF QUINN’S TQFT +34 +This is nearly the same as the bicategory, vProf, of R-linear profunc- +tors, which we will be using in the case that R = κ, and for which see the +next section. It is considered, for instance, in [10] as Definition 2.8. We +will treat a related example later, but note that this is another example of +a symmetric monoidal bicategory. +The tensor product of two R-linear categories, C and D, is defined to +have as its objects, the pairs (C, D), with C ∈ C0 and D ∈ D0, i.e., objects +from the relevant categories, and, following Mitchell, [90], +(C ⊗ D)((C, D), (C′, D′)) := C(C, C′) ⊗R D(D, D′). +For bimodules CMC′, and DND′, their tensor product is the tensor prod- +uct over R of the various parts of the data. We note that the structural +isomorphisms, braidings, etc., are defined in an evident natural way. +Many of the above examples are linked by symmetric monoidal bifunctors, but +we will not give a full formal detailed definition of such here, and will merely sketch +one, directing the reader to Definition 2.5 of [106], and [61, 60, 62], for a more +detailed version. +Let A and B be two symmetric monoidal bicategories, in which the structures +involved, such as the tensor, will be provided with a suffix to show to which of A +and B they relate. +Definition 30. (Sketch only) A symmetric monoidal bifunctor, F : A → B, con- +sists of the following data: +• a homomorphism, (i.e. a bifunctor), F : A → B, between the underlying +bicategories; +• a transformation, (i.e. a pseudo-natural transformation), +χ: ⊗B ◦(F × F) ⇒ F ◦ ⊗A, +of bifunctors, from A × A to B, so we have, given objects, A0 and A1, of +A, a 1-morphism, in B, +χA0,A1 : F(A0) ⊗B F(A1) → F(A0 ⊗A A1), +and given 1-morphisms, f0 : A0 → A′ +0 and f1 : A1 → A′ +1, in A, we have a +natural 2-cell in B, +F(A0) ⊗B F(A1) +F (f0)⊗BF (f1) � +χ(A0,A1) +� +✖✖✖✖� χ(f0,f1) +F(A′ +0) ⊗B F(A′ +1) +χ(A′ +0,A′ +1) +� +F(A0 ⊗A A1) +F (f0⊗Af1) +� F(A′ +0 ⊗A A′ +1), +which is compatible with compositions and horizontal identities, and we also +have a transformation, +i: IB ⇒ F ◦ IA, +together with corresponding adjoint equivalence transformations, χ∗ and i∗, +(and with the relevant adjunction data); +• invertible modifications, ω, γ and δ, measuring compatibility with the rele- +vant associators and unitors (see Fig 2.5. of [106]); +and +• an invertible modification, u, giving compatibility with the braiding (see Fig +2.6 of [106], or [60, page 4239], for the relevant diagram, so that +u : F(RA) ◦ χ ⇒ χ ◦ RB : F(B) ⊗B F(A) → F(A ⊗A B). + +A CATEGORIFICATION OF QUINN’S TQFT +35 +This data is to satisfy certain axioms, which we omit, referring to the discussion +in [106], Definition 2.5., for further details including further references. In par- +ticular the compatibility conditions in [61, §4.3] / [58, page 17] hold, dealing with +the preservation of the monoidal structure by F, and those of [60, page 4239] hold, +similarly describing the symmetric monoidal structure of the bifunctor F. +We will sketch the construction of the entire structure of a symmetric monoidal +bifunctor when we prove, in §6.6, that the once-extended Quinn TQFT, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +is symmetric monoidal. +2.8. Conventions on profunctors. The detailed theory of profunctors can be +found in many texts and on-line sources, for instance, [17, Chapter 7], [81, Section +5] and the nLab [96]. +When searching for such theory, it is important to note +that the terms ‘distributor’ and ‘bimodule’ are often used alternative names for +profunctors. We will give a very minimal sketch here and include a bit of a ‘crib- +sheet’, whilst we are doing that. +2.8.1. Some background and basic definitions. In a general context, given two small +categories, A and B, we have: +Definition 31. A (Set-valued) profunctor from A to B is a functor, +F : Aop × B → Set. +This will be sometimes written F : A ↛ B. +With a suitable notion of composition, these profunctors, for varying domain and +codomain categories, together with their natural transformations, form a bicategory +Prof, whose construction will be briefly recalled below. +Warning: There are several different conventions used in the literature as to +the ‘direction’ of the profunctor. One of the most current, but not the one that we +will use, is to say a profunctor, F : A ↛ B, is a functor, Bop × A → Set. This can +be confusing, but makes no essential difference to the resulting theory. The choice +between them is influenced by the context of their use, but this means that, when +referring to a source on profunctors, the reader is advised to check the convention +being used in that source. +In our case, with a geometry-inspired concatenation composition rule for cobor- +disms, and a compatible algebra-inspired rule for composition (tensor product) of +bimodules, our conventions will not be the same as many of the sources. This may +cause some slight difficulty when looking at the classical source material on pro- +functors / distributors, but is, we think, a good and consistent mix of conventions +for our use of that theory. +Any such profunctor, F : Aop × B → Set, determines a functor, F: Aop × B → +Vect, by composing F with the free vector space functor, Lin: Set → Vect. The +resulting linear, or Vect-valued, profunctor will quite often also be denoted, by +abuse of notation, by the same symbol, F : A ↛ B, or sometimes (as above) by +the symbol for the functor, converted to a boldface type, so F, if both F and its +‘linearisation’ are needed in a discussion. +We will usually work with the Vect-enriched analogue, vProf, [10, 63], of the +bicategory Prof, and, within that, the sub-bicategory, vProfGrp, whose objects +are derived from groupoids, G = (s, t: G1 → G0), each made into a linear category, + +A CATEGORIFICATION OF QUINN’S TQFT +36 +which we may occasionally write as Lin(G), by applying the free vector space +functor, Lin: Set → Vect, to the hom-sets of G. +Given groupoids, G and G′, 1-morphisms from G to G′, in vProf, will be con- +sidered to be functors, H: Gop × G′ → Vect, from now on called Vect-profunctors +from G to G′, although sometimes, no doubt, we will abbreviate that term, just +saying ‘profunctors’. +Example 32. Let A be a (small) category, then we have the bivariant hom-functor, +A(−, −) : Aop×A → Set, which is a Set-valued profunctor from A to itself. If A is +a Vect-enriched category, the natural analogue of the above is A(−, −) : Aop×A → +Vect, and so is a Vect-valued profunctor. We denote the latter by IdA, and call it +the identity profunctor on the (linear) category, A. We may shorten this to IdG, +if A is the linearisation, Lin(G), of a groupoid, G. +Example 33. Given a functor, F : A → B, we can define two profunctors, +ϕF : A ↛ B +and +ϕF : B ↛ A, +by +ϕF (A, B) = B(F(A), B), +whilst +ϕF (B, A) = B(B, F(A)). +In case F is the identity functor on A, the two profunctors coincide and are the +same as that in the previous example. A profunctor that is isomorphic to one of +these two forms is said to be representable. These profunctors are, in fact, adjoint +1-cells in the bicategory, Prof, whose structure we are sketching here. +Suppose, now, that η : F ⇒ G is a natural transformation of functors from A to +B, then, for each A in A, we have a morphism, η(A) : F(A) → G(A), and there +are induced 2-arrows, +ϕη : ϕG ⇒ ϕF , +given by ϕη(A) : ϕG(A) ⇒ ϕF (A), is +ϕη(A) := (η(A))∗ : B(G(A), B) → B(F(A), B), +if B is an object of B, and +ϕη : ϕF ⇒ ϕG, +given, dually, by ϕη(A) := (η(A))∗. +That these do give 2-morphisms / natural +transformations is easy to check. This process also respects composition of natural +transformations, so if η′ : G ⇒ G′ is another natural transformation, then ϕη′η +and ϕη′ϕη are equal as, for an object A of A, they both induce the composite, +(η′η)(A) = η′(A)η(A) and so give the same induced morphism from B(G′(A), B) to +B(F(A), B). +This means that the 2-category of small categories, considered as a bicategory +‘bifunctorially embeds19’ in the bicategory, Prof. This will be important when we +consider the monoidal bicategory structure on Prof. In particular, if F is an equiva- +lence of categories, then ϕF is an equivalence in the bicategorical sense within Prof. +Other similar statements hold for adjointness, etc., but we will not be needing them +as much. +19or ‘pseudo-functorially embeds’ + +A CATEGORIFICATION OF QUINN’S TQFT +37 +Given Vect-profunctors, H, H′ : Gop × G′ → Vect, a 2-morphism, or 2-cell, +η: H =⇒ H′, between them, is a natural transformation of functors, Gop × G′ → +Vect. Hence, we have, given x ∈ G0 and y ∈ G′ +0, a linear map, ηx,y : H(x, y) → +H′(x, y), which is natural in both x and y. +Let G, H, K be groupoids, or more generally small categories. +Given Vect- +profunctors, H: G ↛ H, and H′ : H ↛ K, their composite, H • H′ : G ↛ K, will +be the Vect-profunctor such that, if x ∈ G0 and z ∈ K0, then20 +(3) (H • H′)(x, z) := +ˆ y∈H0 +H(x, y) ⊗ H′(y, z) = +� � +y∈H0 +H(x, y) ⊗ H′(y, z) +� +/ ≃ . +Here, fixing x ∈ G and z ∈ K, the equivalence relation, ≃, is generated (as a linear +equivalence relation21) by +for y, y′ ∈ Y , vx,y ∈ H(x, y) and v′ +y′,z ∈ H(y′, z) and an arrow, y +h−→ y′, in H, +vx,y ⊗ H′(y +h−→ y′, z +1z +−→ z)(v′ +y′,z) ≃ H(x +1x +−→ x, y +h−→ y′)(vx,y) ⊗ v′ +y′,z, +or, more informally, +(4) +vx,y ⊗ h · v′ +y′,z ≃ vx,y · h ⊗ v′ +y′,z. +We note the convention on the order of composition that we are using, and +would remind the reader of the Warning that we placed a short while back. This +convention is used because it reflects the geometric intuition, being a concatenation +order of composition. It also reflects a useful convention for the bicategory, Mor, +of algebras, bimodules and bimodule maps, to which Prof is closely related. +If we just have Set-valued profunctors, this formula for composition still make +sense by interpreting ⊗ as ×, and we note that Lin preserves that composition in +the evident way. +Of course, there is a projection, +(5) +proj : +� +y∈H0 +H(x, y) ⊗ H′(y, z) → (H • H′)(x, z), +which we will need later on. Given any element in (H • H′)(x, z), we can represent +it by an element in H(x, y) ⊗ H′(y, z), for some y, but, working with that, just +as in the setting of bimodules, any resulting calculation has to be shown to be +independent of the y chosen, that is, it must be invariant under the action of the +arrows of H. +Example 34. Suppose we have functors, A +F−→ B +G +−→ C, then we have corresponding +profunctors, ϕF and ϕG, so can form their composite, ϕF • ϕG. It is not hard to +check that this composite is isomorphic to ϕGF . More precisely we have a natural +isomorphism, ϕF • ϕG =⇒ ϕGF , and this is part of the data that says that ϕ(−) +is a pseudo-functor. We will use this later in section 6.6.1, especially on page 160. +Similarly we have ϕG : C ↛ B and ϕF : B ↛ A. Composing these to form +ϕG • ϕF , we find that this is isomorphic to ϕGF : C ↛ A. +The composition of general profunctors and so, in particular, of Vect-profunctors +has left and right (lax) identities. +Suppose we have H: G ↛ H, then we can +compose it with the (enriched) hom-functor, G(−, −) : Gop × G → Set, or to +20Note that this coend is, a priori, defined only up to isomorphism. In this paper, we always +implicitly choose a natural realisation for all limits and colimits appearing, as we have done below. +21i.e. as an equivalence relation whose quotient is a vector space. + +A CATEGORIFICATION OF QUINN’S TQFT +38 +Vect after applying Lin. This acts like a left identity on G. There is a natural +isomorphism, +(6) +λH +G : G(−, −) • H =⇒ H, +often called the ‘left unitor’, and similarly another natural isomorphism, +ρH +H : H • H(−, −) =⇒ H, +the ‘right unitor’. These are easy to write down, for example, using equation (4). +Given natural transformations, η: H1 +=⇒ +H2, between Vect-profunctors, +G ↛ H, and η′ : H′ +1 +=⇒ +H′ +2, between Vect-profunctors, H ↛ K, we have a +natural transformation, (η • η′): H1 • H′ +1 → H2 • H′ +2. Explicitly, given x ∈ G0 and +z ∈ K0, then (η•η′)(x,z) sends the equivalence class of vx,y ⊗v′ +y,z to the equivalence +class of η(x,y)(vx,y)⊗η′ +(y,z)(v′ +y,z). Here y ∈ H0, vx,y ∈ H1(x, y) and v′ +y,z ∈ H2(y, z). +In other words, given x ∈ G0 and z ∈ K0, the linear map, (η •η′)(x,z), is the unique +map that makes the diagram below commute: +(7) +� +y∈H0 +H1(x, y) ⊗ H′ +1(y, z) +� +y∈H0 +η(x,y) ⊗ η′ +(y,z) +� +proj +� +ˆ y∈H0 +H1(x, y) ⊗ H′ +1(y, z) +(η • η′)(x,z) +� +� +y∈H0 +H2(x, y) ⊗ H′ +2(y, z) +proj +� +ˆ y∈H0 +H2(x, y) ⊗ H′ +2(y, z) . +2.8.2. Matrix elements for natural transformations. Let G and H be groupoids. In +most examples appearing in this paper, Vect-profunctors, Gop × H → Vect, arise +from set-valued profunctors, F : Gop × H → Set, by applying the free vector space +functor Lin: Set → Vect. However the natural transformations between profunc- +tors will be full fledged natural transformations between Vect-valued profunctors. +Given a profunctor, F : Gop × H → Set, we thus have that its linearisation, +F = Lin ◦ F : Gop × H → Vect, comes with given bases on each of its constituent +vector spaces, which means that for the purposes of the presentation of the encoded +data, or for calculation, we can use matrices and other tools and insights from +classical representation theory22. +An object of Gop × H is, of course, a pair, (x, y) ∈ G0 × H0, and if we denote +a typical element of F(x, y) by f, we can consider f as an element of the natural, +given basis for F(x, y). +For g : x → x′ in G1, we then have, for each y ∈ H0, +a linear map, F(g, y): F(x′, y) → F(x, y), and so a matrix with matrix elements, +⟨f ′ | F(g, y) | f⟩, and +F(g, y)(f ′) = +� +f∈F (x,y) +⟨f ′ | F(g, y) | f⟩f. +Similarly, if h: y → y′ ∈ H1, we have F(x, h): F(x, y) → F(x, y′). This mean that +the evident actions of G and H on the family of vector spaces, F(x, y), for x ∈ G0 +and y ∈ H0, come with a given matrix representation. This looks like a ‘many +object’ bimodule, and we will recall the relationship with the theory of bimodules +over algebras more fully in §6.4.1. +22In what follows, we will make the assumption that all the F (x, y) are finite sets, as that is +true in the situations that will be of interest to use later on. + +A CATEGORIFICATION OF QUINN’S TQFT +39 +For the moment, we need to examine the way of describing the natural transfor- +mations between such profunctors. Suppose F and F′ : Gop × H → Vect are two +Vect-valued profunctors, linearised from some Set-valued ones, as above. +Fur- +ther suppose ϕ: F ⇒ F′ is a natural transformation from F to F′. +For each +(x, y) ∈ G0 × H0, we then have a linear mapping, ϕ(x, y): F(x, y) → F′(x, y), +and hence, for each f ∈ F(x, y) and f ′ ∈ F ′(x, y), once again a matrix element, +⟨f | ϕ(x, y) | f ′⟩, so that we have a ‘state sum’ description, +ϕ(x, y)(f) = +� +f ′∈F ′(x,y) +⟨f | ϕ(x, y) | f ′⟩f ′. +The fact that ϕ is a natural transformation means that it must be compatible +with changes along any g : x′ → x and h: y → y′, and so must satisfy equations +involving the various F(g, h): F(x, y) → F(x′, y′), and the corresponding matrix +representations. +When we deal with the categorified ‘once-extended’ version of Quinn TQFT, in +Section 5, and in particular §5.4.2, our methods, will give naturality from general +constructions, and then a description of the natural transformations in terms of +these matrices and state sums, rather than starting with the families of matrices +and doing quite complex manipulations to show that they define natural transfor- +mations. +2.8.3. Some lemmas on coends of functors from groupoids. For the convenience of +the reader, we collect a few elementary lemmas, whose explicit formulation can be +difficult to find in the literature. They will be useful when giving some explicit +details in the proof that the once-extended Quinn TQFT is indeed a bifunctor, +especially when it comes to the preservation of horizontal compositions, see §5.1.2 +and Subsection 5.5. +Let G = (s, t: G1 → G0) be a groupoid. Consider a functor F : Gop × G → Set. +The coend of F is a universal wedge, making the diagram, +F(y, x) +F (g,idx) � +F (idy,g) +� +F(x, x) +px +� +F(y, y) +py +� +ˆ z∈G0 +F(z, z), +commute, for all choices of morphisms g : x → y in G. Therefore, as written above, +ˆ z∈G0 +F(z, z) = +� � +z∈G0 +F(z, z) +� +� +∼, +where ∼ is the smallest equivalence relation that makes the diagram above commute +for all choices of g : x → y. +As G is a groupoid, given any g : x → y and g′ : x′ → y′, the map, +F(g, g′): F(y, x′) → F(x, y′), +is a bijection. This gives: +Lemma 35. We have +ˆ z∈G0 +F(z, z) = +� � +z∈G0 +F(z, z) +� +� +≃, +where ≃ is the equivalence relation in which +ux ∈ F(x, x) ≃ uy ∈ F(y, y) + +A CATEGORIFICATION OF QUINN’S TQFT +40 +if there exists g : x → y such that: +F +� +g−1 : y → x, g : x → y +� +(ux) = uy. +□ +In fact, this shows that the equivalence relations ∼ and ≃ are really the same. +Note that any groupoid G comes with a contravariant functor (−)−1 : G → G, +that is the identity on objects and sends g : x → y to g−1 : y → x. There is also the +diagonal functor, ∆: G → G × G, sending x ∈ G0 to (x, x) ∈ G0 × G0, and with +∆ +� +x +g−→ y +� += +� +(x, x) +(g,g) +−−−→ (y, y) +� +. +Hence we have a functor, +F ◦ +� +(−)−1 × id +� +◦ ∆: G → Set. +The previous lemma gives: +Lemma 36. There is a canonical bijection +ˆ z∈G0 +F(z, z) ∼= colim +� +F ◦ +� +(−)−1 × id +� +◦ ∆ +� +. +□ +Let A and B be sets. Consider a linear map, f : κ(A) → κ(B), between free +vector spaces and equivalence relations, ∼A and ∼B, on A and B, such that f +descends to a map f ′ : κ(A/ ∼A) → κ(B/ ∼B), then, given a ∈ A and b ∈ B, the +matrix elements of f ′ satisfy +� +[a]|f ′|[b] +� += +� +b′∈[b] +� +a|f|b′� +. +Combined with the previous discussion, this gives the following. +Lemma 37. Let F, F ′ : Gop × G → Set be functors. Consider, for each (x, y) ∈ +G0 × G0, a linear map, η(x,y) : κ +� +F(x, y) +� +→ κ +� +F ′(x, y) +� +, such that putting all of +the η(x,y) together gives a natural transformation, η: Lin ◦ F =⇒ Lin ◦ F ′, where +Lin: Set → Vect is the free vector space functor. Further consider the induced +map (see [81, Notation 1.1.15]) between coends, +ˆ x∈G0 +η(x,x) : +ˆ x∈G0 +κ(F(x, x)) → +ˆ x∈G0 +κ(F ′(x, x)), +that is (since the free vector space functor preserves colimits, and with a minor +abuse of notation), +ˆ x∈G0 +η(x,x): κ +� ˆ x∈G0 +F(x, x) +� +→ κ +� ˆ x∈G0 +F ′(x, x) +� +. +Its matrix elements satisfy, for each z ∈ G0, uz ∈ F(z, z) and wz ∈ F ′(z, z), +� +[uz] +����� +ˆ x∈G0 +η(x,x) +����� [wz] +� += +� +w′ +z∈Orb(wz) +� +uz|η(z,z) |w′ +z +� +. +Here Orb(wz) is the orbit of wz, under the action of the group homG(z, z), on +F ′(z, z), defined as , +vz ⊳ (g−1 : z → z) := F ′(g−1 : z → z, g : z → z)(vz), +where vz ∈ F ′(z, z). + +A CATEGORIFICATION OF QUINN’S TQFT +41 +On the other hand, if z′ ∈ G0 belongs to a different connected component from z +in G, and if tz′ ∈ F ′(z′, z′), then +� +[uz] +����� +ˆ x∈G0 +η(x,x) +����� [tz′] +� += 0. +Proof. This follows from the previous discussion, and the fact that the diagram +below commutes, where the vertical arrows are the canonical projections, +� +x∈G0 +κ(F(x, x)) +� +� +x∈G0 +η(x,x) +� � +x∈G0 +κ(F ′(x, x)) +� +κ +� ˆ x∈G0 +F(x, x) +� +ˆ x∈G0 +η(x,x) +� κ +� ˆ x∈G0 +F ′(x, x) +� +. +□ +2.8.4. Prof and its relatives as symmetric monoidal bicategories. We next need +to link up these bicategories of profunctors with considerations of our previous +discussions on symmetric monoidal bicategories. +To start with, there is a natural monoidal bicategory structure on Prof itself. +This is explicitly given by Cattani and Winskel, [38]. Define ⊗ on objects to be +simply the product, ×, of categories, and then on 1-morphisms, F : A0 ↛ B0 and +G : A1 ↛ B1, take F ⊗ G to be the composite +(A0 × A1)op × (B0 × B1) +∼ += +−→ (Aop +0 × B0) × (Aop +1 × B1) +F ×G +−−−→ Set × Set +×Set +−−−→ Set, +so on an object, ((A0, A1), (B0, B1)), it takes the value F(A0, B0)×G(A1, B1) given +by the product in Set of the two image sets. On 2-morphisms, if α : F ⇒ F ′ and +β : G ⇒ G′, then +(α ⊗ β)(A0,A1),(B0,B1) = α(A0,B0) × β(A1,B1). +Together with this, one can prove that Prof supports a symmetric monoidal +bicategory structure in which the associativity and braiding profunctors are in- +herited from the associativity and braiding morphisms of the 2-category of small +categories, functors and natural transformation, by applying the ϕ(−)-construction +of Examples 33 and 34. In particular, the braiding in Prof is given explicitly as +follows. +Suppose G and H are small categories, (which for us will usually be groupoids), +then there is an isomorphism of categories, +R : G × H +∼ += +−→ H × G. +We therefore have a profunctor, ϕR : G × H ↛ H × G. This is an equivalence, +and gives the required braiding, RGH := ϕR. This gives a symmetric monoidal +bicategory structure to Prof. +Passing to the κ-linear setting and Vect-enriched profunctors, then the above +constructions adapt well, and vProf is a symmetric monoidal bicategory, whose +objects are the κ-linear categories, [63, Corollary 6.6.]. On objects, we take A ⊗ B +to be the tensor product (over κ) of such things, as was discussed in the previous +section (page 34). The tensor product of two profunctors is obtained by almost +the same composite as above except that, of course, we replace Set by Vect and + +A CATEGORIFICATION OF QUINN’S TQFT +42 +×Set = − × − by ⊗Vect = − ⊗κ −. The 2-morphisms look after themselves. The +braiding structure is as in the Prof case, above. +The sub-bicategories, vProfGrp, vProfGrphf, and vProfGrpfin, are easily +seen all to inherit symmetric monoidal bicategory structures from the larger bicat- +egory, where we use the terminology on groupoids from Subsection 2.2. In partic- +ular if we think of the tensor product of two groupoids, G and H, in vProfGrp +as being the usual cartesian product, G × H, of groupoids, then, after linearising +the groupoids, that product is sent to the tensor product of the two linearised +categories. +All these bicategories are variants of the basic bicategory of bimodules, Bimodκ, +approached from a different direction. +The linearisation functor induces a symmetric monoidal bifunctor from Prof to +vProf. If we restrict attention to profunctors defined on groupoids, this functor +restricts to a symmetric monoidal bifunctor from ProfGrp, the sub-bicategory of +Prof defined on groupoids, to vProfGrp, and, clearly, this further restricts to the +homotopy finite, and finite subcases. +2.8.5. Summary of conventions and notation for bicategories of profunctors. To +summarise the terminology and notation for some of the instances that we will be +needing later on in one place, we have (for terminology, see also Subsection 2.2). +• Prof: the (symmetric monoidal) bicategory of categories, set-valued pro- +functors, and their natural transformations. +• vProf: the (symmetric monoidal) bicategory of Vect-enriched categories, +enriched profunctors and enriched natural transformations. This is also a +symmetric monoidal bicategory; see [63]. +• vProfGrp: the sub-bicategory of vProf, whose objects are groupoids, +each made into a Vect-enriched category by applying the free vector space +functor to the sets of morphisms. Here 1-morphisms, G ↛ H, are functors, +Gop × H → Vect, and are called Vect-profunctors, and 2-morphisms are +natural transformations of such functors. The horizontal compositions of +1- and 2-morphisms are explained in equations (3) and (7), respectively. +This inherits a symmetric monoidal structure from vProf, in which the +tensor product of the groupoids, G and H, is the usual cartesian product. +• vProfGrphf: the sub-bicategory of vProf, whose objects are the homo- +topy finite groupoids. +• vProfGrpfin: the sub-bicategory of vProfGrp whose objects are the finite +groupoids. +Comments: +(i) As is well known, any group can be considered as a groupoid having just one +object, so let G be a (finite) group. When we consider G in vProf, the resulting +linear category, as it has just one object, is effectively just a κ-algebra. This is +precisely the group κ-algebra of G. +From this viewpoint, for G and H (finite) +groups, a Vect-profunctor, G ↛ H, is the same as a G-H bimodule over κ. The +composition of such ‘bimodules’ is exactly that used in more classical treatments, +and is given by a tensor product in the usual way. +We thus take the viewpoint that the profunctors that we are considering here, +are many object versions of bimodules and that the κ-linear categories obtained +from the groupoids, are many object groupoid κ-algebras. +(ii) The relationship between these Vect-enriched categories and the κ-algebras +that are sometimes called ‘groupoid algebras’ is classical, being one of the themes +of Mitchell’s 1972 paper, [90]. We will expand on this later on in §6.4.1. + +A CATEGORIFICATION OF QUINN’S TQFT +43 +Part 2. The homotopy theoretical underpinning of Quinn’s finite total +homotopy TQFT +3. Homotopically finite spaces and the category HFspan +3.1. Homotopically finite (HF) spaces. +Definition 38 (Homotopically finite (HF) space). A space, B, is called homotopi- +cally finite (abbr. HF) if B is CGWH23 and, moreover, B has only a finite set of +path-components, each of which has only a finite set of non-trivial homotopy groups, +all of which are finite. +Clearly finite disjoint unions and finite products of HF-spaces are HF24. Each +path-component of a HF-space is also HF (after possibly applying the k-ification +functor in order to make it a CGWH space). +Remark 39. In the literature, one finds some alternative terminology used for +homotopy finite spaces. Lurie, [83, Appendix E], calls them ‘π-finite spaces’ whilst +Anel, [1], uses the term ‘truncated coherent space’. +The following is essentially in [56, Lemma 3.4], albeit stated in the context of +∞-groupoids. +Lemma 40. Let p: E → B be a fibration. Given b ∈ B, we let Eb := p−1(b). +(1) Suppose that B is path-connected, and that p is surjective. If any two of B, +E and Eb are homotopy finite, then so is the third. +(2) Let B be any space. If B and E are homotopically finite, then so is Eb for +each b ∈ B. If B and each Eb are homotopically finite (for each b ∈ B), +then so is E. +In particular, if p: E → B is a fibration, and E and B are HF, then each fibre of +p is HF. +Proof. Follows from the homotopy long exact sequence of p: E → B; see Equation +(8) below. +□ +The second point of the following result will be crucial for what follows. This is +stated in [56, Lemma 3.13] for ∞-groupoids. +Lemma 41. Consider a pullback diagram of spaces, where p: E → B is a fibration, +X ×B E +q +� +� E +p +� +X +f +� B, +then q is a fibration. If X, E and B are HF, then X ×B E is HF. +Proof. That q is a fibration follows from the standard fact that pullbacks of fi- +brations are fibrations; see e.g. [87, §6.1 Lemma]. Let us prove that X ×B E is +HF. +We use the previous lemma. By assumption, X is HF. We only need to prove +that the fibres of q are HF. By the lemma below, given x ∈ X, the fibre of q at +x is homeomorphic to p−1(f(x)), which is HF since B and E are. (We are using +Lemma 40 here.) +□ +23Recall that this is an abbreviation for “compactly generated and weak Hausdorff”; see Sub- +section 2.3. +24Equally clearly infinite disjoint unions and products may not be! + +A CATEGORIFICATION OF QUINN’S TQFT +44 +Note that the following result is not immediate, given that k-ification was applied +to both product and induced topologies, see Subsection 2.3. +Lemma 42. Given x ∈ X, the fibre of q: X ×B E → X at x is homeomorphic to +p−1(f(x)). +Proof. We have a pullback diagram in CGWH, where the inc denote the obvious +inclusion maps, +q−1(x) +� +inc � X ×B E +q +� +{x} +inc +� X. +By applying the pullback pasting lemma, the outer rectangle of the diagram below +on the left is a pullback in CGWH. The diagram on the right below is a pullback +in CGWH as well, +q−1(x) +� +inc � X ×B E +q +� +� E +p +� +{x} +inc +� X +f +� B, +p−1(f(x)) +� +inc +� E +p +� +{x} +f◦inc +� B. +In particular, we get that p−1(f(x)) is homeomorphic to q−1(x). +□ +3.1.1. The homotopy content of a HF-space. +Definition 43 (Homotopy content). Let B be a path connected HF-space. The +homotopy content of B is defined as , +χπ(B) = +��π2(B, x) +�� ��π4(B, x) +�� ��π6(B, x) +�� . . . +��π1(B, x) +�� ��π3(B, x) +�� ��π5(B, x) +�� . . . ∈ Q, +where x ∈ B is any point. +In general, if X is a HF space, and using the notation in item (15) on page 16, +define +χπ(X) = +� +B∈�π0(X) +χπ(B). +We also put χπ(∅) = 0. (Note that for all other HF spaces F, we have χπ(F) > 0.) +We note that what we have called ‘homotopy content’ is called ‘homotopy order’ +in [101, Lecture 4], ‘homotopy cardinality’ in [4], and also, more recently, in [56, +§3]. +The homotopy content of a space also appeared in [49], without being given a +name, and, there, was also considered for crossed complexes. We will consider that +form separately a bit later on here. The case of ∞-groupoids is treated in [56], +which proves similar results to the one below, in that context. +Note that homotopic HF spaces, and, more generally, weakly homotopic HF +spaces, have the same homotopy content. +Example 44. The customary examples are (i) when X is a finite set, thought of +as a discrete space, then χπ(X) is the usual cardinality of X, and (ii) when X is +the classifying space of a finite groupoid, G, then χπ(X) = � +[x]∈π0(G) +1 +|G(x)|, which +is the groupoid cardinality of G, in the sense of Baez and Dolan, [4]. We will +generalise the previous formula for the case of crossed complexes of groupoids in +§7.6.1. + +A CATEGORIFICATION OF QUINN’S TQFT +45 +3.1.2. The homotopy content of a HF-space: properties. +Lemma 45. If B and B′ are HF-spaces, then +χπ(B ⊔ B′) = χπ(B) + χπ(B′) and χπ(B × B′) = χπ(B)χπ(B′). +Proof. The first equation is straightforward. The second follows from the fact that +�π0(B × B′) ∼= {A × A′|A ∈ �π0(B), A′ ∈ �π0(B′)}, +and that, if x ∈ B and x′ ∈ B′, then πn +� +B × B′, (x, x′) +� ∼= πn(B, x) × πn(B′, x′). +Explicitly, we have: +χπ(B × B′) = +� +A∈�π0(B), A′∈�π0(B′) +��π2 +� +A × A′��� ��π4 +� +A × A′��� ��π6 +� +A × A′��� . . . +��π1 +� +A × A′��� ��π3 +� +A × A′��� ��π5 +� +A × A′��� . . . += +� +A∈�π0(B), A′∈�π0(B′) +��π2 +� +A +��� ��π4 +� +A +��� ��π6 +� +A +��� . . . +��π1 +� +A +��� ��π3 +� +A +��� ��π5 +� +A +��� . . . +��π2 +� +A′��� ��π4 +� +A′��� ��π6 +� +A′��� . . . +��π1 +� +A′��� ��π3 +� +A′��� ��π5 +� +A′��� . . . += χπ(B) χπ(B′). +□ +More generally, +Lemma 46 (Quinn, [101], Baez–Dolan, [4], and Galv´ez-Carillo–Kock–Tonks, [56]). +Suppose that p: E → B is a fibration of HF-spaces and that B is path-connected. +Let b ∈ B be arbitrary, then, recalling Eb = p−1(b) is the fibre at b, +χπ(E) = χπ(B) χπ(Eb). +The proof we give below is as hinted at in the above references, with some crucial +technical details added. +Proof. If E is empty, then so is Eb, and in this case there is nothing to prove. If B +is empty, then so are E and Eb, and there is nothing to prove either. +We are left with the case that E, B ̸= ∅. In this case, it follows that p: E → B +is surjective, as B is path-connected. More generally, if E′ is a path-component of +E, the restriction p′ : E′ → B of p is also surjective. +Suppose, firstly, that E is path-connected. Let x ∈ E and b = p(x), then, cf. +[87, p. 52] or [64, p. 376], the homotopy long exact sequence of p: E → B, at b +and x reads +(8) +→ πi(Eb, x) +ι→ πi(E, x) +∂→ πi(B, b) +δ→ πi−1(Eb, x) → . . . +ι→ π1(E, x) +∂→ π1(B, b) +δx +→ π0(Eb) +ι−→ π0(E) = {0}. +Here, for the last stages of the sequence, the exactness means the following: +• we have a left-action, ⊲, of π1(B, b) on π0(Eb) (reviewed in Lemma 97), whose +stabiliser subgroup at the path-component, PCx(Eb), of x ∈ Eb, is ∂(π1(E, x)); +• the map δx: π1(B, b) → π0(Ex), which is defined as δx(g) = g ⊲ PCx(Eb), is +surjective. +Also note that, by the orbit-stabiliser theorem, |π0(Eb)| = |π1(B, b)|/|∂(π1(E, x))|. +The exactness of the sequence (8) yields that: +|πi(E, x)| = |∂(πi(E, x))| |ι(πi(Eb, x))|, if i ≥ 1, +|πi(B, b)| = |∂(πi(E, x))| |δ(πi(B, b))|, if i ≥ 2, +|π1(B, b)| = |∂(π1(E, x))| |π0(Eb)|, +|πi(Eb, x)| = |δ(πi+1(B, b))||ι(πi(Eb, x))|, if i ≥ 1. + +A CATEGORIFICATION OF QUINN’S TQFT +46 +Therefore, noting that B and E are by assumption path-connected, +χπ(B) = +1 +|∂(π1(E, x))| |π0(Eb)| ++∞ +� +k=2 +���∂(πk(E, x)) +�� ��δ(πk(B, b)) +�� +�((−1)−k) +, +χπ(E) = ++∞ +� +k=1 +���∂(πk(E, x)) +�� ��ι(πk(Eb, x)) +�� +�((−1)−k) +, +and also, +χπ(Eb) = +��π0(Eb) +�� ++∞ +� +k=1 +���i(πk(Eb, x)) +�� ��δ(πk+1(B, b)) +�� +�((−1)−k) +. +Crucially, in the last equation, we also used the fact that given that p: E → B +is a fibration, and E is path-connected, all path-components of Eb = p−1(b) are +homotopy equivalent, [48, Proposition 3]. This is reviewed in Lemma and Definition +94. +We thus have +χπ(Eb)χπ(B) = ++∞ +� +k=1 +���i(πk(Eb, x)) +�� ��∂(πk(E, x)) +�� +�� +(−1)k� += χπ(E). +Suppose now that E may have more that one path-component (but recall that +we still take B to be path-connected). Let E1, . . . , En be the path-components of +E. Let pk : Ek → B be the restriction of p to Ek, for each k = 1, . . . , n. Each pk is +itself a fibration, and is surjective. Let Fk = p−1 +k (b) = Eb ∩ Ek. Note that we have +an obvious continuous bijection ⊔n +k=1Fk → Eb, which is always a weak homotopy +equivalence. We therefore have: +χπ(E) = +n +� +k=1 +χπ(Ek) = +n +� +k=1 +χπ(Fk) χπ(B) += χπ� +n +� +k=1 +Fk +� +χπ(B) = χπ(Eb) χπ(B). +□ +We have the following, which is very useful later on. +Theorem 47. Let p: E → B be a fibration, where B and E are HF. If b ∈ B, and +Eb = p−1(b), then25 +χπ(E) = +� +[b]∈π0(B) +χπ(Eb) χπ(PCb(B)). +(Here we have chosen a representative of each path-component of B, noting that if +b and b′ are in the same path-component then Eb is homotopic to Eb′.) +Proof. If B is empty, then so is E, so the result follows trivially, so we suppose that +B ̸= ∅. +That if b ∈ B, then Eb is HF follows from Lemma 40. +Given [b] ∈ π0(B), +put E[b] = p−1(PCb(B)). The restriction, pb : E[b] → PCb(B), of p: E → B, is a +fibration. We have weak homotopy equivalences, +� +[b]∈π0(B) +PCb(B) → B +and +� +[b]∈π0(B) +E[b] → E, +25Recall that, if b ∈ B, then the path-component of b, in B, with the induced CGWH topology, +is denoted PCb(B); see Subsection 2.3. + +A CATEGORIFICATION OF QUINN’S TQFT +47 +therefore +χπ(E) = χπ� +� +[b]∈π0(B) +E[b] +� += +� +[b]∈π0(B) +χπ� +E[b] +� += +� +[b]∈π0(B) +χπ(Eb) χπ(PCb(B)). +□ +3.2. Fibrant spans of HF spaces and their composition. Before we intro- +duce fibrant spans in detail, we should briefly motivate why we are going to use +them. The objects considered in basic TQFTs are manifolds of some type, and the +cobordisms between them. Such a set-up gives a cospan of CGWH spaces, +Σ +i +�❘ +❘ +❘ +❘ +❘ +❘ +Σ′ +j +�❧❧❧❧❧❧ +S +and we have that the induced map, Σ ⊔ Σ′ → S, is an inclusion, and furthermore +a cofibration; see later in Subsection 4.1, starting on page 61, for a more detailed +discussion. We note that such cofibrant cospans of spaces are studied in detail in +[117, 118]. +To study the state spaces associated to the manifolds, we form the space of maps +from such manifolds to a ‘classifying space’ B, which we will take in Section 4 to +be homotopy finite, but, in so doing, we convert a cospan, as above, to a span, +BS +i∗ +�❦❦❦❦❦❦ +j∗ +�❚ +❚ +❚ +❚ +❚ +❚ +BΣ +BΣ′, +where i∗ and j∗ denote the obvious restriction maps, and we note that the induced +map from BS to BΣ × BΣ′ is a fibration. To study that type of situation, we need +to understand fibrant spans and we will examine them in some generality, not just +in this particular function space set-up. +From the point of view of the dual language of cofibrations, the duals of most +results in this subsection can be found in [117, 118] The techniques used there are +very similar, but, of course, dual to those used here. +3.2.1. Fibrant spans. Let p: A → B be a continuous map of CGWH spaces. If +b ∈ B, recall (Subsection 2.3) that we topologise p−1(b) with the k-ification of the +topology that B induces on A. Since p−1(b) is closed, p−1(b) is CGWH, so the +k-ification will not, in fact, alter the topology. +Definition 48 (Fibrant span). Let B, B′ and M be CGWH-spaces. +A fibrant +span, B +(p,M,p′) +−−−−−→ B′, from B to B′, is a diagram in CGWH of form, +(9) +M +p +�❥❥❥❥❥❥ +p′ +�❚ +❚ +❚ +❚ +❚ +❚ +B +B′, +where the induced map +� +p, p′� +: M → B × B′ is a fibration. If all three spaces are +HF, we will say this is a HF fibrant span or a fibrant span of HF spaces. +Remark 49. Consider the Hurewicz / Strøm model structure on CGWH; see +[112]. Let Λ be the category {−1 ← 0 → 1}. This is an inverse category in the +sense used in, for instance, [66, §5.1]. If we give the injective model structure to +CGWHΛ, then weak equivalences and cofibrations are given objectwise, see [66, +Theorem 5.1.3], whilst the fibrant spans are precisely the fibrant objects in that +category, CGWHΛ. + +A CATEGORIFICATION OF QUINN’S TQFT +48 +Example 50. To obtain lots of examples of such fibrant spans, we can use the +classical fibrant replacement process for turning an arbitrary map into a fibration, +and then apply that in a suitable way to an arbitrary span. +We first recall that fibrant replacement construction; cf. +[87, Chapter 7, §3]. +This replaces an arbitrary map, f : A → B, by a fibration. We first form the path +space, BI, with e0(B) : BI → B being ‘evaluation at 0’. This is a fibration, and we +can form the pullback along f, +Nf +πf +� +pr1 +� +BI +e0(B) +� +A +f +� B, +so Nf ∼= {(a, γ) | a ∈ A, γ : I → B, f(a) = γ(0)}. We set ρ(f) : Nf → B to be +e1(B) ◦ πf, so given by ρ(a, γ) = γ(1). This map is a fibration, and the natural +maps between Nf and A, induced by the constant path map from B to BI, make +them homotopy equivalent; see [87, Chapter 7, §3], or [64, §4.3, page 407]. +We need to apply this to a span, +X +f +�❦❦❦❦❦❦ +g +�❚ +❚ +❚ +❚ +❚ +❚ +A +B, +where no assumption is made about the induced map, ϕ := ⟨f, g⟩ : X → A×B. We +want to replace it by a fibration, +ρ(ϕ) : Nϕ → A × B, +and will thus get a fibrant span, +Nϕ +p +�❥❥❥❥❥❥ +p′ +�❚ +❚ +❚ +❚ +❚ +❚ +A +B, +where Nϕ ∼= {(x, γ1, γ2) | f(x) = γ1(0), g(x) = γ2(0)}, p(x, γ1, γ2) = γ1(1), whilst +p′(x, γ1, γ2) = γ2(1). +Example 51. Let X be a space. The trivial or identity span on X is then +X +idX +�❥❥❥❥❥❥ +idX �❯ +❯ +❯ +❯ +❯ +❯ +X +X. +This is clearly not a fibrant span if X is non-empty, so we want to replace it by a +fibrant replacement, e.g., by using the process sketched above. +Once that is done, we note that, in fact, that fibrant replacement has a simpler +formulation. We have a fibrant span of spaces given by +(10) +XI +sX +�❥❥❥❥❥❥ +tX �❚ +❚ +❚ +❚ +❚ +❚ +X +X, +where, if γ : I → X, then sX(γ) = γ(0), corresponds to e0(X), and tX(γ) = γ(1), +corresponding to e1(X). We can see directly that we have a fibration, +⟨sX, tX⟩: XI → X × X. +This follows, for instance, from the fact that the inclusion, ι: {0, 1} → I, is a +cofibration, and hence the induced map, ι∗ : XI → X{0,1}, is a fibration. Note that +X×X ∼= X{0,1}, as the category CGWH is cartesian closed and {0, 1} ∼= {0}⊔{1}. +We think of the span in Equation (10) as being a fibrantly ‘resolved’ replacement +for the identity span on X. +If X is HF, then so is XI, as it is homotopy equivalent to X. + +A CATEGORIFICATION OF QUINN’S TQFT +49 +Lemma 52. Consider a fibrant span, B +(p,M,p′) +−−−−−→ B′, as in Equation (9). Both +maps, p: M → B and p′ : M → B′, are fibrations, and moreover, given b ∈ B and +b′ ∈ B′, both of the induced maps, p−1(b) → B′ and p′−1(b′) → B, are fibrations. +Proof. For the first point, given the fact that both projections, B × B′ → B and +B ×B′ → B′, are fibrations, and also that the composite of fibrations is a fibration, +it follows that both p and p′ are fibrations. +The second point follows from the fact that pullbacks of fibrations are fibrations. +Specifically, consider a map f : X → B′ and a lifting, ˆf : X → p−1(b), of f (i.e., +p′ ◦ ˆf = f). Consider a homotopy, h: X × I → B′, with h(x, 0) = f(x), where +x ∈ X. and the homotopy, h′ : X × I → B × B′, given by h′(x, t) = (b, h(x, t)), +where x ∈ X and t ∈ I, then ⟨p, p′⟩( ˆf(x)) = h′(x, 0), for x ∈ X. A lifting of h′ to +M (given from the fact that ⟨p, p′⟩: M → B × B′ is a fibration) gives the desired +lifting of h. +□ +Spans can be composed in a well known way, but we will only need this when +working with HF fibrant spans. The calculations related to Quinn’s TQFT are, from +the perspective of this paper, calculations on homotopy invariants of intersections +of fibres of fibrant spans and how they react to composition, but this only in the +case in which the spaces are HF. We start the study of such in the next section. +3.2.2. HF fibrant spans. For the moment we will mostly concentrate on the prop- +erties of fibrant spans that depend on the spaces being HF. +Lemma 53. Let B +(p,M,p′) +−−−−−→ B′ be a fibrant span of HF-spaces from B to B′. Given +any (b, b′) ∈ B × B′, then the fibre, ⟨p, p′⟩−1(b, b′) ⊂ M, is HF. +Proof. This follows from Lemma 40, given that M and B × B′ are both HF. +□ +Lemma 54. Suppose that the fibrant span, B +(p,M,p′) +−−−−−→ B′, is HF. Let b ∈ B and +b′ ∈ B′. The spaces p−1(b), and p′−1(b′) and also the fibre of ⟨p, p′⟩: M → B × B′, +over (b, b′), i.e., +� +p, p′�−1(b, b′), are all HF. +Proof. By Lemma 52, we have a fibration p′ : p−1(b) → B′. The fibres have the +form +� +p, p′�−1(b, b′). These are also fibres for the fibration, ⟨p, p′⟩: M → B × B′, +so they must be HF, as M and B × B′ both are, by Lemma 40. +□ +Lemma 55. Let B, B′ and B′′ be HF-spaces. Consider HF fibrant spans, B +(p,M,p′) +−−−−−→ +B′, and B′ +(p′′,M′,p′′′) +−−−−−−−→ B′′. We form the obvious pullback, as in the diagram, +(11) +M ×B′ M ′ +P +� +q +�❣❣❣❣❣ +q′ +�❲ +❲ +❲ +❲ +❲ +M +p +�❦❦❦❦❦❦ +p′ +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +M ′ +p′′ +�❣❣❣❣❣❣❣❣❣ +p′′′ +�❯ +❯ +❯ +❯ +❯ +❯ +B +B′ +B′′, +where P = p′ ◦ q = p′′ ◦ q′, then the span, denoted +B +(p,M,p′)•(p′′,M′,p′′′) +−−−−−−−−−−−−−−→ B′′, +defined to be +B +(p◦q,M×B′ M′,p′′′◦q′) +−−−−−−−−−−−−−−→ B′′, +is a fibrant span of HF spaces. +We also have that +� +p ◦ q, P, p′′′ ◦ q′� +: M ×B′ M ′ → B × B′ × B′′ is a fibration. + +A CATEGORIFICATION OF QUINN’S TQFT +50 +Proof. That +� +p ◦ q, P, p′′′ ◦ q′� +is a fibration is clear from the fact that +� +p, p′� +and +� +p′′, p′′′� +are fibrations, and from the universal property of pullbacks. It follows +that +� +p ◦ q, p′′′ ◦ q′� +is also a fibration, for the projection, B × B′ × B′′ → B × B′′, +is a fibration. +To prove that M ×B′ M ′ is homotopically finite, it suffices (by Lemma 40) to +observe that B×B′×B′′ is HF and that the fibres of the fibration, +� +p◦q, P, p′′′◦q′� +, +have the form ( +� +p, p′�−1(b, b′)) × ( +� +p′′, p′′′�−1(b′, b′′)). Each fibre is thus HF, since +both components of the product are, by Lemma 53. +□ +Definition 56 (Composition of HF fibrant spans). The HF fibrant span, +B +(p,M,p′)•(p′′,M′,p′′′) +−−−−−−−−−−−−−−→ B′′, +is called the composite of (p, M, p′): B → B′ and (p′′, M ′, p′′′): B′ → B′′. +3.2.3. The category HFspan. We note that this will be a non-locally small category, +as we have a class of maps between objects. This will, however, not cause any +difficulties. +The class of objects of HFspan is the class of all HF spaces. Given HF-spaces, +B and B′, the class of morphisms from B to B′ is given by equivalence classes of +HF fibrant spans, (p, M, p′): B → B′, as we now explain. We will make use of the +materials on fibre homotopy equivalence recalled in §2.5. +Definition 57 (Equivalent HF fibrant spans). Let B and B′ be HF spaces. Two +HF fibrant spans, +(p, M, p′): B → B′ +and +(q, N, q′): B → B′, +are said to be equivalent if there exist fibred maps, Ψ: M → N and Ψ′ : N → M, +i.e. maps making the diagrams below commute, +(12) +M +Ψ +� +p +�♥♥♥♥♥♥ +p′ +�◗ +◗ +◗ +◗ +◗ +◗ +B +B′ +N +q +�PPPPPP +q′ +�♠ +♠ +♠ +♠ +♠ +♠ +and +M +p +�♦♦♦♦♦♦ +p′ +�◗ +◗ +◗ +◗ +◗ +◗ +B +B′, +N +Ψ′ +� +q +�❖❖❖❖❖❖ +q′ +�♥ +♥ +♥ +♥ +♥ +♥ +realising a fibre homotopy equivalence, with respect to the fibrations, ⟨p, p′⟩: M → +B × B′ and ⟨q, q′⟩: N → B × B′. This means that homotopies, H : M × I → M +and H′ : N × I → N, exist such that: +(1) H(m, 1) = Ψ′(Ψ(m)) and H(m, 0) = m for each m ∈ M; +(2) p(H(m, t)) = p(m) and p′(H(m, t)) = p′(m), for each m ∈ M and t ∈ I; +(3) H′(n, 1) = Ψ(Ψ′(n)) and H′(n, 0) = n, for each n ∈ N; +(4) q(H(n, t)) = q(n) and p′(H(n, t)) = q′(n), for each n ∈ N and t ∈ I. +If Ψ and Ψ′ are inverses of each other, then the HF fibrant spans are said to be +isomorphic. +Standard arguments prove that indeed this defines an equivalence relation on the +class of all HF fibrant spans, from B to B′. An equivalence class of HF fibrant +spans, from B to B′, will usually be denoted [(p, M, p′)]: B → B′. +Using the context and notation of Definition 57, we recall that pullbacks along +fibrations are homotopy limits. Given that ⟨p, p′⟩: M → B × B′ and ⟨q, q′⟩: N → +B × B′ are fibrations, several conditions in the definition of equivalence between +HF fibrant spans are, in fact, redundant. By Lemma 4, it follows that: +Lemma 58. Two HF fibrant spans, (p, M, p′): B → B′ and (q, N, q′): B → B′, +are equivalent if there exists a map, Ψ: M → N, making the left-most diagram of +(12) commute and such that Ψ: M → N is a homotopy equivalence of spaces. +□ + +A CATEGORIFICATION OF QUINN’S TQFT +51 +Definition 59. Given a HF space, B, put, +idHFspan +B += [(sB, BI, tB)], +to be the equivalence class of +(13) +BI +sB +�❦❦❦❦❦❦ +tB �❚ +❚ +❚ +❚ +❚ +❚ +B +B, +(see Example 51), under the equivalence relation in Definition 57. +Lemma 60 (The category HFspan). The composition of HF fibrant spans in Defi- +nition 55 descends to the quotient under the equivalence relation in Definition 57, +and, with this, the identities satisfy the evident rules. +Of course, we thus have a non-locally small category, HFspan, whose objects are +the HF-spaces, and in which the morphisms from B to B′ are the equivalence classes +of HF fibrant spans, connecting B and B′. Given a HF space, B, the identity in B +is given by [(s, BI, t)]: B → B. +Proof. That the composition descends to the quotient follows from the univer- +sal property of pullbacks26. +More precisely, suppose that Ψ1 : M1 → N1 and +Ψ′ +1 : N1 → M1 realise a fibred homotopy equivalence between the HF fibrant spans, +(p1, M1, q1): B1 → B and (p′ +1, N1, q′ +1): B1 → B. Suppose that Ψ2 : M2 → N2 and +Ψ′ +2 : N2 → M2 realise a fibred homotopy equivalence between the HF fibrant spans +(p2, M2, q2): B → B2 and (p′ +2, N2, q′ +2): B → B2. This is as in the diagram, +(14) +M1 +p1 +�④④④④④④④④ +q1 +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +Ψ1 +� +� +Ψ′ +1 +M1×BM2 +� +� M2 +Ψ2 +� +� +Ψ′ +2 +p2 +�tttttttttt +q2 +�❉ +❉ +❉ +❉ +❉ +❉ +❉ +❉ +B1 +B +B2 . +N1 +p′ +1 +�❈❈❈❈❈❈❈❈ +q′ +1 +�t +t +t +t +t +t +t +t +t +t +t +N1×BN2 +� +� N2 +p′ +2 +�❏❏❏❏❏❏❏❏❏❏❏ +q′ +2 +�③ +③ +③ +③ +③ +③ +③ +③ +The universal property of pullbacks gives maps, (Ψ1×BΨ2): M1×BM2 → N1×BN2, +co-gluing Ψ1 and Ψ2, and (Ψ′ +1×BΨ′ +2): N1×BN2 → M1×BM2 doing the same for +the other pair. +Choose fibred homotopies (using the notation in §2.5): +Ψ′ +1 ◦ Ψ1 +H1 +====⇒ +B1×B +idM1, +Ψ1 ◦ Ψ′ +1 +H′ +1 +====⇒ +B1×B +idN1, +Ψ′ +2 ◦ Ψ2 +H2 +====⇒ +B×B2 +idM2, +Ψ2 ◦ Ψ′ +2 +H′ +2 +====⇒ +B×B2 +idN2. +Conditions 1 to 4 of Definition 57 imply that these homotopies can be (co)glued to +homotopies, J : (M1×BM2) × I → M1×BM2 and J′ : (N1×BN2) × I → N1×BN2. +By construction, they are such that +(Ψ′ +1×BΨ′ +2) ◦ (Ψ1×BΨ2) +J +=====⇒ +B1×B2 +idM1×BM2, +(Ψ1×BΨ2) ◦ (Ψ′ +1×BΨ′ +2) +J′ +=====⇒ +B1×B2 +idN1×BN2. +26The arguments are essentially identical to those proving that cospans of spaces and maps +between them can be arranged into a bicategory; see [42]. + +A CATEGORIFICATION OF QUINN’S TQFT +52 +To handle the point about identities, let B and B′ be HF spaces. Consider a +HF fibrant span, (p, M, q): B → B′. Let us prove that we have maps Ψ and Ψ′, as +below, realising an equivalence of HF fibrant spans, +(15) +M +Ψ +� +p +�❦❦❦❦❦❦❦❦❦ +q +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +B +B′, +BI×BM +p′ +�❙❙❙❙❙❙ +q′ +�❦ +❦ +❦ +❦ +❦ +❦ +and +M� +Ψ′ +p +�❥❥❥❥❥❥❥❥ +q +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +B +B′. +BI×BM +p′ +�❙❙❙❙❙❙ +q′ +�❥ +❥ +❥ +❥ +❥ +❥ +Here we consider the obvious pull-back, appearing as the diamond in the diagram: +BI×BM +�❥❥❥❥❥❥ +�❚ +❚ +❚ +❚ +❚ +❚ +q′ +� +p′ +� +BI +sB +�❧❧❧❧❧❧ +tB +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +M +p +�✐✐✐✐✐✐✐✐✐ +q +�❘ +❘ +❘ +❘ +❘ +❘ +B +B +B′. +We put Ψ(m) = (� +p(m), m), where � +p(m) is the constant path at p(m) ∈ B. Clearly +Ψ is fibred. By Lemma 4, in the context of Lemma 58, we only need to prove that +Ψ: M → BI×BM is a homotopy equivalence of spaces (as opposed to a homotopy +equivalence of fibred spaces). A homotopy inverse of Ψ: M → BI×BM is given by +the map Φ: BI×BM → M such that Φ(γ, m) = m. (Note that this is not a fibred +map.) We have that Φ ◦ Ψ = idM. On the other hand Ψ(Φ(γ, m)) = ( � +γ(1), m), +for each (γ, m) ∈ BI×BM. (Here � +γ(1)) is the constant path at γ(1) ∈ B.) The +following homotopy connects Ψ ◦ Φ and idBI×BM: +(γ, m, t) ∈ (BI×BM) × I �→ +� +s �→ γ +� +t + (1 − t)s +� +, m +� +∈ BI×BM, +for s ∈ I. +That the resolved identity spans given by the mapping spaces are also identities +on the right is dealt with similarly. +□ +Remark 61. A ‘dual’ category to HFspan, whose objects are spaces, and morphisms +are cofibred homotopy equivalence classes of cofibrant cospans was constructed in +[117, 118]. +Our methods of proofs here are very similar, but, of course, needed +switching from cofibred to fibred homotopy equivalences. +Definition 62. We let HFiso be the subcategory of CGWH with objects the HF +spaces, and homeomorphisms of HF spaces as morphisms. +Lemma 63. We have a functor, I : HFiso → HFspan, given by, if X is a HF space, +then I(X) = X, and if f : X → Y is a homeomorphism of HF spaces, then I(f) is +the equivalence class of the span, +XI +sX +�❥❥❥❥❥❥ +f◦tX�❚ +❚ +❚ +❚ +❚ +❚ +X +Y. +(It is likely that a similar functor will map the category with objects the HF spaces, +and morphisms the homotopy classes of homotopy equivalences of HF spaces, to +HFspan, but we will not consider this, nor do we need it.) +Proof. It is clear that if f : X → Y is a homeomorphism of HF spaces, then I(f) = +(sX, XI, f ◦ tY ): X → Y is a HF fibrant span, since (sX, XI, tX): X → X is a HF +fibrant span. By construction, I sends the identities in HFiso to the identities in +HFspan. + +A CATEGORIFICATION OF QUINN’S TQFT +53 +Let f : X → Y and g : Y → Z be homeomorphisms of HF spaces. We check +that I(g ◦ f) = I(f) • I(g). To see this, look at the diagram below, where the top +diamond is a pullback, defining the composite I(f) • I(g): +XI ×Y Y I +proj1 +�tttttttttt +proj2 +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +XI +sX +�⑤⑤⑤⑤⑤⑤⑤⑤ +f◦tX +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +Y I +sY +�sssssssssss +g◦tY +�❇ +❇ +❇ +❇ +❇ +❇ +❇ +❇ +X +Y +Z. +XI +sX +�❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ +g◦f◦tX +�❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +❦ +The map, Ψ: XI ×Y Y I → XI, such that, +Ψ(γ, γ′)(t) = +� +γ(2t), +if t ∈ [0, 1/2], +f −1(γ′(2t − 1)), +if t ∈ [1/2, 1], +is a homeomorphism that makes the obvious diagram commute. This shows that +I(g ◦ f) = I(f) • I(g). +□ +Remark 64. There is a more general version of the notion of equivalence of (HF) +spans, as given in Definition 57, that will be useful slightly later on. Recall that +spans form a category, CGWHΛ, as noted in Remark 49, in which a morphism is +simply a natural transformation, +(p, M, p′) +(f−1,f0,f1) +� +(q, N, q′) +B +f−1 +� +M +p +� +f0 +� +p′ +� B′ +f1 +� +C +N +q +� +q′ +� C′. +As before, we take the Hurewicz / Strøm model structure on CGWH, and the +injective model structure on CGWHΛ. A morphism, such as (f−1, f0, f1), is thus +a cofibration, in that model structure, if each of f−1, f0, and f1 is a cofibration in +CGWH, and is a weak equivalence if each of these maps is a weak equivalence. +As, in the Hurewicz / Strøm model category structure, the weak equivalences are, +in fact, ‘strong’ homotopy equivalences, we make the following definition: +Definition 65. Two fibrant spans, +(p, M, p′): B → B′ +and +(q, N, q′): C → C′, +are said to be homotopy equivalent if there is a morphism, +(f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′), +in which each fi is a homotopy equivalence. +Of course, in this case, (f−1, f0, f1) is a homotopy equivalence27, and, in the +setting in which f−1 and f1 are the respective identities, we retrieve the notion +of equivalence given in Definition 57. These conditions28 do not depend on the +27which would be called a weak equivalence in the injective model structure on the category +of spans. +28We note that much of this depends on using homotopy equivalences and not just weak +equivalences. Although these coincide on the spaces that we are handling, it is, in practice, the +stronger form of the definition that is being used here. + +A CATEGORIFICATION OF QUINN’S TQFT +54 +spaces involved being homotopy finite, but it is only in that case that we will be +using them. We also note that fibrant spans are cofibrant-fibrant objects in the +injective model structure in CGWHΛ, so a homotopy equivalence of fibrant spans +will actually be a strong homotopy equivalence. +Given a map, (f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′), of fibrant spans, we get an +induced map, +(f0, ⟨f−1, f1⟩) : ⟨p, p′⟩ → ⟨q, q′⟩, +of fibrations (in the sense of Definition 6), so, if (f−1, f0, f1) is a homotopy equiva- +lence, then (f0, ⟨f−1, f1⟩) will be a homotopy equivalence of fibrations, (Definition +7), and by Proposition 8, there will be a homotopy inverse. +Of course, if b ∈ B and b′ ∈ B′, then by Corollary 9, or by using the fact that +(f−1, f0, f1) is a homotopy equivalence, we have +Proposition 66. If (f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′) is a homotopy equivalence +of fibrant spans, then, for any (b, b′) ∈ B × B′, the induced map on fibres, +⟨p, p′⟩−1(b, b′) → ⟨q, q′⟩−1(f−1(b), f1(b′)), +is a homotopy equivalence. +□ +We will return to this result slightly later on, both in the particular case of +an equivalence of fibrant spans going between B and B′ and in this more general +setting. +3.3. A family of functors, R(s) : HFspan → Vect, derived from the homo- +topy content. The results in this subsection are closely related to some given in +[56], where they are stated in the language of ∞-groupoids. They were, in fact, +essentially implicit in [101, Section 4]. The indexation of the family of functors is, +however, a generalisation of the α-degroupoidification set-up introduced by Baez, +Hoffnung and Walker in [5], Proposition 3.3, which we will very briefly recall shortly. +The setting here is particularly suited to constructing Quinn’s finite total homo- +topy TQFT, and explicitly to compute it in a number of cases, as well as moving +towards extended versions of Quinn’s finite total homotopy TQFT. The point about +the parameter, s, is then that, for s = 0, one has Quinn’s theory, as we will see +shortly, but, for other values of s, one also gets functors linked to other TQFTs, in +the normalisations in which they were initially constructed. +3.3.1. Quinn’s finite total homotopy TQFT and degroupoidification. We pause in +the development of the background theory to set the scene for the more detailed +treatment of Quinn’s theory, which is needed if we are to categorify that theory +to an extended form. We will revisit Quinn’s Lecture 4, [101], ‘once over lightly’, +and then look at parts of the paper, [5], on the ‘groupoidification’ and consequent +‘degroupoidification’ of linear algebra, mentioned above. +Quinn considers a fixed space, B, which is often a classifying space of a finite +group, but in any case has to be HF, in our terminology29. The TQFT, Z, has to +assign a ‘state space’ to each space, Y , and Quinn takes +Z(Y ) +abbr. += ZB(Y ) := Q[Y, B], +that is the rational vector space with basis the set, [Y, B], of homotopy classes of +maps from Y to B. +The rˆole of cobordisms in his theory is, then, played by what he calls CW- +triads30. This he takes to mean a CW-complex, X, together with two (disjoint) +29which is, of course, adapted from Quinn’s +30The term ‘CW-triad’ is used more generally in the literature to mean a triple of CW- +complexes as with Quinn, except that the disjointness of the two subcomplexes is not required. + +A CATEGORIFICATION OF QUINN’S TQFT +55 +subcomplexes, Y1 and Y2. Given such a triple, he wants to define a linear map, +ZX : Z(Y1) → Z(Y2), +so, if [f1] ∈ [Y1, B], then we will have +ZX([f1]) = +� +[f2] +µX,f1,f2[f2], +(see [101, p. 340] for more details and interpretation). The condition that Z is to +be a monoidal functor imposes conditions on how the assignment µ must behave. +He says that these conditions essentially determine µ, but does not elaborate on +this point. The formula he gives, in his notation31, is +µX,f1,f2 = #πMapf1(X, B)[f2]. +This is the homotopy content of the space of maps from X to B that restrict to f1 +on Y1, and which are homotopic to f2 on Y2. (Note the asymmetric treatment of the +two boundaries, which is mentioned [101, page 340]. In this paper, we will treat the +two boundaries symmetrically.) This definition works, but, in [101], many details +are only sketched or left as an exercise. Those ‘details’, although fairly ‘clear’ are, +in fact, quite tricky to check in full detail. (Quinn makes reference to a preprint +from 1991, but that does not seem to be readily available now.) +From our viewpoint, the CW-triad defines a cofibrant cospan and so, on passing +to mapping spaces, yields a fibrant span of spaces. In that context, Mapf1(X, B)[f2] +is an example of a ‘spatial slice’ in the terminology that we will introduce below. +Passing to ‘groupoidification’ and ‘homotopy linear algebra’, in the paper by +Baez, Hoffnung and Walker, [5], the setting is now the linear algebra corresponding +to spans of groupoids. Again we give a brief ‘recall’ and, again, adopt or adapt the +terminology of that paper. +Given a groupoid, X, they assign to it the real vector space with the set of +connected components of X as basis, (so that is R[X], in their notation or, for us, +R[π0(X)]). They also consider RX, defined to be the vector space {ψ : X → R}, +i.e., the space of maps from π0(X) to the reals. Of course, RX ∼= R[X]∗. +Now, given a span (of groupoids), +S +q +�❦❦❦❦❦❦ +p +�❚ +❚ +❚ +❚ +❚ +❚ +Y +X, +which is required to satisfy a ‘tameness’ condition32, in their Theorem 5.7, they +show how such a (q, S, p) yields a linear operator, +S +� +: RX → RY , +(and note the contravariance). This operator is defined by +(S +� +ψ)([y]) = +� +[x]∈X +� +[s] +|Aut(x)| +|Aut(s)| ψ([x]), +(and again we refer the reader to [5, p. 501] for the full explanation of the notation +and the limits on the ‘index variable’ [s]33). We again note the lack of symmetry +in the formula and, in fact, quoting from [5, p. 513]: +“one might wonder why this formula uses information about Aut(x), +but not Aut(y). The answer is that we made an arbitrary choice +of conventions. There is another equally nice choice and, in fact, +an entire family of choices interpolating between these two.” +31so his #π is our χπ, +32which we will not be needing so we omit here, +33as this involves the ‘tameness’ criterion that we mentioned before. + +A CATEGORIFICATION OF QUINN’S TQFT +56 +They then show how, for each α ∈ R, one gets a linear operator, +Sα +� +: RX → RY , +on replacing |Aut(x)| by |Aut(x)|1−α|Aut(y)α, and that the case α = 1 +2 has nicer +symmetry properties than that of α = 0, which they had introduced earlier. +We note that, in the context of TQFTs, we can similarly define a family of linear +operators associated to a cobordism, and indexed by a parameter, for us, s in C. +This will be done shortly. In the symmetric case, in which s = 1 +2, the resulting +TQFT is essentially that of Yetter in [124], and for, in that paper, B being the +classifying space of a crossed module. We will also see that these TQFTs, thus +obtained for different choices of s, are all naturally isomorphic. +We now return to our context of HF-spans, generalising the tame spans of +groupoids of [5] and adapting the α-groupoidification to that setting. +3.3.2. Spatial slices. Some terminology and a bit of notation will be useful. +It +will be widely used later on. The basic idea of what we will call a ‘spatial slice’ +is that of the fibre, ⟨p, p′⟩−1(b, b′), for an HF fibrant span, as before, and with +b ∈ B, and b′ ∈ B′, but given the way that we will have to use these, we need a +somewhat simpler notation, as it will be used in formulae which, in any case, are +quite complicated enough! +Definition 67 (Spatial slice). Let B +(p,M,p′) +−−−−−→ B′ be a HF fibrant span. Let b ∈ B, +b′ ∈ B′. Define the following (possibly empty) space, +{b +��(p, M, p′) +��b′} +abbr. += {b +��M +��b′} := {m ∈ M : p(m) = b and p′(m) = b′}, +which will be called here the spatial slice of B +(p,M,p′) +−−−−−→ B′, at b ∈ B and b′ ∈ B′. +We note that the abbreviated notation, {b|M|b′}, does not show the dependence +on p: M → B and p′ : M → B′, but we will use it more often than the more +complete {b +��(p, M, p′) +��b′}, so as not to overload the various formulae. +We also define the following spaces, also called spatial slices, but over the various +subsets of B, or B′, as indicated: +{b +��M +��PCb′(B′)} := {m ∈ M : p(m) = b and p′(m) ∈ PCb′(B′)}, +{PCb(B) +��M +��b′} = {m ∈ M : p(m) ∈ PCb(B) and p′(m) = b′}, +and +{PCb(B) +��M +��PCb′(B′)} = {m ∈ M : p(m) ∈ PCb(B) and p′(m) ∈ PCb′(B′)}. +Remark 68. Let B +(p,M,p′) +−−−−−→ B′ be a HF fibrant span. We collect some useful facts +about its spatial slices that will play important rˆoles later on. +• Since ⟨p, p′⟩: M → B × B′ restricts to a fibration, +{PCb(B) +��M +��PCb′(B′)} → B × B′, +and, by Lemma 52, p: M → B and p′ : M → B′ restrict to fibrations +{PCb(B) +��M +��PCb′(B′)} → B and {PCb(B) +��M +��PCb′(B′)} → B′, +the fibres, or more generally, the inverse images are, respectively, the spaces, +{b +��M +��b′}, {b +��M +��PCb′(B′)} and {PCb(B) +��M +��b′}. +• In particular, the homotopy type of the spaces, {b +��M +��b′}, {b +��M +��PCb′(B′)} and +{PCb(B) +��M +��b′}, depends only on the path-components of b in B and b′ in B′. + +A CATEGORIFICATION OF QUINN’S TQFT +57 +• Lemma 52 also gives that p and p′ restrict to fibrations, +{b|M|PCb′(B′)} → PCb′(B′) and {PCb(B)|M|b′} → PCb(B). +The fibres, again, have the form {b|M|b′}. +• All the spaces, +{b +��M +��b′}, +{b +��M +��PCb′(B′)}, +{PCb(B) +��M +��b′} +and +{PCb(B) +��M +��PCb′(B′)}, +are HF. This follows from Lemma 54. We can therefore take their homotopy +content. +Lemma 69. Let (p, M, p′) be a HF fibrant span. We have +χπ({PCb(B) +��M +��PCb′(B′)}) = χπ({b +��M +��PCb′(B)}) χπ(PCb(B)) += χπ({PCb(B) +��M +��b′}) χπ(PCb′(B′)) += χπ({b +��M +��b′}) χπ(PCb(B)) χπ(PCb′(B′)), +in Q. +Proof. This follows from Lemma 46 applied to the fibrations in Remark 68. +□ +3.3.3. Matrix elements. First recall Definition 67 and Lemma 69, and, as we want to +define some linear maps, recall also the General Notation and terminology relating +to matrix elements that we mentioned near the start of the paper on page 13. +Let B +(p,M,p′) +−−−−−→ B′ be a HF fibrant span. +We introduce a matrix over C, +parametrised by a complex valued index, s. +Definition 70. Let s ∈ C. Given (non-empty) path-components, PCb(B), PCb′(B′), +of B and B′, define the following complex valued ‘matrix-elements’, +� +PCb(B) +��R +(s)(p, M, p′) +��PCb′(B′) +� += χπ� +{b +��M +��b′} +� � +χπ(PCb(B)) +�s � +χπ(PCb′(B′)) +�1−s += χπ({PCb(B) +��M +��PCb′(B′)})) +� +χπ(PCb(B)) +�s−1 � +χπ(PCb′(B′)) +�−s ∈ C. +(As usual, we have written {b|M|b′} for {b|(p, M, p′)|b′}.) +We note, as well, that the homotopy type of {b|(p, M, p′)|b′} depends only on +the path-components, in B and B′, that b and b′, respectively, belong to, so +χπ({b +��M +��b′}) is indeed a function of PCb(B) and PCb′(B′), only. +The following result is essentially in [101]. A version of this result for groupoids +and spans appears in [5, Theorem 41], whilst a version for ∞-groupoids is in [56, +Proposition 8.2]. +Lemma 71. Let s ∈ C. The matrix elements corresponding to R +(s) are multi- +plicative with respect to composition of HF fibrant spans. Explicitly, consider HF +fibrant spans, (p, M, p′): B → B′, (p′′, M ′, p′′′): B′ → B′′, and their composition, +connecting B to B′′, +(PL, M ×B′ M ′, PR) = (p, M, p′) • (p′′, M ′, p′′′), +defined from the diagram below, where the middle diamond is a pullback, +(16) +M ×B′ M ′ +PL +� +PR +� +P +� +q +�❤❤❤❤❤❤ +q′ +�❱ +❱ +❱ +❱ +❱ +❱ +M +p +�♠♠♠♠♠♠ +p′ +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +M ′ +p′′ +�❤❤❤❤❤❤❤❤❤ +p′′′ +�❙ +❙ +❙ +❙ +❙ +❙ +B +B′ +B′′, + +A CATEGORIFICATION OF QUINN’S TQFT +58 +then, given b ∈ B and b′′ ∈ B′′, we have +� +PCb(B) +��R +(s)(PL, M ×B′ M ′, PR) +��PCb′′(B′′) +� += +� +[b′]∈π0(B′) +� +PCb(B) +��R +(s)(p, M, p′) +��PCb′(B′) +� � +PCb′(B′) +��R +(s)(p′, M ′, p′′) +��PCb′′(B′′) +� +. +Proof. We apply Theorem 47 to the fibration, +Pb,b′′ : {b +��M ×B′ M ′��b′′} → B′, +obtained by restricting P : M ×B′ M ′ → B′ to {b +��M ×B′M ′��b′′} = ⟨PL, PR⟩−1(b, b′). +This map, Pb,b′′, is a fibration, since, by Lemma 55, +� +PL, P, PR +� +: M ×B′ M ′ → B × B′ × B′′ +is a fibration34. +We, then, have +� +PCb(B) +��R +(s)(PL, M ×B′ M ′, PR) +��PCb′′(B′′) +� += χπ(PCb(B))s χπ(PCb′′(B′′))1−s χπ({b +��M ×B′ M ′��b′′}) += χπ(PCb(B))s χπ(PCb′′(B′′))1−s +� +[b′]∈π0(B′) +χπ(P −1 +b,b′′(b′)) χπ(PCb′(B′)). +Now note that, as spaces, +P −1 +b,b′′(b′) = {b +��(p, M, p′) +��b′} × {b′��(p′, M ′, p′′) +��b′′}, +so +χπ(P −1 +b,b′′(b′)) = χπ({b +��(p, M, p′) +��b′}) χπ({b′��(p′, M ′, p′′) +��b′′}). +This yields the main formula in the statement of the lemma. +□ +Lemma 72. Let B be a path-connected HF space. +Choose a base-point ∗. Let +Ω∗(B) denote the pointed loop space of (B, ∗), that is, the space of all loops in B +starting and ending in ∗, then Ω∗(B) is HF, and χπ(Ω∗(B)) = 1/χπ(B). +This is the analogue of [56, Lemma 3.10], which is for ∞-groupoids. +Proof. Let F∗(B) denote the space of all paths, α : [0, 1] → B, starting at ∗. +There is, of course, a fibration, F∗(B) → B, sending a path to its second end- +point, and we note that F∗(B) is contractible. +Since both F∗(B) and B are +HF, it follows that the fibre at ∗, which is Ω∗(B), is HF; see Lemma 40. Since +F∗(B) is contractible, we have that χπ(F∗(B)) = 1, and Lemma 46 gives that +χπ(Ω∗(B)) χπ(B) = χπ(F∗(B)) = 1. +□ +Lemma 73. Suppose B is a HF space and let b, b′ ∈ B. Let s ∈ C, then the matrix +element, +� +PCb(B) +��R +(s)(sX, B[0,1], tY ) +��PCb′(B) +� += δ +� +PCb(B), PCb′(B) +� +. +More generally, let f : B → B′ be homeomorphism of HF spaces, then, +� +PCb(B) +��R +(s)(sX, B[0,1], f ◦ tY ) +��PCb′(B′) +� += δ +� +PCb(B), PCf −1(b′)(B) +� +. +Here, if X is a set, which we will need to be the set of path-components of B, we +take δ: X × X → {0, 1}, to be such that δ(x, y) is 0, if x ̸= y, and δ(x, x)) = 1. +34cf. the proof of the second part of Lemma 52. + +A CATEGORIFICATION OF QUINN’S TQFT +59 +Proof. We prove the most general case. First of all note that if PCb(B) ̸= PCf −1(b′)(B), +then +� +PCb(B) +��R +(s)(sX, B[0,1], f ◦ tY ) +��PCb′(B′) +� += 0, +as in this case {b +��(sB, B[0,1], f ◦ tB) +��b′} is empty. On the other hand, +� +PCb(B) +��R +(s)(sX, B[0,1], f ◦ tY ) +��PCf(b)(B′) +� += χπ� +{b +��(sB, B[0,1], f ◦ tB) +��f(b)} +� +χπ(PCb(B))s χπ(PCf(b)(B′))1−s += χπ� +{b +��(sB, B[0,1], tB) +��b} +� +χπ(PCb(B))s χπ(PCb(B))1−s += χπ� +Ωb(PCb(B)) +� +χπ(PCb(B)) = 1, +where we have used Lemma 72. +□ +We thus have35 +Theorem 74. Let s ∈ C. There is a functor, R(s) : HFspan → VectC, such that: +• if B is a HF space, then R(s)(B) = C(�π0(B)), the free vector space over the set +of all path-components of B; +• for [(p, M, p′)]: B → B′, a morphism in HFspan, the matrix elements for the +linear map, +R(s)� +[(p, M, p′)] +� +: C(�π0(B)) → C(�π0(B′)), +are given by +� +PCb(B) +��R(s)� +[(p, M, p′)] +���PCb′(B′) +� +:= +� +PCb(B) +��R +(s)(p, M, p′) +��PCb′(B′) +� +, +for given path-components PCb(B) ∈ �π0(B) and PCb′(B′) ∈ �π0(B′). +Proof. Compatibility of R(s) with the composition and identities in HFspan follows +from Lemmas 71 and 73, respectively. What we have not yet shown is that the +matrix elements R +(s) are invariant under equivalence of spans. This follows by +using lemma 5, combined with Definition 57, as made explicit in Proposition 66. +□ +Lemma 75. We have a functor, ×: HFspan × HFspan → HFspan, sending the +equivalence class of, +� +M +p +�♠♠♠♠♠♠ +q +�◗ +◗ +◗ +◗ +◗ +◗ +X +Y +, +M ′ +p′ +�♠♠♠♠♠♠ +q′ +�◗ +◗ +◗ +◗ +◗ +◗ +X′ +Y ′ +� +, +to the equivalence class of, +M × M ′ +p×p′ +�❤❤❤❤❤ +q×q′ +�❱ +❱ +❱ +❱ +❱ +X × X′ +Y × Y ′. +(Note that the latter span is fibrant, since the product of two fibrations is a fibration.) +Proof. This follows from straightforward calculations. +□ +It can furthermore be proved that HFspan is a monoidal category, with this tensor +product, but we will not need this here. For details in the dual case of cofibrant +cospans, see [117, 118]. +Lemma 76. If ⊗: VectC × VectC → VectC denotes the tensor product functor +in VectC, then we have a natural isomorphism of functors from HFspan × HFspan +to VectC, +η: ⊗ ◦ +� +R(s) × R(s)� +=⇒ R(s) ◦ ×. +35The notation �π0(B) is explained at the end of Subsection 2.3 + +A CATEGORIFICATION OF QUINN’S TQFT +60 +It is such that, given HF spaces X and X′, and x ∈ X, x′ ∈ X′, then +ηX,X′� +PCx(X)⊗PCx′(X′) +� += PC(x,x′)(X × X′). +Proof. If X and X′ are spaces, then we have a natural bijection from �π0(X)×�π0(X′) +to �π0(X × X′), sending +� +PCx(X), PCx′(X′) +� +to PC(x,x′)(X × X′). The naturality +of η then follows from the calculation below. +Consider two HF fibrant spans, +M +p +�♥♥♥♥♥♥ +q +�P +P +P +P +P +P +X +Y, +and +M ′ +p′ +�♠♠♠♠♠♠ +q′ +�❘ +❘ +❘ +❘ +❘ +❘ +X′ +Y ′. +If x ∈ X, x′ ∈ X′, y ∈ Y and y′ ∈ Y ′, we have: +� +PC(x,x′)(X × X′) |R +(s)� +(p, p′), M × M ′, (q, q′) +� +|PC(y,y′)(Y × Y ′) +� += χπ �� +(x, x′) | +� +(p, p′), M × M ′, (q, q′) +� +| (y, y′) +�� +χπ(PC(x,x′)(X × X′) +�s χπ� +PC(y,y′)(Y × Y ′) +�1−s. +Note that we have homeomorphisms, +� +(x, x′) | +� +(p, p′), M × M ′, (q, q′) +� +| (y, y′) +� ∼= {x | (p, M, q) | y} × {x′ | (p′, M ′, q′) | y′}, +PC(x,x′)(X × X′) ∼= PCx(X) × PCx′(X′), +and +PC(y,y′)(Y × Y ′) ∼= PCy(Y ) × PCy(Y ′). +It therefore follows that, +� +PC(x,x′)(X × X′) | R +(s)� +(p, p′), M × M ′, (q, q′) +� +| PC(y,y′)(Y × Y ′) +� += +� +PCx(X) | R +(s)� +(p, M, q) +� +|PCy(Y ) +� � +PCx′(X′) | R +(s)� +(p′, M ′, q′) +� +|PCy′(Y ′) +� +. +The last equation follows from Lemma 45. +□ +Finally, note the following result, that follows from a straightforward calculation +using the conventions in Definition 70. +Proposition 77. Let s, t be complex numbers. We have a natural isomorphism, +ηs,t : R(s) =⇒ R(t), of functors, from HFspan to VectC, which is such that, if X +is a space, and x ∈ X, then, +ηs,t +X (PCx(X)) = χπ (PCx(X))s−t PCx(X). +4. A more detailed review of Quinn’s finite total homotopy TQFT +First let us recall some basic definitions. Most of the time, we will be able to +do the necessary constructions without this level of detail, just as Quinn does in +the primary source, [101], where he discusses a version of the theory using just CW +complexes, but very occasionally, these results, or their consequences, are needed. +They also help to tie Quinn’s theory into the general theory of TQFTs. + +A CATEGORIFICATION OF QUINN’S TQFT +61 +4.1. Cobordism categories. For a positive integer n, let Cob(n,n+1) be the +monoidal category of compact smooth manifolds and cobordisms between them. +Details are discussed in many places in the literature, e.g. [37, 89]. We note, once +again, that we make no assumption that orientations on manifolds and cobordisms +are given, or even that they exist36. +The class of objects of Cob(n,n+1) is given by all compact smooth-manifolds +of dimension n. +Given compact smooth n-manifolds, Σ and Σ′, morphisms in +Cob(n,n+1) from Σ to Σ′, are equivalence classes of cobordisms, (i, S, j): Σ → Σ′. +Here a cobordism is a cospan of compact smooth manifolds and smooth maps, +Σ +i +�◗ +◗ +◗ +◗ +◗ +◗ +Σ′ , +j +�❧❧❧❧❧❧ +S +where i and j are smooth maps inducing a diffeomorphism, ⟨i, j⟩: Σ ⊔ Σ′ → ∂(S). +Two cobordisms, (i, S, j), (i′, S′, j′): Σ → Σ′, are considered equivalent if a smooth +diffeomorphism, f : S → S′, exists, making the diagram, +S +f +� +Σ +i +�s +s +s +s +s +s +i′ +�❑ +❑ +❑ +❑ +❑ +❑ +Σ′ , +j +�▼▼▼▼▼▼ +j′ +�qqqqqq +S′ +commute. +We will use a hopefully evident notation for the equivalence classes of cobordisms, +but may later forget to use it if no ambiguity should arise from its omission. +The composition of morphisms, [(i, S, j)]: Σ → Σ′ and [(i′, S′, j′)]: Σ′ → Σ′′, +is done as follows. +We first consider the pushout, S ⊔Σ′ S′, in CGWH, as in +the diagram below (the nodes in the first and second rows contain the underlying +topological manifolds of the corresponding smooth manifolds): +Σ +i +�❄ +❄ +❄ +❄ +❄ +❄ +❄ +❄ +Σ′ +j +�✈✈✈✈✈✈✈✈✈✈ +i′ +�❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +Σ′′ +j′ +�⑤⑤⑤⑤⑤⑤⑤⑤ +S +k +�● +● +● +● +● +● +● +● +● +S′ +k′ +�✈✈✈✈✈✈✈✈✈ +S ⊔Σ′ S′ +. +The topological space, S ⊔Σ′ S′, is a topological manifold, see [89, §1] or [65, +§8.2], and ⟨k ◦i, k′ ◦j′⟩: Σ⊔Σ′′ → ∂(S ⊔Σ′ S′) is a homeomorphism. This yields the +cospan below in CGWH, where again the nodes, as yet, only denote topological +manifolds, +Σ +k◦i +�▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +Σ′′ +k′◦j′ +�♣♣♣♣♣♣♣ +S ⊔Σ′ S′ +. +As is well known, S ⊔Σ′ S′ can be given a smooth structure, which ‘restricts’ to +the smooth structures in S and in S′. This smooth structure, despite not being +unique, as it depends on the choice of a collar of Σ′ in S and in S′, is unique up to +a diffeomorphism, which is the identity on ∂(S ⊔Σ′ S′); for discussion see [89, §3]. +36Quinn’s finite total homotopy TQFT makes no assumption of orientability of manifolds. + +A CATEGORIFICATION OF QUINN’S TQFT +62 +The composition, •, of morphisms, [(i, S, j)]: Σ → Σ′ and [(i′, S′, j′)]: Σ′ → Σ′′, +in Cob(n,n+1) is then given as +� +[(i, S, j)]: Σ → Σ′� +• +� +[(i′, S′, j′)]: Σ′ → Σ′′� +:= +� +[(k ◦i, S ⊔Σ′ S′, k′ ◦j′)]: Σ → Σ′′� +. +Given an n-manifold, Σ, the identity cobordism, idCob(n,n+1) +Σ +: Σ → Σ, is the +equivalence class of the cospan, +Σ +ιΣ +0 +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +Σ +ιΣ +1 +�sssssss +Σ × I +. +(Here ιΣ +i (x) = (x, i), for all i ∈ {0, 1}.) +4.1.1. The symmetric monoidal structures in Diffn and Cob(n,n+1). Let Diff n +denote the category of closed n-manifolds and diffeomorphism between them. We +have a functor, I′ : Diffn → Cob(n,n+1), which is the identity on objects, and such +that I′(f : Σ → Σ′) is the equivalence class of the cospan, +Σ +ιΣ +0 +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +Σ′ . +ιΣ′ +1 ◦f −1 +�qqqqqqq +Σ × I +The proof of this fact is dual to that of Lemma 63. This, of course, implies that +each I′(f) will be an invertible morphism in Cob(n,n+1), i.e., that the cobordism +is invertible up to equivalence. +It is well known that this functor descends to +the category with objects the closed n-manifolds and morphisms isotopy classes of +diffeomorphisms. This will, however, not be used in the following. +Remark 78. There is another way of obtaining a cobordism from a diffeomorphism. +Instead of using f −1 in the right (co)leg, we could have used f in the left one, that +is, +Σ +ιΣ′◦f +0 +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +Σ′ . +ιΣ′ +1 +�qqqqqqq +Σ′ × I +This gives an equivalent cobordism as f × I : Σ × I → Σ′ × I fits in a commutative +diagram +Σ × I +f×I +∼ += +� +Σ +ιΣ′ +0 ◦f �◆ +◆ +◆ +◆ +◆ +◆ +ιΣ +0 +�♦ +♦ +♦ +♦ +♦ +♦ +♦ +Σ′ , +ιΣ′ +1 ◦f −1 +�PPPPPPP +ιΣ′ +1 +�♥♥♥♥♥♥ +Σ′ × I +so gives the same functor as the previous construction. +We will later on ‘categorify’ this second I′-construction, when we are considering +the symmetric monoidal bicategory structure on 2Cob(n,n+1,n+2) in section 6.5.2. +Recall that both Cob(n,n+1) and Diff n are symmetric monoidal categories, +where the tensor product on objects is given by the disjoint union, Σ⊔Σ′, of closed +n-manifolds, Σ and Σ′. For both symmetric monoidal bicategories, the unit object +is the empty manifold, ∅. In Diffn, the tensor product of morphisms is achieved as +in (CGWH, ⊔), i.e. by performing the disjoint union of diffeomorphisms, namely +(f1 : Σ1 → Σ′ +1) ⊔ (f2 : Σ2 → Σ′ +2) = (f1 ⊔ f2): Σ1 ⊔ Σ2 → Σ′ +1 ⊔ Σ′ +2. +The associativity constraints, braiding, etc., in Diffn are also as those in (CGWH, ⊔). + +A CATEGORIFICATION OF QUINN’S TQFT +63 +In Cob(n,n+1), the monoidal structure is based on the functor, +⊔: Cob(n,n+1) × Cob(n,n+1) → Cob(n,n+1), +so is obtained from the disjoint union of cobordisms, which descends to their equiv- +alence classes. (This is dual to the construction in Lemma 75.) The associativity +and unit constraints, and the braiding in Cob(n,n+1), are obtained from those of +Diffn by applying I′ : Diff n → Cob(n,n+1). +4.2. Quinn’s results on HF function spaces. As mentioned earlier, in the ‘once +over lightly’ description of Quinn’s finite total homotopy TQFT, on page 55, Quinn +uses various results on mapping spaces. +Let B be a HF space. +Lemma 79 (Quinn). Let X be a finite CW-complex, then BX is a HF-space. +Proof. The proof follows from an induction on the number of cells of X, by making +use of the following lemma in each induction step; cf. [101, Chapter 4]. +□ +Lemma 80. Let i be a non-negative integer. Let a space, Y , be obtained from the +CW-complex X by attaching an i-cell. Suppose BX is HF, then so is BY . +Proof. Let Y be obtained from X by attaching an i-cell along f : Si−1 → X. We +hence have a pushout diagram, +Si−1 +� +f +� X +� +Di +� Y , +where both vertical arrows are induced by inclusion of subcomplexes, hence they +are cofibrations. +Passing to function spaces, and using the fact that CGWH +is monoidal closed, we have a pullback diagram, where the vertical arrows are +moreover fibrations, given that they are ‘dual’ to cofibrations, +BY +� +� +BDi +� +BX +f ∗ +� BSi−1 . +Concretely, each vertical arrow is obtained by restricting a function defined on a +CW-complex to a subcomplex. +We now apply Lemma 41. Since BX and BDi are HF, the first by assumption, +the second since BDi is contractible, the proof is reduced to proving that BSi−1 is +HF. This is proved in the following lemma. +□ +Lemma 81. Given any positive integer n, the space, BSn, is HF. +Proof. Again the proof is by induction in n. The base case follows from the fact +that BS0 = B{0,1} ∼= B × B. The induction step follows by observing that we have +the following pullback diagram, where the vertical arrows are fibrations: +BSn +� +� +BDn +� +BDn +� BSn−1 +and noting, once again, that BDn is HF, since it is contractible. +□ + +A CATEGORIFICATION OF QUINN’S TQFT +64 +Lemma 82. Let n be a non-negative integer, and B be a HF-space. +(1) There is a functor, F0 +B : (Diffn)op → HFiso37, that sends a closed and smooth +n-manifold, Σ, to BΣ and a diffeomorphism, f : Σ → Σ′, to the induced map38, +f ∗ : BΣ′ → BΣ. +(2) There is a functor, FB : Cob(n,n+1) → HFspan, that sends the equivalence class +of a cobordism, +Σ +i +�◗ +◗ +◗ +◗ +◗ +◗ +Σ′, +j +�❧❧❧❧❧❧ +S +to the equivalence class of the HF fibrant span, +(17) +BΣ � +i∗ ❘ +❘ +❘ +❘ +❘ +❘ +BΣ′. +� +j∗ +❦❦❦❦❦❦ +BS +Proof. If Σ is a compact smooth manifold, then Σ has a finite triangulation, and, +in particular, it can be given the structure of a finite CW-complex, [94], so, by +Lemma 79, BΣ is HF. The rest of (1) follows from the fact that CGWH is a +cartesian closed category. +For the second point, also note that if S is a smooth manifold with boundary, +then we can find, again by, for instance, [94], a triangulation of the pair (S, ∂(S)) +making ∂(S) a subcomplex of S, so the inclusion ι: ∂(S) → S is a cofibration39. As +a consequence, the induced map, ι∗ : BS → B∂(S), is a fibration. To see that the HF +span in (17) is fibrant, note that B∂(S) ∼= BΣ⊔Σ′ ∼= BΣ × BΣ′, where we used the +fact that CGWH is cartesian closed in the last step. Hence FB sends (equivalence +classes of) cobordisms of manifolds to (equivalence classes of) HF fibrant spans. +That FB preserves the compositions of Cob(n,n+1) and HFspan follows again from +the fact that CGWH is cartesian closed, and, in particular, that the contravariant +functor, B( ) : CGWH → CGWH, sends colimits to limits. +Finally units are +preserved by definition. +□ +4.3. Quinn’s finite total homotopy TQFT. In the following, we let n be a +non-negative integer, B be a HF space, and fix some s ∈ C. We will work over the +complex number field C. +As we mentioned in the introduction, in [101, Lecture 4], Quinn defined what he +called the finite total homotopy TQFT, which we will denote by +QB : Cob(n,n+1) → Vect, +depending on an arbitrary, but fixed, homotopy finite space, B. As we will see, this +is one of a family of such constructions, one for each s ∈ C, albeit all related by +natural isomorphisms, as in Proposition 77. +A particular case of the general s case appeared in [5, 3.3 Proposition, page 513] +in the context of TQFT-like functors for spans of groupoids. +Definition 83 ((The s-indexed form of) Quinn’s finite total homotopy TQFT). +Quinn’s finite total homotopy TQFT, +Q(s) +B : Cob(n,n+1) → Vect, +is defined to be the composite of the functors, +FB : Cob(n,n+1) → HFspan +and +R(s) : HFspan → Vect. +37See Definition 62. +38I.e. the map sending φ: Σ′ → B to φ ◦ f : Σ → B. +39An alternative proof of this fact is in [117, 118]. + +A CATEGORIFICATION OF QUINN’S TQFT +65 +We will write QB for Q(0) +B . (This was Quinn’s original normalisation.) +For the second of these functors, see Theorem 74, and use Lemma 82 to allow +its application here. +Taking this apart, and in more detail, +• given a closed n-manifold, Σ, then +Q(s) +B (Σ) = C(�π0(BΣ)); +• given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, between the closed n- +manifolds, Σ1 and Σ2, we have that the matrix elements of the resulting +linear operator are given by the equation below, for continuous functions +f : Σ1 → B and f ′ : Σ2 → B, +� +PCf(BΣ1) | Q(s) +B ([(i1, S, i2)]) | PCf ′(BΣ2) +� += χπ� +{f|(i∗ +1, BS, i∗ +2)|f ′} +� � +χπ(PCf(BΣ1)) +�s� +χπ(PCf ′(BΣ2)) +�1−s. +We are using the notation of Definition 67, so +{f|(i∗ +1, BS, i∗ +2)|f ′} +abbr. += {f|BS|f ′} = + + + + + +H : S → B | +B +Σ1 +f +�❧ +❧ +❧ +❧ +❧ +❧ +i1 +�❙ +❙ +❙ +❙ +❙ +❙ +Σ2 +f ′ +�❙❙❙❙❙❙ +i2 +�❦❦❦❦❦❦ +S +H +� +commutes + + + + + +. +The following elementary lemma will be used later in the proof that the functor +Q(s) +B : Cob(n,n+1) → Vect, is symmetric monoidal. We use the notation in §4.1.1. +Lemma 84. Let B be a homotopy finite space. Let n be a non-negative integer. +(1) There is a symmetric monoidal functor, TB : Diff n → Vect, such that +• TB(Σ) = Q(s) +B (Σ) = C +� +�π0(BΣ) +� +, +and +• given f : Σ → B and f ′ : Σ′ → B, and a diffeomorphism, φ: Σ → Σ′, +then the matrix elements satisfy +� +PCf(BΣ) | TB(φ: Σ → Σ′) | PCf ′(BΣ′) +� += +� +1, if PCf(BΣ) = PCf ′◦φ(BΣ), +0, otherwise. +(2) If φ: Σ → Σ′ is a diffeomorphism, then +Q(s) +B (I′(φ)) = TB(φ). +Proof. The existence of TB, and that it can be upgraded to be a symmetric monoidal +functor, follows from standard results from algebraic topology. The crucial point is +that the functor π0 : CGWH → Set is a symmetric monoidal functor, since, given +CGWH spaces X and Y , we have a natural bijection, η′′ +X,Y : π0(X) × π0(Y ) → +π0(X ×Y ), such that, for x ∈ X and y ∈ Y , (PCx(X), PCy(Y )) �→ PC(x,y)(X ×Y ). +Furthermore, the natural isomorphism, η′′ : ×◦(π0×π0) =⇒ π0◦×, is associative, +which means that the diagrams below commutes, for all CGWH spaces X, Y and + +A CATEGORIFICATION OF QUINN’S TQFT +66 +Z, +� +π0(X) × π0(Y ) +� +× π0(Z) +η′′ +X,Y ⊗π0(Z) +� +αSet +π0(X),π0(Y ),π0(Z) +� π0(X) × +� +π0(Y ) × π0(Z) +� +π0(X)⊗η′′ +Y,Z +� +� +π0(X × Y ) +� +⊗ π0(Z) +η′′ +X×Y,Z +� +π0(X) ⊗ +� +π0(Y × Z) +� +η′′ +X,Y ×Z +� +π0 +� +(X × Y ) × Z) +� +π0 +� +αCGWH +X,Y,Z +� +� π0 +� +X × (Y × Z) +� +. +If we consider the bijection, ǫ: {∗} → π0({∗}), we can see that, given a CGHW +space X, the two diagrams pertaining to the unitality of η′′, commute, for instance, +π0(X) × {∗} +π0(X)×ǫ +� +ρSet +π0(X) +� +� +π0(X) +π0(X) × π0({∗}) +η′′ +X,{∗} +� π0(X × {∗}), +π0(ρCGWH +X +) +� +so the triple (TB, η′′, ǫ) is a (strong) monoidal functor. Given that the diagram, +π0(X) × π0(Y ) +η′′ +X,Y +� +τ Set +π0(X),π0(Y ) � π0(Y ) × π0(X) +η′′ +Y,X +� +π0(X × Y ) +π0(τ CGWH +X,Y +) +� π0(Y × X). +clearly commutes, (TB, η′′, ǫ) is also a symmetric monoidal functor. +The free vector space functor, Lin: Set → Vect, is symmetric monoidal. By us- +ing the fact that CGWH is a monoidal closed category, the functor F0 +B : (Diffn)op → +HFiso (for notation see Lemma 84) is also symmetric monoidal, in a natural way. +Now note that TB is given by the following composition of functors: +Diffn +(−)−1 +−−−−→ (Diffn)op +F 0 +B +−−→ HFiso +inc +−−→ CGWH +π0 +−→ Set +Lin +−−→ Vect. +The second point of the lemma follows from the second part of Lemma 73. +□ +Note that a part of the monoidal structure of TB is a natural isomorphism, +⊗ ◦ (TB × TB) =⇒ TB ◦ ⊔, +which, by tracking the sequence of compositions above, explicitly is such that, given +closed smooth n-manifolds, Σ and Σ′, and maps, f : Σ → B and f ′ : Σ′ → B, +PCf(BΣ) ⊗ PCf ′(BΣ′) �→ PC⟨f,f ′⟩ +� +BΣ⊔Σ′� +. +4.4. A discussion of the monoidality of Quinn’s finite total homotopy +TQFT. Let, once again, B be a homotopy finite space. The functor, +Q(s) +B : Cob(n,n+1) → Vect, +can be upgraded to be a symmetric monoidal functor, which we will also denote by +Q(s) +B : Cob(n,n+1) → Vect. + +A CATEGORIFICATION OF QUINN’S TQFT +67 +The key part of the construction is the natural isomorphism, +η′ : ⊗ ◦ +� +Q(s) +B × Q(s) +B +� +=⇒ Q(s) +B ◦ ⊔, +of functors from Cob(n,n+1) × Cob(n,n+1) to Vect defined as the composite, +⊗ ◦ +� +Q(s) +B × Q(s) +B +� += ⊗ ◦ +� +R(s) ◦ FB × R(s) ◦ FB) +η◦(FB×FB) +========⇒ R(s) ◦ × ◦ (FB × FB) +∼ += +=⇒ Q(s) +B ◦ FB ◦ ⊔. +Here η is defined in Lemma 76, and the last natural isomorphism follows from the +fact that CGWH is cartesian closed. Explicitly, given compact smooth n-manifolds +Σ and Σ′, +η′ +Σ,Σ′ : Q(s) +B (Σ) ⊗ Q(s) +B (Σ′) → Q(s) +B (Σ ⊔ Σ′) +is such that, given f : Σ → B and f ′ : Σ′ → B, +PCf(BΣ) ⊗ PCf ′(BΣ′) �→ PC⟨f,f ′⟩ +� +BΣ⊔Σ′� +. +This natural isomorphism can easily be proved to be ‘associative’, meaning that, +given closed smooth n-manifolds Σ, Σ′, Σ′′, the following diagram commutes (where +we omitted the labels in the associativity constraints in Vect), +� +Q(s) +B (Σ) ⊗ Q(s) +B (Σ′) +� +⊗ Q(s) +B (Σ′′) +η′ +Σ,Σ′⊗Q(s) +B (Σ′′) +� +αVect � Q(s) +B (Σ) ⊗ +� +Q(s) +B (Σ′) ⊗ Q(s) +B (Σ′′) +� +Q(s) +B (Σ)⊗η′ +Σ′,Σ′′ +� +� +Q(s) +B (Σ ⊔ Σ′) +� +⊗ Q(s) +B (Σ′′) +η′ +Σ⊔Σ′,Σ′′ +� +Q(s) +B (Σ) ⊗ +� +Q(s) +B (Σ′ ⊔ Σ′′) +� +η′ +Σ,Σ′⊔Σ′′ +� +Q(s) +B +� +(Σ ⊔ Σ′) ⊔ Σ′′) +� +Q(s) +B +� +αCob(n,n+1) +Σ,Σ′,Σ′′ +� � Q(s) +B +� +Σ ⊔ (Σ′ ⊔ Σ′′) +� +. +That this diagram commutes, follows from the fact that, using the definition of +the monoidal structure of Cob(n,n+1) sketched in §4.1.1, we have +Q(s) +B +� +αCob(n,n+1) +Σ,Σ′,Σ′′ +� += Q(s) +B +� +I′� +αDiff n +Σ,Σ′,Σ′′ +�� += TB +� +αDiff n +Σ,Σ′,Σ′′ +� +. +Here we have used the second point of Lemma 84 in the last step. We have that +the diagram above commutes, since the functor TB is monoidal, by the first point +of Lemma 84. Note that TB and Q(s) +B coincide on objects. +The remaining bits of the proof of the fact that Q(s) +B : Cob(n,n+1) → Vect can +be turned into a symmetric monoidal functor follow exactly the same pattern. The +details are left to the reader. +4.5. Some examples and properties of Quinn’s finite total homotopy TQFT. +The ground field for Quinn’s finite total homotopy TQFT can be taken to be Q, for +s = 1 or s = 0, or the Galois closure of Q, for s = 1/2. The case s = 0 was the one +developed in [101], whilst the case s = 1/2 coincides with the conventions in [124]. +Note that we have monoidal natural isomorphisms connecting all normalisations of +Quinn’s finite total homotopy TQFT, obtained by applying Proposition 77. Again +the remaining details are left to the reader. +Methods for concrete calculations of Quinn’s TQFT will be addressed in Part 4, +where B will be the classifying space of a homotopy finite crossed complex. There +we will also discuss the methods of calculation in the extended version. For the + +A CATEGORIFICATION OF QUINN’S TQFT +68 +moment, we will restrict our attention to some simple examples and observations. +These are essentially ‘well known’, and easy to calculate, but are included to lay +the ground for the corresponding examples and results for the extended case. +Example 85. A trivial, but sometimes useful, example is from the case in which +B = {∗}, i.e., a singleton space. +For any Σ, BΣ is also a singleton space, as +there is a unique map from Σ to {∗}. We thus have π0(BΣ) is a singleton set, and +Q(s) +B (Σ) ∼= C. +Suppose that (i1, S, i2): Σ1 → Σ2, then BS is also a singleton, FB(S) is the +terminal span and, not surprisingly, the 1 × 1 matrix Q(s) +B ([(i1, S, i2)] is just the +identity 1 × 1 ‘matrix’. We thus have that Q(s) +B is the trivial TQFT. +This trivial example is related to a simple construction for when B is a product +space. +Suppose that B = B1 × B2, then BΣ ∼= BΣ +1 × BΣ +2 , so we have Q(s) +B (Σ) ∼= +Q(s) +B1(Σ) ⊗ Q(s) +B2(Σ), and similarly for the operation on the cobordisms40. The fact +that {∗} is the unit for the product, ×, is reflected here by the fact that, as C is +the unit for the ⊗ in Vect, the operation of tensoring a general Quinn type TQFT +with Q(s) +∗ +will be trivial. +Example 86. Quite often, it is assumed that B is a path-connected space, but it is +interesting to see what happens if it is not. We start in a very simple case, namely +that in which B = {0, 1}, i.e., a two point discrete space. We can think of a map +Σ → B as ‘labelling’ or ‘colouring’ the path components of the space, Σ, by elements +from B, so we may refer to a B-colouring of a space in more generality41. +We first look at Q(s) +B (Σ) for Σ a connected manifold, then BΣ ∼= BΣ +0 ⊔ BΣ +1 , +where B0 = {0}, and B1 = {1}. Of course, Q(s) +B0(Σ) ∼= C, and the same holds for +Q(s) +B1(Σ), and so Q(s) +B (Σ) ∼= Q(s) +B0(Σ) ⊕ Q(s) +B1(Σ) ∼= C ⊕ C. +If, now, Σ is not connected, then we can write it as Σ ∼= Σ′ ⊔ Σ′′, with both parts +non-empty, and, as Q(s) +B is a monoidal functor, Q(s) +B (Σ) ∼= Q(s) +B (Σ′) ⊗ Q(s) +B (Σ′′). +By induction on the size of π0(Σ), suppose Σ is a disjoint union of connected +n-manifolds, ⊔k +i=1Σi, then Q(s) +B (Σ) ∼= (C ⊕ C)⊗k, the tensor product of k-copies of +C2 = C ⊕ C. +Next, suppose that S is a cobordism, (i, S, j): Σ1 → Σ2, and that the (n + 1)- +manifold, S, is connected, then BS = BS +0 ⊔BS +1 , and the linear map, Q(s) +B (i, S, j), is +simply Q(s) +B0(i, S, j) ⊕ Q(s) +B1(i, S, j), the direct sum of the corresponding linear maps, +so giving a block decomposition of the corresponding matrices. +If S is disconnected, then Q(s) +B (i, S, j) will be the tensor product of the values on +the connected components in the evident way. +The argument used in this second example adapts well to handle the general +case of a not necessarily path-connected B. To make this explicit, we recall the +idea of direct sum as defined initially in [45] in the case of 2-D TQFTs, but later +applied, in [105], to general TQFTs. We will repeat the definition in the form used +by Sawin in the above cited [105]. +Suppose that Z1 and Z2 are two TQFTS. +40The link with the example is that the singleton space is the unit space to the (cartesian) +monoidal structure on spaces, and C is the unit for the monoidal structure on Vect. +41Later on, in §7.4.2, and in a more general situation, we will look at the case in which +B = BA, the classifying space of a crossed complex, and then we will talk of A-colourings of +(regular) CW-complexes and related structures rather than BA-colourings. + +A CATEGORIFICATION OF QUINN’S TQFT +69 +Definition 87. The direct sum, Z1 ⊕ Z2, of Z1 and Z2 is the theory which: +• associates, to each connected Σ, the vector space Z1(Σ) ⊕ Z2(Σ); +• associates, to each disconnected Σ, the tensor product of the vector spaces +associated to its components; +• associates, to each connected cobordism, (i, S, j): Σ1 → Σ2, the linear map, +Z1(i, S, j)⊕Z2(i, S, j), interpreted as an operator on the appropriate vector +spaces, +and +• associates to each disconnected cobordism, the tensor product of the values +on the components. +On generalising the argument used in the above example, we clearly have: +Proposition 88. (i) If B = B1 ⊔ B2, then Q(s) +B ∼= Q(s) +B1 ⊕ Q(s) +B2. +(ii) In general, for B a HF space, Q(s) +B +has a direct sum decomposition as +� +x∈π0(B) Q(s) +Bx, where Bx is the connected component labelled by x. +□ +4.6. Changing B. Given these constructions and their properties, one might think +that there was some possible functoriality of Q(s) +B +with B itself, but recall, for +instance from [37, §2.5 and Appendix A2], that if ϕ : Z1 =⇒ Z2 is a (monoidal) +natural transformation between TQFTs, then it is a natural isomorphism. If f : +B1 → B2 is a general continuous map, we therefore should not expect that there +would be some sort of induced ‘morphism’ between Q(s) +B1 and Q(s) +B2. We can explore +the failure of such na¨ıve functoriality with a simple example42. +Example 89. We put ourselves in Cob(1,2), and denote by Σi, i = 1, 2, two 1- +manifolds with S : Σ1 → Σ2, a 2-manifold with boundary, given as a cobordism +between the two 1-manifolds. For B1, we take {0, 1}, for B2, the singleton space, +{∗}, and f : B1 → B2, the unique such map. +The particular case that we will examine will have Σ1 = S1 +a ⊔ S1 +b , the disjoint +union of two circles, where the suffixes are the labels by which we will refer to +the components of Σ1, i.e., π0(Σ1) = {a, b}. Similarly, we want Σ2 = S1 +c and +π0(Σ2) = {c}. Taking the previous discussion in Examples 85 and 86, but applying +it to our simple case here, we see that π0(BΣ1 +1 ) consists of 4 elements, which we +will denote ( a b +0 0 ), ( a b +1 1 ), ( a b +0 1 ) and ( a b +1 0 ), where, for instance, in the last of these, +S1 +a is sent to 1 in {0, 1}, whilst S1 +b is sent to 0. Similarly, π0(BΣ2 +1 ) = {( c +0 ) , ( c +1 )}. +We next look at the ‘pair of trousers’ cobordism, T : Σ1 → Σ2, going from the +two circles ‘near the ankles’ to the circle at the ‘waist’, and we note that BT +1 , also +consists of just two points. Since T is connected, any continuous map, β : T → B1, +must either be constant with value 0, or constant with value 1. We write BT +1 = +{( t +0 ) , ( t +1 )}, and note that, in the induced fibrant span, +BT +1 +i∗ +�❦❦❦❦❦❦ +j∗ +�❙ +❙ +❙ +❙ +❙ +❙ +BΣ1 +1 +BΣ2 +1 , +whilst j∗ is an isomorphism, i∗ sends ( t +0 ) to ( a b +0 0 ), and ( t +1 ) to ( a b +1 1 ), so the fibres +of i∗ over the other two points of BΣ1 +1 +are empty43. +If we now replace B1 by B2 in the above, then BΣ1 +2 += {( a b +∗ ∗ )}, BΣ2 +2 += {( c +∗ )}, +and BT +2 = {( t +∗ )}, with the corresponding fibrant span being the terminal span. +42This example is very simple, but it shows what the problem is in general, so we will give the +calculation in a lot of detail. +43Recall that fibrations are not assumed to be surjective, so can have empty fibres as mentioned +in Remark 2. + +A CATEGORIFICATION OF QUINN’S TQFT +70 +The map, f : B1 → B2, of course, sends anything with a 0 or 1 to the analogous +one with ∗ in the same place. +Turning to the corresponding vector spaces, the +diagram, +Q(s) +B1(Σ1) +f∗ +� +Q(s) +B1(T ) +� +Q(s) +B2(Σ1) +Q(s) +B2(T ) +� +Q(s) +B1(Σ2) +f∗ +� Q(s) +B2(Σ2) +is, in fact, just +C4 ++ +� +proj1,2 +� +C +id +� +C2 ++ +� C, +where proj1,2(x1, x2, x3, x4) = (x1, x2), and the two maps labelled + just sum the +coordinates. +This clearly does not commute! The problem, here, arises with the colourings of +the disconnected Σ1, which do not extend to colourings of the cobordism, T . +We thus have, in general, that there is no functoriality in the B. Under certain +circumstances, however, a map, f : B1 → B2, does induce a natural transformation +between Q(s) +B1 and Q(s) +B2, which as we noted must, then, be a natural isomorphism. +Theorem 90. If f : B1 → B2 is a homotopy equivalence, then f induces a +monoidal natural isomorphism, f∗ : Q(s) +B1 =⇒ Q(s) +B2, between Q(s) +B1 and Q(s) +B2. This +natural isomorphism is defined in the following way: if Σ is a closed n-manifold, +then the linear map, (f∗)Σ : Q(s) +B1(Σ) → Q(s) +B2(Σ), is such that, given g : Σ → B1, +then +(f∗)Σ +� +PCg +� +BΣ +1 +�� += PCf◦g(BΣ +2 +� +, +i.e., is post-composition with f. +Proof. We always have such a family of mappings, f∗ : Q(s) +B1 → Q(s) +B2, induced by +post-composition with f, but, as we saw, in general, this need not define a natural +transformation, due to possible incompatibility with the cobordisms. +Suppose (i, S, j) : Σ1 → Σ2 is a cobordism, thus giving us fibrant spans, +BS +i +i∗ +�❦❦❦❦❦❦ +j∗ +�❙ +❙ +❙ +❙ +❙ +❙ +BΣ1 +i +BΣ2 +i , +for i = 1, 2. The function f : B1 → B2 gives us a commutative diagram, +BΣ1 +1 +f Σ1 +� +BS +1 +i∗ +� +f S +� +j∗ +� BΣ2 +1 +f Σ2 +� +BΣ1 +2 +BS +2 +i∗ +� +j∗ � BΣ2 +2 , +in which the vertical maps are all homotopy equivalences. (We note that this is not +a morphism in the category HFspan, but does relate to a higher category structure + +A CATEGORIFICATION OF QUINN’S TQFT +71 +on the class of HF spans.) The commutative diagram induces a map of fibrations, +(18) +BS +1 +⟨i∗,j∗⟩ +� +f S +� +BΣ1 +1 +× BΣ2 +1 +f Σ1×f Σ2 +� +BS +2 +⟨i∗,j∗⟩ +� BΣ1 +2 +× BΣ2 +2 , +where the vertical arrows are homotopy equivalences. +We also have a diagram of vector spaces and linear maps, +Q(s) +B1(Σ1) +(f∗)Σ1 � +Q(s) +B1 (S) +� +Q(s) +B2(Σ1) +Q(s) +B2(S) +� +Q(s) +B1(Σ2) +(f∗)Σ2 +� Q(s) +B2(Σ2). +To check that this diagram commutes, in this context, we pick basis elements in +Q(s) +B1(Σ1) and Q(s) +B1(Σ2), and compare the matrices corresponding to the left-hand +side, with those on the right-hand side, with respect to the image basis. +First we note that �π0(BΣ +1 ) and �π0(BΣ +2 ) are related by the bijection induced from +f. Consider arbitrary maps g : Σ1 → B1 and g′ : Σ2 → B1. To prove that the +diagram commutes, it suffices to prove that +� +PCg(BΣ1 +1 ) | Q(s) +B1([(i, S, j)]) | PCg′(BΣ2 +1 ) +� += +� +PCf◦g(BΣ1 +2 ) | Q(s) +B2([(i, S, j)]) | PCf◦g′(BΣ2 +2 ) +� +. +Unpacking the notation, this amounts to comparing the corresponding fibres of +the horizontal fibrations in diagram in (18). These fibres are homotopy equivalent +by Corollary 66, applied to the map of fibrations given in (18). +We note that these isomorphisms respect the monoidal structure and also the +composition, which completes the proof. +□ +This result is very useful as it, for instance, implies that any contractible B +gives a TQFT isomorphic to the trivial one Q(s) +{∗}. That is easy to see directly of +course, but equally well we can sometimes simplify the calculations of some Q(s) +B , +by replacing a given B by a homotopy equivalent one, for instance by using a +presentation of its homotopy type that is smaller than that given initially. +More interestingly, perhaps, we note, from the proof, how the isomorphism f∗ +depends only on the homotopy class of the homotopy equivalence f. They are also +compatible with composition of such homotopy equivalences. This gives: +Theorem 91. Given any HF-space, B, there is an action of the group, E(B), of +homotopy classes of self homotopy equivalences of B on the TQFT Q(s) +B , by natural +isomorphisms. +□ +We leave aside, for the moment, the application of this result within specific +calculations. +Finally in this section, we should mention the possibility of twisting the TQFT +by a cohomology class. +Remark 92 (Cohomology twisting of Quinn’s TQFT). We note that if we restrict +to oriented n-manifolds and oriented cobordisms Quinn’s TQFT, Q(s) +B , can be also + +A CATEGORIFICATION OF QUINN’S TQFT +72 +be twisted by a cohomology class in Hn(B, U(1)). Again this is as explained in [43], +for B the classifying space of a finite group, or, more generally, as done in [53], +in the closed case, where B is the classifying space of a finite crossed module. The +details are left to the reader. We hope to give some explicit formulae in a future +publication. +Part 3. Once-extended versions of Quinn’s finite total homotopy TQFT +Part 3 of this paper consists of two sections. +The first, section 5, looks at +the homotopy-theoretical underpinning of the once-extended Quinn TQFT. The +second, section 6, gives the detailed construction of that extended TQFT. +Throughout Part 3, we will work with an arbitrary, but fixed, subfield, κ, of the +complex field44. +Again we let B be a homotopy finite space, and n be a non-negative integer. In +this part of the paper, we will see how the s = 0 case, +QB : Cob(n,n+1) → Vect = Vectκ, +of the Quinn finite total homotopy TQFT (abbr. +the Quinn TQFT), given in +Definition 83, can be ‘categorified’ to a once-extended Quinn TQFT, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +in Definition 149. +Here, as introduced in Subsection 2.8, and, in particular, in +§2.8.5, the bicategory, vProfGrphf, has objects the homotopy finite groupoids, +the 1-morphisms being Vect-valued profunctors between groupoids, and the 2- +morphisms natural transformations of profunctors, and 2Cob(n,n+1,n+2) is the bi- +category with objects the closed (and, by convention, smooth) n-manifolds, the +1-morphisms being the (n + 1)-cobordisms between closed n-manifolds, and the 2- +morphisms the equivalence classes of (n+2)-cobordisms between (n+1)-cobordisms +(by convention, with corners); see [106, 91, 92]. +Let Σ be a closed and smooth n-manifold. Typically, the groupoid 2QB(Σ), +despite being homotopy finite, is still uncountable. In order to reduced the size +of the target groupoids, we will consider a bicategory, 2Cob +(n,n+1,n+2) +B +, with ob- +jects B-decorated, closed, and smooth n-manifolds, with the rest of the bicategory +structure induced by that of 2Cob(n,n+1,n+2), as discussed in Subsection 6.3. We +will then consider another once-extended TQFT, called the finitary once-extended +Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin; +see Definition 154. +The bicategory, vProfGrpfin, is the full sub-bicategory of +vProfGrphf, with objects the finite groupoids; see §2.8.5. This, in turn, gives rise +to another once-extended TQFT, called the Morita valued once-extended Quinn +TQFT, in Definition 168, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor, +where Mor is the bicategory of κ-algebras, bimodules and bimodule maps. +The algebraic construction showing how to go from the bicategory vProfGrpfin +to the bicategory Mor, starting from groupoid algebras, may be of independent +interest. This is laid out in Subsection 6.4. +Depending on which setting is chosen, the groupoids, or algebras that 2QB and +2Q +Mor +B +assign to a closed manifold, Σ, with a B-decoration, explicitly depend on +the B-decoration of Σ. However this dependence is up to a canonically defined, +44For instance, we can take κ = Q. + +A CATEGORIFICATION OF QUINN’S TQFT +73 +and invertible, profunctor or bimodule, which is functorial with respect to further +changes in the decoration (up to natural isomorphism), and natural with respect +to the profunctors, or bimodules, assigned to cobordisms. +This is discussed in +Subsection 6.3 and §6.4.5 +We will show explicit examples of calculations of these once-extended TQFTs +later on in Part 4, Section 8, for the case in which B is the classifying space of a +finite crossed complex. This includes the case of classifying spaces of finite 2-groups, +as appear in higher gauge theory, see e.g. [3, 6, 52, 31]. +As we will further see in §8.4.1, when n = 0, and for the case where B is +the classifying space of a finite groupoid G, the extended Quinn TQFT gives a +homotopy theoretical proof of the existence of the Morita valued once-extended +(0,1,2)-TQFT, derived, elsewhere, from the fact that the groupoid algebra of G is +a ‘separable symmetric Frobenius algebra’; see [106, §3.8] and [75, Example 5.2]. +Remark: It is possible to define a one-parameter categorification of the Quinn +TQFT QB, but we will not deal with that here. This would considerably increase +the complexity of our formulae, without adding much more generality to our con- +struction. +Similarly to our exposition of Quinn’s finite total homotopy TQFT, we will factor +its once-extended version through a homotopy theoretical bicategorical object, later +denoted 2span(HF), whose objects are HF spaces, 1-morphisms are fibrant HF +spans, and 2-morphisms consist of HF-fibrant resolved 2-spans. This will be done +in Section 5. The once-extended versions of the Quinn finite total homotopy TQFT +will then be treated in Section 6. In Subsection 6.5, we will sketch the symmetric +monoidal structure of the once-extended versions of Quinn TQFT. +5. The homotopy-theoretical underpinning of the once-extended +Quinn TQFT +5.1. Notation and some more basic results about fibrations. We will need +some additional results and notation about fibrations, as defined in §2.4. +5.1.1. Holonomy maps and the functor, FM : π1(B, B) → CGWH/ ≃, associated +to a fibration p: M → B. +Recall from Subsection 2.3 that given a CGWH space, B, we defined the maps, +sB = s, tB = t: BI → B, +such that s(γ) = γ(0) and t(γ) = γ(1). We have a fibration, ⟨s, t⟩: BI → B × B, +from which we constructed the identities in HFspan; see Lemma 59 and Definition +60. Given γ ∈ BI, the reverse path to γ will be denoted γ, so γ(u) = γ(1 − u), for +each u ∈ [0, 1]. We also consider the map, const: B → BI, sending x ∈ B to the +constant path, constx, at x. +Let p: M → B be a fibration. Recall that the fibre of x ∈ B is denoted +Mx := p−1(x). +Let M ×BBI denote the pullback of the maps, p: M → B and s: BI → B. Consider +also the canonical projections, +proj1 : M ×B BI → M and proj2 : M ×B BI → BI, + +A CATEGORIFICATION OF QUINN’S TQFT +74 +so we have a pullback diagram, +M ×B BI +proj1 +�✉✉✉✉✉✉✉✉✉✉ +proj2 +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +M +p +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +BI. +sB +�sssssssssss +B +We have a continuous function, λM : I × (M ×B BI) → M, arising from the +diagram below and the homotopy lifting property of p: M → B, +M ×B BI +proj1 +� +{0}×( ) +� +M +p +� +I × (M ×B BI) +λM +�❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +❢ +idI×proj2 +� I × BI +(u,γ)�→γ(u) +� B. +(Here +� +{0} × ( ) +� +(m, γ) = (0, m, γ).) +Definition 93 (Holonomy Map). A function, λM, making the diagram above com- +mute will be called a holonomy map on the fibration p: M → B. +The nomenclature “holonomy map” is borrowed from differential geometry. We will +frequently write “holonomy” rather than “holonomy map”. The holonomy maps +considered here are equivalent to the “path-lifting functions” in [87, Chapter 7], +and the “lifting functions” in [48]. +The following string of classical results are to be found, essentially, in [48] or +[87, Chapter 7]. They follow from simple application of the appropriate homotopy +lifting property, and are ‘well known’, but we give a reference for each one. +Lemma 94. Let p: M → B be a fibration. Consider a fixed holonomy map, λM, +on p: M → B. Let γ be a path, in B, from x ∈ B to y ∈ B. Consider the map +ΓM +γ : Mx → My, defined by +ΓM +γ (m) := λM(1, m, γ), for all m ∈ Mx. +Up to homotopy of maps from Mx to My, the map, ΓM +γ : Mx → My, then, depends +only on the homotopy class of γ (and, in particular, not on the chosen holonomy +map, λM). In fact, ΓM +γ +: Mx → My is a homotopy equivalence between the fibres +Mx and My, with a homotopy inverse to ΓM +γ : Mx → My given by ΓM +γ . +Proof. See [87, §7.6 (Change of fiber)]. +□ +Lemma 95. Let φ be a path in M, and γ = p ◦ φ, its image path in B, then +ΓM +γ (φ(0)) and φ(1) are in the same path-component of Mγ(1). +In particular, if M is path-connected and x, y ∈ B, then if A ∈ �π0(Mx) and +A′ ∈ �π0(My) are path components of the chosen fibres, it follows that A and A′ are +homotopy equivalent. Concretely, choose m ∈ A and m′ ∈ A′, and a path, φ, in M +connecting m to m′, then ΓM +γ : Mx → My restricts to a map, A → A′, giving the +desired homotopy equivalence. (Here γ = p ◦ φ.) +Proof. See [48, page 3]. +□ + +A CATEGORIFICATION OF QUINN’S TQFT +75 +Suppose that we have paths, x +γ−→ y and y +γ′ +−→ z, then ΓM +γγ′ is homotopic to +ΓM +γ′ ◦ ΓM +γ , as maps from Mx to Mz; see [87, §7.6]. Moreover, given x ∈ B, the map, +ΓM +constx : Mx → Mx, is homotopic to the identity. +Recall that CGWH/ ≃ denotes the category with objects the CGHW spaces, +with morphisms being homotopy classes of maps; see §2.3. +Lemma 96. There is a functor, +FM : π1(B, B) → CGWH/ ≃, +(where π1(B, B) is the fundamental groupoid of B). Given x ∈ B, FM(x) := Mx, +and given a path, x +γ−→ y, in B, then, +FM(x +[γ] +−→ y) := [ΓM +γ ]: Mx → My. +Here [ΓM +γ ] is the homotopy class of ΓM +γ : Mx → My. +This functor, FM, depends only on the fibration, p: M → B, and not on the +chosen holonomy map, λM. +Proof. See [87, §7.6]. +□ +We, thus, have a functor � +π0 ◦ FM : π1(B, B) → Set. It sends x ∈ B to �π0(Mx). +(For notation see Subsection 2.3.) Given a path, x +γ−→ y, in B, the functor is such +that, if m ∈ Mx, +� +(�π0 ◦ FM)(x +[γ] +−→ y) +� +(PCm(Mx)) = PCλM(1,m,γ)(My) = PCΓM +γ (m)(My), +where we recall that PCm(Mx) denotes the path-component of m in Mx; see Sub- +section 2.3. This functor depends only on the fibration, p: M → B, and not on the +chosen holonomy map, λM. +Finally in this string of lemmas, we have +Lemma 97. Let x ∈ B. There are left and right actions of π1(B, x) on �π0(Mx). +These are such that, if m ∈ Mx, γ ∈ Ωx, the loop space of B based at x, and [γ] is +the associated element of π1(B, x), then: +[γ]⊲PCm(Mx) = PCΓM +γ (m)(Mx), +and +PCm(Mx) ⊳ [γ] = PCΓM +γ (m)(Mx). +□ +The following result will be needed when addressing why the once-extended +Quinn TQFT can be given the structure of a symmetric monoidal bifunctor. +Lemma 98. Let p: E → X and p′ : E′ → X′ be fibrations. +We thus have a +fibration, (p × p′): E × E′ → X × X′. The functor, +FE×E′ : π1(X × X′, X × X′) → CGWH/ ≃, +provided by (p × p′): E × E′ → X × X′, is given by the composition of the functors +below, +π1(X × X′, X × X′) ∼= π1(X, X) × π1(X′, X′) +F E×F E′ +−−−−−−→ (CGWH/ ≃) × (CGWH/ ≃) +×CGWH +−−−−−−→ CGWH/ ≃ . +Here ×CGWH is the product monoidal structure on CGWH. We will revisit +this in §5.1.2. + +A CATEGORIFICATION OF QUINN’S TQFT +76 +Proof. On objects, this follows from the fact that, if x ∈ X and x′ ∈ X′, then +(p×p′)−1(x, x′) = p−1(x)×p′−1(x′). On morphisms, this follows from the fact that +a holonomy map for the fibration, (p × p′): E × E′ → X × X′, can be obtained by +doing the “product” of those of p: E → X and p: E′ → X′, namely, +I×(E×X XI)×(E′×X′X′I) ∋ (t, e, γ, e′, γ′) �→ +� +λE(t, e, γ), λE′(t, e′, γ′) +� +∈ E×E′. +□ +Definition 99. Let p: M → B be a fibration. Choose a subset, xB, of B. The +functor, +FM +xB : π1(B, xB) → CGWH/ ≃, +is defined by restricting the functor, FM : π1(B, B) → CGWH/ ≃, to π1(B, xB), +the full subgroupoid of π1(B, B), with set of objects xB. +We have, thus, also defined a functor, �π0 ◦ FM +xB : π1(B, xB) → Set. +Remark 100. This latter functor, �π0 ◦ FM +xB : π1(B, xB) → Set, is one of the basic +ingredients for the reduction of the once-extended Quinn TQFT to give something +more amenable. +The set, xB, is a choice of a set of base-points, typically, but +not exclusively, for the components of B, or, at the other extreme, we could take +xB to be the set of all the elements of B. We can choose. If we pick xB finite, +this can be used to reduce the groupoids, π1(B, B), and their action (interpreted as +a functor) to the more classical setting of a set of fundamental groups and their +action. This allows one to reduce a Vect-enriched category to something nearer to +a finite group algebra together with categories of bimodules over them, in fact to the +Morita bicategory that we will be recalling later. +We will need such many pointed extensions of quite a few otherwise classical +results, which are not that easy to find given in an explicit form in the literature, +and so will give them in a bit of detail. +Recall that, +Definition 101. A pair, (X, xX), of topological spaces is said to be 0-connected if +the set, xX, has at least one point in each path-component of X. +Lemma 102. Let p: M → B be a fibration. Choose a subset, xB, of B such that +(B, xB) is 0-connected. We have a natural bijection, +F : colim(�π0 ◦ FM +xB) = +� � +x∈xB +�π0(Mx) +� � +∼ → �π0(M). +Given x ∈ xB and m ∈ Mx, this sends the equivalence class of PCm(Mx) to +PCm(M). +Proof. By construction, the map, +F ′ : +� +x∈xB +�π0(Mx) → �π0(M), +such that, if x ∈ xB and m ∈ Mx, then PCm(Mx) �→ PCm(M), descends to +colim(�π0 ◦ FM +xB). Let us explain this a bit more. Given x +[γ] +−→ y in π1(B, xB), if +m ∈ Mx and n ∈ Ny, and we have that +� +�π0 ◦ FM +xB([γ]) +� +(PCm(Mx)) = PCn(My), +then this means that λM(1, m, γ) is in PCn(My). From this, it follows that m and n +are in the same path-component in M. (Note that the 0-connectedness of (B, xB) +was not used.) + +A CATEGORIFICATION OF QUINN’S TQFT +77 +Now to prove that F is injective. Suppose that, given x, x′ ∈ xB, and m ∈ Mx +and m′ ∈ Mx′, we have PCm(M) = PCm′(M). Choose any path, φ, in M, starting +in m and ending in m′. Let γ = p ◦ φ, then, by using Lemma 95, it follows that +PCΓM +γ (m)(M) = PCm′(M), so +� +(�π0 ◦ FM +xB)(x +[γ] +−→ y) +� +(PCm(Mx)) = PCm′(My). +(Again note that the 0-connectedness of (B, xB) was not used.) +That F is surjective follows analogously, but here we use the fact that (B, xB) is +0-connected. If we are given m ∈ M, there is a path, γ, in B connecting p(m) ∈ B +to some x ∈ xB. Put m′ = ΓM +γ (m). We then have F([PCm′(Mx)]) = PCm(M). +□ +5.1.2. The path components of pullbacks along fibrations. Let p: M → B and +q: N → B be fibrations and consider the pullback diagram in CGWH given by +the diamond in the diagram below, where we put P = q ◦ proj2 = p ◦ proj1, +(19) +M×BN +proj2 +�❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +proj1 +�✈✈✈✈✈✈✈✈✈ +P +� +M +p +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +N. +q +�✉✉✉✉✉✉✉✉✉✉ +B +It is clear, e.g., by the universal property of pullbacks, that P is a fibration. +Let λM and λN be holonomy maps for the fibrations, p: M → B and q: N → B, +then a holonomy map, λM×BN, for the fibration, P : M ×B N → B, can be given +such that, if we have a path x +γ−→ y in B, and a point, (m, n) ∈ Mx×Nx ⊂ M ×B N, +(20) +λM×BN(t, m, n, γ) = +� +λM(t, m, γ), λN(t, n, γ) +� +. +Given b ∈ B, recall that we put Mb = p−1(b), and Nb = q−1(b). +The fi- +bre, P −1(b), of P at b, is homeomorphic to Mb × Nb. We have a bijection, from +�π0(Mb) × �π0(Nb) to �π0(P −1(b)). +This bijection sends +� +PCm(Mb), PCn(Nb) +� +to +PC(m,n)(P −1(b)), where m ∈ Mb and n ∈ Nb. +We have functors, FM, FN : π1(B, B) → CGWH/ ≃, given by the fibrations +p: M → B and q: N → B. By construction, and using (20), the functor, +FM×BN : π1(B, B) → CGWH/ ≃, +given by the fibration, P : M ×B N → B, is naturally isomorphic to the composition +of the functors below, where +� +FM, FN� +is given by the universal property of a +product, +π1(B, B) ⟨F M,F N⟩ +−−−−−−→ (CGWH/ ≃) × (CGWH/ ≃) +×CGWH +−−−−−−→ CGWH/ ≃, +and where ×CGWH : CGWH × CGWH → CGWH denotes the product functor +in CGWH, which descends to a functor, also denoted +×CGWH : (CGWH/ ≃) × (CGWH/ ≃) → CGWH/ ≃ . +(Explicitly, ×CGWH sends a pair, (X, Y ), of CGWH spaces to their product, X×Y , +and analogously for maps.) +The functor, �π0 ◦ FM×BN : π1(B, B) → Set, is, thus, naturally isomorphic to +×Set ◦ +� +�π0 ◦ FM, �π0 ◦ FN� +: π1(B, B) → Set, +where ×Set : Set × Set → Set is the product functor in Set. +Given xB ⊂ B, it also follows that the functor, +�π0 ◦ FM×BN +xB +: π1(B, xB) → Set, + +A CATEGORIFICATION OF QUINN’S TQFT +78 +is naturally isomorphic to +×Set ◦ +� +�π0 ◦ FM +xB, �π0 ◦ FN +xB +� +: π1(B, xB) → Set. +Lemma 103. Suppose that the pair (B, xB) is 0-connected. There is a bijection, +colim +� +×Set ◦ +� +�π0 ◦ FM +xB, �π0 ◦ FN +xB +� � +−→ �π0(M ×B N). +This bijection is such that, if x ∈ xB, m ∈ Mx and n ∈ Nx, then, +[(PCm(Mx), PCn(Nx))] �−→ PC(m,n)(M ×B N). +Proof. This follows from Lemma 102 combined with the previous discussion. +□ +Any groupoid, G, comes with a contravariant functor ( )−1 : G → G, that is the +identity on objects and sends each morphism to its inverse. In particular, we have +a functor, +×Set ◦ +� +�π0 ◦ FM +xB ◦ ( )−1 × �π0 ◦ FN +xB +� +: π1(B, xB)op × π1(B, xB) → Set. +This gives us the following. (We note that a generalisation of this lemma, written +in the context of ∞-groupoids, is in [56, Lemma 3.8].) +Lemma 104. Let p: M → B and q: N → B be fibrations. Choose xB ⊂ B such +that the pair (B, xB) is 0-connected. There is a bijection, +ˆ x∈xB � +(�π0 ◦ FM) ◦ ( )−1(x) +� +× +� +(�π0 ◦ FN)(x) +� +→ �π0(M ×B N). +Noting that, +ˆ x∈xB � +(�π0 ◦ FM) ◦ ( )−1(x) +� +× +� +(�π0 ◦ FN)(x) +� += +� � +x∈xB +(�π0 ◦ FM)(x) × (�π0 ◦ FN)(x) +� � +∼, +given x ∈ xB, the bijection sends the equivalence class of (PCm(Mx), PCn(Nx)) to +PC(m,n)(M ×B N), where m ∈ Mx and n ∈ Nx. +Proof. This follows from the previous lemma, since any arrow in π1(B, xB) is in- +vertible. (We note that, here, Lemma 36, in §2.8.3, is useful to translate between +languages, that of coends and the more classical form.) +□ +5.1.3. The homotopy content of path-components of pullbacks along fibrations. This +lemma will be implicitly used below. +Lemma 105. Let f : E → X be a fibration, with E ̸= ∅. Let e ∈ E and x = f(e). +(1) The induced map, fe : PCe(E) → PCx(X), is a fibration. +(2) If k ∈ f −1 +e +(x), then PCk(f −1 +e +(x)) = PCk(f −1(x)). +Proof. The first point is immediate from the homotopy lifting property. For the +second point, note that clearly PCk(f −1 +e +(x)) ⊆ PCk(f −1(x)) as sets. The reverse +inclusion also holds. This is because a path in f −1(x), starting in k ∈ f −1 +e +(x), +cannot leave PCe(E), so it is a path in f −1 +e (x). That the topologies in the path +components coincide follows from item (8) on page 15. +□ +Consider two fibrations, p: M → B, and q: N → B, and the resulting fibration, +P : M ×B N → B, of diagram (19), and suppose that M, N and B are homotopy +finite. +Lemma 106. The space, M ×B N, is homotopy finite, so P : M ×B N → B is a +fibration of homotopy finite spaces. + +A CATEGORIFICATION OF QUINN’S TQFT +79 +This lemma is a particular case of Lemma 55, and of [56, Lemma 3.8], but here +is a direct proof. +Proof. Let b ∈ B be arbitrary. The fibre of the fibration, P : M ×B N → B, at +b ∈ B, is homeomorphic to Mb × Nb, a product of homotopy finite spaces (by +Lemma 40), so each fibre of P is homotopy finite. Since B is homotopy finite, +it follows that the total space of P : M ×B N → B is homotopy finite, again by +Lemma 40. +□ +Let b ∈ B. By Lemma 97, we have a right action of π1(B, b) on �π0(Mb)×�π0(Nb) ∼= +�π0(P −1(b)). This action is such that, if γ : I → B connects b to b, then, given +m′ ∈ Mb and n′ ∈ Nb, so (m′, n′) ∈ M ×B N, we have, +� +PCm′(B) × PCn′(B) +� +⊳ [γ] = +� +PCΓM +γ (m′)(Mb), PCΓN +γ (n′)(Nb) +� +. +Note that, given the form of the induced holonomy map, (20), for P : M ×B N → B, +it follows that +� +ΓM +γ (m′), ΓN +γ (n′) +� +is in the same path component of (m′, n′) in +M ×B N. +Now fix b ∈ B, and elements, m ∈ Mb, n ∈ Nb, in the fibres of the two fibrations. +The fibration, P : M ×B N → B, restricts to a map, +P(m,n) : PC(m,n)(M ×B N) → PCb(B), +which is a fibration by Lemma 105. Assuming that M, N and B are homotopy +finite, then P(m,n) : PC(m,n)(M ×B N) → PCb(B) is a fibration of homotopy finite +spaces. In particular, the fibre of P(m,n) : PC(m,n)(M ×B N) → PCb(B) has only a +finite number of path-components. +Until the end of this subsection45, consider two fibrations, p: M → B and +q: N → B, of homotopy finite spaces, and also the induced fibration, P : M ×BN → +B, therefore, again of homotopy finite spaces. +Notation 107 (T M×BN +(m,n) +). Let b ∈ B, and also m ∈ Mb, n ∈ Nb. We write T M×BN +(m,n) +for the number of path-components of the fibre of P(m,n) : PC(m,n)(M ×B N) → +PCb(B) at b ∈ B. +Almost by definition it follows that: +Lemma 108. Let b ∈ B, m ∈ Mb and n ∈ Nb, then T M×BN +(m,n) +equals the cardinality +of the orbit of PC(m,n)(P −1(b)) under the right-action of π1(B, b) on �π0(P −1(b)), +derived from the fibration P : M ×B N → B. +Proof. We prove that �π0(P −1 +(m,n)(b)) ⊂ �π0(P −1(b)) coincides with the π1(B, b)-orbit, +of PC(m,n)(P −1(b)), inside �π0(P −1(b)). +Let m′ ∈ Mb and n′ ∈ Nb. +If PC(m′,n′)(P −1(b)) ∈ �π0(P −1 +(m,n)(b)), then, in +particular, (m′, n′) ∈ Mb × Nb ⊂ M ×B N is in the same path-component as (m, n) +in M ×B N. Choose a path, φ, in M ×B N connecting (m, n) and (m′, n′). Applying +Lemma 95, it follows that PC(m,n)(P −1(b))⊳ [p(φ))] = PC(m′,n′)(P −1(b)). The rest +follows by construction. +□ +Lemma 109. Let b ∈ B, m ∈ Mb, and n ∈ Nb, then +χπ� +PC(m,n)(M ×B N) +� += T M×BN +(m,n) +χπ(PCb(B)) χπ(PCm(Mb)) χπ(PCn(Nb)). +45We will not always repeat this for each result. + +A CATEGORIFICATION OF QUINN’S TQFT +80 +Proof. The decomposition of P −1(b) ∼= Mb × Nb, into path-components, gives a +weak homotopy equivalence, +� +(A,A′)∈�π0(Mb)×�π0(Nb) +A × A′ → Mb × Nb. +Each path component of P −1 +(m,n) is also a path-component of P −1(b) ∼=Mb×Nb. (We +are using Lemma 105.). A priori, however, there may be fewer path-components in +P −1 +(m,n)(b) than there are in Mb × Nb. +If we use Lemma 46, applied to the HF fibration, +P(m,n) : PC(m,n)(M ×B N) → PCb(B), +we obtain +χπ� +PC(m,n)(M ×B N) +� += χπ(PCb(B)) χπ� +P −1 +(m,n)(b) +� +. +Now, by Lemma 95, all path-components of P −1 +(m,n)(b) are homotopy equivalent. +The result follows from the fact that, by definition, we have T M×BN +(m,n) +such path +components, each of which is homotopic to +PC(m,n) +� +P −1 +(m,n)(b) +� += PC(m,n) +� +P −1(b) +� +∼= PC(m,n)(Mb × Nb) +∼= PCm(Mb) × PCn(Nb). +(This, of course, uses that the homotopy content of a product of HF spaces is the +product of the homotopy contents, and that homotopy content is additive with +respect to disjoint union.) +□ +5.2. The profunctor construction. Using the results of the previous sections, +we will show that each fibrant span gives us a profunctor with Vect-values, and +that the composition of fibrant spans in Definition 56 translates under this to +composition of profunctors, as described in Subsection 2.8. +We note that our results are related to those of [56, 8. Cardinality as a functor], +which were written in the language of ∞-groupoids. +5.2.1. The profunctor associated to a fibrant span. Consider a fibrant span from X +to Y , then we have a diagram in CGWH of form, +M +p +�❥❥❥❥❥❥ +p′ +�❚ +❚ +❚ +❚ +❚ +❚ +X +Y, +where the induced map, +� +p, p′� +: M → X × Y , is a fibration. Given x ∈ X and +y ∈ Y , recall, from Definition 67, that we defined the spatial slice at x and y as +being +{x|(p, M, p′)|y} := ⟨p, p′⟩−1(x, y). +Also recall that we will frequently abbreviate {x|(p, M, p′)|y} to {x|M|y}, when no +ambiguity arises. +We have a holonomy map, λM, for the fibration, +� +p, p′� +: M → X × Y , of form, +λM : I × (M ×X×Y (X × Y )I) → M. +(We are using the notation of §5.1.1.) Let x, x′ ∈ X and y, y′ ∈ Y . Given paths, +x +γX +−−→ x′ in X, and y +γY +−−→ y′ in Y , the holonomy map, λM, induces a homotopy +equivalence, +ΓM +⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′}, +m �→ λM(1, m, γX, γY ). + +A CATEGORIFICATION OF QUINN’S TQFT +81 +Here ⟨γX, γY ⟩ is the path in X × Y such that I ∋ u �→ +� +γX(u), γY (u) +� +∈ X × Y . +The homotopy class of ΓM +⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′} depends only on the +fibration, ⟨p, p′⟩: M → X × Y , and not on the chosen holonomy map. This yields, +by Lemma 96, a functor, +FM : π1 +� +X × Y, X × Y )) ∼= π1(X, X) × π1(Y, Y ) → CGWH/ ≃, +where FM(x, y) := {x|M|y} and FM� +x +[γX] +−−−→ x′, y +[γY ] +−−−→ y′� +:= [ΓM +(γX,γY )]. +For convenience, we will repeat our conventions, from Subsection 2.8, for pro- +functors between groupoids. Fix a subfield, κ, of the complex field, C. Recall that +Vectk = Vect denotes the category of κ-vector spaces and linear maps. +Definition 110 (Set-profunctor and Vect-profunctor). Let G and G′ be groupoids. +A Set-profunctor, H : G ↛ G′, is a functor, +H : Gop × G′ → Set. +A Vect-profunctor, or Vect-valued profunctor, H: G ↛ G′, is a functor, +H: Gop × G′ → Vect. +The free vector space functor, from Set to Vect, is denoted +Lin = Linκ : Set → Vect. +Each Set-profunctor, H : Gop × G′ → Set, gives rise to a Vect-valued profunctor, +H := Lin ◦ H : Gop × G′ → Vect. Given that Lin preserves colimits, this operation +preserves the composition of profunctors (in Set and in Vect, see Equation (3)), +up to canonical natural isomorphism. +Definition 111 (The Vect-profunctor associated to a fibrant span). Consider a +fibrant span, of GCWH spaces +X +(p,M,p′) +−−−−−→ Y, that is, +� +M +p +�✐✐✐✐✐✐ +p′ +�❯ +❯ +❯ +❯ +❯ +❯ +X +Y +� +. +Its associated Vect-profunctor, denoted, +H(X +(p,M,p′) +−−−−−→ Y ): π1(X, X) ↛ π1(Y, Y ), +which we will frequently abbreviate to +HM : π1(X, X) ↛ π1(Y, Y ), +is, by definition, +HM = Lin ◦ �π0 ◦ FM ◦ +� +( )−1 × id +� +. +Taking this apart, given x ∈ X and y ∈ Y , HM(x, y) is the free vector space +over �π0({x|M|y}), the set of path-components of the fibre of ⟨p, p′⟩: M → X × Y , +at (x, y). Given morphisms, in π1(X, X) and π1(Y, Y ), say +x +[γX] +−−−→ x′ and y +[γY ] +−−−→ y′, +the linear map, +HM� +[γX], [γY ] +� +: HM(x′, y) → HM(x, y′), +is induced by the homotopy equivalence, between fibres, +ΓM +⟨γX,γY ⟩ : {x′|M|y} → {x|M|y′}, +by applying �π0 : CGWH → Set, and then Lin: Set → Vect. Here ⟨γX, γY ⟩ is the +path in X × Y , such that I ∋ u �→ +� +γX(1 − u), γY (u) +� +∈ X × Y . + +A CATEGORIFICATION OF QUINN’S TQFT +82 +The following result will implicitly be used a number of times. +Lemma 112. Suppose that X +(p,M,p′) +−−−−−→ Y is homotopy finite, so that X, Y and M +are homotopy finite spacesm then the profunctor, +HM : π1(X, X) ↛ π1(Y, Y ), +is a 1-morphism in the bicategory, vProfGrphf, of homotopy finite groupoids and +Vect-profunctors between them46. +Given x ∈ X and y ∈ Y , the vector space, HM(x, y), is finite dimensional. +Proof. Since X and Y are homotopy finite, it follows that the groupoids, π1(X, X) +and π1(Y, Y ), each are homotopy finite. Given x ∈ X and y ∈ Y , then by Lemma +40, the fibre {x|M|y} = ⟨p, p′⟩−1(x, y) is homotopy finite, and thus it only has a +finite number of path-components. +□ +Notation 113. More generally, choose subsets, xX and yY , of X and Y , respec- +tively. The restriction, of HM : π1(X, X)op × π1(Y, Y ) → Vect, to π1(X, xX) × +π1(Y, yY ), is a Vect-profunctor, +π1(X, xX) ↛ π1(Y, yY ). +We will use three different notations for it: +H(xX,yY )(p, M, p′): π1(X, xX)op × π1(Y, yY ) → Vect, +H(xX,yY ) +� +X +(p,M,p′) +−−−−−→ Y +� +: π1(X, xX)op × π1(Y, yY ) → Vect, +and finally, +H +M +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, +depending on the context and the amount of detail needed. +We will be mainly interested in the case when X +(p,M,p′) +−−−−−→ Y is homotopy fi- +nite, and furthermore both sets, xX and yY , are finite. In this case, we therefore +have that H +M +(xX,yY ) : π1(X, xX) ↛ π1(Y, yY ) is a 1-morphism in the bicategory +vProfGrpfin, of finite groupoids and Vect-profunctors between them. +5.2.2. The symmetric monoidal-like structure of H +M +(−,−). The following result will +be used when proving that the constructions of once-extended Quinn TQFTs, given +here, do indeed give bifunctors, which are symmetric monoidal. +Lemma 114. Consider two fibrant spans of homotopy finite spaces, +(p, M, q): X → Y and (p′, M ′, q′): X′ → Y ′. +Let xX ⊂ X, x′ +X′ ⊂ X′, yY ⊂ Y and y′ +Y ′ ⊂ Y ′, and form the product HF fibrant +span, +(p × p′, M × M ′, q × q′): X × X′ → Y × Y ′. +There is a natural isomorphism from the profunctor, +H(xX×x′ +X′ ,yY ×y′ +Y ′ ) +� +p × p′, M × M ′, q × q′� +: +π1(X × X′, xX × x′ +X′)op × π1(Y × Y ′, yY × y′ +Y ′) → Vect, +46See §2.8.5 + +A CATEGORIFICATION OF QUINN’S TQFT +83 +to the profunctor obtained from the following composition of functors, +π1(X × X′, xX × x′ +X′)op × π1(Y × Y ′, yY × y′ +Y ′) +∼ += +−→ +� +π1(X, xX)op × π1(Y, yY ) +� +× +� +π1(X′, x′ +X′)op × π1(Y, y′ +Y ′) +� +H(xX ,yY )(p,M,q)×H(x′ +X′ ,y′ +Y ′ )(p′,M′,q′) +−−−−−−−−−−−−−−−−−−−−−−−−−−→ Vect × Vect +⊗Vect +−−−−→ Vect. +Let x ∈ xX, x′ ∈ x′ +X′, y ∈ yY and y′ ∈ y′ +Y ′. This natural isomorphism is such that, +if m ∈ {x|M|y} and m′ ∈ {x′|M ′|y′}, then +PCm +� +{x|M|y} +� +⊗ PCm′� +{x′|M ′|y′} +� +←→PC(m,m′) +� +{(x, x′)|M × M ′|(y, y′)} +� +. +Proof. This follows from Lemma 98. +□ +In order to prove that the once-extended Quinn TQFT, in its various forms, gives +a symmetric monoidal bifunctor, it is convenient to change slightly the language of +the previous result, approximating that of Definition 30. (This point will be made +concrete later, in Subsection 6.5.) Consider the canonical natural isomorphisms, of +groupoids, +m(X,X′) : π1(X, xX) × π1(X′, x′ +X′) → π1(X × X′, xX × x′ +X′), +m(Y,Y ′) : π1(Y, yY ) × π1(Y ′, y′ +Y ′) → π1(Y × Y ′, yY × y′ +Y ′). +By using the notation of Example 33, they yield profunctors, +ϕm(X,X′) : π1(X, xX) × π1(X′, x′ +X′) ↛ π1(X × X′, xX × x′ +X′), +ϕm(Y,Y ′) : π1(Y, yY ) × π1(Y ′, y′ +Y ′) ↛ π1(Y × Y ′, yY × y′ +Y ′). +Continuing the notation of Lemma 114, we have +Lemma 115. There is a diagram of Vect-profunctors, and a natural isomorphism, +χ(M,M′), or in full χ� +(p,M,q),(p′,M′,q′) +�, of Vect-profunctors, as shown below, +π1(X, xX) × π1(X′, x′ +X′) +H(xX ,yY )(p,M,q)⊗H(x′ +X′ ,y′ +Y ′ )(p′,M′,q′) +� +ϕ +m(X,X′) +� +✚✚✚✚� +χ(M,M′) +π1(Y, yY ) × π1(Y ′, y′ +Y ′) +ϕ +m(Y,Y ′) +� +π1(X × X′, xX × x′ +X′) +H(xX ×x′ +X′ ,yY ×y′ +Y ′ )(p×p′,M×M′,q×q′) +� π1(Y × Y ′, yY × y′ +Y ′). +Let x ∈ xX, x′ ∈ x′ +X′, (y1 +γ−→ y2) ∈ π1(Y, yY ) and (y′ +1 +γ′ +−→ y′ +2) ∈ π1(Y, yY ). +This natural isomorphism is such that, referring to the notation in Equation (3), if +m ∈ {x|M|y1} and m′ ∈ {x′|M ′|y′ +1}, then the equivalence class of +� +PCm +� +{x|M|y1} +� +⊗ PCm′� +{x′|M ′|y′ +1} +�� +⊗ m(Y,Y ′)(γ, γ′), +is sent to the equivalence class of +id(x,x′)⊗ +H +M×M′ +(xX×x′ +X′ ,yY ×y′ +Y ′ ) +� +id(x,x′), m(Y,Y ′)(γ, γ′) +�� +PC(m,m′) +� +{(x, x′)|M ×M ′|(y2, y′ +2)} +�� +. +Proof. This follows from the previous lemma, and elementary properties of pro- +functors. +□ + +A CATEGORIFICATION OF QUINN’S TQFT +84 +5.2.3. Is H a functor, or perhaps a bifunctor? Now we know how to construct +profunctors and Vect-profunctors from fibrant spans, we should ask how that con- +struction behaves with respect to composition of fibrant spans, and also what does +it do to identity spans. We will examine preservation of composition here, whilst +preservation of identities will be discussed later, being a consequence of Lemma +145, as it is more convenient to package it with similar results later on. +Consider HF fibrant spans, +(p1, M1, p′ +1): X → Y and (p2, M2, p′ +2): Y → Z. +Recall the definition of the composition47, +(p1, M1, p′ +1) • (p2, M2, p′ +2) = (p1, M1 ×Y M2, p′ +2): X → Z, +which is, itself, a HF fibrant span. +To recall and extend notation from earlier, we repeat the relevant commutative +diagram, in Equation (11), +M1×Y M2 +P +� +�❧❧❧❧❧❧❧ +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +p1 +� +p′ +2 +� +M1 +p1 +�rrrrrr +p′ +1 +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +M2 +p2 +�❧❧❧❧❧❧❧❧❧ +p′ +2 +�▲ +▲ +▲ +▲ +▲ +▲ +X +Y +Z, +in which the middle diamond is a pullback. +We recall, from Lemma 55, that ⟨p1, P, p′ +2⟩: M1 ×Y M2 → X × Y × Z is a +fibration. We also note that, given holonomies, for ⟨p1, p′ +1⟩: M1 → X × Y and +⟨p2, p′ +2⟩: M2 → Y × Z, denoted48, +λM1 : I × +� +M1 ×X×Y (XI × Y I) +� +→ M1, +and +λM2 : I × +� +M2 ×Y ×Z (Y I × ZI) +� +→ M2, +(respectively), then a holonomy, +λM1×Y M2 : I × +� +(M1 ×Y M2) ×X×Y ×Z (XI × Y I × ZI) +� +→ M1 ×Y M2, +for ⟨p1, P, p′ +2⟩: M1 ×Y M2 → X × Y × Z, is obtained from the holonomies, λM1 and +λM2, in the obvious way, namely, +� +t, (m1, m2), (γX, γY , γZ) +� +�→ +� +λM1� +t, m1, (γX, γY ) +� +, λM2� +t, m2, (γY , γZ) +�� +. +The following result shows that the profunctor construction is compatible with +composition of fibrant spans. +Proposition 116. Choose subsets, xX, yY and zZ, of X, Y and Z, respectively. +Suppose that (Y, yY ) is 0-connected. We have a canonical isomorphism of Vect- +profunctors from π1(X, xX) to π1(Z, zZ), +ηM1,M2 +(xX,yY ,zZ) : H +M1 +(xX,yY ) • H +M2 +(yY ,zZ) = +ˆ y∈yY +H +M1 +(xX,yY )(−, y) ⊗ H +M2 +(yY ,zZ)(y, −) +=⇒ H +M1×Y M2 +(xX,zZ) , +47see Lemma 55 and Definition 56, +48Note that (X × Y )I ∼ += XI × Y I and (Y × Z)I ∼ += Y I × ZI, canonically, since CGWH is +cartesian closed. + +A CATEGORIFICATION OF QUINN’S TQFT +85 +such that, if x ∈ xX and z ∈ zZ, then, given any y ∈ yY , m1 ∈ {x|M1|y} and +m2 ∈ {y|M2|z}, we have that the linear map49, +(ηM1,M2 +(xX,yY ,zZ))(x,z) : +ˆ y∈yY +H +M1 +(xX,yY )(x, y) ⊗ H +M2 +(yY ,zZ)(y, z)→H +M1×Y M2 +(xX,zZ) (x, z), +sends the equivalence class of +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ∈ +� +y∈yY +H +M1 +(xX,yY )(x, y) ⊗ H +M2 +(yY ,zZ)(y, z) +to PC(m1,m2)({x|M1 ×Y M2|z}). +Proof. This follows by combining the previous discussion with Lemma 104. Here +it may help to note the comments at the beginning of the proof of Lemma 71. +□ +Note that, in a situation in which we have two pairs of composable fibrant spans, +say, +(p1, M1, p′ +1): X → Y and (p2, M2, p′ +2): Y → Z, +and also, +( ˆp1, ˆ +M1, ˆp′ +1): ˆX → ˆY and ( ˆp2, ˆ +M2, ˆp′ +2): ˆY → ˆZ, +then the natural isomorphisms in Lemma 116 are compatible with those of Lemma +115. We leave it to the reader to write down the corresponding commutative dia- +gram of natural transformations between profunctors. +We still have to handle what H does to (resolved) identities, for which see Lemma +145. +Even with the preservation of identities, we have only been addressing the prop- +erties of the ‘category’ of fibrant spans, but the category of fibrant spans, say of HF +spaces, should have some more bicategorical aspect, and without that in evidence +the full question asked in the title of this section cannot be answered. +5.3. HF fibrant resolved 2-spans connecting fibrant spans. So far we have, +in the main, been handling only the 1-categorical structure related to fibrant spans. +We now introduce ‘spans between spans’, that is ‘2-spans’, or, as we will call them +‘windows’. We will later be investigating how this second level of structure on the +spatial side is reflected, via the profunctor construction, in the ‘linear algebra’, and, +of course, this is the beginning of the extension of Quinn’s theory. +5.3.1. HF fibrant windows. +Definition 117 (Window). By a window, W, we will mean a diagram of, as usual, +CGWH spaces of the form below, +(21) +W = +X +M +p +� +p′ +� Y +Z +Pl +� +Ql +� +L +l +� +r +� +P +� +Q +� +W +Pr +� +Qr +� +X′ +N +q +� +q′ +� Y ′, +so it is a ‘span of spans’. +49We are using the notation in Equation (3). + +A CATEGORIFICATION OF QUINN’S TQFT +86 +The boundary, bd(W), of the above window, is the following diagram, +(22) +bd(W) = +X +M +p +� +p′ +� Y +Z +Pl +� +Ql � +W +Pr +� +Qr +� +X′ +N +q +� +q′ +� Y ′. +By the frame, fr(W), of the window, W, above, we will mean the following limit, +(23) +fr(W) = M +× +X×Y (Z × W) +× +X′×Y ′ N ∼= lim(bd(W)), +and the filler, PL, of the window, W, is given by the naturally defined map, +(24) +PL : L → fr(W). +The restrictions of the diagram, (21), to each of its four boundary spans, are +called the top, bottom, left and right boundary spans of W. Respectively, these +are: +(25) +M +p +�⑤⑤⑤⑤ +p′ +�❇❇❇❇ +X +Y, +N +q +�⑤⑤⑤⑤ +q′ +�❈❈❈❈ +X′ +Y ′, +Z +Pl�⑧⑧⑧⑧ +Ql�❇❇❇❇ +X +X′, +W +Pr +�⑤⑤⑤⑤ +Qr�❊❊❊❊ +Y +Y ′. +There are also the middle horizontal and middle vertical spans of the window +W, defined to be +L +P�③③③③ +Q +�❈❈❈❈ +M +N +and +L +l +�⑤⑤⑤⑤ +r�❉❉❉❉ +Z +W. +In this paper, we will see windows as being “oriented” from top to bottom and +from left to right. +Definition 118 (HF fibrant window). A fibrant window is a window, W, as in +(21), such that: +(1) the filler, PL : L → fr(W), is a fibration, +and, +(2) the four boundary spans (top, bottom, left and right) of W are all fibrant. +If, in addition, all the spaces appearing in diagram (21) are HF, then the window, +W, will be called a HF fibrant window. +The following are some immediate consequences of the definition of HF fibrant +windows. +(1) Suppose that a fibrant window, W, as in (21), is HF, then its frame +fr(W) = M +× +X×Y (Z × W) +× +X′×Y ′ N +is a HF space. This follows by applying Lemma 41 to the pullback diagram, +(26) +M +× +X×Y (Z × W) +× +X′×Y ′ N +� +� +M × N +⟨p,p′⟩×⟨q,q′⟩ +� +Z × W +τ◦(⟨Pl,Ql⟩×⟨Pr,Qr⟩) +� (X × Y ) × (X′ × Y ′), +where τ is the obvious transposition. This uses that the top and bottom +boundary spans, of W, are fibrant. + +A CATEGORIFICATION OF QUINN’S TQFT +87 +(2) There are two naturally defined maps, fr(W) → Z × W and fr(W) → +M × N. Both are fibrations. For the first map, fr(W) → Z × W, this +follows from the pullback diagram (26) above together with Lemma 41, +and similarly for the map fr(W) → M × N, by symmetry. +(3) Composing with the filler, PL : L → fr(W), of W, which by definition is a +fibration, this gives: +Lemma 119. If a window, W, is fibrant, then so are the middle horizontal and +middle vertical spans, namely, +L +P�④④④④ +Q +�❈ +❈❈❈ +M +N, +and +L +l +�⑤⑤⑤⑤ +r�❉❉❉❉ +Z +W. +□ +5.3.2. Isomorphic windows and equivalent fibrant windows. +Definition 120 (Isomorphic windows and equivalent fibrant windows). Two win- +dows, W1 and W2, as below, so with the same boundary, +(27) +W1 = +X +M +p +� +p′ +� Y +Z +Pl +� +Ql +� +L1 +l1 +� +r1 +� +P1 +� +Q1 +� +W +Pr +� +Qr +� +X′ +N +q +� +q′ +� Y ′ +and +W2 = +X +M +p +� +p′ +� Y +Z +Pl +� +Ql +� +L2 +l2 +� +r2 +� +P2 +� +Q2 +� +W +Pr +� +Qr +� +X′ +N +q +� +q′ +� Y ′, +and thus, in particular, fr(W1) = fr(W2), are said to be isomorphic if there exists +a homeomorphism, F : L1 → L2, making the obvious three dimensional diagram +commute. +This is equivalent to saying that there exist maps, F : L1 → L2 and F ′ : L2 → L1, +making the diagrams below commute, +L1 +F +� +PL1 +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +L2, +PL2 +�✇✇✇✇✇✇✇✇ +fr(W2) +and +L1 � +F ′ +PL1 +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +L2, +PL2 +�✇✇✇✇✇✇✇✇ +fr(W2) +such that F ◦ F ′ = idL2 and F ′ ◦ F = idL1. +More generally, if W1 and W2 are fibrant, then W1 and W2 are called equiv- +alent if there exist F : L1 → L2 and F ′ : L2 → L1 making the diagrams above +commute, together with fibred homotopies (see Subsection 2.5), +F ′ ◦ F +H1 +=====⇒ +fr(W2) +idL1 +and +F ◦ F ′ +H2 +=====⇒ +fr(W2) +idL2. +It is easy to see that equivalence between fibrant windows is an equivalence relation. +We will, of course, be mostly interested in the situation in which all the spaces +involved are HF. +5.3.3. HF fibrant resolved 2-spans. + +A CATEGORIFICATION OF QUINN’S TQFT +88 +Definition 121 (HF fibrant resolved 2-span). Given HF fibrant spans, from X to +Y , which we picture as follows, +X +(p,M,p′) +� +(q,N,q′) +� Y, +and also in full, +M +p +�③③③③ +p′ +�❉ +❉ +❉ +❉ +X +Y, +N +q +�④④④④ +q′ +�❈ +❈ +❈ +❈ +X +Y, +a HF fibrant resolved 2-span, W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′), also +written +W = (lX, P, L, Q, rY ): +� +(p, M, p′): X → Y +� +=⇒ +� +(q, N, q′): X → Y +� +, +or, simply, +X +(p,M,p′) +� +(q,N,q′) +� +⇓ W +Y, +is a HF fibrant window of form, as below50, +(28) +W = +X +M +p +� +p′ +� Y +XI +sX +� +tX +� +L +lX +� +rY +� +P +� +Q +� +Y I +sY +� +tY +� +X +N +q +� +q′ +� Y. +This means that X, Y, M, N, and L are HF spaces, (and, thus, so are XI and +Y I), and the filler of W (see Definition 117), below, is a fibration, +(29) +L +PL +−→ fr(W) = M +× +X×Y (XI × Y I) +× +X×Y N. +(Note that the top and bottom boundary spans in (28) are already fibrant, by +assumption. The left and right boundary spans are also fibrant, by construction, +see Example 51.) +We will frequently identity a HF-fibrant resolved 2-span with its filler, L +PL +−→ +fr(W). +Crucially for our constructions later on, the middle horizontal and middle vertical +spans in (28) are fibrant, by Lemma 119. +The ‘resolved’ terminology arises from the fact that we allow the left and right +boundaries of a 2-span to take values in the respective path spaces. For 2-spans +in a usual setting, the left and right vertical edges would be identity spans, but +to ensure the result is fibrant, we must ‘resolve’ those vertical edges, replacing +them with their ‘fibrant replacements’, as in Example 51. This is in line with the +definition of extended cobordisms between cobordisms of manifolds. Our definition +was also designed so that the horizontal composition of resolved 2-spans, defined in +Subsection 5.5, is a resolved 2-span. +50note the notation in item (9) in page 16, and also Definition 59 + +A CATEGORIFICATION OF QUINN’S TQFT +89 +Definition 122 (Equivalent and isomorphic HF fibrant resolved 2-spans). Let X +and Y be HF spaces. Two HF resolved 2-spans, +W1, W2 : +� +(p, M, p′): X → Y ) =⇒ +� +(q, N, q′): X → Y ), +will be said to be equivalent if they are equivalent as HF fibrant windows, and, +similarly, W1 and W2 are isomorphic if they are isomorphic as HF fibrant windows. +(We are using the nomenclature of Definition 120.) +As usual, the above definition is given in the case that we will be using, but we +note that it nowhere uses that fact that the spaces are homotopy finite. +5.4. HF resolved fibrant 2-spans and natural transformations of profunc- +tors. The next step is to study these HF resolved fibrant 2-spans before turning +to how their properties are reflected by the profunctor construction. +5.4.1. The spatial 2-slices of a HF resolved fibrant 2-span. Let X and Y be HF- +spaces. Consider HF fibrant spans, +(p, M, p′), (q, N, q′): X → Y, +and a HF fibrant resolved 2-span, +W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′), +connecting them. Its underlying HF fibrant window, W, is the commutative di- +agram of solid arrows in Equation (30), just below. (The dashed arrows showing +the inclusion, x �→ constx, of a space, X, into the corresponding path space, XI, +via constant paths, do not necessarily commute with the rest of the diagram. They +will, however, be used later.) +(30) +W = + + + + + + + + + + + + + +X +const +� +♣ +① +✤ +❋ ◆ +M +p +� +p′ +� Y +const +� +◆ ❋ +✤ +① +♣ +XI +sX +� +tX +� +L +lX +� +rY +� +P +� +Q +� +Y I +sY +� +tY +� +X +const +� +◆ +❋ +✤ ① ♣ +N +q +� +q′ +� Y +const +� +♣ ① +✤ +❋ +◆ + + + + + + + + + + + + + +. +We repeat that, X, Y, M, N, L are HF spaces, hence XI and Y I are HF-spaces, +and that the filler, PL : L → fr(W), of W, below, is a fibration, +(31) +L +PL +−→ fr(W) = M +× +X×Y (XI × Y I) +× +X×Y N. +Let x ∈ X and y ∈ Y . Recall, from Definition 67, that the spatial slices of the +fibrant spans, (p, M, p′) and (q, N, q′), are defined as +{x|M|y} = ⟨p, p′⟩−1(x, y) = {m ∈ M : p(m) = x and p′(m) = y}, +{x|N|y} = ⟨q, q′⟩−1(x, y) = {n ∈ N : q(n) = x and q′(n) = y}, +and these spatial slices will be homotopy finite; see Lemma 54. +Definition 123 (Spatial 2-slices). Let W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′) +be a HF resolved 2-span, as in (30). Let x ∈ X, y ∈ Y . We define51 the following +space, which we call the spatial 2-slice, of W, at (x, y), +[x|L|y] := ⟨lX, rY ⟩−1(constx, consty) += +� +l ∈ L : lX(l) = constx, rY (l) = consty +� +⊂ L. +51and note the square brackets rather than braces here. That distinction will be needed shortly. + +A CATEGORIFICATION OF QUINN’S TQFT +90 +Given m ∈ {x|M|y} and n ∈ {x|N|y}, also consider the following space, which we +call the spatial 2-slice, of W, at (m, x, y, n), + + +m +x +L +y +n + + := P −1 +L (m, constx, consty, n) += +� +l ∈ L : P(l) = m, lX(l) = constx, rY (l) = consty, Q(l) = n +� +. +More generally, given paths, x +γX +−−→ x′ and y +γY +−−→ y′, in X and Y , m ∈ {x|M|y} +and n ∈ {x′|N|y′}, we define the following space, also called a spatial 2-slice of W, +but at (m, γX, γY , n), + + +m +γX +L +γY +n + + := P −1 +L (m, γX, γY , n) += +� +l ∈ L : P(l) = m, lX(l) = γX, rY (l) = γY , Q(l) = n +� +. +In the context of this definition, note that, given x ∈ X and y ∈ Y , the maps, +P : L → M and Q: L → N, canonically restrict to maps, denoted +Px,y : [x |L| y] → {x|M|y} +and +Qx,y : [x |L| y] → {x|N|y}. +Moreover, given m ∈ {x|M|y} and n ∈ {x|N|y}, we have the following, + + +m +x +L +y +n + + = ⟨Px,y, Qx,y⟩−1(m, n). +Lemma 124. Let x ∈ X, and y ∈ Y . +(1) The induced map, +⟨Px,y, Qx,y⟩: [x |L| y] → {x|M|y} × {x|N|y}, +is a fibration. +(2) If m ∈ {x|M|y} and n ∈ {x|N|y}, the homotopy type of the space, + + +m +x +L +y +n + + , +depends only on the path-components, in {x|M|y}, resp. {x|N|y}, contain- +ing m, resp. n. +Proof. The first item follows by direct application of the homotopy lifting property +of the fibration, PL : L → fr(W). The second follows from the fact that all fibres +of a fibration over a path-connected space are homotopy equivalent. +□ +Recalling that we are assuming that X, Y, L, M, N are HF, we have: +Lemma 125. All spaces appearing in Definition 123 are HF spaces. +Proof. For the spatial 2-slice, + + +m +x +L +y +n + + = P −1 +L (m, constx, consty, n), +this follows from the fact that both L and fr(W) are HF, (for the latter fact, see +the discussion just after Definition 118), and Lemma 40, applied to the fibration +PL : L → fr(W). + +A CATEGORIFICATION OF QUINN’S TQFT +91 +The same sort of argument holds for the general spatial 2-slices, namely, + + +m +γX +L +γY +n + + = P −1 +L (m, γX, γY , n). +We have a fibration, ⟨Px,y, Qx,y⟩: [x |L| y] → {x|M|y} × {x|N|y}, in which the +spaces, {x|M|y} and {x|N|y} are both HF (see Lemma 54), and all of whose fibres, +i.e., all of the +P −1 +L (m, constx, consty, n), +are HF. We thus have [x |L| y] is also HF by Lemma 40. +□ +We define: +Definition 126 (Vertical span of slices). Let W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ +(q, N, q′) be a HF resolved 2-span, as in (30). Let x ∈ X and y ∈ Y . The fibrant +span, +[x|W|y] := + + + +[x|L|y] +Px,y +�♦♦♦♦♦ +Qx,y +�◆ +◆ +◆ +◆ +◆ +{x|M|y} +{x|N|y} + + + , +will be called the vertical span of slices, of W, at x and y. +By the discussion just given, [x|W|y] is a HF fibrant span. Clearly we have + + +m +x +L +y +n + + = ⟨Px,y, Qx,y⟩−1 (m, n). +We continue to fix a HF resolved 2-span, W, as in (30) and prove that, in several +important cases, the spatial 2-slices of W are homotopy equivalent. +Firstly consider holonomy maps for the fibrations, ⟨p, p′⟩: M → X × Y and +⟨q, q′⟩: N → X × Y , which will be denoted +λM : I × (M ×X×Y (X × Y )I) → M and λN : I × (N ×X×Y (X × Y )I) → N, +respectively. +Given paths x +γX +−−→ x′ in X, and y +γY +−−→ y′, in Y , these holonomy maps induce +homotopy equivalences (where we are using the notation of Lemma 94), +ΓM +⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′} and ΓN +⟨γX,γY ⟩ : {x|N|y} → {x′|N|y′}. +Here ⟨γX, γY ⟩ is the path, in X × Y , such that, for u ∈ I, +u �→ +� +γX(u), γY (u) +� +∈ X × Y. +Lemma 127. Consider a HF resolved 2-span as in (30). Let x, x′ ∈ X and y, y′ ∈ +Y . Suppose that γX : I → X is a path in X, connecting x to x′, and that γY : I → Y +is one in Y , connecting y to y′. Let m ∈ {x|M|y}, n ∈ {x|N|y} and n′ ∈ {x′|N|y′}. +(1) The two spaces, + + +m +x +L +y +n + + and + + +ΓM +⟨γX,γY ⟩(m) +x′ +L +y′ +ΓN +⟨γX,γY ⟩(n) + + , +are homotopically equivalent. + +A CATEGORIFICATION OF QUINN’S TQFT +92 +(2) The three spaces, + + +m +γX +L +γY +n′ + + , + + +m +x +L +y +ΓN +⟨γX,γY ⟩(n′) + + , and + + +ΓN +⟨γX,γY ⟩(m) +x′ +L +y′ +n′ + + , +are homotopically equivalent. +(We recall that given a path γ, then γ denotes its reverse.) +Proof. The first point follows from the fact that the two spaces are fibres of the +fibration, PL : L → fr(W), over points in the same path component of fr(W). +Indeed, the two spaces are +P −1 +L (m, constx, consty, n), and P −1 +L +� +ΓM +⟨γX,γY ⟩(m), constx′, consty′, ΓN +⟨γX,γY ⟩(n) +� +, +respectively, for points in fr(W). +A path, in fr(W), connecting +(m, constx, consty, n), +and +� +ΓM +⟨γX,γY ⟩(m), constx′, consty′, ΓN +⟨γX,γY ⟩(n) +� +, +is given by +t �→ +� +λM� +t, m, ⟨γX, γY ⟩ +� +, constγX(t), constγY (t), λN� +t, n, ⟨γX, γY ⟩ +�� +. +The second point follows again by identifying the spaces with fibres of the fibra- +tion PL : L → fr(W), and examining different points of the same path component. +For instance, a path, in fr(W), connecting +(m, γX, γY , n′) +and +� +m, constx, consty, ΓN +⟨γX,γY ⟩(n′) +� +, +is +s �→ +� +m, γX +s , γY +s , λN� +s, n′, ⟨γX, γY ⟩ +�� +. +Here, given a map γ : I → B, where B is a space, and s ∈ [0, 1], we have written +γs : I → B for the path t �→ γ +� +(1 − s)t +� +. +□ +5.4.2. The natural transformation of profunctors associated to a HF resolved fibrant +2-span. As usual, let κ be a subfield of C. +We need to recap a little on the notation, etc. +Let +W = (lX, P, L, Q, rY ): +� +(p, M, p′): X → Y +� +=⇒ +� +(q, N, q′): X → Y +� +be a HF fibrant resolved 2-span, as in (30), connecting the HF fibrant spans, +(p, M, p′) and (q, N, q′). We assume given, or chosen, subsets xX ⊂ X and yY ⊂ Y . +We have Vect-profunctors, as defined in §5.2.1, in particular in Notation 113, +H +M +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, +and +H +N +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect. +These are the restrictions, to π1(X, xX)op × π1(Y, yY ), of the profunctors, +HM : π1(X, X)op × π1(Y, Y ) → Vect, +and +HN : π1(X, X)op × π1(Y, Y ) → Vect. + +A CATEGORIFICATION OF QUINN’S TQFT +93 +We want to define a natural transformation of profunctors, which will be denoted +2H +W +(xX,yY ) : H +M +(xX,yY ) =⇒ H +N +(xX,yY ). +This natural transformation will, itself, be the restriction of a natural transforma- +tion, denoted +2HW : HM +=⇒ HN, +which we will define first. +Of course, given x ∈ X and y ∈ Y , HM(x, y) and HN(x, y) are both κ-vector +spaces, so to specify a natural transformation, as required, we have to specify a +linear transformation, 2HW +(x,y), from HM(x, y) to HN(x, y), depending on x and +y in a ‘natural’ way. Given that, by Lemma 112, both vector spaces are finite +dimensional, we may specify this linear map by giving its matrix elements with +respect to the evident bases. +Definition 128. Consider a HF resolved 2-span, W: (p, M, p′) =⇒ (q, N, q′), as +in (30). Given x ∈ X and y ∈ Y , we define the linear map, +2HW +(x,y) : HM(x, y) → HN(x, y), +where +HM(x, y) = Lin(�π0({x|M|y})) +and +HN(x, y) = Lin(�π0({x|N|y})). +This is to have the following matrix elements, with respect to the usual bases. +Given m ∈ {x|M|y} and n ∈ {x|N|y}, +(32) +� +PCm({x|M|y}) | 2HW +(x,y) | PCn({x|N|y}) +� +:= χπ + + + + +m +x +L +y +n + + + + χπ� +PCn({x|N|y}) +� +. +We refer to Definition 123 for notation. Note that, by Lemma 125, all the spatial +2-slices met here are HF spaces, so we can consider their homotopy content. +Lemma 129. Consider a HF resolved 2-span, +W: +� +(p, M, p′): X → Y +� +=⇒ +� +(q, N, q′): X → Y +� +, +as in (30). +• Let x ∈ X and y ∈ Y . Given m ∈ {x|M|y} and n ∈ {x|N|y}, the value of +the right-hand-side of Equation (32) depends only on the path-components, +in {x|M|y} and in {x|N|y}, respectively, to which m and n belong. +• The family of linear maps, 2HW +(x,y): HM(x, y) → HN(x, y), for all x ∈ X +and y ∈ Y , together defines a natural transformation of Vect-profunctors, +2HW : +� +HM : π1(X, X) ↛ π1(Y, Y ) +� +=⇒ +� +HN : π1(X, X) ↛ π1(Y, Y ) +� +, +and, therefore (by Lemma 112) a 2-morphism in the bicategory vProfGrphf. +Proof. The first statement follows directly from item (2) of Lemma 124, and the +fact that the homotopy content of a homotopy finite space is a homotopy invariant. +The second statement follows from point (1) of Lemma 127, given the explicit +forms of HM and HN in Definition 111. We also use the fact that, given paths +γX : x → x′ in X, and γY : y → y′ in Y , the holonomy map, λN, for ⟨q, q′⟩: N → +X × Y , gives rise to a homotopy equivalence, between fibres, +ΓN +⟨γX,γY ⟩ : {x|N|y} → {x′|N|y′}, + +A CATEGORIFICATION OF QUINN’S TQFT +94 +and, in particular, induces a bijection between the sets of path components. More- +over, given n ∈ {x|N|y}, ΓN +⟨γX,γY ⟩ therefore restricts to a homotopy equivalence, +PCn +� +{x|N|y} +� ∼= PCΓN +⟨γX ,γY ⟩(n) +� +{x′|N|y′} +� +. +□ +Definition 130. Choose xX ⊂ X and yY ⊂ Y . The natural transformation, +2H +W +(xX,yY ) : H +M +(xX,yY ) =⇒ H +N +(xX,yY ), +is defined by restricting 2HW : HM =⇒ HN to π1(X, xX)op × π1(Y, yY ). +Explicitly, given x ∈ xX and y ∈ yY , the linear map, +(2H +W +(xX,yY ))(x,y) : H +M +(xX,yY )(x, y) → H +N +(xX,yY )(x, y), +is, therefore, 2HW +(x,y): HM(x, y) → HN(x, y). +Remark 131. When κ = C, there is a 1-parameter version of 2HW : HM =⇒ HN, +which is denoted 2H(W,s) : HM =⇒ HN, where s ∈ C. This has as matrix elements +� +PCm({x|M|y}) | 2H(W,s) +(x,y) | PCn({x|N|y}) +� +:= χπ + + + + +m +x +L +y +n + + + + χπ� +PCn({x|N|y}) +� +χπ(PCx(X))1−sχπ(PCy(Y ))s. +All results go through with this extra generality. This is a special case of a more +general 2-parameter version, also involving the vertical direction of the spatial 2- +slices. This is left to the reader to explore. +5.4.3. The symmetric monoidal like structure of 2H +W +(−,−). The following string of +results will be used later on (Subsection 6.5) to prove that the constructions of +the once-extended Quinn TQFTs, defined in this paper, give bifunctors which, +furthermore, can be given symmetric monoidal structures. Our starting point is +Lemmas 114 and 115 of §5.2.2. +Lemma 132. Consider HF fibrant resolved 2-spans, denoted +W: +� +(p, M, q): X → Y +� +=⇒ +� +(f, N, g): X → Y +� +, +and +W′ : +� +(p′, M ′, q′): X′ → Y ′� +=⇒ +� +(f ′, N ′, g′): X′ → Y ′� +. +The diagrams for W and W′ will be +(33) +X +M +p +� +q +� Y +XI +sX +� +tX +� +L +l +� +r +� +P +� +Q +� +Y I +sY +� +tY +� +X +N +f +� +g +� Y +and +X′ +M ′ +p′ +� +q′ +� Y ′ +X′I +sX′ +� +tX′ +� +L′ +l′ +� +r′ +� +P ′ +� +Q′ +� +Y ′I +sY ′ +� +tY ′ +� +X′ +N ′ +f ′ +� +g′ +� Y ′. +The following hold. + +A CATEGORIFICATION OF QUINN’S TQFT +95 +(1) The window, W × W′, below, is a HF fibrant resolved 2-span, +� +(p × p′, M × M ′, q × q′): X × X′ → Y × Y ′� +=⇒ +� +(f × f ′, N × N ′, g × g′): X × X′ → Y × Y ′� +, +X × X′ +M × M ′ +p×p′ +� +q×q′ +� Y × Y ′ +(X × X′)I ∼= XI × X′I +sX×sX′ +� +tX×tX′ +� +L × L′ +l×l′ +� +r×r′ +� +P ×P ′ +� +Q×Q′ +� +Y I × Y ′I ∼= (Y × Y ′)I +sY ×sY ′ +� +tY ×tY ′ +� +X × X′ +N × N ′ +f×f ′ +� +g×g′ +� Y × Y ′. +(2) Let x ∈ X, x′ ∈ X′, y ∈ Y and y′ ∈ Y ′. Given m ∈ {x|M|y}, n ∈ {x|N|y}, +m′ ∈ {x′|M ′|y′} and n′ ∈ {x′|N ′|y′}, we have that, +� +PC(m,m′)({(x, x′)|M × M ′|(y, y′)}) | 2HW×W′ +((x,x′),(y,y′)) | PC(n,n′)({(x, x′)|N × N ′|(y, y′)}) +� +equals +� +PCm({x|M|y}) | 2HW +(x,y) | PCn({x|N|y}) +� � +PCm′({x′|M ′|y′}) | 2HW′ +(x′,y′) | PCn′({x′|N ′|y′}) +� +. +Proof. The first point follows from the fact that the product of fibration is a fibra- +tion. The second follows from the fact that, clearly, + + +(m, m′) +(x, x′) +L × L′ +(y, y′) +(n, n′) + + ∼= + + +m +x +L +y +n + + × + + +m′ +x′ +L′ +y′ +n′ + + , +and +PC(n,n′) +� +{(x, x′)|N × N ′|(y, y′)} +� ∼= PCn +� +{x|N|y} +� +× PCn′� +{x′|N ′|y′} +� +. +The formula therefore follows from the fact that the homotopy content of HF spaces +is multiplicative with respect to their product. +□ +In order to prove that the once-extended Quinn TQFT is a symmetric monoidal +bifunctor, it is convenient to change the language of the previous result, to a lan- +guage closer to that of Definition 30. In particular, combining Lemma 132 with +Lemma 115, whose notation we follow, gives the following. +Lemma 133. Let xX ⊂ X, x′ +X′ ⊂ X′, yY ⊂ Y and y′ +Y ′ ⊂ Y ′. The two natural +transformations of Vect-profunctors, i.e. the two 2-morphisms in vProfGrphf, +obtained by pasting the diagrams in vProfGrphf, below, coincide. +π1(X, xX) × π1(X′, x′ +X′) +H(xX ,yY )(f,N,g)⊗H(x′ +X′ ,y′ +Y ′ )(f ′,N ′,g′) +� +H(xX ,yY )(p,M,q)⊗H(x′ +X′ ,y′ +Y ′ )(p′,M′,q′) +� +⇓ +� +2H +W +(xX ,yY )⊗2H +W′ +(x′ +X′ ,y′ +Y ′ ) +� +ϕ +m(X,X′) +� +✘✘✘✘� χ(N,N′) +π1(Y, yY ) × π1(Y ′, y′ +Y ′) +ϕ +m(Y,Y ′) +� +π1(X × X′, xX × x′ +X′) +H(xX ×x′ +X′ ,yY ×y′ +Y ′ )(f×f ′,N×N ′,g×g′) +� π1(Y × Y ′, yY × y′ +Y ′), + +A CATEGORIFICATION OF QUINN’S TQFT +96 +and +π1(X, xX) × π1(X′, x′ +X′) +H(xX ,yY )(p,M,q)⊗H(x′ +X′ ,y′ +Y ′ )(p′,M′,q′) +� +ϕ +m(X,X′) +� +✘✘✘✘� +χ(M,M′) +π1(Y, yY ) × π1(Y ′, y′ +Y ′) +ϕ +m(Y,Y ′) +� +π1(X × X′, xX × x′ +X′) +H(xX ×x′ +X′ ,yY ×y′ +Y ′ )(p×q,M×M′,q×q′) +� +H(xX ×x′ +X′ ,yY ×y′ +Y ′ )(f×f ′,N×N ′,g×g′) +� +⇓ +� +2H +W×W′ +(xX ×x′ +X′ ,yY ×y′ +Y ′ ) +� +π1(Y × Y ′, yY × y′ +Y ′). +□ +We continue to follow the notation in Definition 30. +Notation 134. Given pairs of spaces, (X, xX), (X′, x′ +X′) and (X′′, x′′X′′), with +X, X′ and X′′ homotopy finite, we have an obvious invertible 2-morphism in vProfGrphf, +as shown in the diagram below. (We have condensed the notation, so π(X) means +π1(X, xX), X × X′ means (X × X′, xX × x′X′), and so on.) +� +π(X) × π(X′) +� +× π(X′′) +ϕ +αGrp +(π(X),π(X′),π(X′′ )) +�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +ϕ +m(X,X′)×idπ(X′′) +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +π(X) × +� +π(X′) × π(X′′) +� +idπ(X)×ϕ +m(X′,X′′) +� +❴❴❴❴� +ω(X,X′,X′′) +π(X × X′) × π(X′′) +ϕ +m(X×X′,X′′) +� +π(X) × +� +π(X′ × X′′) +� +ϕ +m(X,X′×X′′) +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +π +� +(X × X′) × X′′� +ϕ +π +� +αCGWH +(X,X′,X′′) +� +�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +π +� +X × (X′ × X′′) +� +. +This 2-morphism of profunctors arises from the fact that, if we switch back the +profunctors in the arrows of the diagram above, to the functors that gave rise to +them, then, applying Example 33, gives rise to a commutative diagram of functors. +Indeed, note that, in general, if F : C → C′ and G: C′ → C′′ are functors, then we +have a canonical natural isomorphism from the profunctor, ϕG◦F : C ↛ C′′, to the +composition of the profunctors, C +ϕF +↛ C′ ϕG +↛ C′′; for more details see Example 34. +The 2-morphisms, ω(X,X′,X′′) in vProfGrphf satisfy an obvious cocycle con- +dition, given pairs of spaces, (X, xX), (X′, x′X′), (X′′, x′′X′′), and (X′′′, x′′′X′′′). +The equations satisfied are those in [61, §4.3] / [58, page 17]. This follows from an +explicit calculation. +These cocycles, ω(X,X′,X′′), are furthermore compatible with the 2H +W +(−,−) and +the χ(−,−). Suppose that we are given three HF spans, +(p, M, q): X → Y, +(p′, M ′, q′): X′ → Y ′, +and +(p′′, M ′′, q′′): X′′ → Y ′′. + +A CATEGORIFICATION OF QUINN’S TQFT +97 +The 2-morphisms, χ(−,−), in vProfGrphf, arising from Lemma 115, can be pasted +in two different ways, as written below (where we have again condensed the nota- +tion, in the obvious way), +� +π(X) × π(X′) +� +× π(X′′) +ϕ +m(X,X′)×idπ(X′′) +� +✘✘✘✘� χ(M,M′)⊗HM′′ +(HM⊗HM′ +)⊗HM′′ +� � +π(Y ) × π(Y ′) +� +× π(Y ′′) +ϕ +m(Y,Y ′)×idπ(X′′) +� +π(X × X′) × π(X′′) +HM×M′ +⊗HM′′ +� +ϕ +m(X×X′,X′′) +� +✘✘✘✘� χ(M×M′,M′′) +π(Y × Y ′) × π(Y ′′) +ϕ +m(Y ×Y ′,Y ′′) +� +π +� +(X × X′) × X′′� +ϕ +π +� +αCGWH +(X,X′,X′′) +� +� +✘✘✘✘� ∼ += +H(M×M′)×M′′ +� π +� +(Y × Y ′) × Y ′′� +ϕ +π +� +αCGWH +(Y,Y ′,Y ′′) +� +� +π +� +X × (X′ × X′′) +� +HM×(M′×M′′) +� π +� +Y × (Y ′ × Y ′′) +� +, +and +� +π(X) × π(X′) +� +× π(X′′) +ϕ +αGrp +(π(X),π(X′ ),π(X′′)) +� +✘✘✘✘� ∼ += +(H +M⊗H +M′ +)⊗H +M′′ +� � +π(Y ) × π(Y ′) +� +× π(Y ′′) +ϕ +αGrp +(π(Y ),π(Y ′),π(Y ′′)) +� +π(X) × +� +π(X′) × π(X′′) +� +HM⊗(HM′ +⊗HM′′ +) +� +π(X)×ϕ +m(X′,X′′) +� +✘✘✘✘� H +M⊗χ(M′,M′′) +π(Y ) × +� +π(Y ′) × π(Y ′′) +� +π(Y )×ϕ +m(Y ′,Y ′′) +� +π(X) × π(X′ × X′′) +ϕ +m(X,X′×X′′) +� +✘✘✘✘� χ(M,M′×M′′) +HM⊗HM′×M′′ +� π(Y ) × π(Y ′ × Y ′′) +ϕ +m(Y,Y ′×Y ′′) +� +π +� +X × (X′ × X′′) +� +H +M×(M′×M′′) +� π +� +Y × (Y ′ × Y ′′) +� +. +We note that the first of these diagrams fits together with the diagram for ω(X,X′,X′′) +on the left, and the various 2-morphisms compose well. +The second lower dia- +gram likewise composes, this time with ω(Y,Y ′,Y ′′) and on the right, again giving +a second composite 2-morphism, which has the same source and target composite +1-morphisms as the first. +Lemma 135. The two composite 2-morphisms obtained as above by pasting, re- +spectively, ω(X,X′,X′′) or ω(Y,Y ′,Y ′′) to the two diagrams, are equal. +□ +Remark 136. The profunctors, +H(xX×x′ +X′ ,yY ×y′ +Y ′ ) +� +X × X′ +(p×q,M×M′,q×q′) +−−−−−−−−−−−−→ Y × Y ′ +� +: +π1(X × X′, xX × x′ +X′)op × π1(Y × Y ′, yY × y′ +Y ′) → Vect, +as well as the natural transformation between them that are obtained from fibrant +resolved 2-spans, are similarly well behaved with respect to swapping the order of +coordinates, and with respect to products with trivial spaces. +We leave it to the +reader to unpack what this means in terms of diagrams similar to those that were +just presented. + +A CATEGORIFICATION OF QUINN’S TQFT +98 +5.5. The horizontal composition of HF resolved 2-spans. HF resolved 2- +spans can be composed both horizontally and vertically. Here we first look at the +horizontal composition before considering the effect of that composition on the +corresponding natural transformations. +5.5.1. The horizontal composition of HF resolved 2-spans in detail. Consider HF +fibrant spans, +(p1, M1, p′ +1): X → Y and (p2, M2, p′ +2): Y → Z. +Recalling the definition of their composition from Lemma 55 and Definition 56, we +have that +(p1, M1, p′ +1) • (p2, M2, p′ +2) = (p1, M1 ×Y M2, p′ +2): X → Z +is itself a HF fibrant span. It is defined by the pullback diagram, appearing as the +diamond in the commutative diagram below, which we repeat from earlier for the +notation being used, +M1×Y M2 +�❣❣❣❣❣ +�❲ +❲ +❲ +❲ +❲ +p1 +� +p′ +2 +� +M1 +p1 +�❥❥❥❥❥❥ +p′ +1 +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +M2 +p2 +�❣❣❣❣❣❣❣❣❣ +p′ +2 +�❚ +❚ +❚ +❚ +❚ +❚ +X +Y +Z, +and from which we extract the HF fibrant span, +M1×Y M2 +p1 +�qqqqqq +p′ +2 +�▼ +▼ +▼ +▼ +▼ +▼ +X +Z, +which is the composite of the given pair of fibrant spans. +Consider a diagram of spaces, HF spans, and HF resolved 2-spans, as below, +X +(p1,M1,p′ +1) +� +(q1,N1,q′ +1) +� +⇓ W1 +Y +(p2,M2,p′ +2) +� +(q2,N2,q′ +2) +� +⇓ W2 +Z , +where the diagrams for W1 and W2 are as shown below, +(34) +X +M1 +p1 +� +p′ +1 +� Y +XI +sX +� +tX +� +L1 +l1 +� +r1 +� +P1 +� +Q1 +� +Y I +sY +� +tY +� +X +N1 +q1 +� +q′ +1 +� Y +and +Y +M2 +p2 +� +p′ +2 +� Z +Y I +sY +� +tY +� +L2 +l2 +� +r2 +� +P2 +� +Q2 +� +ZI +sZ +� +tZ +� +Y +N2 +q2 +� +q′ +2 +� Z. +We will also need to consider the fillers, Definition 117, of W1 and W2, as usual +denoted by +PL1 : L1 → fr(W1) +and +PL2 : L2 → fr(W2). + +A CATEGORIFICATION OF QUINN’S TQFT +99 +We will define the horizontal composite, W1#0W2, of W1 and W2, in such a +way that W1#0W2 is a HF resolved 2-span, which fits inside the diagram below, +(35) +X +(p1,M1×Y M2,p′ +2) +� +(q1,N1×Y N2,q′ +2) +� +⇓ W1#0W2 +Z += X +(p1,M1,p′ +1)•(p2,M2,p′ +2) +� +(q1,N1,q′ +1)•(q2,N2,q′ +2) +� +⇓ W1#0W2 +Z. +This is done by considering the obvious pullback along the common vertical HF +fibrant span in (34). Explicitly the horizontal composition of W1 and W2 will be +given by the window +(36) +W1#0W2 := + + + + + + + + + + + + + + +X +M1 ×Y M2 +p1 +� +p′ +2 +� Z +XI +sX +� +tX +� +L1 ×Y I L2 +lX +� +rZ +� +P +� +Q +� +ZI +sZ +� +tZ +� +X +N1 ×Y N2 +q1 +� +q′ +2 +� Z + + + + + + + + + + + + + + +. +Here we will need to consider the pullback diagram included (as the middle dia- +mond) in the commutative diagram below, in which, +(i) ΨL1,L2 : L1 ×Y I L2 → Y I is ΨL1,L2 = l2 ◦ proj2 = r1 ◦ proj1, +(ii) lX = l1 ◦ proj1, +and, +(iii) rZ = r2 ◦ proj2, +(37) +L1 ×Y I L2 +lX +� +rZ +� +ΨL1,L2 +� +proj1 +�tttttttttt +proj2 +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +L1 +r1 +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +l1 +�⑤⑤⑤⑤⑤⑤⑤ +L2 +l2 +�tttttttttt +r2 +�❈ +❈ +❈ +❈ +❈ +❈ +❈ +XI +Y I +ZI . +We have obvious maps, ¶: L1 ×Y I L2 → M1 ×Y M2, induced by P1 : L1 → M1 +and P2 : L2 → M2, and Q: L1 ×Y I L2 → N1 ×Y N2, induced by Q1 : L1 → M1 and +Q2 : L2 → N2. +We will briefly explain why W1#0W2, in Equation (36), is a HF fibrant resolved +2-span, fitting inside diagram (35). This follows by a sequence of observations. +(1) Lemma 55, together with the fact that the top, bottom and middle hori- +zontal (Lemma 119) spans of the diagrams in (34) are HF fibrant, implies +that all spaces appearing in diagram (36) are HF. + +A CATEGORIFICATION OF QUINN’S TQFT +100 +(2) Secondly, we note that the naturally defined map, below, which is the filler +of W1#0W2, is a fibration, +PL1,L2 : L1 ×Y I L2 → lim + + + + + + + + + + + + + +X +M1 ×Y M2 +p1 +� +p′ +2 +� Z +XI +sX +� +tX +� +ZI +sZ +� +tZ +� +X +N1 ×Y N2 +q1 +� +q′ +2 +� Z + + + + + + + + + + + + + +. +The argument is very similar to that in the proof of Lemma 55. +(a) First note that we can put the fillers, PL1 : L1 → fr(W1) and PL2 : L2 → +fr(W2), together, to obtain a map, as below, +(38) +PL1,L2 : L1 ×Y I L2 → lim + + + + + + + + + + + + + +X +M1 +p1 +� +p′ +1 +� Y +M2 +p2 +� +p′ +2 +� Z +XI +sX +� +tX +� +Y I +sY +� +tY +� +ZI +sZ +� +tZ +� +X +N1 +q1 +� +q′ +1 +� Y +N2 +q2 +� +q′ +2 +� Z + + + + + + + + + + + + + +. +The fact that both the fillers, PL1 and PL2, are fibrations, together +with the universal property of pullbacks, then gives that PL1,L2 is a +fibration. +(b) We also have a naturally defined map, +(39) +Pout : lim + + + + + + + + + + + + + +X +M1 +p1 +� +p′ +1 +� Y +M2 +p2 +� +p′ +2 +� Z +XI +sX +� +tX +� +Y I +sY +� +tY +� +ZI +sZ +� +tZ +� +X +N1 +q1 +� +q′ +1 +� Y +N2 +q2 +� +q′ +2 +� Z + + + + + + + + + + + + + +−→ lim + + + + + + + + + + + + + +X +M1 ×Y M2 +p1 +� +p′ +2 +� Z +XI +sX +� +tX +� +ZI +sZ +� +tZ +� +X +N1 ×Y N2 +q1 +� +q′ +2 +� Z + + + + + + + + + + + + + +. +From the fact that ⟨sY , tY ⟩: Y I → Y × Y is a fibration, it follows that +Pout is a fibration. +(c) We thus have that PL1,L2 = Pout ◦ PL1,L2 is a fibration, for it is given +by the composition of fibrations. + +A CATEGORIFICATION OF QUINN’S TQFT +101 +We now need to analyse the composite window, W1#0W2, via its spatial 2-slices, + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + , +with, of course, x ∈ X, z ∈ Z, (m1, m2) ∈ M1 ×Y M2 and (n1, n2) ∈ N1 ×Y +N2. Taking this apart a bit, just listing elementary properties, we get, with the +conventions as in (34): +x = p1(m1), z = p′ +2(m2), and there is some y = p′ +1(m1) = p2(m2) ∈ Y ; +x = q1(n1), z = q′ +2(n2), and there is some y′ = q′ +1(n1) = q2(n2) ∈ Y . +We note that the two elements, y and y′, in Y , need not be the same here. However +note that +m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′} and n2 ∈ {y′|N2|z}. +Moreover, if ℓ = (ℓ1, ℓ2) ∈ + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + +, then it is required to satisfy: +– P(ℓ) = (m1, m2); +– Q(ℓ) = (n1, n2); +– ℓ = (ℓ1, ℓ2) ∈ L1 ×Y I L2, so r1(ℓ1) = l2(ℓ2), which is in Y I, of course, and +so it is some path, γ : [0, 1] → Y ; see diagram (34). +– Referring again to that diagram, the path γ = r1(ℓ1) starts at sY +� +r1(ℓ1) +� += +p′ +1(P1(ℓ1)) = p′ +1(m1) = p2(m2) = y, and, similarly, it ends at q2(n2) = y′. +Summarising this for future reference, we have: +Lemma 137. If we have +ℓ = (ℓ1, ℓ2) ∈ + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + , +then r1(ℓ1) is a path in Y , from y := p2(m2) to y′ := q2(n2), and so y and y′ will +be in the same path component of Y . +□ +The discussion and results above have several important consequences. +a) If m2 ∈ M2 and n2 ∈ N2 are such that p2(m2) and q2(n2) are in different +path components of Y , then the spatial 2-slice, + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + , +is empty. +b) If the spatial 2-slice, + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + , +is non-empty, then it has the same homotopy type as a spatial 2-slice, + + +(m1, m2) +x +L1 ×Y I L2 +z +(n′ +1, n′ +2) + + , +having m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n′ +1 ∈ {x|N1|y} and n′ +2 ∈ {y|N2|z}, +i.e., with the same y in both expressions. +More precisely, + +A CATEGORIFICATION OF QUINN’S TQFT +102 +c) using the notation of Lemma 137, if y and y′ are in the same path-component +of Y , we can always find a representative, (n′ +1, n′ +2), of the path-component, +PC(n1,n2)({x|N1 ×Y N2|z}), with n′ +1 ∈ {x|N1|y} and n′ +2 ∈ {y|N2|z}. (This +uses the last point of Lemma 55, and applying the homotopy lifting prop- +erty of the fibration, +{x|N1 ×Y N2|z} → Y, +to a path connecting y′ to y.) Since +PC(n1,n2)({x|N1 ×Y N2|z}) = PC(n′ +1,n′ +2)({x|N1 ×Y N2|z}), +and using Lemma 124, we then have + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + ∼= + + +(m1, m2) +x +L1 ×Y I L2 +z +(n′ +1, n′ +2) + + . +With that choice, namely that we take ‘the same y’, as in the above, there is an +action of the space, Ωy(Y ), of loops of the pair, (Y, y), on the corresponding spatial +2-slice. We have: +Lemma 138. Let x ∈ X, y ∈ Y and z ∈ Z. Let m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, +n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}. The map, ΨL1,L2 : L1 ×Y I L2 → Y I in diagram +(37), induces a fibration, +ΨL1,L2 : + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + −→ Ωy(Y ). +Proof. This follows from the fact that the map, PL1,L2, in (38) is a fibration. +□ +We will use this fact shortly. +5.5.2. The natural transformation associated to the horizontal composition of HF +resolved 2-spans. We assume given a diagram of spaces, HF spans, and HF resolved +2-spans, and their horizontal composite, as shown52, +X +(p1,M1,p′ +1) +� +(q1,N1,q′ +1) +� +⇓ W1 +Y +(p2,M2,p′ +2) +� +(q2,N2,q′ +2) +� +⇓ W2 +Z, +and +X +(p1,M1×Y M2,p′ +2) +� +(q1,N1×Y N2,q′ +2) +� +⇓ W1#0W2 +Z . +Let xX ⊂ X, yY ⊂ Y and zZ ⊂ Z be subsets. Definition 130 gives natural +transformations of profunctors, therefore 2-morphisms in vProfGrphf, +2H +W1 +(xX,yY ) : H +M1 +(xX,yY ) =⇒ H +N1 +(xX,yY ), +2H +W2 +(yY ,zY ) : H +M2 +(yY ,zY ) =⇒ H +N2 +(yY ,zZ), +and +2H +W1#W2 +(xX,zZ) : H +M1×Y M2 +(xX,zZ) +=⇒ H +N1×Y N2 +(xX,zZ) , +where, in more notational detail, H +M1 +(xX,yY ) : π1(X, xX) ↛ π1(Y, yY ), etc. +52The underlying windows, W1 and W2, are as in (34), and their horizontal composite is as +in (36). + +A CATEGORIFICATION OF QUINN’S TQFT +103 +We will now prove the important fact that, with the assumption that (Y, yY ) is +0-connected, and noting Proposition 116, the natural transformation arising from +W1#0W2 is obtained by horizontally composing the natural transformations given +by W1 and W2. We leave the reader to explore what might happen when the +condition on (Y, yY ) does not hold. +A crucial fact that we use is the conclusion of Lemma 138, i.e., that +ΨL1,L2 : + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + → Ωy(Y ), +is a fibration of HF spaces, (of course, subject to the conditions required there). +This will be used together with Theorem 47. The notation and results in §5.1.3 will +also play a key role. Our conventions on profunctors are laid out in Subsection 2.8. +Proposition 139. Let xX ⊂ X, yY ⊂ Y and zZ ⊂ Z be subsets with (Y, yY ) +0-connected, then +2H +W1#0W2 +(xX,zZ) += 2H +W1 +(xX,yY ) • 2H +W2 +(yY ,zZ), +as natural transformations of Vect-profunctors, +H +M1 +(xX,yY ) • H +M2 +(yY ,zY ) =⇒ H +N1 +(xX,yY ) • H +N2 +(yX,zY ). +Here we have abused notation, and noted that, by Proposition 116, we have +canonical natural isomorphisms, of Vect-profunctors, +H +M1 +(xX,yY )•H +M2 +(yY ,zZ) = +ˆ y∈yY +H +M1 +(xX,yY )(−, y)⊗H +M2 +(yY ,zZ)(y, −) +ηM1,M2 +(xX ,yY ,zZ ) +========⇒ H +M1×Y M2 +(xX,zZ) , +and +H +N1 +(xX,yY )•H +N2 +(yY ,zZ) = +ˆ y∈yY +H +N1 +(xX,yY )(−, y)⊗H +N2 +(yY ,zZ)(y, −) +ηN1,N2 +(xX ,yY ,zZ ) +========⇒ H +N1×Y N2 +(xX,zZ) . +For instance, the natural isomorphism, ηM1,M2 +(xX,yY ,zZ), is such that, if x ∈ xX and +z ∈ zZ, and given any y ∈ yY , m1 ∈ {x|M1|y} and m2 ∈ {y|M2|z}, then the linear +map, +� +ηM1,M2 +(xX,yY ,zZ) +� +(x,z), sends the equivalence class of +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) +to +PC(m1,m2)({x|M1 ×Y M2|z}). +Proof. We will prove that the following diagram of natural transformations com- +mutes53, +ˆ y∈yY +H +M1 +(xX,yY )(−, y) ⊗ H +M2 +(yY ,zZ)(y, −) +2HW1 +(xX ,yY )•2HW2 +(yY ,zZ ) +� +ηM1,M2 +(xX ,yY ,zZ ) +� H +M1×Y M2 +(xX,zZ) +2HW1#0W2 +(xX ,zZ ) +� +ˆ y∈yY +H +N1 +(xX,yY )(−, y) ⊗ H +N2 +(yY ,zZ)(y, −) +ηN1,N2 +(xX ,yY ,zZ ) +� H +N1×Y N2 +(xX,zZ) . +53The argument in the proof gives that the diagram commutes even if (Y, yY ) is not 0- +connected. However in this case the horizontal natural transformations are not necessarily natural +isomorphisms. + +A CATEGORIFICATION OF QUINN’S TQFT +104 +To this end, we prove that, given x ∈ xX and z ∈ zZ, the following diagram of +linear maps commutes, +(40) +� +y∈yY +HM1(x,y) ⊗ HM2(y, z) +proj +� +ˆ y∈yY +H +M1 +(xX,yY )(x, y) ⊗ H +M2 +(yY ,zZ)(y, z) +� +2H +W1 +(xX ,yY )•2H +W2 +(yY ,zZ ) +� +(x,z) +� +� +ηM1,M2 +(xX ,yY ,zZ ) +� +(x,z) +� H +M1×Y M2 +(xX,zZ) (x, z) +� +2H +W1#0W2 +(xX ,zZ ) +� +(x,z) +� +ˆ y∈yY +H +N1 +(xX,yY )(x, y) ⊗ H +N2 +(yY ,zZ)(y, z) � +ηN1,N2 +(xX ,yY ,zZ ) +� +(x,z) +� H +N1×Y N2 +(xX,zZ) (x, z), +where proj is the projection mentioned earlier, in Equation (5) on page 37. +Let y, y′ ∈ yY . Let also m1 ∈ {x|M1|y}, and m2 ∈ {y|M2|z}, and then n1 ∈ +{x|N1|y′}, and n2 ∈ {y′|N2|z}. +We first consider the linear map, +F tr : +� +y∈yY +HM1(x,y) ⊗ HM2(y, z) → H +N1×Y N2 +(xX,zZ) (x, z), +obtained from the path in diagram (40) passing through the top right corner. The +corresponding matrix elements, +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) +��F tr�� PC(n1,n1)(N1 ×Y N2) +� +, +are defined, by Equation (32), to be +(41) +χπ + + + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + + + χπ� +PC(n1,n2)({x|N1 ×Y N2|z}) +� +. +As we noted in part a) of the discussion after Lemma 137, the spatial 2-slices, + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + +are empty if, in the notation from that lemma, y and y′ are not in the same +path component of Y , so if y and y′ are in different components of Y , the matrix +elements of F tr in (41) have value 0. On the other hand, using part c) of that same +discussion, if y and y′ are in the same path-component of Y , we can always find a +representative, (n′ +1, n′ +2), of the path-component, PC(n1,n2)({x|N1 ×Y N2|z}), with +n′ +1 ∈ {x|N1|y} and n′ +2 ∈ {y|N2|z}. Consequently, when the matrix elements in (41) +are non zero, we can suppose that y′ = y. +We therefore let y ∈ yY , m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y} and +n2 ∈ {y|N2|z}, and compute the value of (41) in this case. We use Lemma 138 and +apply Theorem 47 to the fibration, +ΨL1,L2 : + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + → Ωy(Y ). + +A CATEGORIFICATION OF QUINN’S TQFT +105 +Let γ1, . . . , γr, where r = |π1(Y, y))| = |π0(Ωy(Y ))|, be representatives of the +different path-components of Ωy(Y ), which we recall is homotopy finite (Lemma +72). We will suppose, with no loss of generality, that γ1 = consty, the constant +path at y. Using the notation in Definition 123, we have +χπ + + + + +(m1, m2) +x +L1 ×Y I L2 +z +(n1, n2) + + + + = +r +� +i=1 +χπ� +Ψ +−1 +L1,L2(γi) +� +χπ(PCγi(Ωy(Y ))) +(42) += +r +� +i=1 +χπ + + + + +m1 +constx +L1 +γi +n1 + + × + + +m2 +γi +L2 +constz +n2 + + + + χπ(PCγi(Ωy(Y ))) += +r +� +i=1 +χπ + + + + +m1 +constx +L1 +γi +n1 + + + + χπ + + + + +m2 +γi +L2 +constz +n2 + + + + χπ(PCγi(Ωy(Y ))). +Using point (2) of Lemma 127, each term in this sum has form +χπ +�� +m1 +x +L1 +y +ΓN1 +⟨constx,γi⟩(n1) +�� +χπ +�� +m2 +y +L2 +z +ΓN2 +⟨γi,constz⟩(n2) +�� +χπ(PCγi(Ωy(Y ))), +which we will note for future use. +Next we apply Lemma 109 to calculate the next term, that is, +χπ� +PC(n1,n2)({x|N1 ×Y N2|z}) +� +, +in expression (41). We use Lemma 52. Consider the commutative diagram, below, +where the middle diamond is a pullback, +N1×Y N2 +P +� +�♠♠♠♠♠♠♠ +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +q1 +� +q′ +2 +� +N1 +q1 +�rrrrrr +q′ +1 +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +N2 +q2 +�❧❧❧❧❧❧❧❧❧ +q′ +2 +�▼ +▼ +▼ +▼ +▼ +▼ +X +Y +Z . +Put {x|N1} = q−1 +1 (x) and {N2|z} = q′ +2 +−1(z), then q′ +1 : N1 → Y and q2 : N2 → Y +induce fibrations, qr : {N2|z} → Y and ql : {x|N1} → Y ; see Lemma 52. Moreover +we have a pullback diagram, as in the diagram below, where Px,z is the unique map +making the diagram commute, +{x|N1 ×Y N2|z} +Px,z +� +proj1 +�♦♦♦♦♦♦♦♦♦♦♦ +proj2 +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +{x|N1} +ql +�P +P +P +P +P +P +P +P +P +P +P +P +P +{N2|z} +qr +�♥♥♥♥♥♥♥♥♥♥♥♥♥ +Y. +We also have that Px,z is a fibration (see the proof of Lemma 71), and its fibre over +y ∈ Y satisfies +P −1 +x,z(y) ∼= {x|N1|y} × {y|N2|z}. +By using Lemma 109, we have, for n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}, an expres- +sion of form +(43) +χπ� +PC(n1,n2)({x|N1 ×Y N2|z} +� += T {x|N1×Y N2|z} +(n1,n2) +χπ(PCy(Y )) χπ(PCn1({x|N1|y})) χπ(PCn2({y|N2|z})). + +A CATEGORIFICATION OF QUINN’S TQFT +106 +Here T {x|N1×Y N2|z} +(n1,n2) +is the number of path-components of the fibre over y of the +fibration, +P n1,n2 +x,y +: PC(n1,n2)({x|N1 ×Y N2|z}) → PCy(Y ), +obtained by restricting Px,z to PC(n1,n2)({x|N1 ×Y N2|z}). +In other words, by +Lemma 108, T {x|N1×Y N2|z} +(n1,n2) +is the cardinality of the orbit of the path-component, +PC(n1,n2)({x|N1|y} × {y|N2|z}), under the right action, ⊳, of π1(Y, y) on the set +�π0 +� +{x|N1|y} × {y|N2|z} +� ∼= �π0({x|N1|y}) × �π0({y|N2|z}). The latter is the set of +path-components of the fibre of the fibration Px,z : {x|N1 ×Y N2|z} → Y at y, and +the action is as in Lemma 97. (This action is derived in the obvious way from the +product action of π1(Y, y) on �π0({x|N1|y}) × �π0({y|N2|z}.) +Going back to (41), we now put (42) and (43) together. Two more observations +are needed. Recall r = |π0(Ωy(Y ))| = |π1(Y, y)|. +(1) We have homotopy equivalences, for each i ∈ {1, . . . , r}, +PCn1({x|N1|y}) ∼= PCΓN1 +⟨constx,γi⟩(n1)({x|N1|y}), +(44) +and +PCn2({y|N2|z}) ∼= PCΓN2 +⟨γi,constz⟩(n2)({y|N2|z}). +(45) +These are induced by the homotopy equivalences, +ΓN1 +⟨constx,γi⟩ : {x|N1|y} → {x|N1|y} +and +ΓN2 +⟨γi,constz⟩ : {y|N2|z} → {y|N2|z}. +(We are using Lemma 95 here.) +(2) By Lemma 140 below, all path-components, PCγi(Ωy(Y )), of the loop +space, of Y at y, are homotopic. We therefore have +χπ(PCγi(Ωy(Y ))) = χπ(Ωy(Y ))/|π1(Y, y)|, for each i ∈ {1, . . . , r}. +Putting everything together, we have: +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,n1)(N1 ×Y N2) +� += T {x|N1×Y N2|z} +(n1,n2) +χπ(PCy(Y ))χπ(Ωy(Y )) +|π1(Y, y)| +� |π1(Y,y)| +� +i=1 +� +PCm1({x|M1|y})|(2H +W1 +(xX,yY ))(x,y)|PCΓN1 +(constx,γi)(n1)({x|N1|y}) +� +� +PCm2({y|M2|z})| +� +2H +W2 +(yY ,zZ) +� +(y,z)|PCΓN2 +(γi,constz)(n2)({y|N2|z}) +�� +. +By the definition of the right action, ⊳, of π1(Y, y) on the set, +�π0 +� +{x|N1|y} × {y|N2|z} +� ∼= �π0({x|N1|y}) × �π0({y|N2|z}), + +A CATEGORIFICATION OF QUINN’S TQFT +107 +the set of path-components of the fibre of the fibration, Px,z : {x|N1×Y N2|z} → Y , +at y, and from the discussion in §5.1.2, we have +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,n1)(N1 ×Y N2) +� += +T {x|N1×Y N2|z} +(n1,n2) +|π1(Y, y)| +� +� +g∈π1(Y,y) +� +PCm1({x|M1|y})|(2H +W1 +(xX,yY ))(x,y)|PCn1({x|N1|y})⊳g +� +� +PCm2({y|M2|z})| +� +2H +W2 +(yY ,zZ) +� +(y,z)|PCn2({y|N2|z}) ⊳ g +�� +. +Note that we also have used that, by Lemma 72, χπ(PCy(Y ))χπ(Ωy(Y )) = 1. +We now let T {x|N1×Y N2|z} +(n1,n2) +denote the orbit of the element below, +� +PCn1({x|N1|y}, PCn2({y|N2|z}) +� +∈ �π0({x|N1|y}) × �π0({y|N2|z}), +under the right action of π1(Y, y) on the set, �π0({x|N1|y} × {y|N2|z}), and thus on +�π0({x|N1|y}) × �π0({y|N2|z}). (Recall again that this is the set of path-components +of the fibre of the fibration, Px,z : {x|N1 ×Y N2|z} → Y at y.) By Lemma 108, +|T {x|N1×Y N2|z} +(n1,n2) +| = T {x|N1×Y N2|z} +(n1,n2) +. +We let S{x|N1×Y N2|z} +(n1,n2) +denote the cardinality of the stabiliser of the same element, +� +PCn1({x|N1|y}, PCn2({y|N2|z}) +� +. +Using the elementary fact that, if a finite group, G, acts on a set, then given any +pairs of elements, k and l, in the same orbit, the cardinality of {g ∈ G : k ⊳ g = l} +is that of the stabiliser subgroup of k, then, on applying this to the case of G being +π1(Y, y), and, invoking the orbit-stabiliser theorem, we have, firstly, that +S{x|N1×Y N2|z} +(n1,n2) +T {x|N1×Y N2|z} +(n1,n2) += |π1(Y, y)|, +and thus +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,n1)(N1 ×Y N2) +� += +S{x|N1×Y N2|z} +(n1,n2) +T {x|N1×Y N2|z} +(n1,n2) +|π1(Y, y)| +× +� � +PCm1({x|M1|y}) | +� +2H +W1 +(xX,yY ) +� +(x,y) | PCn′ +1({x|N1|y}) +� +� +PCm2({y|M2|z}) | +� +2H +W2 +(yZ,zZ) +� +(y,z) | PCn′ +2({y|N2|z}) +� += +� � +PCm1({x|M1|y})| +� +2H +W1 +(xX,yY ) +� +(x,y)|PCn′ +1({x|N1|y}) +� +� +PCm2({y|M2|z})| +� +2H +W2 +(yZ,zZ) +� +(y,z)|PCn′ +2({y|N2|z}) +� +, +where each sum is indexed by the set of elements of form +� +PCn′ +1({x|N1|y}), PCn′ +2({y|N2|z}) +� +in T {x|N1×Y N2|z} +(n1,n2) +. +Recall that x ∈ xX, z ∈ zZ, and that we took y, y′ ∈ yY , where y and y′ are +in the same path-component in Y (so without loss of generality we take y = y′), +and also m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′}, and n2 ∈ {y′|N2|z}. The +latter formula gives exactly the corresponding matrix element, +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) +��F bl�� PC(n1,n1)(N1 ×Y N2) +� +, + +A CATEGORIFICATION OF QUINN’S TQFT +108 +of the linear map associated to the path in (40) passing through the bottom left +corner. This follows from (7), because if x ∈ xX and z ∈ zZ, the linear bijection, +ˆ y∈yY +H +N1 +(xX,yY )(x, y) ⊗ H +N2 +(yY ,zZ)(y, z) +� +ηN1,N2 +(xX ,yY ,zZ ) +� +(x,z) +� H +N1×Y N2 +(xX,zZ) (x, z), +is such that given y ∈ yY , n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}, it sends the equivalence +class of +PCn1({x|N1|y}) ⊗ PCn2({y|N2|z}) ∈ +� +y∈yY +H +N1 +(xX,yY )(x, y) ⊗ H +N2 +(yY ,zZ)(y, z) +to PC(n1,n2)({x|N1 ×Y N2|z}). Lemma 37 in §2.8.3 is useful to translate between +the categorical and the combinatorial languages. +Now suppose that y, y′ ∈ yY are not in the same path-component in Y . +If +m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′}, and n2 ∈ {y′|N2|z}, we already +saw that +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) +��F tr�� PC(n1,n1)(N1 ×Y N2) +� += 0. +Applying the second point of Lemma 37, it also follows that, +� +PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) +��F bl�� PC(n1,n1)(N1 ×Y N2) +� += 0. +Therefore, diagram (40) commutes as required. +□ +In the proof, we used the following lemma. +Lemma 140. Let Y be any CGWH space54. All path components of Ωy(Y ) = {γ ∈ +Y I : sY (γ) = tY (γ) = y} are homotopic. +Proof. Let Py = {γ ∈ Y I : sY (γ) = y}. This is a path-connected space. We have +a fibration, ty : Py → Y , induced by tY : Y I → Y . Clearly Ωy(Y ) is the fibre of +ty at y. Hence all path-components of Ωy(Y ) are homotopy equivalent. (This uses +Lemma 95.) +□ +5.6. The vertical composition of HF resolved 2-spans. Checking that hori- +zontal composition translates via the profunctor construction to horizontal compo- +sition of the corresponding natural transformations required some delicate count- +ing arguments, the corresponding checks for the vertical composition require other +methods. +5.6.1. Preliminaries for the vertical composition of HF resolved 2-spans. Let X be +a CGWH space. By Example 51, there is a fibrant span, (sX, XI, tX): X → X, +from which we constructed the identity of X in the category HFspan. +The composite, (see Definition 56), (sX, XI, tX) • (sX, XI, tX): X → X is the +fibrant span, (sX, XI ×X XI, tY ): X → X. As before, +XI ×X XI = {(γ, γ′) ∈ XI × XI | γ(1) = γ′(0)}, +and we recall that sX(γ, γ′) = γ(0) and tX(γ, γ′) = γ′(1). +We consider the homeomorphism, FX : XI ×X XI → XI, defined as +(46) +FX(γ, γ′)(t) = +� +γ(2t), +t ∈ [0, 1/2], +γ′(2t − 1), +t ∈ [1/2, 1]. +54which need not be HF + +A CATEGORIFICATION OF QUINN’S TQFT +109 +Clearly FX makes the diagram below commute, +XI×XXI +FX +� +sX +�✉✉✉✉✉✉✉✉✉✉ +tX +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +X +X. +XI +sX +�■■■■■■■■■■ +tX +�t +t +t +t +t +t +t +t +t +t +We thus have that FX is an isomorphism (of fibrations) over X×X, or, equivalently, +of fibrant spans. +5.6.2. The vertical composition of HF resolved 2-spans. Let X and Y be HF spaces. +Consider a diagram of fibrant HF resolved 2-spans as shown below, +⇓ W2 +X +(p2,M2,p′ +2) +� +(p1,M1,p′ +1) +� +(q1,N1,q′ +1) +�Y. +⇓ W1 +Here we have HF fibrant resolved 2-spans of the form: +W2 : +� +(p2, M2, p′ +2): X → Y +� +=⇒ +� +(p1, M1, p′ +1): X → Y +� +, +and +W1 : +� +(p1, M1, p′ +1): X → Y +� +=⇒ +� +(q1, N1, q′ +1): X → Y +� +. +Explicitly the windows, W1 and W2, have the form: +(47) +W1 = +X +M1 +p1 +� +p′ +1 +� Y +XI +sX +� +tX +� +L1 +l1 +� +r1 +� +P1 +� +Q1 +� +Y I +sY +� +tY +� +X +N1 +q1 +� +q′ +1 +� Y +and W2 = +X +M2 +p2 +� +p′ +2 +� Y +XI +sX +� +tX +� +L2 +l2 +� +r2 +� +P2 +� +Q2 +� +Y I +sY +� +tY +� +X +M1 +p1 +� +p′ +1 +� Y . +We want to construct a vertical composite, W2#1W1, which should be a HF +resolved 2-span, such that W2#1W1 : (p2, M2, p′ +2) =⇒ (q1, N1, q′ +1). We do this in +two steps. +The first step is exactly as when we constructed the horizontal composition of +HF fibrant resolved 2-spans. Namely, we perform the obvious pullback along the +common horizontal spans of W1 and W2. This yields the following HF fibrant + +A CATEGORIFICATION OF QUINN’S TQFT +110 +window, +W2#′ +1W1 = + + + + + + + + + + + + + +X +M2 +p2 +� +p′ +2 +� Y +XI ×X XI +sX +� +tX +� +L2 ×M1 L1 +l1 +� +r2 +� +P2 +� +Q1 +� +Y I ×Y Y I +sY +� +tY +� +X +N1 +q1 +� +q′ +1 +� Y + + + + + + + + + + + + + +. +Here, given (ℓ2, ℓ1) ∈ L2 ×M1 L2, we have written +l1 +� +(ℓ2, ℓ1) +� += +� +l2(ℓ2), l1(ℓ1) +� +, +r2 +� +(ℓ2, ℓ1) +� += +� +r2(ℓ2), r1(ℓ1) +� +, +and also, +P2 +� +(ℓ2, ℓ1) +� += P2(ℓ2), +Q1 +� +(ℓ2, ℓ1) +� += Q1(ℓ1). +To prove that W2#′ +1W1 is a HF fibrant window, we can use the same argument +that we used for the horizontal composition of HF resolved 2-spans; see §5.5.1. +We now need to ‘adjust’ the left and right vertical spans of W2#′ +1W1. To this +end, we use the homeomorphisms, below, of fibrant spans; as in §5.6.1, +XI×XXI +FX +� +sX +�✉✉✉✉✉✉✉✉✉✉ +tX +�■ +■ +■ +■ +■ +■ +■ +■ +■ +■ +X +X +XI +sX +�■■■■■■■■■■ +tX +�✉ +✉ +✉ +✉ +✉ +✉ +✉ +✉ +✉ +✉ +and +Y I×Y Y I +FY +� +sY +�✈✈✈✈✈✈✈✈✈✈ +tY +�❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +❍ +Y +Y, +Y I +sY +�❍❍❍❍❍❍❍❍❍❍ +tY +�✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +✈ +and the commutative diagram, +X +M2 +p2 +� +p′ +2 +� Y +XI +sX +�r +r +r +r +r +r +r +r +r +r +r +r +tY +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +XI ×X XI +sX +� +tX +� +FX +� +L2 ×M1 L1 +l1 +� +r2 +� +P2 +� +Q1 +� +Y I ×Y Y I +sY +� +tY +� +FY +� Y I. +sY +�▲▲▲▲▲▲▲▲▲▲▲▲ +tY +�rrrrrrrrrrrr +X +N1 +q1 +� +q′ +1 +� Y +This yields what will be called the vertical composite of the fibrant resolved 2-spans, +W2 and W1, as displayed below, +W2#1W1 := +X +M2 +p2 +� +p′ +2 +� Y +XI +sX +� +tX +� +L2 ×M1 L1 +FX◦l1 +� +FY ◦r2 +� +P2 +� +Q1 +� +Y I +sY +� +tY +� +X +N1 +q1 +� +q′ +1 +� Y. +By construction, the window, W2#1W1, is a HF fibrant resolved 2-span such that +W2#1W1 : +� +(p2, M2, p′ +2): X → Y +� +=⇒ +� +(q1, N1, q′ +1): X → Y +� +. + +A CATEGORIFICATION OF QUINN’S TQFT +111 +Given a HF resolved 2-span, +W = (lX, P, L, Q, rY ): +� +(p, M, p′): X → Y +� +=⇒ +� +(q, N, q′): X → Y +� +, +and (x, y) ∈ X × Y, we have its vertical HF span of slices at x and y, as defined in +Definition 126, which idenoted +[x|W|y] = +� +P W +x,y, [x|L|y], QW +x,y +� +: {x|M|y} → {x|N|y} += + + + +[x|L|y] +P W +x,y +�♦♦♦♦♦ +QW +x,y +�◆ +◆ +◆ +◆ +◆ +{x|M|y} +{x|N|y} + + + . +The following simple lemma will be essential later on. We continue to use the +notation introduced earlier in this section and are using the composition of HF +fibrant spans as in Definition 56. +Lemma 141. There exists an isomorphism55 of HF spans, from {x|M2|y} to +{x|N1|y} +[x|W2#1W1|y] ∼= [x|W2|y] • [x|W1|y]. +Proof. This follows from the fact that the concatenation of two paths, a +γ−→ a and +a +γ′ +−→ a, in any space, see (46), is a constant path at a if, and only if, γ, γ′ = +consta. +□ +As before, suppose that we have HF spaces, X and Y , and vertically composable +HF fibrant resolved 2-spans, +W2 : +� +(p2, M2, p′ +2): X → Y +� +=⇒ +� +(p1, M1, p′ +1): X → Y +� +, +W1 : +� +(p1, M1, p′ +1): X → Y +� +=⇒ +� +(q1, N1, q′ +1): X → Y +� +. +Let xX and yY be subsets of X and Y . Consider the corresponding Vect-profunctors, +H +M2 +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, +H +M1 +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, +and +H +N1 +(xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect. +The HF fibrant resolved 2-spans W2 and W1 give rise to natural transformations +of Vect-profunctors, as in Definition 130. +Lemma 142. The composite of the natural transformations, +H +M2 +(xX,yY ) +2HW2 +(xX ,yY ) � H +M1 +(xX,yY ) +2HW1 +(xX ,yY ) � H +N1 +(xX,yY ) , +is +H +M2 +(xX,yY ) +2H +W2#1W1 +(xX ,yY ) +� H +N1 +(xX,yY ) . +55See Definition 57. + +A CATEGORIFICATION OF QUINN’S TQFT +112 +Proof. Let x ∈ xX and y ∈ yY . We claim that the composite of the linear maps, +H +M2 +(xX,yY )(x, y) +� +2HW2 +(xX ,yY ) +� +(x,y) +� H +M1 +(xX,yY ) +� +2HW1 +(xX ,yY ) +� +(x,y) +� H +N1 +(xX,yY ) , +is +H +M2 +(xX,yY ) +� +2H +W2#1W1 +(xX ,yY ) +� +(x,y) +� H +N1 +(xX,yY ) . +This follows by combining Lemma 141 with the s = 0 case of Lemma 71, and using +Definition 130 and Equation (32). +□ +5.7. Towards horizontal and vertical identities. We still have to examine if +the suggested compositions, both horizontal and vertical, have identities. This is +needed, also, to partially answer the query left over from §5.2.3. +5.7.1. The vertical identity. Let (p, M, q): X → Y be an HF fibrant span. +We +define the following window56, +id2 +(p,M,q) := + + + + + + + + + + + + +X +M +p +� +q +� Y +XI +sX +� +tX +� +M I +lX +� +rY +� +sM +� +tM +� +Y I +sY +� +tY +� +X +M +p +� +q +� Y + + + + + + + + + + + + +. +Here, given γ : I → M, we put lX(γ) = p ◦ γ and rX(γ) = q ◦ γ. +This definition is motivated by the construction of the bicategory 2Cob(n,n+1,n+2), +below in Subsection 6.1. In particular, the diagram above is a function space coun- +terpart of the vertical identity of a cobordism, as given in item (7) on page 120. +Remark 143. We do not know whether id2 +(p,M,q) is, in general, a fibrant window +or not. Whenever id2 +(p,M,q) is a fibrant window (which holds in all cases required in +the construction of the once-extended Quinn TQFT, in Section 6), we note that it +will be a HF fibrant resolved 2-span, connecting (p, M, q) to itself. This is because +M I, XI and Y I are all HF, as they are homotopic to M, X and Y , respectively. +Lemma 144. Let (p, M, q): X → Y be a HF fibrant span. Suppose that id2 +(p,M,q) +is a HF fibrant resolved 2-span, therefore connecting (p, M, q) to itself. +In this +situation, given any subsets xX ⊆ X and yY ⊆ Y , the natural transformation, +2H +id2 +(p,M,q) +(xX,yY ) : H +M +(xX,yY ) =⇒ H +M +(xX,yY ), +is the identity natural transformation. +Proof. Let x ∈ xX and y ∈ yY . If m, n ∈ {x|M|y}, then +� +PCm({x|M|y}) | (2H +id2 +(p,M,q) +(xX,yY ))(x,y) | PCn({x|M|y}) +� += χπ + + + + +m +x +M I +y +n + + + + χπ� +PCn({x|M|y}) +� +. +56As usual, given a path, γ, in a space, X, sX(γ) = γ(0) and tX(γ) = γ(1), and the similarly +for Y , etc. + +A CATEGORIFICATION OF QUINN’S TQFT +113 +Now note that, by Example 51 and Remark 68, we have an HF fibrant span, +{x|M|y}I +s{x|M|y} +�❣❣❣❣❣ +t{x|M|y}�❲ +❲ +❲ +❲ +❲ +{x|M|y} +{x|M|y}, +and that + + +m +x +M I +y +n + + = +� +s{x|M|y}, t{x|M|y} +�−1(m, n), +so we only need to apply the first part of Lemma 73, for the case s = 0. +□ +5.7.2. Horizontal identities and unitors. Let X be a HF space and consider the HF +fibrant span, (sX, XI, tX): X → X. Let xX ⊆ X. +The associated Vect-profunctor, +H +XI +(xX,xX) : π1(X, xX)op × π1(X, xX) → Vect, +is such that, given x ∈ xX and y ∈ xX, +(x, y) +H +XI +(xX ,xX ) +�−−−−−−−−−−→ Lin +� +�π0({x|XI|y}) +� ∼= Lin +� +homπ1(X,xX)(x, y) +� +. +A holonomy map, λXI, see §5.1.1, for the fibration, ⟨sX, tX⟩: XI → X × X, can +be constructed so that, given paths in X, γ : x → y, γl : x′ → x and γr : y → y′, +ΓXI +(γl,γr)(γ) = γl ∗ γ ∗ γr, +the concatenation of three paths, each fitting into a third of [0, 1]. In particular, this +means that, as a morphism, from x′ to y′, in π1(X, xX), (applying the comments +just after Definition 111), +HXI +(xX,xX)([γl], [γr])([γ]) = [γl][γ][γr]. +The profunctor associated with the identity span, (sX, XI, tX): X → X, is, there- +fore, canonically isomorphic to the horizontal identity, Idπ1(X,xX), of π1(X, xX), in +the bicategory vProfGrp; see Example 32. +Continuing this approach, we now discuss a type of “would be” unitor for HF +fibrant spans, given by certain HF fibrant resolved 2-spans, and also how the bona +fide unitors in the bicategory vProfGrp can be obtained from the former by +computing the associated natural transformations of profunctors. +This will be +crucial for constructing the once-extended Quinn TQFT in Section 6. We will only +discuss left unitors, as the case of right unitors is analogous. +Let X and Z be HF spaces. Consider a HF fibrant span, (p, M, q): X → Z. We +suppose, and this will be the case in all settings that we need for constructing the +once-extended Quinn TQFT in Section 6, that the following conditions, (1) – (3), +are satisfied. (These conditions may seem a bit mysterious at this stage. However, +as we will see later, they arise naturally from the construction of the unitors of a +cobordism in the bicategory 2Cob(n,n+1,n+2), when looking at their function space +counterparts; see Subsection 6.1, especially item (9), starting on page 120.) +(1) We have a homeomorphism, Φ: XI ×X M → M, making the diagram +(48) +M +p +�✐✐✐✐✐✐✐✐✐ +q +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +X +Z. +XI×XM +Φ +� +p′ +�❚❚❚❚❚❚ +q′ +�❥ +❥ +❥ +❥ +❥ +❥ +commute. + +A CATEGORIFICATION OF QUINN’S TQFT +114 +(As we will see in the following section, this homeomorphism is an ana- +logue of a collar of the boundary of a manifold, when considering spaces +of functions on manifolds.) Here we used the pullback diamond inside the +commutative diagram +XI×XM +�❤❤❤❤❤ +�❱ +❱ +❱ +❱ +❱ +q′ +� +p′ +� +XI +sX +�❥❥❥❥❥❥ +tX +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +M +p +�❣❣❣❣❣❣❣❣❣ +q +�❚ +❚ +❚ +❚ +❚ +❚ +X +X +Z. +(2) Let x, x′ ∈ X and z ∈ Z. Given a path, x +γ−→ x′, in X and m ∈ {x|M|z}, +then Φ(γ, m) ∈ {x′|M|z} is in the same path-component as ΓM +⟨γ,constz⟩(m) ∈ +{x′|M|z}. Here, recalling the notation in Lemma 94, +ΓM +⟨γ,constz⟩ : {x|M|z} → {x′|M|z} +is defined from the fibration ⟨p, q⟩: M → X × Z. +(3) The following window is fibrant (and further note that all spaces appearing +are HF), +λ(p,M,q) +X +:= + + + + + + + + + + + + +X +XI ×X M +p′ +� +q′ +� Z +XI +sX +� +tX +� +M I +lX +� +rZ +� +Φ−1◦sM +� +tM +� +ZI +sZ +� +tZ +� +X +M +p +� +q +� Z + + + + + + + + + + + + +. +As before, lX(γ) = p ◦ γ and rZ(γ) = q ◦ γ, if γ ∈ M I. +Lemma 145. Let X and Z be HF spaces. Let a HF fibrant span, (p, M, q): X → Z, +satisfy the conditions, (1) – (3), just outlined. Suppose xX ⊂ X and zZ ⊂ Z, and +that (X, xX) is 0-connected. +Given x, x′ ∈ xX, γ ∈ {x|XI|x′}, z ∈ zZ, and m ∈ {x′|M|z}, m′ ∈ {x|M|z}, +then +� +PC(γ,m) +�� +x|XI ×X M|z +�� ��� +� +2H +λ(p,M,q) +X +(xX,zZ) +� +(x,z) +��� PCm′� +{x|M|z} +�� += +� +1, if PCΓM +⟨γ,constz⟩(m) +� +{x|M|z} +� += PCm′� +{x|M|z} +� +, +0, otherwise. +In particular, the natural transformation, 2H +λ(p,M,q) +X +(xX,zZ), of profunctors gives the ap- +propriate left-unitor, +λ +HM +(xX ,zZ ) +π1(X,xX), +for π1(X, xX), in the bicategory vProfGrp. More precisely, the composite of the +natural transformations of profunctors, below, from π1(X, xX) to π1(Z, zZ), +Idπ1(X,xX) • H +M +(xX,zZ) +∼ += +=⇒ H +XI +(xX,xX) • H +M +(xX,zZ) +ηXI ,M +(xX ,xX ,zZ ) +========⇒ H +XI×XM +(xX,zZ) +2H +λ(p,M,q) +X +(xX ,zZ ) +========⇒ H +M +(xX,zZ), + +A CATEGORIFICATION OF QUINN’S TQFT +115 +is +Idπ1(X,xX) • H +M +(xX,zZ) +λ +HM +(xX ,zZ ) +π1(X,xX ) +=======⇒ H +M +(xX,zZ). +We note that here the first equivalence is discussed earlier in this section, and +the second is in Lemma 116. +Proof. The proof of the first statement is exactly as in the proof of Lemma 144. +The second statement follows by passing to the language of profunctors. +□ +5.8. Comment and Summary. In this short summary and commentary, we ask +again Is H a bifunctor? as in §5.2.3, and add Is it symmetric monoidal? There +have been a lot of fairly technical results in this section and it is easy to lose track +of what they say in toto, so we will step back to look at why they were necessary +in the form we gave, using the above questions as a guide. +The answer is in two parts: (i) it is almost, but not quite, and then (ii) from the +point of view of the composite ‘constructions’ that we will examining in the next +section, the lack of that property does not make any difference. +For H to be a (symmetric monoidal) bifunctor, an elementary prerequisite would +be that it had a symmetric monoidal bicategory as its domain. In this section, we +have constructed a bicategorical type of object, though not quite a bicategory, that +we will from now on denote by 2span(HF), following the description starting in +page 6 of the Introduction. The objects of 2span(HF) are homotopy finite spaces. +Given homotopy finite spaces, X and Y , the 1-cells, from X to Y , are homotopy +finite fibrant spans, (p, M, q): X → Y . We have a non-associative composition, •, +of 1-cells, obtained via the obvious pullback, see Lemma 55 and Definition 56. Each +homotopy finite space X has a ‘horizontal identity’, given by the path-space fibrant +span, (sX, XI, tX): X → X. +Given 1-cells (p, M, p′), (q, N, q′): X → Y , the 2-cells, in 2span(HF), connecting +them, consist of homotopy finite resolved 2-spans (see §5.3.3), as below, +W = (lX, P, L, Q, rY ): +� +(p, M, p′): X → Y +� +=⇒ +� +(q, N, q′): X → Y +� +, +or, in full, +W = + + + + + + + + + + + + + +X +M +p +� +p′ +� Y +XI +sX +� +tX +� +L +lX +� +rY +� +P +� +Q +� +Y I +sY +� +tY +� +X +N +q +� +q′ +� Y + + + + + + + + + + + + + +. +Again, homotopy finite resolved 2-spans can be composed horizontally and ver- +tically, as described in detail in §5.5.1 and §5.6.2. None of these compositions is +associative. (The vertical composition can be made associative by considering fi- +brant resolved 2-spans, up to the equivalence relation in Definition 122, similarly +to the construction of the category HFspan in Subsection 3.2. We will not develop +this further, as we do not need it here.) +As discussed in §5.7.1 and §5.7.2, if we apply certain restrictions on the 1-cells +(p, M, p′): X → Y that we allow (which will be automatically satisfied in the cases +arising in the construction of the once-extended Quinn TQFT, in Section 6), we +then have ‘vertical identities’, +id(p,M,p′) : +� +(p, M, p′): X → Y +� +=⇒ +� +(p, M, p′): X → Y +� +, + +A CATEGORIFICATION OF QUINN’S TQFT +116 +as well as ‘unitor 2-cells’, whenever a 1-cell comes equipped with the function space +analogue of a collar neighbourbood of the boundary of a manifold, +ρ(p,M,q) +X +: (X +(p,M,p′) +−−−−−→ Y ) • (Y +(sY ,Y I,tY ) +−−−−−−−→ Y ) =⇒ (X +(p,M,p′) +−−−−−→ Y ), +λ(p,M,q) +X +: (X +(sX,XI,tX) +−−−−−−−→ X) • (X +(p,M,p′) +−−−−−→ Y ) =⇒ (X +(p,M,p′) +−−−−−→ Y ). +It is quite possible that, by considering instead equivalence classes of homotopy +finite resolved 2-spans, we could, in this way, obtain a bicategory from 2span(HF), +categorifying the category HFspan, but this will not be needed for this paper. Similar +constructions are in [59, 91] and in [117, 118]. +Throughout Section 5, we constructed an ‘assignment’, from now on denoted +H = +� +π1(−, −), H, 2H +� +: 2span(HF) → vProfGrphf, +more precisely a map of 2-truncated globular sets, that gives the following. +(1) Each homotopy finite space, X, is sent to its fundamental groupoid, π1(X, X). +(2) Given a homotopy finite fibrant span, X +(p,M,p′) +−−−−−→ Y , we have a Vect- +profunctor, as defined in §5.2.1, +H +� +X +(p,M,p′) +−−−−−→ Y +� +: π1(X, X)op × π1(Y, Y ) → Vect. +(3) Given a homotopy finite fibrant resolved 2-span, W: (p, M, p′) =⇒ (q, N, q′), +as above, we have a natural transformation, of functors π1(X, X)op × +π1(Y, Y ) → Vect, as discussed in §5.4.2, +2HW : H +� +X +(p,M,p′) +−−−−−→ Y +� +=⇒ H +� +X +(q,N,q′) +−−−−−→ Y +� +. +It follows from the sequence of results in Section 5 that the assignment, +H: 2span(HF) → vProfGrphf, +preserves all various compositions, and the horizontal identities in 2span(HF) and +in vProfGrphf, up to applying appropriate natural isomorphisms, and that vertical +identities, and unitors likewise are preserved by H, whenever they exist. Therefore +H is as close to being a bifunctor as it can be. +There is a relative variant of 2span(HF), from now on denoted 2span(HF), +where homotopy finite spaces, X, come equipped with subsets, xX ⊂ X, such +that (X, xX) is 0-connected (meaning that xX has at least one point in each path- +component of X), and the rest of the ‘bicategorical’ structure of 2span(HF) is +induced by that of 2span(HF). We also saw in this section that H can be modified +to give a ‘bifunctor’, +H: 2span(HF) → vProfGrphf, +that gives the following. +(1) Each pair, (X, xX), is sent to the corresponding fundamental groupoid, +π1(X, xX). (We will, in the following section, furthermore suppose that xX +is finite, so given that X is homotopy finite, it follows that π1(X, xX) is +then a finite groupoid.) +(2) Given a homotopy finite fibrant span, X +(p,M,p′) +−−−−−→ Y , xX ⊂ X, and yY ⊂ Y , +we thus have a 1-cell, (p, M, p′): (X, xX) → (Y, yY ), in 2span(HF), and a +Vect-profunctor, as defined in Notation 113, +H(xX,yY ) +� +X +(p,M,p′) +−−−−−→ Y +� +: π1(X, xX)op × π1(Y, yY ) → Vect, + +A CATEGORIFICATION OF QUINN’S TQFT +117 +obtained by restricting H +� +X +(p,M,p′) +−−−−−→ Y +� +: π1(X, X)op × π1(Y, Y ) → Vect +to the subgroupoid π1(X, xX)op × π1(Y, yY ). +(3) Finally, if we are given xX ⊂ X and yY ⊂ Y , and a 2-cell in 2span(HF), +W: (p, M, p′) =⇒ (q, N, q′), we have 2-cell in 2span(HF), +W: +� +(p, M, p′): (X, xX) → (Y, yY ) +� +=⇒ +� +(q, N, q′): (X, xX) → (Y, yY ) +� +. +By using Definition 130, we, then, have a natural transformation of pro- +functors, +2H +W +(xX,yY ) : H(xX,yY ) +� +X +(p,M,p′) +−−−−−→ Y +� +=⇒ H(xX,yY ) +� +X +(q,N,q′) +−−−−−→ Y +� +, +obtained by restricting 2HW : H +� +X +(p,M,p′) +−−−−−→ Y +� +=⇒ H +� +X +(q,N,q′) +−−−−−→ Y +� +, +to π1(X, xX)op × π1(Y, yY ). +In this section, we also proved that, just as for H: 2span(HF) → vProfGrphf, +the relative version +H: 2span(HF) → vProfGrphf, +preserves all the various compositions, plus the unitors and identities (when they +exist) up to natural isomorphisms. The most challenging calculation concerned the +fact that the natural transformations 2H +W +(−,−) send the horizontal composition of +fibrant resolved 2-spans of homotopy finite spaces to that of natural isomorphisms +between profunctors, which was dealt with in §5.5.2. +We also saw in §5.4.3, buiding from lemmas 114 and 115, that H, and similarly +H, takes the product / cartesian monoidal structure in 2span(HF) to the tensor +product in vProfGrphf. This will be discussed further in Subsection 6.6, where it +will be furthermore written down in the language of symmetric monoidal bifunctors, +as in Definition 30. +In order to define our once-extended versions of Quinn’s TQFT, we will only need +to use 2span(HF) as a ‘half-way house’ between the bicategory, 2Cob(n,n+1,n+2), +of 2-cobordisms, that we will introduce in detail in the next section, and vProfGrphf, +the second part of this process being given by H. In the symmetric monoidal bi- +category, vProfGrphf, the images of our possible problem ‘equations’, that do not +hold in 2span(HF), are satisfied. In other words, in defining H, and similarly H, +and checking that it preserves the horizontal and vertical compositions and identi- +ties, we have that the composite assignment will be a bifunctor, and, later on, will +similarly have that it is symmetric monoidal. As usual, however, we are getting a +bit ahead of ourselves and do need to define and study the once-extended TQFT +and its variants in some more detail. What we can claim to know for certain at +this stage of the paper is that the resulting basic form of that extended TQFT will +be a bifunctor. More will be revealed shortly. +6. Once-extended versions of Quinn’s TQFT +Let X be a space. +In this section, we will frequently abbreviate ιX +k : X → +X × [0, 1] to ιk, hence ιk(x) = ιX +k (x) = (x, k), usually for k = 0 or k = 1. +6.1. Conventions for the bicategory 2Cob(n,n+1,n+2). Let n be a non-negative +integer. The bicategory, 2Cob(n,n+1,n+2), is that of closed smooth n-manifolds, +(n + 1)-cobordisms between manifolds, and equivalence classes of (n + 2)-extended +cobordisms connecting (n + 1)-cobordisms. The details of the construction are in +[106, §3.1.2] and [91]. Some other important definitions are here in Subsection 2.6, + +A CATEGORIFICATION OF QUINN’S TQFT +118 +and we will give an overview in what follows, so as to set out the conventions we +will be using57. +The bicategory, 2Cob(n,n+1,n+2), is thus defined as follows. +(1) The class of objects of 2Cob(n,n+1,n+2) is the class of all smooth closed +(i.e. compact and with empty boundary) n-dimensional manifolds. +(2) Given closed smooth n-manifolds, Σ and Σ′, a 1-morphism, +(i, S, j): Σ → Σ′, +is a cospan, in the category of smooth manifolds and smooth maps, as +below, and will be called a (n + 1)-cobordism, +Σ +i +�■ +■ +■ +■ +■ +■ +Σ′ +j +�tttttt +S +. +This should be such that S is a compact smooth (n + 1)-manifold, possibly +with a non-empty boundary, and the universally defined map, +⟨i, j⟩: Σ ⊔ Σ′ → S, +gives a diffeomorphism, Σ ⊔ Σ′ ∼= ∂(S), the boundary of S. +(3) The composition of the 1-morphisms (i, S, j): Σ → Σ′ and (i′, S′, j′): Σ′ → +Σ′′, denoted (i, S • S′, j′) := (i, S ⊔Σ S′, j′), is given by considering the +pushout, in CGWH, included as the diamond in the commutative diagram, +Σ +i +� +i +�❉ +❉ +❉ +❉ +❉ +❉ +❉ +Σ′ +j +�sssssssss +i′ +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +Σ′′. +j′ +� +j′ +�✇✇✇✇✇✇✇ +S +k +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +S′ +k′ +�ssssssss +S ⊔Σ′ S′ +(As already mentioned, in this paper, we implicitly choose a natural reali- +sation for all limits and colimits. In this case, we took the obvious choice, +S ⊔Σ′ S′ = +� +(S × {0}) ∪ (S′ × {1}) +� +/j(s) ∼ i′(s), for all s ∈ Σ′, with the +quotient topology.) +We note that the pushout is formed in CGWH, so initially we forget the +smooth structure on the given manifolds, and consider just their underlying +structure as topological spaces. The smooth structure on S ⊔Σ′ S′ is then +inserted afterwards. +As recalled in Subsection 4.1, in order to give a smooth structure to +S ⊔Σ S′, we could, for instance, consider collars of Σ′ in S and S′. How- +ever the collars are not part of the structure given here to cobordisms. +This issue can be resolved as in [106, §3.1.2 and §3.2], either by consid- +ering “halations”, where collars essentially become part of the cobordism +information, or applying the axiom of choice for classes, to endow each +cobordism with appropriate collars. We will not say more on this issue +(and essentially will ignore it when we come to compose extended cobor- +disms, below). We can safely do this as our constructions depend only on +the underlying topological manifolds of the smooth manifolds. +(4) Given closed smooth n-manifolds, Σ1 and Σ2, and cobordisms, +(i1, S, i2): Σ1 → Σ2 +and +(i′ +1, S′, i′ +2): Σ1 → Σ2, +57As in previous sections, we make no assumption that orientations exist on the manifolds, +cobordisms, nor now on the extended cobordisms. + +A CATEGORIFICATION OF QUINN’S TQFT +119 +the 2-morphisms, +[K]: +� +(i1, S, i2): Σ1 → Σ2 +� +=⇒ +� +(i′ +1, S′, i′ +2): Σ1 → Σ2 +� +, +between them, are given by equivalence classes of diagrams, in the category +of manifolds and smooth maps, of the form (49) below, called (n + 2)- +extended cobordisms, +K: +� +(i1, S, i2): Σ1 → Σ2 +� +=⇒ +� +(i′ +1, S′, i′ +2): Σ1 → Σ2 +� +, +(49) +K = + + + + + + + + + + + + + +Σ1 +i1 +� +ιΣ1 +0 +� +S +iS +� +Σ2 +i2 +� +ιΣ2 +0 +� +Σ1 × I +iE +� K +Σ2 × I +iW +� +Σ1 +ιΣ1 +1 +� +i′ +1 +� S′ +iS′ +� +Σ2 +i′ +2 +� +ιΣ2 +1 +� + + + + + + + + + + + + + +. +Here K is a compact smooth (n + 2)-manifold with corners, called the +support of K. (The E on the middle right pointing map is there to indicate +that the arrow is ‘pointing’ east in the diagram, and the W, similarly, is +pointing west.) +Dually to the ideas of windows and fibrant resolved 2-spans, see Defini- +tion 117 and §5.3.3, the frame, fr(K), of an extended cobordism, K, as in +(49), is defined to be +(50) +fr(K) := colim + + + + + + + + + + + + + +Σ1 +i1 +� +ιΣ1 +0 +� +S +Σ2 +i2 +� +ιΣ2 +0 +� +Σ1 × I +Σ2 × I +Σ1 +ιΣ1 +1 +� +i′ +1 +� S′ +Σ2 +i′ +2 +� +ιΣ2 +1 +� + + + + + + + + + + + + + +. +We have a canonically defined map, fK : fr(K) → K, as we had for HF +resolved 2-spans, hence similarly called the filler of K. It is required that +fK provides a diffeomorphism fr(K) ∼= ∂(K), the boundary of the manifold +with corners, K; cf., page 21. +(5) Two extended cobordisms, +K, K′ : +� +(i1, S, i2): Σ1 → Σ2 +� +=⇒ +� +(i′ +1, S′, i′ +2): Σ1 → Σ2 +� +, +so with the same frame, are called equivalent if there exists a diffeomor- +phism, f : K → K′, between the supports of K and K′, that makes the +diagram below commute, +fr(K) +id +� +fK +� K +f +� +fr(K′) +fK′ +� K′. +(6) The horizontal and vertical compositions of extended cobordisms are done +via the obvious horizontal and vertical pushouts, dually to the case of HF +resolved 2-spans, as treated in §5.5.1 and §5.6.2; see also [106, §3.1.2] and +[91, 92]. (As already mentioned, we are omitting the details on how to + +A CATEGORIFICATION OF QUINN’S TQFT +120 +construct smooth structures on the resulting topological manifolds. +We +will skip these, and refer to [106, §3.1.2] for details. +Our constructions +of once-extended Quinn TQFTs do not take into account smooth struc- +tures on manifolds, but we note that the construction of the bicategory +2Cob(n,n+1,n+2) does make use of their existence.) These two compositions +of extended cobordisms descend to their equivalence classes, which defines +the horizontal and vertical compositions of 2-morphisms in 2Cob(n,n+1,n+2). +(7) Given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, its vertical identity is the +equivalence class of the extended cobordism, +id2 +(i1,S,i2) := + + + + + + + + + + + + +Σ1 +i1 +� +ιΣ1 +0 +� +S +ιS +0 +� +Σ2 +i2 +� +ιΣ2 +0 +� +Σ1 × I +iE +� S × I +Σ2 × I +iW +� +Σ1 +ιΣ1 +1 +� +i1 +� S +ιS +1 +� +Σ2 +i2 +� +ιΣ2 +1 +� + + + + + + + + + + + + +. +Here iE(s, t) = (i1(s), t), for s ∈ Σ1 and t ∈ I, and, similarly, iW (s, t) = +(i2(s), t). +(8) Given a smooth compact n-manifold, Σ, the horizontal identity of Σ is +id1 +Σ := (ιΣ +0 , Σ × I, ιΣ +1 ): Σ → Σ. +(9) Given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, we have left and right +unitor (n + 2)-extended cobordisms, +λ′(i1,S,i2) +Σ1 +: (ιΣ1 +0 , Σ1 × I, ιΣ1 +1 ) • (i1, S, i2) =⇒ (i1, S, i2), +and +ρ′(i1,S,i2) +Σ2 +: (i1, S, i2) • (ιΣ2 +0 , Σ2 × I, ιΣ2 +1 ) =⇒ (i1, S, i2). +The support of both is S × I. We will explain the construction of the left +unitor extended cobordism, λ′(i1,S,i2) +Σ1 +. The construction of the right unitor +extended cobordism is similar. +Consider the (n + 1)-cobordism, +� +i′ +1, (Σ1 × I) ⊔Σ1 S, i′ +2 +� += (ιΣ1 +0 , Σ1 × I, ιΣ1 +1 ) • (i1, S, i2), +and also an explicit isomorphism of cospans, +S +Φ +� +Σ1 +i1 +�♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +i′ +1 +�▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +▼ +Σ2 +i2 +�◆◆◆◆◆◆◆◆◆◆◆◆◆ +i′ +2 +�qqqqqqqqqqq +(Σ1 × I) ⊔Σ1 S. +(Note that to construct such a homeomorphism, Φ: S → (Σ1 × I) ⊔Σ1 S, +we need a collar for the inclusion of Σ1 in S.) The left unitor extended +cobordism is defined by tweaking the vertical identity of (i1, S, i2): Σ1 → + +A CATEGORIFICATION OF QUINN’S TQFT +121 +Σ2, as shown in the diagram58, +λ′(i1,S,i2) +Σ1 +:= + + + + + + + + + + + + + +Σ1 +i′ +1 � +ιΣ1 +0 +� +(Σ1 × I) ⊔Σ1 S +ιS +0 ◦Φ−1 +� +Σ2 +i′ +2 +� +ιΣ2 +0 +� +Σ1 × I +iE +� S × I +Σ2 × I +iW +� +Σ1 +ιΣ1 +1 +� +i1 +� S +ιS +1 +� +Σ2 +i2 +� +ιΣ2 +1 +� + + + + + + + + + + + + + +. +The equivalence classes of the left and right unitor extended cobordisms +give the left and right unitors in 2Cob(n,n+1,n+2), as denoted below, +λ(i1,S,i2) +Σ1 += [λ′(i1,S,i2) +Σ1 +]: (ιΣ1 +0 , Σ1 × I, ιΣ1 +1 ) • (i1, S, i2) =⇒ (i1, S, i2), +and +ρ(i1,S,i2) +Σ2 += [ρ′(i1,S,i2) +Σ2 +]: (i1, S, i2) • (ιΣ2 +0 , Σ2 × I, ιΣ2 +1 ) =⇒ (i1, S, i2). +In addition to the above basic structure, we note that, in the classical setting, the +category, Cob(n,n+1), has the structure of a symmetric monoidal category with the +coproduct / disjoint union, ⊔, as the tensor product, as recalled in §4.1.1, and that +the extended form, 2Cob(n,n+1,n+2), similarly, has a symmetric monoidal bicate- +gory structure, again having ⊔ as its tensor product. This follows from Theorem +2.15 of Carboni, Kelly, Walters and Wood, [35]. An explicit proof is in [106, §3.1.4]. +We will revisit this structure in Subsection 6.5, particularly §6.5.2. +6.2. A once-extended version of Quinn’s TQFT. As before, n will be a non- +negative integer, and B a homotopy finite space. These will be the standard as- +sumptions throughout this section. +Consider an (n, n + 1)-cobordism, between closed smooth n-manifolds, as in +Subsection 4.1, viewed as a cospan in the category CGWH, +(i, S, j) := + + +Σ +i +�■ +■ +■ +■ +■ +■ +Σ′ +j +�tttttt +S + + , +so the nodes only encode the data of the underlying topological manifolds. +Applying the contravariant mapping space functor, B(−) : CGWH → CGWH, +sends this cospan to a span in CGWH, whose nodes contain the corresponding +spaces of maps from the topological manifolds into B, +(i∗, BS, j∗) := + + + +BS +i∗ +�ssssss +j∗ +�▲ +▲ +▲ +▲ +▲ +▲ +BΣ +BΣ′ + + + . +Lemma 146. This span, (i∗, BS, j∗), of function spaces is a fibrant span in which +all the spaces appearing are homotopy finite. +58We will see, later on in section 6.5.2, several instances of this type of construction, which +will be studied for use in defining further structure on 2Cob(n,n+1,n+2), and verifying that that +structure satisfies the required equations. + +A CATEGORIFICATION OF QUINN’S TQFT +122 +Now consider an extended (n, n + 1, n + 2)-cobordism with 2-cospan diagram59 +as follows, +K := + + + + + + + + + + + + + +Σ1 +i1 +� +ι0 +� +S +iN +� +Σ2 +i2 +� +ι0 +� +Σ1 × I +iE +� K +Σ2 × I +iW +� +Σ1 +ι1 +� +i′ +1 +� S′ +iS +� +Σ2 +i′ +2 +� +ι1 +� + + + + + + + + + + + + + +. +Applying the same contravariant functor, B(−), to K gives a dual ‘window’, +BK := + + + + + + + + + + + + + +BΣ1 � +i∗ +1 +� +ι∗ +0 +BS� +i∗ +N +BΣ2 +� +i∗ +2 +� +ι∗ +0 +BΣ1×I � +i∗ +E +BK +BΣ2×I +� +i∗ +W +BΣ1 +� +ι∗ +1 +� +i′∗ +1 +BS′� +i∗ +S +BΣ2 +� +i′∗ +2 +� +ι∗ +1 + + + + + + + + + + + + + +. +Recalling the definition of fibrant resolved 2-spans in §5.3.3, we get the following. +Lemma 147. The window, BK, of mapping spaces is a fibrant resolved 2-span, +in which all the spaces appearing are homotopy finite. Furthermore, the applica- +tion of B(−) preserves the compositions of all cobordisms and extended cobordisms, +sending them to the corresponding compositions of fibrant spans, as in Definition +56, and of fibrant resolved 2-spans, as in §5.5.1 and §5.6.2. The vertical units in +2Cob(n,n+1,n+2), as well as the horizontal identities, and unitors, are also sent to +those of §5.7.1 and §5.7.2, when passing to the mapping spaces. +We note that CGWH is cartesian closed, so BΣ1×I ∼= (BΣ1)I, canonically. +Proof. We will prove Lemmas 146 and 147 together as the proofs are related. It +may be useful to compare with the proof of our earlier Lemma 82. We will continue +to refer to the mapping space picture as being ‘dual’ to the other one. +The fibrancy of the dual span follows from the fact that the inclusion of Σ⊔Σ′ ∼= +∂(S) into S is a cofibration and, similarly, for the dual window, the inclusion of +fK : fr(K) ∼= ∂(K) into K is a cofibration, so the dual map, f ∗ +K : BK → Bfr(K) ∼= +fr(BK), is a fibration. (In that last step, we again used the fact that CGWH +is cartesian closed, so the mapping space contravariant functor, B( ) : CGWH → +CGWH, send colimits to limits.) For the latter reason, all compositions are pre- +served (up to isomorphism) when going from cobordisms and extended cobordisms +to fibrant spans and fibrant resolved 2-spans. +In order to prove that all spaces in BK are homotopy finite, we use the fact that +all compact smooth manifolds can be given the structure of a finite CW-complex, +and use, once again, Lemma 79. +Vertical units, horizontal units, and unitors, are preserved by construction. +□ +We also note the following, that once again follows from the fact that CGWH +is cartesian closed. +59As before this is a diagram in CGWH, and this will be the same for all subsequent diagrams. + +A CATEGORIFICATION OF QUINN’S TQFT +123 +Lemma 148. Given two manifolds, Σ1 and Σ2, in CGWH, we have a natural +isomorphism, +BΣ1⊔Σ2 ∼= BΣ1 × BΣ2, +and this is true also for 1- and 2-cobordisms. +□ +This interprets as saying that ‘taking the mapping space’ converts the ⊔-monoidal +structure of CGWH to the − × − one. +Recall, now, the construction of the bicategory, vProfGrphf, defined in Subsec- +tion 2.8, particularly §2.8.5. +Definition 149 (The once-extended Quinn TQFT). The once-extended Quinn +TQFT, denoted +2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, +is defined to be the bifunctor given by: +• if Σ is a closed n-manifold, then 2Q0 +B(Σ) := π1(BΣ, BΣ), which defines 2QB +on objects; +• if (i, S, j): Σ → Σ′ is an (n + 1)-cobordism, then: +2Q1 +B +� +Σ +(i,S,j) +−−−−→ Σ′ +� +:= H +� +BΣ +(i∗,BS,j∗) +−−−−−−−→ BΣ′� +: π1(BΣ, BΣ) ↛ π1(BΣ′, BΣ′), +where we are using the notation, H, from Subsection 5.2, particularly Definition +111; +and +• the equivalence class of an extended cobordism, as in Equation (49), +K: +� +(i1, S, i2): Σ1 → Σ2 +� +=⇒ +� +(i′ +1, S′, i′ +2): Σ1 → Σ2 +� +, +is sent to the natural transformation of functors, +π1(BΣ1, BΣ1)op × π1(BΣ2, BΣ2) → Vect, +defined as (using the construction in §5.4.2, particularly Lemma 129), +2Q2 +B([K]) := 2HBK : H +� +BΣ1 +(i∗ +1,BS,i∗ +2) +−−−−−−→ BΣ2 +� +=⇒ H +� +BΣ1 +(i′∗ +1 ,BS′,i′∗ +2 ) +−−−−−−−−→ BΣ2 +� +. +It should perhaps be noted that the name we have used here needs justifying. We +have not as yet shown that the structure outline above does give a once extended +TQFT as that will require a proof that the bifunctor is symmetric monoidal. That +will be shown later (see Theorem 176 in Subsection 6.6). +From the constructions60 in Section 5, combined with the previous lemmas, it +follows that we do indeed have a bifunctor, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf. +The fact that 2QB preserves the composition of cobordisms is in Proposition 116, +that 2QB preserves the horizontal composition of extended cobordisms follows +from Proposition 139, and that 2QB preserves the vertical composition of extended +cobordisms is dealt with by Lemma 142. Preservation of vertical identities follows +from Lemma 144. Finally, preservation of horizontal identities and unitors follows +from the discussion in §5.7.2, particularly Lemma 145. +We note also that if Σ is a smooth closed manifold, then the groupoid, 2Q0 +B(Σ) = +π1(BΣ, BΣ), is homotopy finite. This follows since the function space, BΣ, is homo- +topy finite (Lemma 79) and so, given a pair of objects, f, f ′ : Σ → B, of π1(BΣ, BΣ), +the set of arrows from f to f ′, in π1(BΣ, BΣ), is finite. +60It may be useful to refer to the summary of Section 5 in Subsection 5.8. + +A CATEGORIFICATION OF QUINN’S TQFT +124 +Remark 150. Let 2Cob′(n,n+1,n+2) be obtained from 2Cob(n,n+1,n+2), by consid- +ering the 2-cells to be extended cobordisms, therefore not considering the latter to be +up to equivalence. Because the vertical composition of extended cobordisms is not +associative, 2Cob′(n,n+1,n+2) is then not a bicategory. However, we still have com- +positions, units and unitors, so 2Cob′(n,n+1,n+2) is, similarly to 2span(HF), a +2-truncated cubical set with compositions, see Subsection 5.8. Moreover, the map- +ping space construction B−, in Lemma 147, gives rise to a map of 2-truncated +globular sets, +B(−) : 2Cob′(n,n+1,n+2) → 2span(HF), +which preserves all compositions, units and unitors up to isomorphism. The once- +extended Quinn TQFT arises from the composite of assignments, below, using the +notation of Subsection 5.8, +2Cob′(n,n+1,n+2) +B(−) +−−−→ 2span(HF) +H +−→ vProfGrphf. +In the remark below, we use, from items (13) and (15) in the discussion on page +16, that, if X is a CGHW space, and x ∈ X, then the path-component in X to +which x belongs, will be denoted PCx(X), and that we denote the set of all such +path-components by �π0(X) = {PCx(X) : x ∈ X}. (Note, again, that different +x ∈ X may induce the same PCx(X).) We also note the definition of the homotopy +content, χπ(B), of a homotopy finite space B, given in Definition 43. +Remark 151. We can give a more explicit description of 2QB. +On objects, 2QB sends a closed n-manifold, Σ, to the fundamental groupoid, +π(BΣ, BΣ), of the function space. +Given a cobordism, (i1, S, i2): Σ1 → Σ2, then, if f1 : Σ1 → B and f2 : Σ2 → B +are continuous functions, we have +2Q1 +B(i1, S, i2)(f1, f2) = Lin +� +�π0({f1|BS|f2}) +� +, +where Lin: Set → Vect is the free vector space61 functor, and {f1|BS|f2}, in full +{f1|B(i∗ +1,S,i∗ +2)|f2}, is the space of maps, H : S → B, such that the diagram, +B +Σ1 +f1 +�❧ +❧ +❧ +❧ +❧ +❧ +i1 +�❙ +❙ +❙ +❙ +❙ +❙ +Σ2. +f2 +�❙❙❙❙❙❙ +i2 +�❦❦❦❦❦❦ +S +H +� +, +commutes, given with the induced CGWH topology. +Given paths, γ1 : f ′ +1 → f1 in BΣ1 and γ2 : f2 → f ′ +2 in BΣ2, the linear map, +2Q1 +B(i1, S, i2)(f ′ +1 +[γ1] +−−→ f1, f2 +[γ2] +−−→ f ′ +2) : Lin +� +�π0({f1|BS|f2}) +� +→ Lin +� +�π0({f ′ +1|BS|f ′ +2}) +� +, +is defined from the functor, +F(BS) : π1(BΣ1 × BΣ2, BΣ1 × BΣ2) → CGWH/ ≃, +obtained from the path-space fibration, +⟨i1, i2⟩∗ : BS → BΣ1 × BΣ2 ∼= BΣ1⊔Σ2, +in the usual way (see [87, Chapter 7], as reviewed in Lemma 96), and then by +applying �π0 : CGWH/ ≃→ Set; finally linearising by applying Lin: Set → Vect. +(Note that the path γ1 : f ′ +1 → f1 must be inverted before applying F(BS).) An explicit +description can be obtained from the comments just after Definition 111. +61Recall we are working over an arbitrary subfield of C. + +A CATEGORIFICATION OF QUINN’S TQFT +125 +Given an extended cobordism, +K := + + + + + + + + + + + + +Σ1 +i1 +� +ι0 +� +S1 +iN +� +Σ2 +i2 +� +ι0 +� +Σ1 × I +iE +� K +Σ2 × I +iW +� +Σ1 +ι1 +� +j1 +� S2 +iS +� +Σ2 +j2 +� +ι1 +� + + + + + + + + + + + + +, +then the natural transformation, of profunctors, +2Q2 +B([K]): 2Q1 +B((i1, S1, i2): Σ1 → Σ2) =⇒ 2Q1 +B((j1, S2, j2): Σ1 → Σ2), +is such that, if f1 : Σ1 → B and f2 : Σ2 → B, given H1 ∈ {f1|BS1|f2} and H2 ∈ +{f1|BS2|f2}, then we have the following formula for the matrix elements, +(51) +� +PCH1 +� +{f1|BS1|f2} +� +| +� +2Q2 +B([K]) +� +(f1,f2) | PCH2 +� +{f1|BS2|f2} +�� += χπ + + + + + + + + + + + +T : K → B +�������� +T ◦ iN = H1, +T ◦ iS = H2 +∀s ∈ Σ1, ∀t ∈ [0, 1] : T (iE(s, t)) = f1(s) +∀s′ ∈ Σ2, ∀t ∈ [0, 1] : T (iW (s′, t)) = f2(s′) + + + + + + + + + + + +χπ� +PCH2 +� +{f1|BS2|f2} +�� +. +(It follows from the construction in Section 5, in particular Lemma 125, that we +are indeed considering homotopy contents only of homotopy finite spaces.) +Note that, unless B is a finite set with the discrete topology, then for Σ, a closed +smooth n-manifold, the groupoid, 2Q0 +B(Σ) = π1(BΣ, BΣ), is uncountable, since +it has an uncountable set of objects. However 2Q0 +B(Σ) is homotopy finite, so we +can extract an equivalent finite subgroupoid from it. Namely, if we choose a finite +subset, f Σ ⊂ BΣ, containing at least one element for each path-component, then +π1(BΣ, f Σ) will be a finite groupoid, equivalent to 2Q0 +B(Σ) = π1(BΣ, BΣ). +This latter fact will be used in the next subsection to construct a finitary version +of the once-extended Quinn TQFT. +6.3. A finitary version of the once-extended Quinn TQFT. For technical +and historical reasons in the applications of the above theory, it is often useful to +replace groupoids that have possibly infinitely many objects, by more finitary, but +equivalent, ones. There are several useful ways of doing this, for instance, using +triangulations of the manifolds, and CW decompositions of B. We will, however, +reduce the size of the groupoids by a different means as follows. +As always, let B be a homotopy finite space, and n be a non-negative integer. +Definition 152 (B-decorated manifold). A B-decorated n-manifold, Σ = (Σ, fΣ), +is given by a closed smooth n-manifold, Σ, called the underlying manifold of Σ, +together with a finite subset, fΣ, of BΣ, containing at least one function, f : Σ → B, +for each path component of the space, BΣ, of functions from Σ to B. +Let Σ be a closed smooth manifold. We recall, [94], that Σ has a finite CW- +decomposition62. Since B is homotopy finite, BΣ is homotopy finite (Lemma 79), +62which can be obtained using various means, + +A CATEGORIFICATION OF QUINN’S TQFT +126 +and hence it has a finite number of path-components. In particular, we can see +that all closed (smooth) manifolds possess a B-decoration. +Definition 153 (2Cob +(n,n+1,n+2) +B +). We define a bicategory, 2Cob +(n,n+1,n+2) +B +, as +follows. +• The objects are B-decorated n-manifolds. +• Given B-decorated n-manifolds, Σ = (Σ, f Σ) and Σ′ = (Σ′, fΣ′), the 1- +morphisms, +(i, S, j): Σ → Σ′, +are given by (n+ 1)-cobordisms, (i, S, j): Σ → Σ′, with no additional struc- +ture on S. (The (n + 1)-cobordism, (i, S, j): Σ → Σ′, associated to a 1- +morphism (i, S, j): Σ → Σ′, will be called the underlying (n+1)-cobordism +of that 1-morphism.) +• Given 1-morphisms, (i, S, j), (i′, S′, j′): Σ → Σ′, the 2-morphisms, +[K]: +� +(i, S, j): Σ → Σ′� +=⇒ +� +(i′, S′, j′): Σ → Σ′� +, +are given by equivalence classes of extended cobordisms, +K: +� +(i, S, j): Σ → Σ′� +=⇒ +� +(i′, S′, j′): Σ → Σ′� +, +with the equivalence relation as in 2Cob(n,n+1,n+2). (As for 1-morphisms, +we define the underlying 2-morphism of 2Cob(n,n+1,n+2), associated to such +a 2-morphism of 2Cob +(n,n+1,n+2) +B +, by forgetting the B-decoration on the n- +manifolds Σ and Σ′.) +The rest of the bicategory structure in 2Cob +(n,n+1,n+2) +B +is induced, in the obvious +way, by that of the undecorated case, i.e. by composing the underlying cobordisms +or underlying 2-morphisms of the 1- and 2-morphisms in 2Cob +(n,n+1,n+2) +B +. For +instance, the composition below, +(Σ, f Σ) +(i,S,j) +−−−−→ (Σ′, fΣ′) +(i′,S′,j′) +−−−−−→ (Σ′′, f Σ′′) +simply gives +(Σ, f Σ) +(i,S•S′,j′) +−−−−−−−→ (Σ′′, fΣ′′), +where (i, S • S′, j′) gives the composite of cobordisms as in item (3) on page 118. +For convenience, we recall that the bicategory, vProfGrpfin, defined in §2.8.5, +is the sub-bicategory of vProfGrp, whose objects are the finite groupoids. +Given a B-decorated manifold, Σ = (Σ, f Σ), the pair (BΣ, fΣ) is, by definition, +0-connected. From Lemmas 116 and 139, it follows that the bifunctor, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +induces a bifunctor, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrphf, +by restricting from π1(BΣ, BΣ) to π1(BΣ, f Σ), and leaving the rest of the structure +unaltered. We will give this structure in more detail shortly. +Since we assume that Σ is a closed smooth manifold, as above, it follows that BΣ +is homotopy finite, and, thus, that the groupoid π1(BΣ, f Σ) is finite. This leads to +the following definition in which we are again using the notation of Definition 130 +and from Subsection 5.2, in particular, from notational comment 113. + +A CATEGORIFICATION OF QUINN’S TQFT +127 +Definition 154 (The finitary once-extended Quinn TQFT). The finitary once- +extended Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +is the bifunctor defined as follows. +• If Σ = (Σ, f Σ) is a B-decorated n-manifold, then +2Q +0 +B(Σ) := π1(BΣ, fΣ). +(This is a finite groupoid as f Σ is finite, B is a HF space, and thus so is +BΣ.) +• If +(i, S, j): Σ = (Σ, f Σ) → Σ′ = (Σ′, f Σ′) +is a 1-morphism of 2Cob(n,n+1,n+2), whose underlying cobordism is +(i, S, j): Σ → Σ′ = +� +Σ +i +�❘ +❘ +❘ +❘ +❘ +❘ +Σ′ +j +�❧❧❧❧❧❧ +S +� +, +then +2Q +1 +B +� +(Σ, f Σ) +(i,S,j) +−−−−→ (Σ′, fΣ′) +� +:= H(f Σ,f Σ′) +� +BΣ +(i∗,BS,j∗) +−−−−−−−→ BΣ′� +: π1(BΣ, fΣ) ↛ π1(BΣ′, fΣ′) +abbr. += H(f Σ,fΣ′ ) +� +(i∗, BS, j∗) +� +: π1(BΣ, f Σ) ↛ π1(BΣ′, fΣ′). +Concretely the functor, +2Q +1 +B +� +(Σ, f Σ) +(i,S,j) +−−−−→ (Σ′, f Σ′) +� +: π1(BΣ, f Σ)op × π1(BΣ′, fΣ′) → Vect, +is the restriction to π1(BΣ, fΣ)op × π1(BΣ′, f Σ′), of the functor, +2Q1 +B +� +Σ +(i,S,j) +−−−−→ Σ′ +� +: π1(BΣ, BΣ)op × π1(BΣ′, BΣ′) → Vect. +• A 2-morphism, +[K]: +� +(i1, S, i2): (Σ1, fΣ1) → (Σ2, f Σ2) +� +=⇒ +� +(i′ +1, S′, i′ +2): (Σ1, fΣ1) → (Σ2, fΣ2) +� +, +arising as the equivalence class of an extended cobordism, +K: +� +(i1, S, i2): Σ1 → Σ2) +� +=⇒ +� +(i′ +1, S′, i′ +2): (S′ : Σ1 → Σ2) +� +, +is sent to the natural transformation, +2HBK +(f Σ1,fΣ2 ) : H(fΣ1 ,fΣ2 )(i∗ +1, BS, i∗ +2) =⇒ H(f Σ1 ,fΣ2 )(i′∗ +1 , BS′, i′∗ +2 ), +of functors, π1(BΣ1, fΣ1)op × π1(BΣ2, fΣ2) → Vect, as in Definition 130. +As before, we point out that as an extended TQFT is defined to be a symmetric +monoidal bifunctor, this definition will be completed only once we have established +the existence of that structure63. That will take a little time and will be completed +with Theorem 177. +The proof that we have indeed defined a bifunctor 2QB, follows as for the +earlier case of 2QB. As before, the crucial fact that 2QB preserves the compo- +sition of the 1-morphisms and the horizontal compositions of the 2-morphisms of +63and we recall that being a symmetric monoidal bifunctor requires extra structure as it is not +just a property. + +A CATEGORIFICATION OF QUINN’S TQFT +128 +2Cob +(n,n+1,n+2) +B +, follows from Proposition 116 and Proposition 139. It is for these +cases that we need to impose that each pair, (BΣ, fΣ), is 0-connected. +Remark 155 (The dependence on decorations). Let, as before, Σ be a closed +smooth n-manifold. +The finitary once-extended Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +does not give a value to Σ itself, except when Σ is given a B-decoration, f Σ. Given +two B-decorations, f Σ and f +′ +Σ, of Σ, then there exists a canonically defined, and +invertible, profunctor, 2Q +0 +B(Σ, f Σ) ↛ 2Q +0 +B(Σ, f +′ +Σ), defined by +2Q +1 +B +� +(Σ, f Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′ +Σ) +� +: 2Q +0 +B(Σ, f Σ) ↛ 2Q +0 +B(Σ, f +′ +Σ). +These profunctors associated to pairs of B-decorations of Σ are “functorial” in +the following sense. +Consider decorations fΣ, f +′ +Σ and f +′′ +Σ of Σ. We have 1-morphisms in 2Cob +(n,n+1,n+2) +B +, +(Σ, fΣ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′ +Σ), +(Σ, f +′ +Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′′ +Σ), +and the following, which is homeomorphic to their composite, +(Σ, f Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′′ +Σ). +Applying Lemma 116, we have a natural isomorphism of profunctors, +2Q +1 +B +� +(Σ, f Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′ +Σ) +� +• 2Q +1 +B +� +(Σ, f +′ +Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′′ +Σ) +� +=⇒ 2Q +1 +B +� +(Σ, f Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′′ +Σ) +� +. +It can be proved that these natural isomorphisms satisfy appropriate relations when +we consider four different B-decorations of Σ. In particular, this shows that the +profunctor associated to a change of B-decoration is always invertible up to 2- +isomorphism, i.e. is an adjoint equivalence. +Given that we have a bifunctor, 2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, the +profunctors associated to moving from one B-decoration of a manifold to another +one are, furthermore, compatible with the profunctors associated to the 1-morphisms +of 2Cob +(n,n+1,n+2) +B +. +6.4. The Morita valued extended Quinn TQFT. We continue working with +our chosen field, κ. +The finitary theory, as given in the previous section, takes values in a bicat- +egory of Vect-valued profunctors between finite κ-linear categories. To be more +easily able to use more usual representation theoretic methods and ideas, it can be +convenient to replace this bicategory by one that is better know within the rep- +resentation theoretic setting, namely that of finite dimensional algebras (with 1), +bimodules and morphisms between them. +Here we will first review the construction of an algebra from a linear category, +as given by Mitchell, [90], §7, and then look at it in detail for Lin(Γ), the linear +category associated to a (finite) groupoid, Γ, obtained by applying the free vector +space functor to the morphism sets of Γ. In this case, the resulting algebra is the well +known groupoid algebra, [123]. We look, in some detail, at the relationship between +bimodules over a category algebra and profunctors. Some of this is folklore, and +is quite difficult to find explicitly in the literature, yet it seems important for the + +A CATEGORIFICATION OF QUINN’S TQFT +129 +understanding of the relationship between the Prof-valued and the Mor-valued +extended TQFTs. +We will define the Morita bicategory, Mor, or, more exactly, Morκ, (also some- +times denoted Alg2), of algebras, bimodules and the bimodule morphisms, these +latter being often known as intertwiners in a representation theoretic context, and +will examine its relation to vProf. We will then see how to define a Morita valued +extended Quinn TQFT. +6.4.1. The algebra of a small linear category. In section 7, (page 33), of the classic +paper, [90], by Mitchell, it was shown how to associate a ring to a small additive +category. That construction easily extends to a κ-linear version for any κ-linear +category. +Let C be a (small) κ-linear category, having C0 as its set of objects. We set [C] to +be the set of C0 × C0 matrices, c, where, for p, q ∈ C0, the (p, q)-entry, denoted cp,q, +is an element of the vector space, C(p, q), of arrows from p to q, and each row and +column has only a finite number of non-zero entries. Using the addition in each +C(p, q), together with the composition from C, we can give [C] the structure of a +κ-algebra, which will not usually be commutative, nor, in general, unital. +We thus have that, as a vector space, +[C] = +� +p,q∈C0 +C(p, q), +and the multiplication is given by +g · f = +� +g ◦ f +if domain(g) = codomain(f) +0 +otherwise. +Although, in general, [C] will not have a multiplicative identity, each object +p ∈ C0 gives an idempotent matrix, 1p, namely the matrix having the identity +morphism on p in the (p, p)-position and zeroes elsewhere. +If C0 is finite, then � +p∈C0 1p is, however, a multiplicative identity for [C]. The +algebra, [C], is an example of a generalised matrix algebra. This algebra is called +the category algebra of C. +Any element, c = (cp,q), in [C] can be written as a sum of matrices of form cp,q, +where the matrix cp,q is to be zero in all positions except the (p, q) position, where +it is, no surprise, cp,q. This sum is finite. The element, cp,q, clearly has domain +equal to p and codomain equal to q, so, in a completely classical way, the product +c · 1p = � +r,s cr,s · 1p = � +s cp,s, whilst 1q · c = � +r cr,q, so 1q · c · 1p = cp,q, and, +in particular, we have the useful equation: 1q · cp,q · 1p = cp,q, which we will use +shortly. +We will usually be considering this when C is the κ-linearisation of a (usually +finite) groupoid, Γ. If Γ is finite, more generally if Γ has a finite set of objects, then +the resulting ‘groupoid algebra’ will be unital. It has a well known description in +terms of the arrows of Γ, which we include in case it is found easier to understand, +as it is written in a slightly less abstract way; see also [15, 16, 95, 123] and [34] for +various versions and the development of further theory. +This groupoid algebra, Lin(2)(Γ), has as its underlying vector space, Lin(Γ1), +that is the free vector space over the set of morphisms of Γ. It thus comes equipped +with a natural choice of basis. The product in Lin(2)(Γ) is given on generators by +(x +g−→ y)(x′ +g′ +−→ y) := δ(y, x′)(x +gg′ +−−→ y), + +A CATEGORIFICATION OF QUINN’S TQFT +130 +where δ(y, x′) is 1, if the two objects, y and x′, are equal, and is zero otherwise. +The multiplicative unit of Lin(2)(Γ) is given by � +x∈Γ0(x +idx +−−→ x), which makes sense +since Γ0 is finite. +6.4.2. Examples: The following examples are well known, but worth recalling. +Example 156. If Γ is a group, G, thought of as a single object groupoid in the +usual way, and then further thought of as (a basis for) the corresponding κ-linear +category, then Lin(2)(Γ) is just the usual group algebra, κ[G]. It is thus also a Hopf +algebra. +Example 157. Returning to the general case for the moment, if P is a (finite) pre- +ordered set, and C is the linearisation of the corresponding small category, then [C] +is the incidence algebra of the poset. As specific examples, if P = {1 < . . . < n}, +then [C] is the algebra of n × n upper triangular matrices over κ. If we replace +the given preorder by the discrete preorder, so p ≤ q here means p = q, then the +corresponding [C] is the algebra of diagonal matrices. If, on the other hand, we +replace the preorder by the codiscrete preorder (in which p ≤ q for every pair of +elements (p, q)), then [C] = Mn(κ), the full algebra of n × n matrices over the field +κ. +Example 158. (The Quantum Double as a groupoid algebra) +Let G be a finite group, then we know that its group algebra, κ[G], is a Hopf +algebra in a natural way. The Drinfel’d double of that Hopf algebra is a groupoid +algebra for a naturally defined groupoid; see, for example, [123]. +There is an action of G on itself by conjugation, g +a−→ aga−1, and we can form the +action groupoid of this action64. This has the elements of G as its objects and the +arrows have form (g, a) : g → aga−1. This is variously written AUT (G) or G � G +in the literature. We will use the former of these for the moment. Composition +is by group multiplication (in reverse order). We write [AUT (G)] for the groupoid +algebra of the linearisation of AUT (G). This is given in detail in, for instance, [34, +§1.10]. The product on the basis elements, as above, is given by +(g, a)(g′, a′) = δ(aga−1, g′)(g, aa′), +where g, g′, a, a′ ∈ G. +If we define a comultiplication, +∆(x, g) = +� +yz=x +(y, g) ⊗ (z, g), +and a counit, ǫ(x, g) = δ(x, 1G), then, for suitable and quite evident definitions of +an antipode and an R-matrix, the resulting object is a quasi-triangular Hopf algebra. +It is clear, from standard descriptions of the ‘double construction’, that this is D(G) +the Drinfel’d double or quantum double of the Hopf algebra, κ[G]. +It is worth noting that this is the untwisted version. The twisted version is ex- +amined in [123]. Another useful reference giving further insight to this construction +is [34], that we mentioned above. +Example 159. The link between groupoid algebras, for groupoids having finitely +many objects, and Hopf algebra-type structure is stronger than just the previous +example. To discuss this briefly, we will need to give the idea of a weak bialgebra. +This is a κ-module, B, having both an algebra and a coalgebra structure, subject to +some compatibility conditions (which can be found in, for example, [15, 16]). +If Γ is a groupoid having finitely many objects, and we take B = Lin(Γ1), the +‘free vector space’ on the set of arrows of Γ, in the terminology of [16], then the +64We will give a more general form of action groupoid later; see page 217. + +A CATEGORIFICATION OF QUINN’S TQFT +131 +groupoid algebra of Γ has a coproduct, ∆ : B → B × B, given by the diagonal, +∆(f) = f ⊗ f, whilst the counit sends all basic generators to the identity element, +1, of κ. +A weak Hopf algebra, as defined in, for example, [16], is a weak bialgebra equipped +with an antipode map, S : H → H, subject to some axioms generalising the proper- +ties required of an antipode in the definition of Hopf algebra. +The groupoid algebra of a groupoid having finitely many objects has an antipode, +given on the basic generators by S(f) := f −1. Because of this, weak Hopf algebras +are sometimes called quantum groupoids. +Weak Hopf algebras have a nice and well understood representation theory, some +references for which can be found in the paper, [15], already cited. +(For more on this see, for instance, [15, 16, 95].) +As this last example applies to the type of groupoid algebras that we will be +producing from the once-extended Quinn TQFTs, it is particularly promising for +future developments of the theory using the tools from the theory of weak Hopf +algebras / quantum groupoids. +6.4.3. Non-functoriality of [−] and Morita equivalences. This subsection examines +more properties of this situation, but they will not be immediately needed for the +main theme of this paper, so can be left aside on first reading, moving on to section +6.4.4, where the bicategory, Mor is discussed. We would recommend that they be +at least skimmed in a later reading as they provide further insights on the algebraic +mechanisms involved later on. +A functor, F : C → D, does induce a linear map, [F] : [C] → [D], but, in +general, this map will not preserve multiplication, so [−] is not a functor. The +more-or-less minimal example for this is to take C to be the (κ-linearisation of the) +discrete category on the set having just two elements, say a and b, and D to be +the corresponding construction on a singleton set, {c}. Take F : C → D to be the +functor induced by the unique map from {a, b} to {c}. If we write 1a for the identity +on a, etc., then [C] is the direct sum of two copies of κ, but with 1a.1b = 0, since the +composite of a and b in C is not defined. Over in [D], which is, itself, isomorphic +to κ, [F](1a).[F](1b) = 1c.1c = 1c, so certainly [F](1a.1b) ̸= [F](1a).[F](1b). +We will examine this slightly odd situation in a bit more detail shortly, as it +is perfectly manageable given the approach that we are using. In any case, the +following result of Mitchell, [90], Theorem 7.1, makes one realise that there is a lot +of power in the category algebra construction. For the statement, we think of the +category algebra as a linear category having just a single object. +Theorem 160. Suppose C is a linear category having only finitely many objects, +then C and [C] are Morita equivalent categories. +We can expand this as follows, if we define C−Mod to be Funcκ(C, Vect), the +set of κ-linear functors from C to Vect, then C−Mod and [C]−Mod are equivalent +categories. This, thus, says that, as far as linear representations are concerned, C +and [C] are encoding the same information. +We will sketch out a proof of this as it contains some ideas that help one to +understand what is happening here, and hence why the ‘linearisation’ versus ‘cat- +egorification’ process is so useful. +(A full proof is given in [90] on page 34. +A +discussion of the ideas can be found online in the n-Category Caf´e, (May 14, 2014), +in a post, Categories vs. Algebras, by Tom Leinster; see [79].) +First we construct a functor, [−], from C −Mod to [C]−Mod, so suppose that +M : C → Vect is a κ-linear functor. We set +[M] = ⊕q∈C0M(q), + +A CATEGORIFICATION OF QUINN’S TQFT +132 +the direct sum of all the image vector spaces of the functor, M. +This is finite +dimensional if each M(q) is, as C0 is finite. +It has a [C]-module structure in a +natural and fairly obvious way. If c = (cp,q) is an element of [C], then cp,q : p → q +is in C, so M(cp,q) : M(p) → M(q) is a linear map. Now, if m = (mp) is an element +of [M], we define c · m = � +p,q M(cp,q)(mp). The linearity and functoriality of M +ensures that this does give a [C]-module structure to [M]. This is easily seen to +define a functor, [−], as claimed. This forms part of the claimed equivalence. +The other direction, starting from a [C]-module and ending up with a κ-linear +functor from C to Vect, is not so obvious, although it is, in fact, a generalisation +of a well known process from elementary linear algebra. +Suppose N is a [C]-module and that p is an object of C. The element, 1p ∈ [C], +is idempotent, so multiplication by it gives an idempotent linear map from N to +itself. We can thus split N as 1pN ⊕ (1 − 1p)N, and we can repeat this with each +object. We get N ∼= ⊕p∈C01pN. We set � +N(p) to be the summand, 1pN, and show +that this is the object part of the required functor. If cp,q : p → q in C, then, +as before, let cp,q ∈ [C] be the matrix having cp,q in position (p, q). If n ∈ � +N(p), +then 1p · n = n, and cp,q · n = 1q · cp,q · 1p · n, which is in 1qN = � +N(q). We +define � +N(cp,q) : � +N(p) → � +N(q) by � +N(cp,q)(n) = cp,q · n. Linearity of this map is +automatic. The proof that � +N : C → Vect is a functor is fairly routine, as is that of +the functoriality of the construction, � +(−) : [C]−Mod → C−Mod. Finally it should +be fairly clear65 that this is the required quasi-inverse for [−]. +Remark: This proof is a mild categorification of that of the elementary result +that the internal and external descriptions of direct sum amount to the same thing. +We have left ‘to the reader’ the detailed verification that the above constructions +do yield an equivalence between C−Mod and [C]−Mod, as it is fairly routine to give +a direct proof. We will, in fact, investigate that equivalence by a separate route. +For this, we recall that [C], as it is a κ-algebra, can be considered as a κ-linear +category in its own right, namely one having a single object, ∗, and with [C](∗, ∗) +being the set of elements of [C] itself, with composition being the multiplication in +[C]. We will not make any notational distinction between the κ-algebra, [C], and +the linear category, [C], at least where no confusion is likely to arise by so doing. +It is well known that two κ-algebras, R and S, are Morita equivalent if there +are bimodules, RAS and SBR, such that the functors, A ⊗S − and B ⊗R −, form +an adjoint equivalence. What is also clear it that this should generalise to κ-linear +categories and it does. It does, however, seem a bit difficult to find a simple pub- +lished proof of this, as it is a special case of some very wide ranging generalisations, +whose generality we do not need, or, in fact, want here as our aim is to justify and +interpret some calculations in a specific case of that general theory. +It does, however, suggest that we try to find ‘bimodules’, CA[C] and [C]BC, with +similar properties. What are such ‘bimodules’ to be? They are just another name +for Vect-valued profunctors, which, in the case of interest, would give A : C ↛ [C] +and B : [C] ↛ C. This observation, and quite a bit of what follows, is adapted +from the n-Category Caf´e discussion, (May 14, 2014), [79], as mentioned before. +(The ideas, there and here, were largely given by Karol Szumi�lo, but with a few +additional features and verifications added here. We should add that any errors +should be attributed to us, and not to him.) +Before that, however, we will give the pair of profunctors as suggested above. +65i.e. this is ‘left to the reader’. + +A CATEGORIFICATION OF QUINN’S TQFT +133 +Earlier, on and around page 36, we saw that, in Prof or vProf, the identity +profunctor on a category, C, was the double Yoneda embedding, +C(−, −) : Cop × C → Set, +or, of course, with codomain Vect if C is a κ-linear category. We need, here, a +functor, A : Cop × [C] → Vect, and an obvious candidate can be derived from +that Yoneda based functor, by applying the [−]-construction to one side of it. We, +therefore, define a functor, A, as required, by +A(p, ∗) = ⊕q∈C0C(p, q). +We recall that, here, ∗ is the unique object of the κ-linear category, [C]. The formula +is clearly (contravariantly) functorial in p, so it remains to see how some ∗ +c−→ ∗, +acts on A(p, ∗), where c is a matrix, (cr,s), and each cr,s ∈ C(r, s). +Let p be an object of C. Let xp be an element of A(p, ∗). As we wrote before, +given another object q, its q component is some xp,q ∈ C(p, q), and then +(xp · c)p,s = +� +q∈C +xp,q · cq,s. +We want to calculate the composite, C +A +↛ [C] +N +−→ Vect, for a [C]-module, N. +We will write N for both the module, and the functor, N : [C] → Vect, although +we should remember that N is also N(∗), i.e. the functor evaluated on the single +object of the algebra (considered as a linear category). +What should this mean? The composite of a profunctor and a functor? We +can interpret this as being A • ϕN : C ↛ Vect, so giving us a profunctor. (The +notation ϕN is explained in Example 33). That would be a first step, thus we want +to examine the corresponding functor, A • ϕN : Cop × Vect → Vect. We evaluate +it on a pair of objects, (p, V ), with p ∈ C0 and V being a vector space over κ. The +formula for the composition in vProf gives +(A • ϕN)(p, V ) = +ˆ ∗ +A(p, ∗) ⊗ Vect(N, V ), +but we note that the coend is ‘integrating’ over the one object category correspond- +ing to [C], so is just the term being ‘integrated’ divided by the diagonal action of +the algebra, i.e., it is +(⊕q∈C0C(p, q) ⊗ Vect(N, V ))/ ≃, +where the action of any cr,s, which is homogeneous with value cr,s, on the right +hand side, Vect(N, V ), is by the action on N, so if v : N → V , then (cr,s · v)(n) = +v(cr,s · n), whilst on the left hand side, it is by post-composition by cr,s. +Any +element, (cp,q, v) in this direct sum is ≃-equivalent to one of the form (1p, w), by +factoring the cp,q as 1p · cp,q, and then shifting the cp,q across to the other side. +If we do this to 1p itself, we find that (1p, w) ≃ (1p, 1p · w), so is determined by +the restriction of the linear map, w, to the direct summand 1pN, which we have +denoted above by � +N(p), as 1p · w is the composition of w with the projection onto +that direct summand. In other words, +(A • ϕN)(p, V ) ∼= Vect( � +N(p), V ). +It is easy to see that this isomorphism is natural in both N and V , so (A • ϕN) +is a representable profunctor, represented by � +N. +To summarise, the composite +profunctor, C +A +↛ [C] +N +−→ Vect, is ‘really’ the functor � +N, as we hoped. + +A CATEGORIFICATION OF QUINN’S TQFT +134 +We now turn to the profunctor, B : [C] ↛ C, so B : [C]op × C → Vect. Given the +success of the formula for A above, the ‘obvious’ formula for B is +B(∗, q) = ⊕p∈C0C(p, q). +This certainly gives a functor, [C]op × C → Vect, (and thus a profunctor) as hoped +for, and, to get the analogue of our earlier calculation, we will think of a functor, +M : [C] → Vect, as a profunctor, ϕM : [C] ↛ Vect. We have +(B • ϕM)(∗, V ) ∼= +ˆ q +⊕p∈C0C(p, q) ⊗ Vect(M(q), V ), +for q ∈ C0 and a vector space, V . This is isomorphic to ⊕p∈C0Vect(M(p), V ) by +one of the forms of the co-Yoneda lemma, and this, in turn, is Vect([M], V ), up to +isomorphism. These isomorphisms are natural, so (B•ϕM) ∼= ϕ[M]. The composite +profunctor, (B • ϕM), is thus representable, and is ‘really’ [M]. +This sets up the two functors, [−] and � +(−), on the categories of ‘modules’, as +being given by the profunctors A and B, respectively. The final steps to explore, in +this investigation of Theorem 160, are to calculate the composites, A • B : C ↛ C +and B • A : [C] ↛ [C]. These are (slightly careful) manipulations involving the +coend formulation of profunctor composition. +Proposition 161. (i) A • B ∼= C(−, −), the unit profunctor on C. +(ii) B • A ∼= [C](∗, ∗) ∼= [C], the unit profunctor / bimodule on [C]. +Proof. (i) We take p, q ∈ C0, then +A • B(p, q) = +ˆ ∗ +A(p, ∗) ⊗ B(∗, q) += +ˆ ∗ +⊕r∈C0C(p, r) ⊗ ⊕s∈C0C(s, q). +As this coend is over (the single object category) [C], it can be calculated as +(⊕r∈C0C(p, r)) ⊗[C] (⊕s∈C0C(s, q)), +so as a tensor product over (the algebra), [C], then, given the form of the mul- +tiplication in [C], it is clear that this tensor product is isomorphic to the vector +space, C(p, q), and as all the isomorphisms are natural in p and q, we thus have +that A • B ∼= C(−, −), the unit profunctor on C as required. +(ii) This part is easier: +B • A(∗, ∗) = +ˆ q +B(∗, q) ⊗ A(q, ∗) += +ˆ q +⊕p∈C0C(p, q) ⊗ ⊕r∈C0C(q, r) +∼= ⊕p,rC(p, r) +∼= [C], +which is, of course, the same as [C](∗, ∗), as required. +□ +Remark 162. This resolves, at least in part, the problem that we noted earlier, +namely that [−] is not a functor as such, at least in the most obvious sense. Suppose, +however, that F : C → D is a κ-linear functor, then there is an ‘induced’ way to +get from [C] to [D]. It can be given by the composite profunctor, +BC • ϕF • AD : [C] ↛ C ↛ D ↛ [D], +where we have indicated the ‘versions’ of the profunctors, A and B, by adding +suitable suffices, e.g., AD being the A profunctor for D, and so on. As this is a + +A CATEGORIFICATION OF QUINN’S TQFT +135 +profunctor between two single object linear categories, it is ‘just’ a left [C]-, right +[D]-bimodule (determined up to isomorphism). +This, in fact, shows clearly that the bicategory of algebras, bimodules, and bimod- +ule morphisms has some better properties than the category of algebras and algebra +homomorphisms. The original functor induces a bimodule, but, in general, not a +homomorphism, between the two category algebras. We could have got to this in- +duced bimodule without going via the Morita context, but the route we have taken +has some advantages for what we will be needing. +Remark 163. We note that the above construction for a functor extends easily +to handling a profunctor, H : C ↛ D. Any such profunctor can be whiskered by +suitable units and counits, A and B, to give +BC • H • AD : [C] ↛ C ↛ D ↛ [D]. +The discussion that we just gave can be used to prove that we have a bifunctor +from the bicategory of linear categories, Vect-enriched profunctors between them, +and enriched natural transformations, to the bicategory of algebras, bimodules and +bimodule maps. It sends C to [C], and H : C ↛ D to the composite profunctor +above. This construction will be clarified in §6.6.3. We will deal with a particular +case of this latter construction in the following section. +We next turn to the bicategory of algebras, and describe it in a bit more detail, +as it and related structures are the target for our next version of the once-extended +TQFT. +6.4.4. The bicategory, Mor. We next give the detailed definition of the bicategory +of algebras, bimodules, and bimodule morphisms / intertwiners, that we have been +using in a fairly sketchy form for some time. In so doing, we will shift our notation +to put the actions of the algebras on the bimodules into a more central role. +This bicategory is sometimes denoted Alg or Alg2 in the literature, but we will +denote it by Mor, and refer to it as the Morita bicategory, as it is the natural and +classical setting for Morita equivalence, an adjoint equivalence in Mor, in the sense +of bicategory theory, being precisely a classical Morita equivalence. +We follow [10, 63, 106], as well as more classical sources on bicategories. +Definition 164. The Morita bicategory, Mor = Morκ, is the bicategory such +that: +• the objects of Mor are unital κ-algebras; +• given algebras A and B, 1-morphisms A ↛ B are (A, B)-bimodules, M. To +give some more detail, and for a notational recall, M will be a κ-vector space +equipped with a left A-representation / action, ⊲, and a right B-representation, +⊳, that are compatible, meaning that given a ∈ A, b ∈ B and m ∈ M, we have +that (a⊲m) ⊳ b = a⊲(m ⊳ b); +• the 2-morphisms, F : (M : A ↛ B) +=⇒ +(N : A ↛ B), are given by (A, B)- +bimodule maps, F : M → N; +• and, finally, the horizontal composite of a compatible pair of 2-morphisms, +A +M1 +� +N1 +� +⇓ F1 +B +M2 +� +N2 +� +⇓ F2 +C , + +A CATEGORIFICATION OF QUINN’S TQFT +136 +is +A +M1⊗BM2 +� +N1⊗BN2 +� +⇓ F1 ⊗B F2 +C , +where ⊗B is the usual tensor product over the algebra, B. +There are also ’well known’ horizontal units and unitors, completing the con- +struction of the bicategory Mor, whose explicit description is left to the reader. +Recall that vProfGrpfin is the sub-bicategory of vProfGrphf, whose objects +are finite groupoids. The constructions of the previous section (§6.4.3) give a bi- +functor, Lin(2) : vProfGrpfin → Mor. This bifunctor sends: +• each finite groupoid, Γ = (s, t: Γ1 → Γ0, id), to its groupoid algebra, +Lin(2)(Γ), see page 129; +and +• if given groupoids, Γ = (s, t: Γ1 → Γ0, id) and Γ′ = (s, t: Γ′ +1 → Γ′ +0, id), and +a profunctor, H: Γ ↛ Γ′, hence a functor H: Γop ×Γ′ → Vect, we consider +the bimodule, Lin(2)(H), to have underlying vector space, +Lin(2)(H) := +� +x∈Γ0, y∈Γ′ +0 +H(x, y). +For the rest of the bimodule structure on Lin(2)(H), we let a ∈ Γ0 and +b ∈ Γ′ +0. Below, we will not distinguish between an element, v(a,b) ∈ H(a, b), +and its image under the obvious inclusion of H(a, b) into Lin(2)(H). The +left and right actions of the algebras, Lin(2)(Γ) and Lin(2)(Γ′), on Lin(2)(H) +are such that, given v(a,b) ∈ H(a, b), we have, given (x +g−→ y) ∈ Γ1 and +(x′ +g′ +−→ y′) ∈ Γ′ +1, +(x +g−→ y)⊲v(a,b) = +� +H +� +x +g−→ y, b +idb +−−→ b +� +(v(a,b)), +if y = a, +0, +if y ̸= a, +hence, if v(y,b) ∈ H(y, b), then (x +g−→ y)⊲v(y,b) ∈ H(x, b). +For the right action, +v(a,b) ⊳ (x′ +g′ +−→ y′) = +� +H +� +a +ida +−−→ a, x′ +g′ +−→ y′� +(v(a,b)), +if x′ = b, +0, +if x′ ̸= b, +so, if v(a,x′) ∈ H(a, x′), then v(a,x′) ⊳ (x′ +g′ +−→ y′) ∈ H(a, y′). +Remark 165. The bimodule, Lin(2)(H): Lin(2)(Γ) ↛ Lin(2)(Γ′), is an instance +of the general construction mentioned at the end of §6.4.3, namely Lin(2)(H) is +isomorphic to the composite +BC • H • AD : [C] ↛ C ↛ D ↛ [D], +where, here, C is Lin(2)(Γ) and D is Lin(2)(Γ′). This can help when checking, for +instance, preservation, up to invertible 2-morphisms, of horizontal composition for +the candidate bifunctor, Lin(2) : vProfGrpfin → Mor, see below. +Remark 166. Note that if a ∈ Γ0 and b ∈ Γ′ +0, then, for v(a,b) ∈ H(a, b), we have +that +(a +ida +−−→ a)⊲v(a,b) = H +� +a +ida +−−→ a, b +idb +−−→ b +� +(v(a,b)) = v(a,b) +and +v(a,b) ⊳ (b +idb +−−→ b) = H +� +a +ida +−−→ a, b +idb +−−→ n +� +(v(a,b)) = v(a,b). + +A CATEGORIFICATION OF QUINN’S TQFT +137 +The remaining details of the verification that the above construction does give +a bifunctor, Lin(2) : vProfGrpfin → Morκ, will mostly be left to the reader. The +key property that Lin(2) preserves horizontal compositions of 1-morphisms, up to +a canonical natural equivalence, is given by the following lemma. +Lemma 167. Consider finite groupoids, Γ, Γ′, and Γ′′, and profunctors, H: Γ ↛ Γ′ +and H′ : Γ′ ↛ Γ′′. We have a canonical isomorphism of +� +Lin(2)(Γ), Lin(2)(Γ′′) +� +- +bimodules, +I : Lin(2)(H • H′) =⇒ Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H). +Proof. As mentioned above, in Remark 165, this follows from the calculations in the +previous section, and in particular on the properties of the composite profunctors, +A • B and B • A, as given in Lemma 161, but we will give a direct proof, so as +to accustom the reader to the links between profunctor and bimodule composition +arguments. +We first see what happens at the level of underlying vector spaces. Let x ∈ Γ0 +and z ∈ Γ′′ +0, then +(H • H′)(x, z) = +ˆ y∈Γ′ +0 +H(x, y) ⊗ H′(y, z) = +� � +y∈Γ′ +0 +H(x, y) ⊗ H′(y, z) +� +/ ≃ . +Here, fixing x ∈ Γ0 and z ∈ Γ′′ +0, the linear equivalence relation66, ≃, is generated +by, for y, y′ ∈ Γ′ +0, v(x,y) ∈ H(x, y) and v′ +(y′,z) ∈ H(y′, z), and an arrow, y +g−→ y′, in +Γ′ +1, +v(x,y) ⊗ H′(y +g−→ y′, z +1z +−→ z)(v′ +(y′,z)) ≃ H(x +1x +−→ x, y +g−→ y′)(v(x,y)) ⊗ v′ +(y′,z). +The latter relation means exactly that, given y, y′ ∈ Γ′ +0, v(x,y) ∈ H(x, y) and +v′ +(y′,z) ∈ H(y′, z), and an arrow, y +g−→ y′ in Γ′ +1, we have +v(x,y) ⊗ ((y +g−→ y′)⊲v′ +(y′,z)) ≃ (v(x,y) ⊳ (y +g−→ y′)) ⊗ v′ +(y′,z). +We also note that +Lin(2)(H • H′) = +� +x∈Γ0,z∈Γ′′ +0 +(H • H′)(x, z). +On the other hand, we have +Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) = +� +� +x∈Γ0,z∈Γ′′ +0 +� +y,y′∈Γ′ +0 +H(x, y) ⊗ H′(y′, z) +� +/ ∼ . +Here the linear equivalence relation, ∼, is such that, given x ∈ Γ0, z ∈ Γ′′ +0, y, y′ ∈ Γ′ +0, +v(x,y) ∈ H(x, y) and v′ +(y′,z) ∈ H(y′, z), we have +v(x,y) ⊗ ((w +g−→ w′)⊲v′ +(y′,z)) ∼ (v(x,y) ⊳ (w +g−→ w′)) ⊗ v′ +(y′,z), +for arbitrary (w +g−→ w′) ∈ Γ′ +1. +Clearly we have a bimodule map, +I : Lin(2)(H • H′) =⇒ Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H), +sending the equivalence class of +v(x,y) ⊗ v′ +(y,z) ∈ +� +x∈Γ0,z∈Γ′′ +0 +� +y∈Γ′ +0 +H(x, y) ⊗ H′(y, z), +66i.e., an equivalence relation whose quotient is a vector space. + +A CATEGORIFICATION OF QUINN’S TQFT +138 +under ≃, to the equivalence class of +v(x,y) ⊗ v′ +(y,z) ∈ +� +x∈Γ0,z∈Γ′′ +0 +� +y,y′∈Γ′ +0 +H(x, y) ⊗ H′(y′, z), +under ∼, and we claim that I is a bijection. +If y, y′ ∈ Γ′ +0 are not equal, and v(x,y) ∈ H(x, y) and v′ +(y′,z) ∈ H(y′, z), then +v(x,y) ⊗ v′ +(y′,z) ∼ 0. This is because, (on using Remark 166), +v(x,y) ⊗ v′ +(y′,z) = +� +v(x,y) ⊳ (y +1y +−→ y) +� +⊗ v′ +(y′,z) +∼ v(x,y) ⊗ +� +(y +1y +−→ y) +� +⊲v(y′,z) +� += 0. +In particular, I is surjective. +We now define a bimodule map, +I′ : Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) =⇒ Lin(2)(H • H′). +There is a bilinear map, I′′ : Lin(2)(H) × Lin(2)(H′) → Lin(2)(H • H′), given by, if +x ∈ Γ0, y, y′ ∈ Γ0 and z ∈ Γ′′ +0, and also v(x,y) ∈ H(x, y) and v′ +(y′,z) ∈ H(y′, z), then, +I′′(v(x,y), v′ +(y′,z)) = +� +[v(x,y) ⊗ v′ +(y′,z)]≃, +if y = y′, +0, +if y ̸= y′. +This is clearly balanced, considering the right and left actions of Lin(2)(Γ′), so I′′ +descends to a linear map, I′ : Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) → Lin(2)(H • H′). By +construction, I′ ◦ I = id, and so, in particular, I is injective as well. +The rest of the details are left to the reader. +□ +6.4.5. The Morita-valued once-extended Quinn TQFT. Using the results of the pre- +vious sections, we can take our finitary version of the once-extended Quinn TQFT +that we gave in subsection 6.3, and reflect it into the bicategory Mor, as follows. +As always, we let n be a non-negative integer and B be a homotopy finite space. +Definition 168 (The Morita-valued once-extended Quinn TQFT). The Morita +valued once-extended Quinn TQFT, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor, +is defined as the following composite of bifunctors, +2Cob +(n,n+1,n+2) +B +2QB +−−−→ vProfGrpfin +Lin(2) +−−−−→ Mor. +In order to simplify the notation, we will use the same notation, 2Q +Mor +B +, for all +components of the bifunctor 2Q +Mor +B +. +Remark 169. (We follow here an approach found in [30, Subsection 10.3]) Let +Σ be a closed smooth n-manifold. Given two B-decorations, f Σ and f +′ +Σ, of Σ, the +argument in Remark 155 gives a canonically defined invertible bimodule connecting +the algebras, 2Q +Mor +B +(Σ, fΣ) and 2Q +Mor +B +(Σ, f +′ +Σ), namely +2Q +Mor +B +� +(Σ, f Σ) +(ιΣ +0 ,Σ×I,ιΣ +1 ) +−−−−−−−−→ (Σ, f +′ +Σ) +� +: 2Q +Mor +B +(Σ, f Σ) ↛ 2Q +Mor +B +(Σ, f +′ +Σ). +All comments in Remark 155 pass over to the Morita setting with the obvious mod- +ifications. +Remark 170. From the previous remark, one can prove that, if Σ is a closed +smooth manifold, then all the algebras, 2Q +Mor +B +(Σ, fΣ), where fΣ is a B-decoration +of Σ, are Morita equivalent. Moreover, such Morita equivalences can be canonically +chosen, given a pair of decorations of Σ. + +A CATEGORIFICATION OF QUINN’S TQFT +139 +6.5. The symmetric monoidal structure in 2Cob(n,n+1,n+2). We fix a non- +negative integer n throughout this subsection and the following as well. The central +result of this paper is that one can categorify the finite total homotopy TQFT of +Quinn, [101], in a sensible way to get a once-extended TQFT, +2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, +where B is a homotopy finite space. From there we have shown that the resulting +theory can be cut down in size to be more finitary by various means such as the +introduction of decorations, and can be linked up with better known ‘algebraic’ +bicategories such as Mor, which are frequently met in representation theoretic +contexts. +Following Schommer-Pries, [106], Lurie, [82], and others, we have taken a once- +extended TQFT to be a symmetric monoidal bifunctor, as above, but we remark +that the existing definitions do not agree on the target / codomain bicategory. +We have defined 2QB, and have shown it to be a bifunctor. There is, however, +one further step to complete the proof that these constructions give once-extended +TQFTs, and that is to prove 2QB, and its cousins, are symmetric monoidal bi- +functors. For this, we have to specify the symmetric monoidal structures on the +cobordism bicategory, 2Cob(n,n+1,n+2), and will also recall that of vProfGrphf, +which was formally proved to exist in [63]. +We note that being a symmetric monoidal bifunctor is a structure, not a property, +and refer the reader to the sketch in Definition 30 and to [106, Definition 2.5] for a +more detailed description. Sometimes the extra structure, i.e., that beyond being +a bifunctor, is ‘evident’, but in our case that extra categorical structure encodes +some of the ‘geometric’ structure, for instance cobordisms, and 2-cobordisms, and +we do need to have the transition between the various contexts made explicit to +allow the naturality of the constructions to be made clear. +6.5.1. A preliminary result towards the construction of the symmetric monoidal +structure in 2Cob(n,n+1,n+2). The details of the construction of the symmetric +monoidal structure in the bicategory 2Cob(n,n+1,n+2), using the language of sym- +metric monoidal pseudo-double categories [63], can be found in [106, §3.1.4]. In +[82, Remark 1.2.7.], it is stated that, in the case of 2Cob(n,n+1,n+2) (or, more ex- +actly, Lurie’s analogue of this), the monoidal structure is straightforward, as “the +tensor product operation is simply given by disjoint union of manifolds”, just as +in the more classical case of Cob(n,n+1). Although correct, this statement hides +some important details. The disjoint union of manifolds, cobordisms and extended +cobordism indeed gives rise to a bifunctor, by abuse of language denoted67 +⊔: 2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) → 2Cob(n,n+1,n+2). +A monoidal bicategory is however not just the tensor product and unit, but also the +associator, unitors and with additional pentagonators, etc., as sketched in Definition +26. Moreover, we need the tensor product to be symmetric, so need to specify a +braiding, etc. This may seem excessive detail to give, but is needed as there is a +slight trap that has to be avoided, as we will now see. +In general, as we saw in Definition 26, in a monoidal bicategory, (A, ⊗, I, . . .), +the structure is given by morphisms in A. For instance, the monoidal associator is +an adjoint equivalence, so for each triple, A, B, C, of objects in A, we have that +αCBA : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A), +67It is important to note that this bifunctor is not strict, in the sense that the natural isomor- +phism, ϕ, in item (3) of Definition 20 is non-trivial. + +A CATEGORIFICATION OF QUINN’S TQFT +140 +whilst +α∗ +CBA : C ⊗ (B ⊗ A) → (C ⊗ B) ⊗ A. +We thus have, in the situation that A = 2Cob(n,n+1,n+2), that we need to specify +cobordisms, +(C ⊔ B) ⊔ A +i +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +C ⊔ (B ⊔ A), +j +�❧❧❧❧❧❧❧❧ +M +for all triples of n-manifolds. This is not difficult, but does involve some techni- +calities. +The two ends of the required cobordisms are not equal, although they +are naturally homeomorphic (and diffeomorphic if we include consideration of the +smooth structure). +The required (n + 1)-manifold, M, will be, topologically, a +cylinder, but some care is needed with the labelling of its ends, and throughout we +need to remember that a cobordism is not just an (n + 1)-manifold, but has extra +structure; see Definition 11. +We note that an analogous situation was encountered when discussing the bicat- +egory Span(C), in the list in Examples 29. +We cannot directly use the diffeomorphism between the two sides, (C ⊔ B) ⊔ A +and C ⊔ (B ⊔ A), as the associator, as that would be a morphism in C = CGWH +or Diff n, but not in 2Cob(n,n+1,n+2). We have to convert that isomorphism to a +cobordism before checking that it works. For this, and for similar later situations, +we need a result which is, in some sense, a dual of Lemma 63, the context for which +we will set up next. +In Section 4, we saw, page 62, that denoting by Diff n, the category of closed +n-manifolds and diffeomorphisms between them, we have a functor, I′ : Diffn → +Cob(n,n+1), which is the identity on objects, and such that I′(f : Σ → Σ′) is the +equivalence class of the cospan, +I′(f : Σ → Σ′) = + + + +Σ +ιΣ +0 +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +Σ′ +ιΣ′ +1 ◦f −1 +�rrrrrrr +Σ × I + + + , +or equally well of the cospan with f used in the left (co)leg, as in Remark 78. +Such a cospan is always cofibrant in the following sense, which is dual to the earlier +definition of fibrant spans. +Definition 171. A cospan, +A +i +�❘ +❘ +❘ +❘ +❘ +❘ +B, +j +�❦❦❦❦❦❦ +M +in CGWH is said to be cofibrant if the induced map, +� i +j +� +: A ⊔ B → M, is a +cofibration. +A thorough discussion on cofibrant cospans can be found in [117, 118]. +Remark 172. (i) In this case, both i and j will be cofibrations as well, and inter- +preting cofibrations more or less as inclusions of locally flat submanifolds, it also +says that the two submanifolds, i(A) and j(B), of M, have empty intersection. +(ii) We note that these cofibrant cospans are the cofibrant objects in the category +CGWHΛop, where, as in Remark 49, CGWH is given the Hurewicz / Strøm model +structure but, now, CGWHΛop is given the projective model category structure, so +that weak equivalences and fibrations are objectwise; cf. Remark 49 for the dual +case of fibrant spans. + +A CATEGORIFICATION OF QUINN’S TQFT +141 +(iii) It is also of note that we could have replaced the role of CGWH with that +Hurewicz / Strøm model category structure, by many other suitably structured model +categories in the above remarks. +From a terminological point of view, it is perhaps also worth noting that the +projection, σ : A × I → A, gives a morphism, +(ι0, A × I, ι1) → (idA, A, idA), +which is a weak equivalence in Cospan(CGWH) := CGWHΛop, but the idenitity +cospan is not a cofibrant, so in passing to cofibrant cospans, we are ‘resolving’ the +cospans by cofibrant ones. +We may sometimes work with cofibrant cospans in CGWH, although the objects +that are central to our study will be the objects coming from cobordisms and ex- +tended cobordisms between them. A stand-alone treatment of Cospan(CGWH), +containings some of the discussion above is in the already mentioned references, +[117, 118]. +We need an adapted categorified version, I : Diff n → 2Cob(n,n+1,n+2), of the +construction, I′ : Diff n → Cob(n,n+1), given in §4.1.1, page 62, that we mentioned +above. +We note that the latter construction works in more generality, e.g., at +the cospan level with homeomorphisms rather than diffeomorphisms, but then the +smoothness of the corresponding cobordism would be in doubt. +We suppose that f is a diffeomorphism from X to Y and write I(f) for the +cobordism represented by the cofibrant cospan below +(52) +I(f) := + + + + +X +ιY +0 f +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +Y. +ιY +1 +�①①①①①①①① +Y × I + + + + . +Note that this is the cobordism and not just the equivalence class determined by it, +so, for instance, the corresponding right (co)leg version, using f −1 is distinct from +this, although there is an invertible 2-cobordism connecting them. +This I does not give a ‘functor’ from Diffn to 2Cob(n,n+1,n+2). The reason is, +essentially, that the horizontal composition in 2Cob(n,n+1,n+2) is that of a bicate- +gory, not a category, since we are now not taking cobordisms up to diffeomorphism. +Therefore I does not preserve composition in a direct way, as is easy to show. How- +ever, we instead have a pseudo-functor I : Diff n → 2Cob(n,n+1,n+2), in the sense +that we now describe. +Suppose that we have diffeomorphisms, +X +f−→ Y +g−→ Z, +and thus two cobordisms / cospans I(f): X → Y and I(g): Y → Z, as well as +I(gf): X → X. We can form I(f)• I(g) by the usual pushout and can put all this + +A CATEGORIFICATION OF QUINN’S TQFT +142 +into a diagram as follows: +(53) +X +ιY +0 f +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +ιZ +0 gf +� +Y +ιZ +0 g +�▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +▲ +ιY +1 +�rrrrrrrrrr +Z +ιZ +1 +�②②②②②②②② +ιZ +1 +� +Y × I +ΨY +g,f +� +ℓ +�❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +❑ +Z × I +ΨZ +g,f +� +r +�ssssssssss +PO(g, f) +� +Ψg,f +�✤ +✤ +✤ +Z × I +The pushout, PO(g, f), can be given, up to isomorphism, by (Y × I) ⊔ (Z × I)/ ∼, +where, for all y ∈ Y , +(y, 1) ∼ (g(y), 0). +We will often abuse notation a bit and write this as (Y × I) ⊔Y (Z × I)/ ∼, leaving +explicit mention of the equivalence relation ∼ from the notation. Similarly we will +usually omit mention of ℓ and r. +Other models for the pushout could be used, but this seems the easiest to handle +so we will use this. The important thing to remember is that the pushout is defined +by a universal property so, as we said, it is only determined up to isomorphism. +However, as before in this paper, we implicitly chose a natural realisation for each +pushout, here by considering the obvious quotient of the disjoint union. +It is easy to see that there is a map, +Ψg,f : PO(g, f) → Z × I, +given by +ΨY +g,f(y, t) = (g(y), t/2), +and +ΨZ +g,f(z, t) = (z, (t + 1)/2). +We note that the pushout in (53) is a pushout in CGWH. In order for this +construction to be usable in the context of 2Cob(n,n+1,n+2), we must put a smooth +structure on PO(g, f), and also possibly modify Ψg,f : PO(g, f) → Z × I, slightly, +in order that it is smooth at the junction, where the cylinders Y × I and Z × I +join. These however can be easily handled using the usual mechanisms of collars, +etc., so we will not concern ourselves more with this aspect. +The map, Ψg,f, is a homeomorphism, with inverse, Ψ′ +g,f, given by +Ψ′ +g,f(z, t) = +� +(g−1(z), 2t), +for 0 ≤ t ≤ 1/2, +(z, 2t − 1), +for 1/2 ≤ t ≤ 1. +Up to now, of course, this construction is very similar to what we used in our +earlier section, §4.1.1, to show that the uncategorified version of the construction +gave a functor from Diffn to Cob(n,n+1), except that, as we already mentioned, we +are now not taking the quotient of cobordisms by diffeomorphism (relative to the +boundary). However it is not yet quite in the right form to be used for extended +cobordisms, so as to give an extended cobordism / 2-cobordism between I(f)•I(g) +and I(gf). For that we use an analogue of the I-construction one dimension up. + +A CATEGORIFICATION OF QUINN’S TQFT +143 +In general, suppose we have an isomorphism of cobordisms, i.e. a diffeomor- +phism, f, making the diagram below commute, +(54) +M +f +∼ += +� +X +i +�① +① +① +① +① +① +① +i′ +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +Y. +j +�❋❋❋❋❋❋❋ +j′ +�①①①①①①① +N +We can expand this out as a map of cospans, +(55) +X +i +� +idX +� +M +f +� +Y +j +� +idY +� +X +i′ +� N +Y, +j′ +� +to which we apply the same idea as in the I-construction to each vertical diffeo- +morphism to get +(56) +J (f) := + + + + + + + + + + + + +X +i +� +ιX +0 � +M +ιN +0 ◦f +� +Y +j +� +ιY +0 +� +X × I +i′×I +� N × I +Y × I +j′×I +� +X +i′ +� +ιX +1 +� +N +ιN +1 +� +Y +ιY +1 +� +j′ +� + + + + + + + + + + + + +. +This corresponds to an (n + 2)-extended cobordism / 2-cospan. Passing to equiva- +lence classes, we get a 2-morphism, +[J (f)] : (i, M, j) =⇒ (i′, N, j′), +in 2Cob(n,n+1,n+2). This is a vertically invertible 2-morphism. To prove this one +uses a categorified version of the argument used to show functoriality of I on page +62, and, in particular, that, if f : M → N, in (54), is a diffeomorphism, then [I(f)], +in (56) is an invertible 2-cobordism class. +Now suppose that we have diffeomorphisms of cospans, f : (i, M, j) → (i′, N, j′) +and g : (i′, N, j′) → (i′′, P, j′′), as below, +M +f +∼ += +� +X +i +�① +① +① +① +① +① +① +i′ +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +Y +j +�❋❋❋❋❋❋❋ +j′ +�①①①①①①① +N +and +N +g +∼ += +� +X +i′ +�① +① +① +① +① +① +① +i′′ +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +Y. +j′ +�❋❋❋❋❋❋❋ +j′′ +�①①①①①①① +P +We can then compose them to get gf : (i, M, j) → (i′′, P, j′′). +The 2-cospans, +J (f): (i, M, j) =⇒ (i′, N, j′) and J (g): (i′, N, j′) =⇒ (i′′, P, j′′), equally well +compose, using the vertical composition given by the obvious pushout diagram, +which fits into a diagram analogous to the diagram, (53), above, but, of course, +replacing X, Y , and Z, with M, N and P, respectively. (We leave the enterprising +reader to extend this diagram to include what happens to the vertical cospans, + +A CATEGORIFICATION OF QUINN’S TQFT +144 +X → X × I ← X, etc.) The composite 2-cospan will be of form, +X +� +� +M +� +Y +� +� +(X × I) ⊔X (X × I) +� L +(Y × I) ⊔Y (Y × I) +� +X +� +� +P +� +Y, +� +� +in which L is given by the pushout, +N +ιP +0 ◦g � +ιN +1 � +P × I +� +N × I +� L. +There are diffeomorphisms, (X × I) ⊔X (X × I) +∼ += +−→ X × I, extending the obvious +one from I ⊔{∗} I → I, and the discussion given after (53) carries over to the setting +here, giving an equivalence between J (f)#1J (g) and J (gf), so +[J (f)]#1[J (g)] = [J (gf)], +in 2Cob(n,n+1,n+2). +Example 173. We note that, when M = N and f is the identity diffeomorphism +on M, [J (f)] is the vertical identity on (i, M, j). +With the functoriality property, we get that if we have an arbitrary diffeomor- +phism, f, the 2-morphism, [J (f)], will be invertible, as claimed. +It should, now, be more-or-less clear that we have a pseudo-functor, +I : Diffn → 2Cob(n,n+1,n+2), +so we refer back to page 27 for a checklist of structure and properties needed. (We +note that I is contravariant due to our notational convention for composition of +cobordisms.) In this setting, +• for each manifold, X, considered as an object of Diff n, we have that I(X) +is that same object considered as an object of 2Cob(n,n+1,n+2), but note +that we will write X instead of I(X) most of the time in this context; +• for each diffeomorphism, f : X → Y , we have a 1-morphism / cobordism, +I(f): X → Y ; +• for each composable pair, +X +f−→ Y +g−→ Z, +an invertible 2-morphism, +[J (Ψg,f)]: I(f) • I(g) =⇒ I(gf); +and +• for each object, X, I(idX) is the chosen identity cobordism on X, so there +is no problem with the identities (and similarly none with the left and right +unitors). +This leaves us just to check compatibility of I with the associator in 2Cob(n,n+1,n+2), +namely that, given a triple of composable diffeomorphisms, +X +f−→ Y +g−→ Z +h−→ W, + +A CATEGORIFICATION OF QUINN’S TQFT +145 +the diagram +(57) +(I(f)#0I(g))#0I(h) +[J (g,f)]#0I(h) � +a +� +I(gf)#0I(h) +[J (h,gf)] +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +I(hgf), +I(f)#0(I(g)#0I(h)) +I(f)#0[J (h,g)] +� I(f)#0I(hg) +[J (hg,f)] +�❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +commutes. Here we have written hgf for the value of (hg)f and h(gf), which, of +course, are equal, and have abbreviated J (Ψg,f) to J (g, f) for ease of labelling +the diagram. Furthermore, to emphasise that, here, it is the horizontal composi- +tion that is being used, we have replaced the convenient, but ‘generic’, symbol for +composition, •, by the more specific one, #0. +We will formalise this in a proposition for ease of reference. +Proposition 174. There is a pseudo-functor, +I : Diffn → 2Cob(n,n+1,n+2), +given as the identity on objects, and, if f : X → Y is a diffeomorphism between +manifolds, then I(f) is given by the cospan, +X +ιY +0 f +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +Y, +ιY +1 +�②②②②②②②② +Y × I +or, equivalently, as the corresponding cobordism. +The rest of the structure of I is explained in some detail below. The proof of +the above will take the form of a reasonably informal exploration of I and J . As it +is quite long and a bit technical, we suggest that, if the reader is willing to accept +the truth of the above proposition, then such an impatient reader who wants to see +the result in action should skip that discussion and go to page 154. +Proof. The associativity 2-morphism, a, in the bicategory 2Cob(n,n+1,n+2), ap- +pearing in the diagram in (57), is the (equivalence class of the) image under J of +the natural homeomorphism68 that comes from the two iterative ways of forming +the colimit of the diagram +(58) +Y +ιY +1 +�②②②②②②②② +ιY +0 g +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +Z +ιZ +1 +�②②②②②②②② +ιZ +0 h +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +Y × I +Z × I +W × I, +i.e. ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I) and (Y × I) ⊔Y ((Z × I) ⊔Z (W × I)), so, in +the left hand biassed one, first constructing the pushout on the left span, followed +by that on the right, and, similarly, for the right hand biassed one, doing the right +one first in the evident way69. +68which needs smoothing +69We note that the notation used, ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I), etc., does not specify how +the equivalence relations used are defined, and is just a shorthand referring back to the diagram +and the maps used. + +A CATEGORIFICATION OF QUINN’S TQFT +146 +Given our triple, +X +f−→ Y +g−→ Z +h−→ W, +however, we can form a colimit of the resulting zig-zag, (58), in a non-biassed way +which leads to a direct triple construction. That non-biassed model of the colimit +is formed from the disjoint union, (Y × I) ⊔ (Z × I) ⊔ (W × I), (which can be given +without reference to any pairwise, and thus iterative, formation of such a disjoint +union) by quotienting by the evident equivalence relation, generated by +(y, 1) ∼ (g(y), 0) +and +(z, 1) ∼ (h(z), 0). +We will write this model for the colimit of the zigzag diagram as +(Y × I) ⊔Y (Z × I) ⊔Z (W × I), +that is, without (extra) parentheses. This and the other two that we have used are +all related by unique diffeomorphisms, given by the fact that they all satisfy the +universal property of colimits with respect to the diagram, (58). +We can use this unbiassed form of the colimit as a means to construct a ‘multiple’ +composite, which it seems reasonable to write as I(f)#0I(g)#0I(h), from X to +W. This is the cospan obtained from +(59) +X +ιY +0 f +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +Y +ιZ +0 g +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +ιY +1 +�②②②②②②②② +Z +ιZ +1 +�②②②②②②②②② +ιW +0 h +�❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +❊ +W, +ιW +1 +�✇✇✇✇✇✇✇✇ +Y × I +� +Z × I +� +W × I +� +Colim +where Colim = (Y × I) ⊔Y (Z × I) ⊔Z (W × I), by composing to get +X → Colim ← W. +In diagrams, we will sometimes abbreviate Colim to just C, or, perhaps, to that +letter with suffices / superfices to indicate which setting we are using it in. This is +merely as this takes less space. +Returning to the associativity constraint, in 2Cob(n,n+1,n+2), appearing in (57), +it is fairly evident that as the diffeomorphisms given by the universal properties of +a colimit are unique with that property, we have that the associativity 2-morphism, +� +I(f)#0I(g) +� +#0I(h) +a=⇒ I(f)#0 +� +I(g)#0I(h) +� +, +in (57), can be factored as the composite +� +I(f)#0I(g) +� +#0I(h) +a +⇐⇒ I(f)#0I(g)#0I(h) +a +⇐⇒ I(f)#0 +� +I(g)#0I(h) +� +, +where we have abused notation by writing a for the equivalence class of the images +under J of each of the relevant unique diffeomorphisms. These 2-morphisms do +very little as they simply remove or add parentheses to objects and morphisms, but +are needed to enable certain diagrams to be constructed without evident clashes +with regard to the notation. We will see shortly that this also helps subdivide +diagram (57) into two smaller more manageable pieces. +As, for various choices of g and f, [J (g, f)] will occur repeatedly in the coming +pages, it will be useful to give it in a lot more detail and to consider generalisations +as well. This is really just a question of plugging the specification of Ψg,f, in (53), + +A CATEGORIFICATION OF QUINN’S TQFT +147 +into the J -construction, but rather than having to work through that each time +we need it and for the various combinations of composites that we will be using, +we will give the basic form here to act as a template. We have diffeomorphisms, +X +f−→ Y +g−→ Z, +as before, and then [J (g, f)] is the equivalence class of the extended cobordism / +2-span, +(60) +X +ιY +0 .f +� +ιX +0 +� +(Y × I) ⊔Y (Z × I) +d +� +Z +ιZ +1 +� +ιZ +0 +� +X × I +(ιZ +0 .(gf))×I � (Z × I) × I +Z × I +ιZ +1 ×I +� +X +ιZ +0 .(gf) +� +ιX +1 +� +(Z × I) +ι(Z×I) +1 +� +Z, +ιZ +1 +� +ιZ +1 +� +where d(y, s) = (g(y), s/2, 0) and d(z, s) = (z, (s + 1)/2, 0). We note that we have +put +� +(ιZ +0 .(gf))×I +� +(x, t) = (gf(x), 0, t), whilst the other maps are hopefully evident. +As we also have to consider composable triples of diffeomorphisms, for the com- +patibility diagram, (57), it is natural also to consider the direct way of getting from +the unbiassed I(f)#0I(g)#0I(h) to I(hgf). Analogously to the case with just two +inputs, we have a diffeomorphism, with domain the unbiassed colimit, +Φh,g,f : (Y × I) ⊔Y (Z × I) ⊔Z (W × I) → W × I, +given by morphisms: +(i) +� +α : Y × I → W × I, +α(y, s) = (hg(y), s/3); +(ii) +� +β : Z × I → W × I, +β(z, s) = (h(z), (s + 1)/3); +and +(iii) +� +γ : W × I → W × I +γ(w, s) = (w, (s + 2)/3). +Adapting diagram (59), we have +(61) +X +ιY +0 f +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +ιW +0 .(hgf) +� +Y +ιZ +0 g +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +ιY +1 +�②②②②②②②② +Z +ιZ +1 +�①①①①①①①①① +ιW +0 h +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +ιY +1 +�①①①①①①①①① +W +ιW +1 +�✇✇✇✇✇✇✇✇✇ +ιW +1 +� +Y × I +α +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +Z × I +β +� +W × I +γ +�❧❧❧❧❧❧❧❧❧❧❧❧❧❧ +W × I +and the induced map, Φh,g,f : Colim → W × I, gives a diffeomorphism of cospans, +from I(f)#0I(g)#0I(h) to I(hgf). We can, thus, form [J (Φh,g,f)], which we will +relabel [J (h, g, f)], and which will be a 2-morphism, again from I(f)#0I(g)#0I(h) +to I(hgf). This fits into a subdivided version of the compatibility diagram / cocycle + +A CATEGORIFICATION OF QUINN’S TQFT +148 +equation, (57): +(62) +(I(f)#0I(g))#0I(h) +[J (g,f)]#0I(h) � +� +a +� +I(gf)#0I(h) +[J (h,gf)] +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +I(f)#0I(g)#0I(h) +[J (h,g,f)] +� I(hgf), +I(f)#0(I(g)#0I(h)) +I(f)#0[J (h,g)] +� +� +a +� +I(f)#0I(hg) +[J (hg,f)] +�❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +❥ +and clearly commutativity of this subdivided form will imply commutativity of (57). +We have that [J (h, g, f)] is the equivalence class of the 2-span, in the following +diagram, where we recall that C is the Colim in (59), +(63) +X +ιY +0 .f +� +ιX +0 � +C +ι(W ×I) +0 +.Φh,g,f +� +W +ιW +1 +� +ιW +0 +� +X × I +(ιW +0 .(hgf))×I� (W × I) × I +W × I +ιW +1 ×I +� +X +ιW +0 .(hgf) +� +ιX +1 +� +(W × I) +ι(W ×I) +1 +� +W. +ιW +1 +� +ιW +1 +� +This is, of course, easy to calculate by analogy with previous diagrams, but it +will be very useful to have it explicitly given to help in the comparison with the +2-morphism that results from the composite around the top of the compatibility +diagram, or, for that matter, that around the bottom half. +In a diagram as here, we will often omit a specific notation for the composite of +an inclusion into a coproduct followed by the quotient to a pushout, or more general +colimit, thus, in the above, we have written ιY +0 .f : X → C for the composite +X +ιY +0 .f +−−−→ Y × I ֒→ (Y × I) ⊔ (Z × I) ⊔ (W × I) +quot +−−−→ C. +Now in order to show the compatibility diagram, (57), or (62), commutes, it will +pay to take at least some of the other main 2-morphisms apart to see what they +are actually doing. (This is really included as an exercise in ‘book-keeping’.) +The idea behind the fact that the top face of the diagram in (62) commutes, +is that both compositions yield manifolds homeomorphic to W × I2, and each +composition is obtained by gluing rectangles, Y × I2, Z × I2 and W × I2, along +the way indicated in the schematic pictures, (64) and (65), just below, and making +use of the homeomorphisms f : X → Y , g : Y → Z and h: Z → Y , to perform the + +A CATEGORIFICATION OF QUINN’S TQFT +149 +identifications: +(64) +• +Y ×I +• +• +Z×I +• +• +W×I +• +• +X×I +• +• +• +• +• +• +W×I +Z × I2 +W × I2 +• +• +• +• +• +• +• +Z×I +• +• +W×I +• +• +X×I +• +• +• +• +W×I +W × I2 +• +• +• +• +• +W×I +• +and +(65) +• +Y ×I +• +• +Z×I +• +• +W×I +• +• +X×I +• +• +• +• +• +• +• +W×I +Y × I2 +Z × I2 +W × I2 +• +• +• +• +• +• +• +• +• +Y ×I +• +• +Z×I +• +• +W×I +• +• +X×I +• +• +• +• +• +W×I +W × I2 +• +• +• +• +• +W×I +• +Looking first at [J (g, f)]#0I(h), in (62), corresponding to the top bit of the +schematic picture in (64), we note that its (vertical) domain is +� +I(f)#0I(g) +� +#0I(h), +which is a cospan from X to W, whose underlying (n + 1)-cobordism is +L := ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I). +We note again, for later comparison purposes, that this means (y, 1) ∼ (g(y), 0), +(z, 1) ∼ (h(z), 0), and the maps making it a cospan are x �→ (f(x), 0) and w �→ +(w, 1). This L is a parenthesised version of C, which we recall is +C = (Y × I) ⊔Y (Z × I) ⊔Z (W × I). +The (vertical) codomain of [J (g, f)]#0I(h), is I(gf)#0I(h), which has as its +middle object, i.e., underlying (n+1)-cobordism, (Z ×I)⊔Z (W ×I), where (z, 1) ∼ +(h(z), 0), and the end maps are x �→ (gf(x), 0) and w �→ (w, 1). +Putting this together, and adding a bit more detail, we get [J (g, f)]#0I(h) is +of the form below, which completes the description of the top bit of the schematic + +A CATEGORIFICATION OF QUINN’S TQFT +150 +picture in (64), +(66) +X +ιY +0 .f +� +ιX +0 � +L +down +� +W +ιW +1 +� +ιW +0 +� +X × I +(ιZ +0 .(gf))×I � M +W × I +ιW +1 ×I +� +X +ιZ +0 .(gf) +� +ιX +1 +� +N +ιN +1 +� +W. +ιW +1 +� +ιW +1 +� +Here +L = ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I), +M = ((Z × I) × I) ⊔Z×I ((W × I) × I), +and +N = (Z × I) ⊔Z (W × I). +The morphism labelled ‘down’ satisfies +down(y, s) = (g(y), s/2, 0), +down(z, s) = (z, (s + 1)/2, 0), +and +down(w, s) = (w, s, 0), +with this last being in the cofactor ((W ×I)×I). (As was said before, we will often, +in this sort of context, omit a specific notation for the composite of an inclusion +into a coproduct followed by the quotient to a colimit such as a pushout. Here, for +instance, we have written ιZ +0 .(gf) for the composite map, +X +ιZ +0 .(gf) +−−−−−→ (Z × I) +inc +−−→ (Z × I) ⊔ (W × I) +quot +−−−→ (Z × I) ⊔Z (W × I), +where the second map, quot, is the quotient map from the coproduct to the pushout.) +The final part of the composite along the top face of (62), which corresponds +to the bottom bit of the sketch in (64), is obtained by vertical composition with +[J (h, gf)]. Again, to include some ‘book-keeping’, we note that [J (h, gf)] is the +equivalence class of the 2-cospan, +(67) +X +ιZ +0 .(gf) � +ιX +0 +� +(Z × I) ⊔Z (W × I) +δ +� +W +ιW +1 +� +ιW +0 +� +X × I +(ιW +0 .(hgf))×I +� (W × I) × I +W × I +ιW +1 ×I +� +X +ιW +0 .(hgf) +� +ιX +1 +� +W × I +ι(W ×I) +1 +� +W, +ιW +1 +� +ιW +1 +� +where δ(z, s) = (h(z), s/2, 0) and δ(w, s) = (w, (s + 1)/2, 0). (Of course, the middle +of the vertical domain / top row is exactly N from earlier.) + +A CATEGORIFICATION OF QUINN’S TQFT +151 +If we now calculate the entire +� +[J (g, f)]#0I(h) +� +#1[J (h, gf)], sketched in dia- +gram (64), we first construct, where the notation is defined in (53), +(68) +X +ℓ.ιY +0 .f +� +ιX +0 � +L +down +� +W +ιW +1 +� +ιW +0 +� +X × I +(ιZ +0 .(gf))×I +� M +W × I +ιW +1 ×I +� +X +ιX +1 +� +ιZ +0 .(gf)� +ιX +0 +� +(Z × I) ⊔Z (W × I) +ιN +1 +� +δ +� +W +ιW +1 +� +ιW +1 +� +ιW +0 +� +X × I +(ιW +0 .(hgf))×I � (W × I) × I +W × I +ιW +1 ×I +� +X +ιW +0 .(hgf) +� +ιX +1 +� +(W × I) +ι(W ×I) +1 +� +W, +ιW +1 +� +ιW +1 +� +and then, taking pushouts ‘vertically in the central belt’, we get +(69) +X +ℓ.ιY +0 .f +� +ιX +0 � +L +λ +� +W +ιW +1 +� +ιW +0 +� +X × I +µ +� P +W × I +ν +� +X +ιW +0 .(hgf) +� +ιX +1 +� +W × I +ι(W ×I) +1 +� +W, +ιW +1 +� +ιW +1 +� +where we have simplified the expressions using X × I ∼= (X × I) ⊔X (X × I) in the +left hand side, and similarly for W × I on the right. These diffeomorphism will be +important later. +We need to explore more thoroughly the description of P here, as the iden- +tifications used in its construction are not all of the same type. We have M = +((Z × I) × I) ⊔Z×I ((W × I) × I), so a point in M is either of form (z, s, t) or +(w, s, t), where (z, 1, t) ∼ (h(z), 0, t). +The straightforward way to form P is to +first take M ⊔ ((W × I) × I) and then form a quotient, so in addition to the +two forms of element we had from M, we have elements in the second copy of +((W ×I)×I), and we will write a typical element there as (w, s, t)2, to indicate this +is in the second copy. To get the quotienting relations, we take typical elements +of N = (Z × I) ⊔Z (W × I). These will be either of form (z, s) or (w, s), and we +have (z, 1) ∼ (h(z), 0). The image of (z, s) in M is (z, s, 1) and similarly for (w, s). +We then recall that δ(z, s) = (h(z), s/2, 0)2, and δ(w, s) = (w, (s + 1)/2, 0)2. The +equivalence relation on M ⊔ ((W × I) × I) is generated by (z, s, 1) ∼ (h(z), s/2, 0) +and (w, s, 1) ∼ (w, (s + 1)/2, 0)2. We will return to this very shortly to provide a +diffeomorphism from P to another space which is easier to handle. Changing P by +that diffeomorphism will not change the class of the 2-cospan and thus just gives a +different representing 2-cospan for that 2-morphism in 2Cob(n,n+1,n+2). + +A CATEGORIFICATION OF QUINN’S TQFT +152 +We have, however, still to describe70 the three arrows, λ, µ, and ν, in diagram +(69). We have the following. +(i) λ is defined on L as being the morphism labelled ‘down’ from L to M, +composed with the inclusion of M = ((Z × I) × I) ⊔Z×I ((W × I) × I) into +M ⊔N ((W × I) × I). Recalling that L = +� +(Y × I) ⊔Y (Z × I) +� +⊔Z (W × I), +this gives +λ(y, s) = (g(y), s/2, 0), +λ(z, s) = (z, (s + 1)/2, 0), +and +λ(w, s) = (w, s, 0). +(ii) µ is defined via the diffeomorphism from X × I to (X × I) ⊔X (X × I), so +has different defining rules on X × [0, 1/2] and on X × [1/2, 1]. +If 0 ≤ t ≤ 1/2, µ(x, t) is given by diagram (66), so is ((ιZ +0 .(gf))×I)(x, 2t) +which is (gf(x), 0, 2t) in the ((Z × I) × I) cofactor of M, then ‘included’ +into P. +If 1/2 ≤ t ≤ 1, then the relevant diagram is (67) and so +µ(x, t) = ((ιW +0 .(hgf)) × I)(x, 2t − 1) = (hgf(x), 0, 2t − 1). +Finally, +(iii) for ν, again we have different descriptions for ν(x, t) in the two evident +cases. +If 0 ≤ t ≤ 1/2, ν(w, t) = (w, 1, 2t), whilst if 1/2 ≤ t ≤ 1, ν(w, t) = +(w, 1, 2t − 1)2, so within the other copy of (W × I) × I. +We now want to change P by a diffeomorphism, ϕ : P → Q. This will result +in an equivalent 2-cospan, as we can compose each of the four maps in (69) which +have codomain P, with this ϕ to get the result we want. The idea is to deform +the 2-cospan in diagram (69) within its equivalence class so as to look more like +J (h, g, f), making comparison with that of diagram (63) easier. +A rough guide to what we want to obtain is given by the schematic picture in +(65), representing the final result of the middle path from +� +I(f)#0I(g) +� +#0I(h) to +I(hgf), in (62). +We return to look again at +M = ((Z × I) × I) ⊔Z×I ((W × I) × I), +whose schematic picture is that of the top bit of the sketch in (64). It is formed in +very much the same way as the middle term (i.e. underlying (n + 1)-cobordism) of +I(f)#0I(g), where we linked that to Z × I; see the comments just after (60). If we +try the same idea here, we have a diffeomorphism, ϕ′, between M and (W × I) × I +given by +ϕ′(z, s, t) = (h(z), s/2, t) +ϕ′(w, s, t) = (w, (s + 1)/2, t). +We note that ϕ′(z, 1, t) = (h(z), 1/2, t) = ϕ′(h(z), 0, t), so this assignment is a +continuous map from M to (W × I)× I. We leave the reader to check that this is a +homeomorphism. That it can then be smoothed to a diffeomorphism is then clear. +We can induce a diffeomorphism, ϕ : P → (W × I) × I, which we can take to be +the ϕ : P → Q of our previous discussion71, by first forming ϕ′ ⊔N ((W × I) × I), +70It may help to look again at the schematic figure in (64). +71thus setting Q = (W × I) × I, + +A CATEGORIFICATION OF QUINN’S TQFT +153 +followed by again using ((W × I) × I) ⊔(W×I) (W × I) × I) ∼= (W × I) × I. That +this is compatible with the identifications is routine and is left to the reader. +We do need now to calculate the deformed model of the 2-cospan of diagram +(69). As ι(W×I) +1 +ends up in a part of ((W × I) × I) left untouched by the diffeo- +morphism, this does not change its specification (although, of course, its codomain +does change). It is also easy to check that ϕµ is ((ιW +0 .(hgf)) × I), whilst ϕν is +ιW +1 ×I. In other words, the deformed version of diagram (69) is almost the same as +diagram (63), differing from it only in the downward pointing arrow, which is ϕλ +in the first of these, and ι(W×I) +0 +.Φh,g,f in (63). +Calculation of ϕλ gives +ϕλ(y, s) = (hg(y), s/4, 0) +ϕλ(z, s) = (h(z), (s + 1)/4, 0) +and +ϕλ(w, s) = (w, (s + 1)/2, 0). +Comparison with the formula for ι(W×I) +0 +.Φh,g,f, referring to page 147 for that for +Φh,g,f, we have that they differ only by a reparametrisation of (W ×I)×{0} within +(W × I) × I, and that can be performed by a diffeomorphism of (W × I) × I, which +is fixed on the other three faces, (W × {0}) × I, (W × {1}) × I, and (W × I) × {1}. +That gives us a diffeomorphism between the supports of the extended cobordism +corresponding to [J (g, f)]#0I(h))#1[J (h, gf)] and that given by [J (h, g, f)], after +inserting of parentheses. We can therefore conclude that +[J (h, g, f)] = ([J (g, f)]#0I(h))#1[J (h, gf)]. +It is clear how to adapt the above argument to show that +[J (h, g, f)] = (I(f)#0[J (h, g)])#1[J (hg, f)], +which concludes the proof of Proposition 174, as well as providing confirmation +that I and J together act as a categorification of the I-construction from the +unextended case. +□ +We note that there is a contravariant version of this Proposition that can also +be useful and in which f is sent to the cospan, +Y +ιY +0 +�❋ +❋ +❋ +❋ +❋ +❋ +❋ +❋ +X. +ιY +1 f +�①①①①①①①①① +Y × I +The proof is more or less the same as that of the above, with some obvious changes. +Remark 175. We note that, if g : Y → X is the inverse diffeomorphism of f : X → +Y , then +I(f) • I(g) ∼= idX, +and +I(g) • I(f) ∼= idY , +in 2Cob(n,n+1,n+2). These can be used to prove that, in the bicategory 2Cob(n,n+1,n+2), +I(f) forms part of an adjoint equivalence. + +A CATEGORIFICATION OF QUINN’S TQFT +154 +Finally, if f : A → B and g : C → D are diffeomorphisms, then we can form +f ⊔ g : A ⊔ C → B ⊔ D, and it is easy to see that I(f ⊔ g) ∼= I(f) ⊔ I(g), again by +the diffeomorphism coming from (B ⊔ D) × I ∼= (B × I) ⊔ (D × I). This implies +that the pseudo-functor, +I : Diffn → 2Cob(n,n+1,n+2), +is compatible with the coproduct monoidal structure. In particular, we will use this +in the case when one of the two diffeomorphisms is the identity on the corresponding +object. +6.5.2. A sketch of the construction of the symmetric monoidal structure in the bi- +category, 2Cob(n,n+1,n+2). After this technical diversion, we can return to the +problem of the monoidal associators in the monoidal bicategory, 2Cob(n,n+1,n+2), +that we started discussing on page 140. We need to formalise things a little more. +For this, it may be helpful to give a reference for a fairly standard form of the +axioms for a monoidal bicategory. We will use Johnson and Yau, [68], §1.2, as a +basic reference and will, in general, use their terminology. +For each triple of objects, A, B, C in 2Cob(n,n+1,n+2), we seek a cobordism, +αCBA, of form +(C ⊔ B) ⊔ A +i +�◗ +◗ +◗ +◗ +◗ +◗ +◗ +◗ +C ⊔ (B ⊔ A) +j +�♠♠♠♠♠♠♠♠ +M +. +We have, for n-manifolds, A, B, C, a diffeomorphism, +aCBA : (C ⊔ B) ⊔ A → C ⊔ (B ⊔ A), +and note that, as (CGWH, ⊔, ∅), forms a monoidal category, these satisfy the +pentagon axiom, so for A, B, C and D, the diagram, +(70) +(D ⊔ (C ⊔ B)) ⊔ A +� D ⊔ ((C ⊔ B) ⊔ A) +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +((D ⊔ C) ⊔ B) ⊔ A +�♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +D ⊔ (C ⊔ (B ⊔ A)). +(D ⊔ C) ⊔ (B ⊔ A) +�❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +commutes. We now write +αCBA := I(aCBA), +using the notation defined in (52). Given that we have a pseudo-functor, +I : Diffn → 2Cob(n,n+1,n+2), +as shown in Proposition 174, whenever we have two composable diffeomorphisms, +f and g, we have a 2-morphism, +[J (Ψg,f)]: I(f) • I(g) =⇒ I(gf), + +A CATEGORIFICATION OF QUINN’S TQFT +155 +which satisfies the cocycle identity in (57). Applying this to the arrows in (70), we +can then derive an expression for the required pentagonator: +(D ⊔ (C ⊔ B)) ⊔ A +� D ⊔ ((C ⊔ B) ⊔ A) +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +((D ⊔ C) ⊔ B) ⊔ A +�♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +♣ +�❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +❲ +⇓πDCBA +D ⊔ (C ⊔ (B ⊔ A)). +(D ⊔ C) ⊔ (B ⊔ A) +�❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +❣ +. +By construction, as I : Diff n → 2Cob(n,n+1,n+2) is a pseudo-functor, this pentag- +onator then satisfies a higher order cocycle identity, as in [61, Page 61] and [58, +Page 10], when we have five (closed and smooth) n-manifolds. +Furthermore, given (n + 1)-cobordisms, (iA, K, jA′): A → A′, (iB, M, jB′): B → +B′, and (iC, N, jC′): C → C′, we have a natural 2-morphism in 2Cob(n,n+1,n+2), +fitting inside the diagram below, +(71) +(C ⊔ B) ⊔ A +αCBA +� +✙✙✙✙� +α2 +NMK +� +(iC,N,jC′)⊔(iB,M,jB′) +� +⊔(iA,K,jA′) +� (C′ ⊔ B′) ⊔ A′ +αC′B′A′ +� +C ⊔ (B ⊔ A) +(iC,N,jC′)⊔ +� +(iB,M,jB′)⊔(iA,K,jA′) +� � C′ ⊔ (B′ ⊔ A′). +We note that, in the diagram above, we have abbreviated the notation, putting +α2 +NMK = α� +(iC,N,jC′),(iB,M,jB′ ),(iA,K,jA′ ) +�. +This 2-morphism, α2 +N,M,K, arises from the obvious diffeomorphism between the +(n+1)-cobordisms obtained from the two paths, from (C ⊔B)⊔A to C′ ⊔(B′ ⊔A′), +in the diagram above, together with the construction in Equation (56). That diffeo- +morphism underpins the naturality of the associativity constraints in Cob(n,n+1), +where the diagram consisting of the 1-dimensional arrows in (71) would commute. +In the monoidal bicategory 2Cob(n,n+1,n+2), this diffeomorphism is unsurprisingly +promoted to being a part of the symmetric monoidal bicategory structure. +Together with the associator 1-morphisms, αCBA, the class of all 2-morphisms, +α2 +NMK, defines a pseudo-natural transformation of bifunctors, +α: +� +2Cob(n,n+1,n+2)�3 → 2Cob(n,n+1,n+2), +called the associator pseudo-natural transformation, as shown below, +� +2Cob(n,n+1,n+2)�3 +⊔× +� +2Cob(n,n+1,n+2)� +� +� +2Cob(n,n+1,n+2)� +×⊔ +� +✗✗✗✗� α +� +2Cob(n,n+1,n+2)�2 +⊔ +� +� +2Cob(n,n+1,n+2)�2 +⊔ +� 2Cob(n,n+1,n+2). +We have two different bifunctors, from +� +2Cob(n,n+1,n+2)�4 to 2Cob(n,n+1,n+2), +defined as ⊔◦(⊔×id)◦ +� +⊔×id×id +� +and as ⊔◦(id×⊔)◦ +� +id×id×⊔ +� +. Two different +pseudo-natural transformations between these bifunctors can be constructed using +the associator pseudo-natural transformation, α, above, by considering the two +different paths in diagram (70). The class of all pentagonators, πDCBA, then defines + +A CATEGORIFICATION OF QUINN’S TQFT +156 +a modification between the corresponding pseudo-natural transformations. This +“pentagonator modification” satisfies its own cocycle identity, where we have five +copies of 2Cob(n,n+1,n+2). The equation satisfied is in [61, Page 61] and [58, Page +10]. +We can similarly use that the unit object in (CGWH, ⊔, ∅), comes with natural +isomorphisms, +∅ ⊔ A +ℓA +−→ A and A ⊔ ∅ +rA +−−→ A, +to obtain cospans, λA := I(ℓA) and ρA := I(rA). These are just the obvious ones, +but linking them with the construction of the pseudo-functor explicitly means that +certain diagrams will immediately do what we need, without further checking. +The Middle Unity Axiom gives that +(A ⊔ ∅) ⊔ B +aA,∅,B� +rA⊔B +� +A ⊔ (∅ ⊔ B) +A⊔ℓB +� +A ⊔ B += +� A ⊔ B, +commutes, so, on applying I, we get a specific modification72, +µA,B : (idA ⊔ λB) ◦ αA,∅,B → ρA ⊔ idB, +in which ◦ stands for the composition of cospans. +The evident commutative diagram, +(∅ ⊔ A) ⊔ B +ℓA⊔B +� +a∅,A,B +�❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +❖ +A ⊔ B +∅ ⊔ (A ⊔ B) +ℓA⊔B +�r +r +r +r +r +r +r +r +r +r +r +after application of I gives a left 2-unitor, and the reverse / adjoint of r, +r∗ +A : A → A ⊔ ∅, +likewise gives the right 2-unitor. +The fact that the pasting diagrams for these modifications work as required +follows from the (trivially commutative) diagrams in (CGWH, ∅, ⊔) itself, on ap- +plication of I. All this works in CGWH, but we note that if the objects are smooth +manifolds, the structure gives corresponding cobordisms as required. +Turning to the braiding, R, on 2Cob(n,n+1,n+2), the structural 1-morphisms +are obtained as the image under I of the braiding, τA,B : A ⊔ B ∼= B ⊔ A, in +CGWH, given by the universal property of the coproduct, so given closed smooth +n-manifolds, A and B, we put +RA,B = I(τA,B). +As for the case of the associator pseudo-natural transformation, given cobordisms, +(iA, M, jA′): A → A′ and (iB, K, jB′): B → B′, we have an extended cobordism, +R2 +M,N = R� +(iA,M,jA′),(iB,K,jB′ ) +�, +72An explicit formula can be written down using the explicit construction of I. In fact, however, +that formula is not that useful in itself. + +A CATEGORIFICATION OF QUINN’S TQFT +157 +fitting into the commutative diagram, +A ⊔ B +RA,B +� +✘✘✘✘� +R2 +M,N +(iA,M,jA′ )⊔(iB,K,jB′ ) +� A′ ⊔ B′ +RA′,B′ +� +B ⊔ A +(iB,K,jB′ )⊔(iA,M,jA′ ) +� B′ ⊔ A′. +Again, this extended cobordism arises from the obvious diffeomorphism between the +two composite cobordisms from A⊔B to B′ ⊔A′, obtained from the two paths from +A⊔B to B′⊔A′ in the diagram above, on applying the construction in diagram (56). +(Similarly to the associator natural transformation, this diffeomorphism underpins +the naturality of the braiding in Cob(n,n+1), but is now promoted to a crucial +bit of structure in the symmetric monoidal bicategory 2Cob(n,n+1,n+2).) Together +with the RA,B, the class of all R2 +M,N defines a pseudo-natural transformation of +bifunctors, R, fitting into the diagram below, +2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) +⊔ +�❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +❙ +τ +� +✞✞✞✞� +R +2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) +⊔ +� +2Cob(n,n+1,n+2). +(Here the bifunctor τ is obtained simply by swapping coordinates.) Moreover, this +is part of an adjoint equivalence, as in [60, page 4234]. +As in the case of the associator natural transformation, to finish constructing +a braiding in the monoidal bicategory 2Cob(n,n+1,n+2), we still need to specify +modifications as in [60, page 4235], which we will not need explicitly here, and +also check the remaining axioms for a braided monoidal bicategory, see loc cit. +Finally, this braiding satisfies the axioms for a braided monoidal bicategory to be +a symmetric monoidal bicategory, which can be found in [62, 1.1. Definitions]. +This finishes the sketch of the construction of the symmetric monoidal structure +on 2Cob(n,n+1,n+2). +6.6. The symmetric monoidal structure of the bifunctor 2QB. As usual, +let B be a homotopy finite space, and recall that we fix a non-negative integer, n, +throughout this section. +6.6.1. The basic case. We now sketch the proof of the fact that the bifunctor, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +can be given the structure of a symmetric monoidal bifunctor, with respect to +the symmetric monoidal structure, ⊔, in 2Cob(n,n+1,n+2), whose construction we +just sketched73, and the symmetric monoidal structure in vProfGrphf, outlined in +§2.8.4. The latter symmetric monoidal structure is a particular case of that of the +bicategory of Vect-enriched productors, which is discussed in [63, Corollary 6.6], +in a more general context, and using the language of symmetric monoidal pseudo +double categories. +We know the structure of Prof (and of vProf, etc.), as bicategories (see Subsec- +tion 2.8), and this induces the bicategory structure in vProfGrphf. The monoidal +structure in vProfGrphf is essentially given as follows: +73see §6.5.2 for a quick outline, + +A CATEGORIFICATION OF QUINN’S TQFT +158 +• on objects (i.e., homotopy finite groupoids) it is given by the usual cartesian +product of groupoids; +• if F : A0 ↛ B0 and G: A1 ↛ B1 are 1-morphisms in vProfGrphf, i.e. +Vect-profunctors, then F ⊗G: A0×A1 ↛ B0×B1 is given by the composite +functor74, +(A0×A1)op×(B0×B1) +∼ += +−→ (Aop +0 ×B0)×(Aop +1 ×B1) +F ×G +−−−→ Vect×Vect +−⊗Vect− +−−−−−−→ Vect; +• on 2-morphisms, the rule is +(α ⊗ β)(A0,A1),(B0,B1) = α(A0,B0) ⊗ β(A1,B1). +With this information, we can construct a bifunctor, +⊗: vProfGrp × vProfGrp → vProfGrp, +which is the starting point for the construction of the symmetric monoidal struc- +ture on the bicategory vProfGrp. +The remaining bits of structure look after +themselves. +A crucial component for our discussion of the symmetric monoidal structure of +the bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, is the discussion in Lem- +mas 114 and 115, and in §5.4.3, which we need to transfer from 2span(HF) to +2Cob(n,n+1,n+2), by using the mapping space construction B(−); see Remark 150 +and Subsection 5.8 for notation. The notation for the additional bits of structure +that we will give to 2QB follows the pattern of the notation of Definition 30, though +we will add a prime to all structure morphisms, to distinguish the notation here +from that already used in the context of 2span(HF). +We first construct a pseudo-natural transformation of bifunctors, fitting into the +diagram, +� +2Cob(n,n+1,n+2)�2 +2QB×2QB +� +⊔ +� +✗✗✗✗� χ′ +� +vProfGrphf)2 +⊗ +� +2Cob(n,n+1,n+2) +2QB +� vProfGrphf. +Given closed smooth n-manifolds, X and X′, the cartesian closed structure of +CGWH gives a natural isomorphism of groupoids, +m′ +(X,X′) : π1(BX, BX) × π1(BX′, BX′) → π1(BX⊔X′, BX⊔X′). +We hence have a profunctor, using the construction in Example 33, +χ′ +(X,X′) : π1(BX, BX) × π1(BX′, BX′) ↛ π1(BX⊔X′, BX⊔X′), +defined as χ′ +(X,X′) := ϕm′ +(X,X′ ). Furthermore, given cobordisms, (i, Σ, j): X → Y +and (i′, Σ′, j′): X′ → Y ′, and hence a cobordism, +(i ⊔ i′, Σ ⊔ Σ′, j ⊔ j′) : X ⊔ X′ → Y ⊔ Y ′, +we have a 2-morphism in vProfGrphf, +π1(BX, BX) × π1(BX′, BX′) +H +� +(i∗,BΣ,j∗) +� +⊗H +� +(i′∗,BΣ′,j′∗) +� +� +χ′ +(X,X′) � +✚✚✚✚� χ′ +((i,Σ,j),(i,Σ′ ,j′)) +π1(BY , BY ) × π1(BY ′, BY ′) +χ′ +(Y,Y ′) +� +π1(BX⊔X′, BX⊔X′) +H +�� +(i⊔i′)∗,BΣ⊔Σ′ ,(j⊔j′)∗�� +� π1(BY ⊔Y ′, BY ⊔Y ′), +74see page 41. + +A CATEGORIFICATION OF QUINN’S TQFT +159 +or changing notation, +2QB(X) ⊗ 2QB(X′) +2QB +� +(i,Σ,j) +� +⊗2QB +� +(i′,Σ′,j′) +� +� +χ′ +(X,X′) +� +✙✙✙✙� χ′ +((i,Σ,j),(i,Σ′ ,j′)) +2QB(Y ) ⊗ 2QB(Y ′) +χ′ +(Y,Y ′) +� +2QB(X ⊔ X′) +2QB +�� +(i⊔i′,Σ⊔Σ′,j⊔j′) +�� +� 2QB(Y ⊔ Y ′). +This natural isomorphism of profunctors is obtained from Lemma 115, together +with the fact that we have a homeomorphism of HF spans, +� +(i ⊔ i′)∗, BΣ⊔Σ′, (j ⊔ j′)∗� +∼= +� +i∗ × i′∗, BΣ × BΣ′, j∗ × j′∗� +, +arising from the cartesian closed structure of CGWH. By applying Lemma 133, +it follows that χ′ is a pseudo-natural transformation, +χ′ : ⊗ ◦(2QB × 2QB) → 2QB ◦ ⊔. +We will sketch the construction of the rest of the symmetric monoidal structure +on 2QB. +We have two bifunctors L, R: +� +2Cob(n,n+1,n+2)�3 → vProfGrphf, defined by +composition along the boundary left and right paths, in the two diagrams below, +as in [61, page 67], +� +2Cob(n,n+1,n+2)�3 +id×⊔ +� +⊔×id +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +✚✚✚✚� χ′×2QB +2QB×2QB×2QB +� � +vProfGrphf +�3 +⊗×id +� +� +2Cob(n,n+1,n+2))2 +⊔ +� +⇐= +α +� +2Cob(n,n+1,n+2))2 +⊔ +�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +❴❴❴❴� +χ′ +2QB×2QB +� � +vProfGrphf +�2, +⊗ +�✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ +2Cob(n,n+1,n+2) +2QB +� vProfGrphf +ω′ ❴� +� +2Cob(n,n+1,n+2)�3 +id×⊔ +� +✚✚✚✚� 2QB×χ′ +2QB×2QB×2QB +� � +vProfGrphf +�3 +⊗×id +� +id×⊗ +�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ +� +2Cob(n,n+1,n+2))2 +⊔ +� +✕✕✕✕� χ′ +2QB×2QB +� � +vProfGrphf)2 +⇐= +α +⊗ +� +� +vProfGrphf +�2, +⊗ +�❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ +2Cob(n,n+1,n+2) +2QB +� vProfGrphf, +as well as two natural transformations, connecting L and R, as defined in the two +diagrams above. (We note that we have used the same notation for the associator +modification of 2Cob(n,n+1,n+2) and of vProfGrphf.) +The constructions, as discussed in §5.4.3, especially in Notation 134 and Lemma +135, give a modification, ω′, as shown above. Explicitly, given manifolds X, X′ and +X′′, and abbreviating, for a topological space X, π(X) = π1(X, X), we have that + +A CATEGORIFICATION OF QUINN’S TQFT +160 +ω′ +(X,X′,X′′) is the obvious natural isomorphism of profunctors in the diagram, +� +π(BX) × π(BX′) +� +× π(BX′′) +D +�❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ +A +�P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +P +π(BX) × +� +π(BX′) × π(BX′′) +� +E +� +❴❴❴❴� +ω′ +(X,X′,X′′) +π(BX⊔X′) × π(BX′′) +B +� +π(BX) × π(BX′⊔X′′) +F +�❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +❘ +π +� +B(X⊔X′)⊔X′′� +C +�♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ +π +� +BX⊔(X′⊔X′′)� +. +Here we have considered the associator, +αCGWH,⊔ +X,X′,X′′ : (X ⊔ X′) ⊔ X′′ → X ⊔ (X′ ⊔ X′′), +in the category CGWH, with the disjoint union monoidal structure. The arrows +in the above diagram are labelled as follows: +• A = ϕm′ +(X,X′) × idπ(BX′′ ); +• B = ϕm′ +(X⊔X′ ,X′′); +• C = ϕπ +� +B +� +αCGWH,⊔ +(X,X′,X′′) +�−1� +; +• D = ϕ +αGrp +(π(BX ),π(BX′ ),π(BX′′ )); +• E = idπ(BX) × ϕm′ +(X′,X′′); +and +• F = ϕm′ +(X,X′⊔X′′). +This diagram is the result of applying the ϕ(−)-construction, given in Example 33 in +which a functor is converted into a profunctor, to an evident commutative diagram +at the groupoid level. +For this reason, the modification, ω′, satisfies the cocycle equation in [61, §4.3], +or [58, page 17], when we are given manifolds X′, X′, X′′ and X′′′. +The bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, is also compatible with +the unitor natural transformations in 2Cob(n,n+1,n+2) and in vProfGrphf, as well +as with the braiding. This follows from considerations analogous to those we have +just given. +We state for the sake of reference: +Theorem 176. The bifunctor, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +i.e. what we have called the once-extended Quinn TQFT75 of Subsection 6.2, can +be upgraded to being a symmetric monoidal bifunctor. +□ +75The name used is now justified by this theorem + +A CATEGORIFICATION OF QUINN’S TQFT +161 +6.6.2. A symmetric monoidal structure for the B-decorated case. The bicategory +2Cob(n,n+1,n+2) induces an obvious symmetric monoidal structure on the bicat- +egory 2Cob +(n,n+1,n+2) +B +, defined in Subsection 6.3. The tensor product of (Σ, f Σ) +with (Σ′, gΣ′) is given by +(Σ, fΣ) ⊗ (Σ′, gΣ′) = (Σ ⊔ Σ′, f Σ ⊗ gΣ′), +where +f Σ ⊗ gΣ′ := +� +⟨φ, φ′⟩ | φ ∈ fΣ and φ′ ∈ gΣ′ +� +. +The discussion in that subsection can easily be adapted for the case where we +have decorated manifolds. This follows again from the calculations in §5.4.3. We +hence have: +Theorem 177. The bifunctor, 2QB, defined in Subsection 6.3, (Definition 154), +i.e., what we have called the finitary once-extended Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +can be upgraded to being a symmetric monoidal bifunctor. +□ +6.6.3. A symmetric monoidal structure for the Morita valued once-extended Quinn +TQFT. So far in this section, we have sketched proofs of the facts that both bi- +functors, +2QB : 2Cob(n,n+1,n+2) → vProfGrphf, +and +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +can be given the structure of a symmetric monoidal bifunctor, hence they are fully +fledged once-extended TQFTs. +We now sketch the construction of the symmetric monoidal structure of the +Morita valued once-extended Quinn TQFT, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor, +defined in §6.4.5. In other words, we show that 2Q +Mor +B +is a fully fledged once- +extended TQFT. +Given that the Morita valued once-extended TQFT is obtained as a composite +of bifunctors, +2Cob +(n,n+1,n+2) +B +2QB +−−−→ vProfGrpfin +Lin(2) +−−−−→ Mor, +it will be sufficient to prove that the latter arrow, from vProfGrpfin to Mor, can +be upgraded to being a symmetric monoidal bifunctor. +That Lin(2) can be given a symmetric monoidal structure is a purely categorical +/ algebraic exercise, so we will just give a sketch of that claim. Our main tool +is Mitchell’s theorem, here Theorem 160 on page 131, for a linear category C, +with finitely many objects, using the approach that we took of constructing a +bifunctor [−] : C−Mod → [C]−Mod as well as the two profunctors, AC and BC, +giving a Morita equivalence; see the discussion in §6.4.3, starting on page 131, +and especially Remark 163. In fact, we will show how that theory allows one to +define a bifunctor from the bicategory vProf, of Vect-enriched categories and +Vect-enriched profunctors between them, to Mor, which is the ‘reflector’ onto +the sub-bicategory corresponding to Mor. +We note that no finiteness or other +restrictions are needed for this, which is why this construction works on the whole +of vProf, and not just in vProfGrpfin. +We define a bifunctor, [−]: vProf → Mor, as follows: + +A CATEGORIFICATION OF QUINN’S TQFT +162 +• remembering that vProf 0 consists of linear categories, we have that +[−]0 : vProf 0 → Mor0 +sends C to the algebra [C]; +• if H : C ↛ D is a Vect-valued profunctor, then [H]: [C] ↛ [D] is the com- +posite bimodule, obtained by the following composition of profunctors76, as +discussed in §6.4.3, particularly in Remark 163, +BC • H • AD : [C] ↛ C ↛ D ↛ [D]. +We recall that this bimodule is determined only up to natural isomor- +phism as it is given by multiple use of coends. The question of the lack of +associativity of the composition is absorbed into that as well, but a standard +form of representing bimodule can be given, namely, +[H] = +� +p∈C,q∈D +H(p, q). +The left and right algebra actions of [C] and [D] are as discussed in §6.4.3. +Here, following our conventions for profunctors77, H : C ↛ D corresponds +to a functor H : Cop × D → Vect. +• We next assume given H : C ↛ D and K : D ↛ E. We do not expect that +composition will be preserved by [−], so look at the two ways of producing +things, namely [H • K] and [H] • [K]. Firstly +[H • K] = BC • H • K • AE, +and then +[H] • [K] = BC • H • AD • BD • K • AE. +(We will ignore any problems arising from composition being non-associative +in vProf, as these can be handled using associators, etc., in a standard +way, completely analogous to handling non-associativity of tensor products +in Mor, which is a special case of this one.) +We recall, from Lemma 161, that the profunctors AD •BD and D(−, −), +from D to itself, are naturally isomorphic, so we have an invertible 2-cell +[H] • [K] ⇒ [H • K], in Mor, as required, and moreover, this satisfies the +appropriate cocycle identity, given a triple of composable bifunctors. We +note that explicit formulae can be given for this 2-cell in terms of the direct +sum over the objects of C and E, with tensoring78 over the algebra [D]. +• We also need an invertible 2-cell, id[H] ⇒ [idH]. The domain of this is +the identity morphism on [H], as a bimodule, and the righthand side is the +whiskered composite, BC •idH •AD. This is easily checked to be isomorphic +to the identity on [H], for instance using that [H] ∼= ⊕p∈C,q∈DH(p, q). +We omit the verification that this structure determines a bifunctor, [−]: vProf → +Mor, as that verification is quite long, whilst not being that full of insight. +We next ask if [−] is monoidal, or more precisely whether it can be given a +monoidal structure. +We start with F : A ↛ B and G: C ↛ D, and also have +F ⊗ G: A × C ↛ B × D, given by +(F ⊗ G)((a, c), (b, c)) = F(a, b) ⊗κ G(c, d), +76Note once again our composition convention for profunctors, cf. Subsection 2.8. +77Again see Subsection 2.8. +78In those terms, the invertibility of the 2-cell is related to the distributivity of tensors and +direct sums. + +A CATEGORIFICATION OF QUINN’S TQFT +163 +so +[F ⊗ G] = ⊕(a,b) ⊕(c,d) F(a, b) ⊗κ G(c, d). +On the other hand, +[F] ⊗ [G] = +� +⊕(a,b) F(a, b) +� +⊗κ +� +⊕(c,d) G(c, d) +� +, +and these two are naturally isomorphic by the standard argument relating tensors +and direct sums. +Interaction of [−] with the monoidal units is easy. +In vProf, the monoidal +unit is the single object linear category having a 1-dimensional vector space as its +endomorphism ring, i.e., it is actually an algebra in its own right, being essentially +a copy of κ itself, and applying [−] does nothing to it! We thus have that [−] is a +normalised monoidal bifunctor. +That [−] respects the symmetric monoidal bicategory structure (up to specified +isomorphisms) is then, once again, a result of the natural isomorphisms linking +tensor products of direct sums with direct sums of tensor products, when that is +suitably interpreted. +We leave it to the reader explicitly to write down the full symmetric monoidal +bifunctor structure of [−]: vProf → Mor, as just outlined, in the language of Defi- +nition 30, similarly to that which we did for the bifunctor 2QB : 2Cob(n,n+1,n+2) → +vProfGrphf in §6.6. +We note that [−] is essentially the identity bifunctor when it is restricted to +Mor (which can be considered as a sub-bicategory of vProf). We will not explore +further this relationship between vProf and Mor as this might distract from the +use we will be making of the basic reflection bifunctor, [−]. +The implication of the above is that the ‘Mor-valued’ once-extended TQFT, +developed in Subsection 6.4, is, as we claimed, actually a fully-fledged once-extended +TQFT, as it can be given the structure of a symmetric monoidal bifunctor from +2Cob +(n,n+1,n+2) +B +to Mor. +This proves: +Theorem 178. The bifunctor, 2QB, defined in §6.4.5, i.e., what we called the +Morita valued once-extended Quinn TQFT, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor. +can be upgraded to being a symmetric monoidal bifunctor. +□ +Part 4. Calculations for classifying spaces of ω-groupoids +We now have a TQFT and a once-extended TQFT that depend on the choice +of a homotopy finite space, B. In this Part, in Section 7, we will show that we +have the means for efficiently calculating the values of such TQFTs and once- +extended TQFTs, if we restrict to homotopy finite spaces B that are classifying +spaces of homotopy finite crossed complexes (the latter being equivalent to strict +ω-groupoids, [27, §13.6], that are themselves homotopy finite). In Section 8, we will +further use these ideas to calculate the values for specific examples. +As before, let n be a non-negative integer, and B be a homotopy finite space. +We have for s, a complex parameter, Quinn’s finite total homotopy TQFT, +Q(s) +B : Cob(n,n+1) → VectC, + +A CATEGORIFICATION OF QUINN’S TQFT +164 +(Definition 83), as well as the finitary once-extended Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin +(Definition 153), and the Morita valued once-extended Quinn TQFT, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor, +(Definition 154). These can all be explicitly computed, combinatorially. This can +be achieved in various ways, for example, by passing to the category of simplicial +sets, or similar combinatorial models for homotopy theory. +The calculations of Q(s) +B , 2QB and 2Q +Mor +B +, using the category of simplicial sets, +require only finite calculations, since, by Ellis’ theorem, [46], path-connected spaces +with a finite number of non-trivial homotopy groups, all of which are finite, can be +represented (up to homotopy) by finite simplicial groups. We will examine this in +a separate paper, as it requires some development of other techniques, especially +around triangulations. +For the remainder of this paper, we will outline the interesting special cases of +such explicit combinatorial computations for the case in which B is the classifying +space, BA, of a homotopy finite crossed complex, A. We apply the tools of the +homotopy theory of crossed complexes, developed for instance in [27, 116], and show +that their use yields explicit formulae for Q(s) +B +and 2QB, and, as a consequence, +for 2Q +Mor +B +, as well. +A particularly simple case is when A is a finite crossed complex, with a unique +0-cell. The formulae we will obtain for Q(s) +B : Cob(n,n+1) → VectC, in this case, +extend those of our previous paper, [53], (which only dealt with the case of closed +manifolds), in the particular case of a trivial homology class. +We will summarise the necessary parts of the theory of crossed complexes and +will also give some new results that will be needed to enable the computations to +proceed without difficulty. +The category of crossed complexes, which strictly includes the category of strict +2-groups, [7], and [27, §2.5], is equivalent to the category of (strict) omega-groupoids. +For a precise statement and proof of this see [27, §13.6]. This latter fact justifies +the title of this paper. Crossed complexes also model strict ∞-groupoids via the +nerve construction. +It should be noted that homotopy finite crossed complexes do not model all +homotopy finite spaces. The specification of the homotopy types thus classified is +slightly complicated, and will not be needed here, so will not be recalled in this +paper. +Such crossed complexes do, however, model all 2-types, X, that is, all +spaces, X, such that πi(X, x) = 0 if i ≥ 3, and for all possible choices of base- +points. The explicit formulae we will construct, thus apply to the TQFTs Q(s) +B , +and the extended TQFTs 2QB and 2Q +Mor +B +, where B is the classifying space of +a finite 2-group (equivalent to a crossed module), as mentioned above, and as in +[53]. In particular, the last part of this paper leads to a construction of TQFTs and +extended TQFTs derived from discrete higher gauge theory based on a 2-group; +see [124], and also [34, 31]. In the extended case, the formulae are similar to those +derived from the ‘tube algebras’ considered in [32] and [33, Section 3], in the context +of excitations of strict 2-group topological phases. + +A CATEGORIFICATION OF QUINN’S TQFT +165 +In more detail, in the coming section, we will review some of the basics of the +homotopy theory of crossed complexes, and their classifying spaces, and then prove +some refinements of well know results in the literature, which will lead to the explicit +formulae for TQFTs and once-extended TQFTs derived from crossed complexes +mentioned above. This latter work will be done in Section 8. +We note that subsections 7.1 – 7.4 contain no new results, and essentially fol- +low [25, 26, 27, 116]. Subsection 7.5, on fibrations of crossed complexes, revisits +definitions and results from [22, 26]. A crucial new result, refining the main theo- +rem in [26], concerns a crossed complex model for the fibre of the restriction map +(BA)|S| → (BA)|T |, where A is a crossed complex, and T is a subcomplex of a sim- +plicial set, S, ( | − | here denoting geometric realisation). This result will be used, +later, to write down, in all detail, TQFTs and extended TQFTs derived from ho- +motopy finite crossed complexes. This is done in Section 8. Before that, the results +in Subsection 7.6 are essentially in [49, 53], and they allow for a simple calculation +of the homotopy content of finite crossed complexes, akin to the well known for- +mula for the Euler characteristic of a finite CW-complex, as the alternating sum of +cardinalities of the sets of i-cells. +7. Crossed complexes: their homotopy theory and their classifying +spaces +The main sources for this section are [27], and / or some of the sources already +listed, which are summarised therein. A new source, that we hope will be available +shortly, is our (pre) preprint, [54], which handles some of those aspects of the theory +of crossed complexes that were not included in the main source, cited above. +This section will, naturally, consist of lots of definitions, with some commentary. +7.1. Definition of crossed complexes, and related notions. We first need +some useful terminology. +• Let X be a set. By a set over X, we will mean a set, Y , together with a +surjective map, β : Y → X. +• A groupoid right-action of a groupoid, Γ = (s, t: Γ1 → Γ0), on a set over Γ0, +(Y, β), is an operation which, given y ∈ Y and an arrow, (β(y) +γ−→ x) ∈ Γ1, +associates y ⊳ γ, also denoted y ⊳ (β(y) +γ−→ x), in Y , with β(y ⊳ g) = x. This +is such that if β(y) = a, we always have: +� +y ⊳ (a +γ−→ b) +� +⊳ (b +γ′ +−→ c) = y ⊳ (a +γγ′ +−−→ c), +y ⊳ (a +1a +−→ a) = y. +A groupoid action of Γ gives rise to a functor Γ → Set. +Various equivalent formulations of the definition of crossed complex can be found +in [12, 13, 20, 27, 116], and many other places in the literature. Note that Baues, +in [12], and [13], prefered to call them crossed chain complexes. For convenience, +we recall one form of the definition here. +Definition 179 (Crossed complex, [27, §7.1.iii]). A crossed complex, A = (An)n∈Z+ +0 , +is given by: +• a set, A0, called the set of objects of A; +• a groupoid, A1 = (s, t: A1 +1 → A0), with object set A0; +• for each integer n ≥ 2, a totally disconnected groupoid, +An = (β : A1 +n → A0), + +A CATEGORIFICATION OF QUINN’S TQFT +166 +with object set A0; +• whenever n ≥ 2, a groupoid map ∂n = ∂ : An → An−1, which is required to +restrict to the identity on the set of objects A0; +and +• a groupoid right-action, ⊳, of A1 on all the underlying sets, β : An → A0, +over A0, for all n ≥ 2, which is required to preserve the composition and +the identities in each An, where n ≥ 2. +Given x, y ∈ A0, a = (x +a−→ x) ∈ A1(x, x), and g = (x +g−→ y) ∈ A1(x, y), we also +write a ⊳ g := g−1ag ∈ A1(y, y). +This data is to satisfy the following additional conditions: +(1) for all n ≥ 3, given (x +a−→ x) ∈ A1 +n with x ∈ A0, then ∂(∂(x +a−→ x)) = 1x; +(2) for n ≥ 2, and given any x, y ∈ A0 and g ∈ A1(x, y), if a ∈ An(x, x), +we have ∂(a ⊳ g) = ∂(a) ⊳ g. (This is sometimes called the first Peiffer +condition.) +(3) for any x ∈ A0 and a, b ∈ A2(x, x), then a ⊳ ∂(b) = b−1ab. +(This is +sometimes called the second Peiffer condition.) +(4) If n ≥ 3, then given any x ∈ A0, and a ∈ A2(x, x), b ∈ An(x, x), we have +b ⊳ ∂(a) = b, and so ∂(A2) ≤ A1 acts trivially on all An for n ≥ 3. +(5) If x ∈ A0 and n ≥ 3, then each group, An(x, x), is abelian. +Our notation, A = (An)n∈Z+ +0 , for a crossed complex leaves the boundary maps, +∂, and the actions of the groupoid, A1, implicit in the notation. +Another useful way of picturing this structure is: +A = +� +. . . +∂−→ A1 +n +∂−→ A1 +n−1 +∂−→ . . . +∂−→ A1 +2 +∂−→ A1 +1 +t +⇒ +s A0 +� +. +We will usually refer to the arrows in An, as morphisms, making clear in which +dimension they are. We may also call them n-morphisms or n-arrows if, in some +context, that will be clearer. +If A0 is a singleton, and hence the groupoid, A1, has just a single object, {∗}, +as will often be the case, we may also write this as: +. . . +∂−→ An +∂−→ An−1 +∂−→ . . . +∂−→ A2 +∂−→ A1 +β→{∗}. +We note that, here, we have identified each groupoid, A1 +i , with its group of mor- +phisms. +This latter type of crossed complex is sometimes referred to as being reduced. +The two sources, [12, 13], restrict attention to such reduced crossed complexes, +calling them ‘crossed chain complexes’, but we will need the non-reduced variety +as we will be considering the crossed complexes corresponding to function spaces, +where the restriction to a reduced case would be very unnatural. +Such a reduced crossed complex is thus a chain complex of groups, such that +Ai is abelian if i ≥ 3, together with actions of the group, A1, on all the groups in +higher dimensions and, of course, satisfying some other axioms, as above. +For later use, we will also need the following. +Definition 180 (Crossed modules of groupoids). A crossed module of groupoids +is given by a 2-truncated crossed complex, that is, one in which each groupoid, An, +for n > 2, consists just of identity loops on each object. +It is easy to give a direct definition of this notion, independently of that of crossed +complex, but, for the moment, we leave that to the reader to find in the sources +or to concoct by throwing away axioms and structure in dimensions greater than 2 + +A CATEGORIFICATION OF QUINN’S TQFT +167 +in the above. For ease of reference, we will give an explicit definition of the group +(i.e. reduced) version later on in §8.4.4. +We will define truncation more formally shortly. +Definition 181. (Maps between crossed complexes, see, for instance, [27, §7.1.iii].) +Let A = (An)n∈Z+ +0 and B = (Bn)n∈Z+ +0 be crossed complexes. A crossed complex +map, f = (fn)n∈Z+ +0 : A → B, is given by a set map, f0 : A0 → B0, and groupoid +maps fi : Ai → Bi, where i ≥ 1, that restrict to f0 on objects. These are required +to preserve the actions of A1 and B1, and the boundary maps in A and B, in the +obvious way. +Crossed complexes and the maps between them form a category which will be +denoted Crs. As is shown in [27, §7.2], Crs is closed under all small limits and +colimits. +The following is essentially in [27, §7.1.vi]. +Definition 182. Given a positive integer n, a crossed complex, A = (Ai)i∈Z+ +0 , is +said to be n-truncated if, for i > n, the groupoids, Ai, have only identity morphisms. +We thus have that, for i > n, each of the groupoids, Ai, is a discrete / trivial +groupoid on the set of objects, A0. +We have inclusion functors, ι1 : Grp → Crs, sending a groupoid to the obvious +1-truncated complex, and ι0 : Set → Crs, sending a set to the obvious 0-truncated +crossed complex, which is thus discrete in all dimensions. More generally, we can +consider the inclusion, ιn, of the subcategory of n-truncated crossed complexes into +Crs. All of these will be treated as if they are inclusions of subcategories, and then +the ιn may, sometimes, be omitted. For instance, we may think of groupoids as 1- +truncated crossed complexes, and so once we have defined the classifying space BA +of a crossed complex, A, then we may treat the construction of a classifying space +of a group or groupoid as being a particular case of that more general construction, +without making additional comments. +These inclusion functors have right adjoints, denoted Tn, so that T1 and T0 +(respectively), send A = (An)n∈Z+ +0 to the groupoid, A1, and to the set, A0. +The functors, ι1 : Grp → Crs and ι0 : Set → Crs, also have very useful left +adjoints79, π1 : Crs → Grp, the fundamental groupoid functor, and π0 : Crs → Set, +the set of components functor. In simple terms, these are given as follows: +Definition 183 (The functors, π1(A, A0) and π0(A)). Let A = (An)n∈Z+ +0 be a +crossed complex. The fundamental groupoid of A is defined as: +π1(A) := A1/∂(A2). +(We will sometimes denote π1(A) by the more suggestive π1(A, A0).) We also put: +π0(A) := π0(A1), +the set of connected components of the groupoid at the base of A. +As the notation indicates, π1(A, A0) is a groupoid with one object for each element +of A0, and we note that π0(A) = π0(π1(A, A0)), the set of components of the +fundamental groupoid of A. +As the functor, π0, is left adjoint to ι0 : Set → Crs, it comes with a unit map, +ηA : A → ι0 +� +π0(A) +� +, +79The other functors ιn also have left adjoints, but we will not be using them here. + +A CATEGORIFICATION OF QUINN’S TQFT +168 +which is a quotient map. This quotient map is, thus, really part of the definition +of π0(A), although it is often not mentioned explicitly. +This map, ηA, is, in fact, also the coequaliser morphism of the evident pair of +maps from AI to A, being evaluation at 0 and 1 respectively, analogously to the +situation in the topological case that we used earlier. Here by AI, we mean the +crossed complex, CRS(I, A), of maps from the unit interval groupoid, which we +denote here by I, thought of as a 1-truncated crossed complex. This groupoid, I, +which we will meet again very shortly, is the fundamental groupoid of the unit inter- +val, [0, 1], based at the two end points. We will see the construction of CRS(A, B) +in general a bit later. As we noted above, we will often, even usually, omit the ι0 +from the notation, thus tacitly thinking of a set as the set of objects of the corre- +sponding 0-truncated crossed complex, which will be denoted by the same letter, +and we will do the analogous thing for ι1. +We will return to this topic below, but before that we need: +Definition 184 (Fibres of a crossed complex map). Let A = (An)n∈Z+ +0 and B = +(Bn)n∈Z+ +0 be crossed complexes. Let f = (fn)n∈Z+ +0 : A → B be a crossed complex +map, and let b ∈ B0. The fibre of f : A → B, at b, is the sub-crossed complex, +f −1(b) = (Cn)n∈Z+ +0 , of A, with object set C0 = f −1 +0 (b), and such that Cn consists +of those elements, a ∈ An, such that fn(a) = 1Bn +b , the identity of the groupoid, Bn, +at b. +As in the above definition, let b ∈ B0, then we let ˆb denote the sub-crossed +complex of B with object set, {b}, and only identity arrows. We clearly have a +pullback diagram +f −1(b) +� +inc +� A +f +� +ˆb +inc +� B. +(Inclusions of crossed complexes, with the obvious meaning, will sometimes be +denoted inc, provided no confusion arises.) +Just as in the topological case, the ‘set’, π0(A), is not just a set. As given above, +it is a quotient of A0, so its ‘elements’ would naturally be thought of equivalence +classes of elements of that set of objects of A. That, however, forgets that π0(A) +is also a quotient of A, and the quotienting map, ηA, is really part of the structure +of this ‘set of connected components’. +We now give some formal definitions. +Let A = (An)n∈Z+ +0 be a crossed complex. +Definition 185 (PCx(A) and � +π0(A)). Given x ∈ A0, we define the crossed com- +plex, PCx(A) := (Bn)n∈Z+ +0 , in the following way. +• The set, B0, of objects of PCx(A) consists of all elements in A0 connected +to x in the groupoid, A1, so if a ∈ B0, there is an arrow a → x. +• For each positive integer n, the set of morphisms in Bn consists of the +morphisms in An connecting elements in B0, so either, if n = 1, connecting +two elements of B0 in the groupoid A1, or, if n > 1, in the set of loops, +An(b, b), for some b ∈ B0. +We call the crossed complex PCx(A), the path-component of x in A. +We also may write � +π0(A) = {PCx(A) | x ∈ A0}, the collection of these path- +components. (Note that different elements x ∈ A0 may induce the same PCx(A)). +A crossed complex is said to be path-connected if it only has one path-component. + +A CATEGORIFICATION OF QUINN’S TQFT +169 +We leave it to the reader to check that indeed PCx(A) is a crossed complex. +We note that, for c ∈ A0, PCc(A) is the fibre of ηA over ηA(c). +Remark 186. Given a crossed complex A, there is an obvious epimorphism of +crossed complexes, +� +B∈� +π0(A) +B → π0(A). +This epimorphism maps the crossed complex, PCc(A), to the element ηA(c). (This +is almost a bit of ‘double think’, but, later on, it will be useful to distinguish between +π0(A) and � +π0(A).) +Remark 187. Let A = (An)n∈Z+ +0 be a crossed complex. +We have, in fact, a +canonical isomorphism of crossed complexes, +� +B∈� +π0(A) +B −→ A, +where on the cofactor, B = (PCc(A) → π0(A)), the map is the inclusion of B. This +says that any crossed complex, A, is a coproduct of its connected components. +Just as with spaces, crossed complexes come with a notion of homotopy groups. +They are defined to be the obvious homology groups. +Definition 188 (Homotopy groups of crossed complexes). Let A = (An)n∈Z+ +0 be +a crossed complex. Let c ∈ A0, and let n ≥ 1. +We define πn(A, c) to be the group, +ker +� +∂ : An(c, c) → An−1(c, c) +� +/im +� +∂ : An+1(c, c) → An(c, c)) +� +, +thus, in particular, π1(A, A0)(c, c) = π1(A, c). +7.2. Fundamental crossed complexes of filtered spaces. Many of the prime +examples of crossed complexes come from filtered spaces, and, in particular, from +CW-complexes considered with their natural skeletal filtration. For a more com- +plete view of this, see [27, §7.1.i] and the development of related ideas there. +We first need to say what we mean by a filtered space, and the appropriate +notion of maps between them. +Definition 189. A filtered space, X∗, is a CGWH space, X, together with an +increasing sequence, X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X, of subspaces of X. +A filtered map, f : X∗ → Y∗, between filtered spaces, is a continuous map, +f : X → Y , of the ambient spaces, such that f(Xi) ⊆ Yi for all i ∈ Z+ +0 . +We +let Fil denote the category of filtered spaces and filtered maps. +We have a crossed complex functor, Π: Fil → Crs, sending a filtered space, X∗, +to its fundamental crossed complex, Π(X∗), defined as follows. +Definition 190. The fundamental crossed complex, Π(X∗), of a filtered space X∗, +is specified by the following: +• the set of objects of Π(X∗) is Π(X∗)0 = X0; +• the groupoid, Π(X∗)1, is given by the fundamental groupoid, π1(X1, X0), of +X1, with set of base-points X0; +• if n ≥ 2 and x ∈ X0, then Π(X∗)n(x) = Π(X∗)n(x, x) := πn(Xn, Xn−1, x), +the usual relative homotopy group based at x80; +80Recall from page 14, that if G is a groupoid, then G(x) is the same as G(x, x), and is the +vertex group of a groupoid at an object x. + +A CATEGORIFICATION OF QUINN’S TQFT +170 +• for each n ≥ 2, and each x ∈ X0, the boundary map, +∂ : Π(X∗)n(x) → Π(X∗)n−1(x), +of Π(X∗) is the map appearing at the relevant position in the long homotopy +exact sequence of the triple (Xn, Xn−1, Xn−2), +and +• the action of π1(X1, X0) is the standard one. +We direct the reader to [27, §7.1.v] for more explanation. +Most filtrations appearing in this paper will be skeletal filtrations of CW-complexes, +X, and, to start with, these may be denoted Xsk := (X0 ⊆ X1 ⊆ . . . ), where Xi is +the i-skeleton of X. In this case, X, and all the spaces, Xi, are CGWH. As usual, +a filtered map, f : X → Y , between CW-complexes with the skeletal filtrations, is +called cellular. +We will give quite a few examples, so as to fix some notation. First we consider +some basic spaces. +Example 191. Let I = [0, 1], with the standard CW-decomposition with two 0- +cells, at 0 and 1, and one 1-cell. +If n ≥ 1, we let Sn have the CW-decomposition with one 0-cell, denoted ∗, for +concreteness at the south pole, and one n-cell. +Let Dn+1 have the CW-decomposition for which Sn is subcomplex, and we have +an additional (n + 1)-cell attaching along the identity map, Sn → Sn. +We have: +• Π(Isk) ∼= ι1(π1(I, {0, 1})). Here, in π1(I, {0, 1}), we have objects 0 and 1, and +only two non-identity morphisms, 0 +(0,1) +−−−→ 1 and 1 +(1,0) +−−−→ 0, so this is exactly the +‘unit interval groupoid,’ that we denoted by I above, but now considered as a +crossed complex in the way that we mentioned earlier. All the higher dimensional +groupoids are discrete. +• Π(S1 +sk) ∼= · · · → 0 → 0 → 0 → Z → ∗. +• Π(D2 +sk) ∼= · · · → 0 → 0 → Z +id +−→ Z → ∗. Here the action of Z ∼= π1(S1, ∗) on +π2(D2, S1, ∗) ∼= Z is the trivial one. +• If n ≥ 2, then Π(Sn +sk)n = Z, Π(Dn+1 +sk +)n = Z and Π(Dn+1 +sk +)n+1 ∼= Z, and all +other groupoids are trivial. Hence: +Π(Sn +sk) ∼= · · · → 0 → 0 −→ Z → 0 → · · · → 0 → ∗, +Π(Dn+1 +sk +) ∼= · · · → 0 → Z +id +−→ Z → 0 → · · · → 0 → ∗. +We will also need the fundamental crossed complex of the n-simplex, with its +skeletal filtration, and, more generally, of the geometric realisation of simplicial +sets, where the crossed complex can also be specified algebraically in terms of the +combinatorial description of the simplicial set, by using the crossed complex version +of the homotopy addition lemma. A good source for the details on these topics is +§9.10 of [27]. +Before we look at those families of examples, we will need to recall a few results on +the more general case of CW-complexes. We note that referring to a CW-complex +means that the space comes with a specified CW-decomposition, and the cells used, +attaching and characteristic maps, etc., are regarded as part of the structure. In +fact, it will be useful, for the sake of the exposition, to be a bit more restrictive, +and to consider what we will call special CW-complexes. + +A CATEGORIFICATION OF QUINN’S TQFT +171 +Definition 192 (Special CW-complex). A special CW-complex is a CW-complex, +X, for which the attaching maps of all n-cells, for n ≥ 2, are such that the unique +0-cell of Sn−1 is sent to a 0-cell of Xn−1. +Note that it follows that the attaching maps and characteristic maps of each cell of +a special CW-complex are cellular. +Let CW be the category of CW-complexes, (each given a specified CW-decomposition), +and cellular maps. +We let sCW be the full subcategory, of CW, whose ob- +jects are the special CW-complexes. The fundamental crossed complex functor, +Π: Fil → Crs, restricts to functors, Π: CW → Crs, and Π: sCW → Crs. +The fundamental crossed complex of a CW-complex is ‘free’ on its cells [20, +Corollary 7.11]. +Let us explain what this means, for the sake of exposition, in +the particular case of special CW-complexes. The following is adapted from [27, +Example 7.3.19 and Corollary 8.3.14]. +First the intuitive idea, before we give a few more details. +Suppose that X +is a CW-complex, then, for each 0-cell / vertex, x, we have Π(X∗)n(x, x) = +πn(Xn, Xn−1, x), which, if n ≥ 3, is well known to be a free module, over the +fundamental group π1(X, x), free on the set of n-cells. Moreover, π1(X1, X0), it- +self, is a free groupoid, over the set of 1-cells, of X. For n = 2, π2(X2, X1, x) is a free +crossed π1(X1, X0)-module, [27, §2.2]. The idea of ‘freeness’ in the crossed complex +case is that we can completely specify a crossed complex map, f : Π(X∗) → A, by +saying where that generating set of n-cells is to be sent in each dimension. Of +course, that assignment should be compatible with all the actions, boundary maps, +etc., so requires quite a bit more work, and a bit more notation, to make this +precise. +Given a special CW-complex, X, and n ∈ Z+ +0 , we let C(X, n) be the set of n-cells +of X. Given an n-cell, c ∈ C(X, n), we let Dn +c = Dn and Sn−1 +c += Sn−1, (the latter +being empty if n = 0), and let ic: Sn−1 +c +→ Dn +c be the inclusion. Supposing that +n ≥ 1, let ψc : Sn−1 → Xn−1 be the attaching map of c. Let φc : Dn +c → Xn be the +characteristic map of c. The inclusion of the (n − 1)-skeleton Xn−1 into Xn, the +n-skeleton of X, will be denoted ιn : Xn−1 → Xn, in the two diagrams below. +We have a pushout diagram in the category CGWH, +� +c∈C(X,n) +Sn−1 +c +� +c∈C(X,c) +ψc +� +� +c∈C(X,n) +ic +� +� +c∈C(X,n) +Dn +c +� +c∈C(X,c) +φc +� +Xn−1 +ιn +� Xn. +(The vertical arrows arise from universal properties of disjoint unions.) Note that +all maps appearing in the diagram above are cellular if X is a special CW-complex. + +A CATEGORIFICATION OF QUINN’S TQFT +172 +The diagram below is a pushout in the category of crossed complexes. +� +c∈C(X,n) +Π(Sn−1 +c,sk ) +� +c∈C(X,c) +Π(ψc) +� +� +c∈C(X,n) +Π(ic) +� +� +c∈C(X,n) +Π(Dn +c,sk) +� +c∈C(X,c) +Π(φc) +� +Π(Xn−1 +sk +) +Π(ιn) +� Π(Xn +sk). +Here Sn−1 +c,sk and Dn +c,sk denote the skeletal filtrations of Sn−1 +c +and Dn +c . Furthermore, +we have a natural isomorphism (of functors from sCW to Crs), +Π(Xsk) ∼= colimn(Π(Xn +sk), Π(ιn)). +This implies the freeness criteria we mentioned before. More exactly, we note the +following. +Remark 193. (For more details, see [27, Definition 7.3.13] and [13, Chapter III], +also reviewed in [49, §2.2.1], and in [50] for fundamental crossed modules.) Let X +be a special CW-complex. A way to state the freeness of Π(Xsk) = (Π(Xsk)n)n∈Z+ +0 +on the cells of X is as follows. +• The groupoid, Π(Xsk)1 = π1(X1, X0), is the free groupoid on the graph +corresponding to the 1-skeleton of X1 of X. In other words, Π(Xsk)1 is the +free groupoid on the set of 1-cells of X, and their attaching maps in X0. +• The totally disconnected groupoid Π(Xsk)2, with X0 as its set of objects, is +the top groupoid of the free crossed π1(X1, X0)-module, cf. [27, 7.3.ii] on +the attaching maps for the 2-cells. For an explicit description of what this +means, see [31, §3.3]. +• If n ≥ 3, then the totally disconnected abelian groupoid, Π(Xsk)n, is a free +π1(X1, X0)-module over the set of n-cells of X, and the boundary map, +∂ : Π(Xsk)n → Π(Xsk)n−1, is derived from the attaching maps of the n- +cells. +For the sake of interpretation, we note that if X is a reduced CW-complex, i.e. +if it has a unique 0-cell, then Π(Xsk) is a reduced crossed complex, which means it +has a unique object, and the group, Π(Xsk)1, is free on the set of loops making up +the 1-skeleton of X. Baues, [13], refers to such a crossed complex as being totally +free. +Remark 194. Let X be a special CW-complex. +Given an (n+1)-cell, c ∈ C(X, n+1), we have an induced map of pointed spaces, +ψc : (Sn, ∗) → (Xn, ψc(∗)), +and we let ι′(c) ∈ πn(X, ψc(∗)) be the element given by the image of the generating +element of πn(Sn, ∗). This gives an element, ι(c) ∈ πn(X, Xn−1, ψc(∗)). +Let, now, +A = (. . . +∂−→ An +∂−→ An−1 +∂−→ . . . +∂−→ A2 +∂−→ A1 +t +⇒ +s A0) +be a crossed complex. +It follows from our previous discussion and, for instance +from [27, page 238], that crossed complex maps, f = (fn)n∈Z+ +0 : Π(Xsk) → A, are + +A CATEGORIFICATION OF QUINN’S TQFT +173 +in one to one correspondence with sequences of maps, +� +f ′ +n : C(X, n) → An)n∈Z+ +0 , +such that, for each n and c ∈ C(X, n), we have fn−1(ι(c)) = ∂f ′ +n(c). +This allows for the inductive construction of maps, Π(Xsk) → A, by giving their +values on the cells of X. +7.3. Homotopy of crossed complexes. A source for much of this review of +the homotopy of maps of crossed complexes is [27, §9.3]. The particular case of +homotopy of crossed modules (of groupoids) is in [28] and, also in [53, §2.6.1]. A +version for reduced crossed complexes, as already mentioned, there called crossed +chain complexes, is given by Baues, [13, page 98]. +We will give a short description of the notion of homotopy of crossed complex +maps, focusing on showing the particular explicit formulae that we will need to write +down the TQFTs and extended TQFTs derived from finite crossed complexes. +7.3.1. Homotopy of crossed complex maps. Throughout this subsection, we fix two +crossed complexes, +A = . . . +∂−→ A1 +n +∂−→ A1 +n−1 +∂−→ . . . +∂−→ A1 +2 +∂−→ A1 +1 +t +⇒ +s A0, +and +B = . . . +∂−→ B1 +n +∂−→ B1 +n−1 +∂−→ . . . +∂−→ B1 +2 +∂−→ B1 +1 +t +⇒ +s B0, +so we have groupoids, A1 = (s, t: A1 +1 → A0), B1 = (s, t: B1 +1 → B0), and totally +disconnected groupoids, An = (β : A1 +n → A0) and Bn = (β : B1 +n → B0), if n ≥ 2, +where β is the map that says where an arrow, in A1 +n or B1 +n, is based. +There are several equivalent ways of defining the notion of homotopy between +morphisms of crossed complexes. Given a crossed complex, A, we can form the +tensor product, I ⊗ A, in a way we will see shortly. (Here, as before, we have used +I to refer to the unit interval crossed complex, Π(Isk).) This gives a model for a +‘cylinder’ on A, so then a homotopy between two maps, f and f ′ : A → B, will be +a morphism, h: I ⊗ A → B, satisfying some fairly obvious conditions as in [27]; see +Theorem 200, below. +For future use, we note that the ‘cylinder’ structure, (e0, e1, σ), on I ⊗ A, is +induced from the morphisms, 0 and 1, from the terminal groupoid, {∗}, to I and +the terminal morphism I +termI +−−−−→ {∗}, so for instance, σ = (termI)A +∗ = term ⊗ A81. +Alternatively, we can use the internal ‘hom’, which we will meet in §7.3.2, and +form AI = CRS(I, A), which we have seen before, and which acts as a cocylinder +on A, so is the analogue of the path space construction. This leads to a homotopy +being seen as a morphism from A to BI, as it can with spaces. +There is another definition of homotopy, which is the crossed complex analogue +of the notion of homotopy of morphisms of chain complexes often given in books +on Homological Algebra. This does not need additional constructions to make it +work and, in fact, is needed to make sense of the construction, CRS, of the internal +hom, and then of the tensor, so we start with this. The idea is that we start with +both a morphism f and an ‘f-homotopy’, i.e. a homotopy that ends at f, and +then obtain the ‘other end’ of the homotopy from that input; see [20, §7.3]. The +restricted version on ‘crossed chain complexes’, i.e., reduced crossed complexes, is +given on page 98 of [13]. +81For more precision on cylinder functors, see [69] and, in this particular case, the hopefully +forthcoming [54]. + +A CATEGORIFICATION OF QUINN’S TQFT +174 +Definition 195. Consider a crossed complex map, f = (fn)n∈Z+ +0 : A → B. An +f-homotopy, H = (hn)n∈Z+ +0 , or 1-fold f-homotopy, is given, in low dimensions, by +• a set map, h0 : A0 → B1 +1, such that t ◦ h0 = f0, so h0 looks like +x ∈ A0 +h0 +�−→ +� +s(h0(x)) +h0(x) +−−−→ f0(x) +� +∈ B1 +1; +• a set map, h1 : A1 +1 → B1 +2, such that β(h1(g)) = f0(t(g)), and h1 looks like +(x +g−→ y) ∈ A1 +1 +h1 +�−→ +� +f0(y) +h1(g) +−−−→ f0(y) +� +∈ B1 +2. +This map, h1, is to be such that, if the morphisms / 1-arrows, g and g′, in A1 +can be composed, then +h1(gg′) = +� +h1(g) ⊳ f1(g′) +� +h1(g′), +so, algebraically, it is a form of derivation. +In higher dimensions, we have, +• if n ≥ 2, a groupoid map, hn : An → Bn+1, which, on objects, restricts to f0, +such that, given x, y ∈ A0, if a ∈ An(x) and g ∈ A1(x, y), then +hn(a ⊳ g) = hn(a) ⊳ f1(g). +We denote the set of 1-fold f-homotopies by CRS1(A, B)f. +In the setting of this definition, given f = (fn)n∈Z+ +0 : A → B, and H = (hn)n∈Z+ +0 ∈ +CRS1(A, B)f, it then follows that we have a crossed complex map, +f ′ = (f ′ +n)n∈Z+ +0 : A → B, +defined by: +f ′ +0(x) = s(h0(x)), if x ∈ A0; +f ′ +1 +� +x +g−→ y) = h0(x) f(x +g−→ y) ∂ +� +f0(y) +h1(g) +−−−→ f0(y) +� +h0(y)−1, if (x +g−→ y) ∈ A1 +1; +f ′ +n(a) = +� +fn(a) hn−1(∂(a)) ∂(hn(a)) +� +⊳ h1(β(a))−1, if n ≥ 2 and a ∈ A1 +n; +see [27, Exercise 7.1.39]. +We put s(H, f) = f ′ and t(H, f) = f, and also write f ′ +(H,f) +−−−→ f. We may say +that (H, f) is a crossed complex homotopy from f ′ to f. +We put CRS0(A, B) = Crs(A, B), the set of crossed complex maps from A to B. +As the notation indicates in its use of s and t, we have a groupoid +CRS1(A, B) = +� +s, t: CRS1(A, B)1 → CRS0(A, B) +� +, +whose objects are the maps, f : A → B, and the morphisms from f ′ to f are +the f-homotopies such that f ′ +(H,f) +−−−→ f. The composition of f ′′ +(H′,f ′) +−−−−→ f ′ and +f ′ +(H,f) +−−−→ f, denoted f ′′ +(J,f) +−−−→ f, with J = (jn)n∈Z+ +0 , is such that, if H = (hn)n∈Z+ +0 +and H′ = (h′ +n)n∈Z+ +0 , then +j0(x) = h′ +0(x) h0(x), if x ∈ A0, +and +jn(x +g−→ y) = hn(x +g−→ y) +� +h′ +n(x +g−→ y) ⊳ h0(y) +� +, if n ≥ 1, and (x +g−→ y) ∈ A1 +n. +Remark 196. If considering possible higher order analogues of this theory, we +note that the proof that J is indeed an f-homotopy uses the second Peiffer condition, +given in Definition 179. In the more general case of 2-crossed complexes, which is in +many ways similar but where the second Peiffer condition need not hold, cofibrancy +conditions are needed, on A, to concatenate homotopies, see [57]. + +A CATEGORIFICATION OF QUINN’S TQFT +175 +7.3.2. The internal hom functor, CRS(−, −), in the category of crossed complexes. +Let, as before, A and B be crossed complexes. +The groupoid, CRS1(A, B) = +� +s, t: CRS1(A, B)1 → CRS0(A, B) +� +, can be ‘ex- +tended’ to a crossed complex, denoted, +CRS(A, B) += +� +. . . +δ−→ CRSn(A, B)1 +δ−→ . . . +δ−→ CRS2(A, B)1 +δ−→ CRS1(A, B)1 +t +⇒ +s CRS0(A, B) +� +, +by considering k-fold homotopies between crossed complex maps for each k ∈ Z+. +This construction is explicitly given in both [25] and [27, §9.3.i]. +With regard to this, we will give some explicit formulae and results that we will +need later in this paper. +Definition 197. Consider a crossed complex map, f = (fn)n∈Z+ +0 : A → B, and let +k ≥ 2. +A k-fold f-homotopy, Hk = (hk +n)n∈Z+ +0 = (hk +0, hk +1, hk +2, . . . ), is given by: +• the choice of an element hk +0(x) ∈ Bk(f0(x)) for each x ∈ A0, so we have a +mapping hk +0(x): A0 → B1 +k, given in the form +x ∈ A0 +hk +0 +�−→ (f0(x) +hk +0(x) +−−−→ f0(x)) ∈ Bk(f0(x)); +• given (x +g−→ y) ∈ A1 +1, the choice of an element, hk +1(x +g−→ y) ∈ Bk+1(f0(y)), +to be such that, if g and g′ can be composed in A1, then +hk +1(gg′) = +� +hk +1(g) ⊳ f1(g′) +� +hk +1(g′); +• given n ≥ 2, and x ∈ A0, a function, hk +n : A1 +n → B1 +n+k, satisfying +β(hk +n(a)) = f0(β(a)), for all a ∈ An. +This mapping, hk +n : A1 +n → A1 +n+k, is to be such that, given any x ∈ A0, the re- +striction of hk +n to An(x) is a group homomorphism, An(x) → Bn+k(f0(x)), +and further, if x, y ∈ A0, a ∈ An(x) and (x +g−→ y) ∈ A1 +1, then +hk +n +� +a ⊳ (x +g−→ y) +� += hk +n +� +a) ⊳ (f0(x) +f1(g) +−−−→ f0(y)). +We let CRSk(A, B)f denote the set of all k-fold f-homotopies. +Let f : A → B be a crossed complex map. Suppose k ≥ 2. By using the obvious +point-wise product of k-fold f-homotopies, as in [27, Definition 9.3.5], we have that +the CRSk(A, B)f has a group structure, and that is abelian if k ≥ 3. +Given k ≥ 2, we have a totally disconnected groupoid, +CRSk(A, B) := + +β : +� +f : A→B +CRSk(A, B)f → CRS0(A, B) + + , +with object set, CRS0(A, B) = Crs(A, B), the set of crossed complex maps, f : A → +B, and with the obvious map, +β : +� +f : A→B +CRSk(A, B)f → CRS0(A, B). +Lemma 198. Let f : A → B be a crossed complex map. Let H2 = (h2 +0, h2 +1, . . . ) be +a 2-fold f-homotopy, then δ(H2) = (δ(h2 +0), δ(h2 +1), . . . ), defined by: +• δ(h2 +0)(x) := ∂ +� +h1 +0(x) +� +, for each x ∈ A0; +• δ(h2 +1)(x +g−→ y) := +� +h2 +0(x) +�−1 ⊳ +� +f0(x) +f1(g) +−−−→ +f0(y) +� +h2 +0(y) ∂(h2 +1(x +g−→ y)), +where (x +g−→ y) ∈ A1 +1, + +A CATEGORIFICATION OF QUINN’S TQFT +176 +and +• given n ≥ 2 and a ∈ A1 +n, +δ(h2 +n)(a) = ∂ +� +h2 +n+1(a) +� +h2 +n(∂(a)) +� +(−1)n� +, +is an f-homotopy, f +(δ(H),f) +−−−−−→ f. +Proof. This is proved by explicit calculations, using the second Peiffer condition +from Definition 179. +□ +Similarly in higher dimensions, for n ≥ 3, we have groupoid maps, +δ: CRSn(A, B) → CRSn−1(A, B), +which, again, restrict to the identity on the set of objects, together with ac- +tions of the groupoid, CRS1(A, B), on all of the totally disconnected groupoids, +CRSn(A, B), for n ≥ 2. This gives rise to a crossed complex, CRS(A, B), the in- +ternal hom in the category of crossed complexes. Again for the details, see [27, +§9.3.i]. +The following groupoid will play a key role later on when we give explicit de- +scriptions of the once-extended Quinn TQFT derived from a finite crossed complex. +Let A and B be crossed complexes. We have the following groupoid, where we +are using the notation of Definition 183, +π1 +� +CRS(A, B), CRS0(A, B) +� += π1 +� +CRS(A, B) +� +. +The set of objects of π1 +� +CRS(A, B), CRS0(A, B) +� +is, thus, the set, CRS0(A, B) = +Crs(A, B), of crossed complex maps, f from A to B, and given f, g : A → B, the set +of arrows from f to g is given by all equivalence classes of homotopies, f +[(H,g]) +−−−−→ g, +connecting f and g, where homotopies are considered up to the equivalence given +by 2-fold homotopies. +7.3.3. Tensor product and homotopies of crossed complexes. A crucial property of +the crossed complexes, CRS(A, B), where A and B are crossed complexes, is that +they vary functorially in both positions, so we have a functor, +CRS(−, −) : Crsop × Crs → Crs, +sending (A, B) to CRS(A, B); see [25]. +This functor, CRS(−, −), acts as an ‘internal hom’, that is to say that CRS(A, B) +behaves like the “object of morphisms” from A to B, so CRS(−, −) is analogous to +the mapping space functor that we mentioned on page 15. We note some explicit +formulae below. +Notation 199. If B is a crossed complex, we have a functor, +CRS(−, B): Crsop → Crs. +It sends a crossed complex, A, to CRS(A, B), and a crossed complex map, f : A′ → +A, to the crossed complex map, φ∗ : CRS(A, B) → CRS(A′, B), such that: +(1) each crossed complex map, φ: A → B, is sent to the composite, +φ ◦ f : A′ → B; +and +(2) given a positive integer k, a crossed complex map, φ: A → B, and a k-fold +φ-homotopy, hk = (hk +0, hk +1, hk +2, . . . ), then f ∗(hk) := (hk +0 ◦f, hk +1◦f, hk +2 ◦f, . . . ), +which is a k-fold (φ ◦ f)-homotopy. This corresponds to ‘pre-composition with f’. +There is also a functor, CRS(A, −), with the ‘opposite’ properties, and which on +morphisms gives ‘post-composition’. + +A CATEGORIFICATION OF QUINN’S TQFT +177 +Given crossed complexes, A and A′, we can also form their tensor product, A⊗A′; +again, for details, see [27, §9.3.iii] and [116, Definition 1.4.]. We have a functor, +⊗CRS : Crs × Crs → Crs, sending (A, A′) to A ⊗ A′, and an exponential law, +Crs(A ⊗ A′, B) ∼= Crs(A, CRS(A′, B)), that holds naturally in A and B, showing +that the functor ‘tensor product with A′’, i.e., −⊗A′, is left adjoint to the functor, +CRS(A′, −), derived from the internal hom. +This gives Crs the structure of a +monoidal closed category, [27, Theorem 9.3.17]. +In fact, the tensor product is +symmetric, so Crs with the above tensor is a symmetric monoidal closed category, +see [27, Theorem 9.3.16], but also take note of the discussion after that result in +that source. +Given a crossed complex, A, we have morphisms, which look like the inclusions +of the ends of a cylinder, and will here be denoted i0, i1 : A → Π(Isk) ⊗ A; see +[116, page 203]. (We note that, as this notation, i0, etc., is overcharged, occurring +in several contexts, often with different meanings, we will sometimes replace i0 +by e0(A), or ιA +0 , etc., depending on the other use of symbols in the setting.) If +k ≥ 2, we also have a canonical inclusion, i: A → Π(Dk +sk) ⊗ A, as Π(Dk +sk) is +a reduced crossed complex. Moreover, there is a morphism from Π(Isk) ⊗ A to +A, which is a partial inverse to the ‘end inclusion’ morphisms. This means that +Π(Isk)⊗ A behaves exactly like a cylinder on A, and can be used to define a notion +of homotopy between morphisms in Crs, which, thankfully, coincides with the one +that we introduced earlier, where we used the abbreviation I for Π(Isk). All this +is very thoroughly discussed in [27, §9.3.i], and we note: +Theorem 200. Let A and B be crossed complexes. Let f : A → B be a crossed +complex map. +• There is a canonical correspondence between homotopies, f ′ +(f,H) +−−−→ f, and +commutative diagrams in Crs of form: +A +f ′ +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +i0 +� Π(Isk) ⊗ A +H′ +� +A. +i1 +� +f +�ttttttttttt +B +• If k ≥ 2, we have a canonical correspondence between k-fold f-homotopies, +Hk, and commutative diagrams in Crs of form +A +f ′ +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +i +� Π(Dk +sk) ⊗ A +H′ +� +B +□ +We could also use H′ in the first bullet point to get a morphism, h: A → +CRS(I, B). This involves first using the symmetry of the tensor product to get a +morphism from A ⊗ I to B and then the adjunction to give a homotopy in that +form, h: A → CRS(I, B). If we write BI for the codomain of that map, then, as +we mentioned earlier, BI has the structure of a ‘cocylinder’, or ‘path space’ object +in Crs. We will not pursue this here, but it is developed further in both [27] and +[13]. +For the geometric interpretation of the tensor product, the following is important. + +A CATEGORIFICATION OF QUINN’S TQFT +178 +Theorem 201. ([27, Theorem 9.8.1]) Let X and Y be CW-complexes. Give X ×Y +the usual structure of a CW-complex. We have a natural isomorphism of crossed +complexes, +Π +� +(X × Y )sk +� ∼ += +−→ Π(Xsk) ⊗ Π(Ysk). +□ +A version of this result, for the slight variant of the tensor product used by +Baues, is also discussed on page 92 of [13]. +7.3.4. Homotopies and totally free crossed complexes. To be able to work fairly +simply with the above notions of homotopy between crossed complex maps, we will +need to be able to construct homotopies in ways analogous to the ‘induction up +the skeleton’ methods used in many topological contexts. It is intuitively clear that +this should be the case, but, as ever, writing out the detailed statement and proof +that matches that intuition needs a bit of care. +The detailed result in Lemma 202, below, is crucial for what follows and is one +such statement. It is given a direct proof in [49, §2.2.6], for the case when the +CW-complex, X, has a single 0-cell, so is ‘reduced’, and in [53, §2.24.1] for general +CW-complexes, but, there, for the particular case in which A is a crossed module. +Related results are also to be found in [27, Corollary 9.6.6], [29, Proposition 7.3 I] +and [13, Chapter III, §4]. +In order to simplify the notation and the exposition, we state the result only +in the case when A has a single object, so A is itself a reduced crossed complex. +Otherwise further compatibility relations are needed. +We will assume that X is a special CW-complex, as in Definition 192. Given an +n ≥ 0, we let C(X, n) be the set of n-cells of X. +Lemma 202 (Free construction of crossed complex homotopies). Suppose that +A = +� +. . . +∂−→ A1 +n +∂−→ A1 +n−1 +∂−→ . . . +∂−→ A1 +2 +∂−→ A1 +1 +β−→ A0 = {∗} +� +is a reduced crossed complex, thus with a single object. Let f : Π(Xsk) → A be a +crossed complex map. +Let k be a non-negative integer, then k-fold f-homotopies are uniquely specified +by their value on the elements of Π(Xsk), defined from the cells of X, the latter in +the sense explained in Remark 193. +Explicitly this means that: +(1) there exists a one-to-one correspondence between 1-fold f-homotopies and +sequences of maps, +(m1 +i : C(X, i) → A1 +i+1)i∈Z+ +0 , +and note that there are no further compatibility conditions between the +maps, m1 +i : C(X, i) → A1 +i+1, and the boundary maps of Π(Xsk) and A. +We will denote this bijection by +(72) +(m1 +i )i∈Z+ +0 �→ Extend1 +X +� +(m1 +i )i∈Z+ +0 , f +� +∈ CRS1(Π(Xsk), A)f. +(2) If k ≥ 2, there exists a one-to-one correspondence, between k-fold f- +homotopies and sequences of maps, +(mk +i : C(X, i) → A1 +i+k)i∈Z+ +0 . +We denote this bijection by: +(mk +i )i∈Z+ +0 �→ Extendk +X +� +(mk +i )i∈Z+ +0 , f +� +∈ CRSk(Π(Xsk), A)f. + +A CATEGORIFICATION OF QUINN’S TQFT +179 +Proof. This follows essentially from Remark 193, whose nomenclature we use. +For k = 1, given (m1 +i : C(X, i) → A1 +i+1)i∈Z+ +0 , then Extend1 +X +� +(m1 +i )i∈Z+ +0 , f +� +is +the unique 1-fold f-homotopy that takes the value, m1 +i , on the set of i-cells of +X (or more precisely on the elements of Π(X)i given by them). +The existence +and uniqueness of Extend1 +X +� +(m1 +i )i∈Z+ +0 , f +� +follows from elementary techniques, since +Π(Xsk) is free on the set of cells of X, in the sense explained in Remark 193. (As +mentioned before, more details can be found in [49, §2.2.6], in the case when X has +a unique 0-cell.) +A similar argument is valid when k ≥ 2. Given a sequence of maps, +(mk +i : C(X, i) → A1 +i+k)i∈Z+ +0 , +then Extendk +X +� +(mk +i )i∈Z+ +0 , f +� +is the unique k-fold f-homotopy that takes the value, +mk +i , on the set of i-cells of X. +□ +Remark 203. A perhaps more conceptual proof of Theorem 202 follows by combin- +ing Theorems 200 and 194, using the fact that the crossed complex, Π +� +(X ×I)sk +� ∼= +Π(Xsk) ⊗ Π(Isk), is free. The same argument works for k ≥ 2. The more combina- +torial approach we have given is, however, often better for use with our presentation +of extended TQFTs. +Theorem 202 naturally leads to the following definition, in which k is a positive +integer. +Definition 204 (k-fold X-homotopy sequence). Let X be a special CW-complex +and let C(X, i) be the set of i-cells of X. Let +A = . . . +∂−→ A1 +n +∂−→ A1 +n−1 +∂−→ . . . +∂−→ A1 +2 +∂−→ A1 +1 +β→ A0 = {∗} +be a reduced crossed complex. +A k-fold X-homotopy sequence is a sequence of maps, (mk +i )i∈Z+ +0 , in the category +of sets, where mk +i : C(X, i) → A1 +i+k. +By Theorem 202, given a crossed complex map, f : Π(X) → A, there is a bijection +(72) between k-fold f-homotopies and k-fold X-homotopy sequences. +We will need a generalisation of Theorem 202 for when (X, Y ) is a CW-pair. We +let X be a special CW-complex, Y be a subcomplex of X, with ι: Y → X denoting +the inclusion map. We also consider a reduced crossed complex, A, as before. +Notation 205. Let f : Π(Xsk) → A be a crossed complex map and let k be a +positive integer. +• Given a k-fold f-homotopy, hk = (hk +j )j∈Z, we define the k-fold Y -homotopy +sequence, denoted +Restrictk +Y (hk) = (mk +j : C(Y, j) → A1 +j+k)j∈Z+ +0 , +to be the restriction of hk to the elements of Π(X)j defined by the j-cells of Y ; +see Remark 193. +• Given a k-fold Y -homotopy sequence, nk = (nk +j : C(Y, j) → A1 +j+k)j∈Z+ +0 , we define +the k-fold X-homotopy sequence, denoted +Expandk(nk, Y, X) = (mk +j : C(X, j) → A1 +j+k)j∈Z+ +0 , +to be such that, given j ∈ Z+ +0 , then mk +j coincides with nk +j over C(Y, j) ⊆ C(X, j), +and otherwise mk +j takes as values the identity element of A1 +j+k. +We use the above notation in the following. + +A CATEGORIFICATION OF QUINN’S TQFT +180 +Lemma 206. Let X be a special CW-complex and A = (An)n∈Z+ +0 be a reduced +crossed complex. Let f : Π(Xsk) → A be a crossed complex map and k be a positive +integer. +(1) Given a k-fold f-homotopy, hk, we have +Extendk +X +� +Restrictk +X(hk), f +� += hk. +(2) Let Y be a subcomplex of X. Let Π(ι): Π(Ysk) → Π(Xsk) be the induced map, +which induces a crossed complex map, going in the other direction, via “restric- +tion”, +Π(ι)∗ : CRS +� +Π(Xsk), A +� +→ CRS +� +Π(Ysk), A +� +, +(see Notation 199). Given a k-fold f ◦ Π(ι)-homotopy, hk, where we note that +f ◦ Π(ι): Π(Ysk) → A, we have that: +Π(ι)∗� +Extendk� +Expandk� +Restrictk +Y (hk), Y, X +� +, f +�� += hk. +Proof. This follows from the freeness of Π(Xsk), that of Π(Ysk), and Lemma 202. +□ +Remark 207. It is well known that inclusions of CW-complexes are cofibrations, +both in the standard (Quillen/Serre) model category structure, and in the Strøm +Hurewicz model category structure on the category of (compactly generated and +weak Hausdorff) topological spaces. More generally one has the notion of a relative +CW complex, in which one has a pair, (X, Y ), of spaces where X is obtained from +Y by attaching cells. There is an analogous notion for crossed complexes, called a +morphism of relatively free type in [27, section 7.3.iii]. The idea is that a morphism, +f : A → B, of crossed complexes, is of relatively free type if it can be constructed +by forming a succession of pushouts, which attach copies of various dimensional +Π(Dn+1 +sk +), by attaching maps, Π(Sn +sk) → A(k), to the kth step of the process. +We note that if A +j−→ B is a crossed complex morphism of relatively free type, +then it is a cofibration in the model category structure on Crs given by Brown and +Golasi´nski, [22]. The corresponding cofibrant objects are the ‘crossed complexes of +free type’ of [27, section 7.3.iii], i.e., the free crossed complexes. More on the Brown- +Golasi´nski model category structure can be found in [54], which is our (hopefully +forthcoming) summary of the homotopy theory of crossed complexes from a point +of view near to that of this paper. +The notion, discussed above, is treated on page 109 of [13], and is related to the +cofibration category structure on Crs that is developed both there, and in [12]. From +either viewpoint, Lemma 206 can be used to prove that the fundamental crossed +complex functor preserves cofibrations. +7.4. The classifying space of a crossed complex. To set the scene for this +section a little, recall that lattice models of quantum field theory are finite ap- +proximations to quantum field theories, which are set up on a ‘lattice’ on a closed +(2-)manifold, X, say; see [71, 31]. Typically in simple cases, one ‘colours’ the edges +of the lattice with elements of a finite group, G, subject to some ‘flatness’ con- +ditions relating to the ‘plaquettes’ of the manifold, i.e., the squares cut out by +the embedded lattice, where we impose that the product of the elements around +a square is the identity. If we replace the lattice by a triangulation, which seems +sensible if given dimensions higher than 2, then again we can consider G-colourings +of the edges of the triangulation, TM, with the flatness conditions coming from the +2-dimensional faces of the triangulation. This interprets, after a little bit of work, +as thinking of a G-colouring as being a simplicial map from TM to the classifying +space, BG, of G. (Here we are skating over quite a few important details, but will +return to them later.) + +A CATEGORIFICATION OF QUINN’S TQFT +181 +We recall that, classically, a ‘classifying space’, BG, of a group, G, classifies +principal G-bundles / G-torsors, so given a space, X, one has a bijection between +isomorphism classes of G-torsors on X and homotopy classes of maps from X to the +space, BG. We note that for a discrete group, G, one method of construction of such +a classifying space is as the geometric realisation of the ‘nerve’ of the corresponding +single object groupoid, sometimes denoted G[1] or BG. The correspondence pulls +back a universal G-bundle over BG along a (representative of a) class of maps in +[X, BG] and thus of maps from X to BG. If, as in the case of interest here, G is +discrete, these are also flat connections on X, and this gives geometric meaning +to a G-colouring, which is a more algebraic / combinatorial analogue of this same +notion. +We will see how to construct an analogue of the nerve for crossed complexes and +will link that to A-colourings for a crossed complex A, in similar way. +In [53], we considered colourings by elements of a finite crossed module / 2-group, +or, more generally, of a (homotopy finite) crossed complex, A, and such colourings +corresponded to simplicial maps from TM to the classifying space, BA, of A. +As this section extends the approach in [53], we will recall some of the theory of +such colourings and of the corresponding classifying spaces, but, in addition, will +explore and refine slightly different aspects of the homotopy theoretic results linking +that construction of classifying spaces with the homotopy theory of simplicial sets, +and of function spaces. In fact, a few of the results that we will prove in detail do +not seem to be published in the literature on crossed complexes, although they are +related to others that are. +7.4.1. The fundamental crossed complex of a simplicial set. We freely use the notion +of a simplicial set; see e.g, [41, 55, 86] and numerous other places in the literature +and on-line. The category of simplicial sets will be denoted by Simp. +Let S be a simplicial set. Given a non-negative integer n, we let Sn denote the +set of n-simplices of S, the face maps by di := dn +i : Sn → Sn−1, for 0 ≤ i ≤ n, +similarly the degeneracy maps by sj := sn +j : Sn → Sn+1, for 0 ≤ i ≤ n, and then +let Snd +n +denote the set of non-degenerate n-simplices of S. +Recall that the geometric realisation, |S|, of S is canonically a special CW- +complex, with one n-cell for each non-degenerate n-simplex of S; see e.g. +[55, +Theorem 4.3.5]. +Let S and S′, now, be simplicial sets, then the geometric realisation of a simplicial +map, f : S → S′, is a cellular82 map, |f|: |S| → |S′|; see [55, Corollary 4.3.7]. +To simplify notation, we will, from now on, tend to insist that given a simplicial +set, S, the notation |S| will stand both for the topological space |S|, the geometric +realisation of S, but also for |S|, with its CW-decomposition, hence considered as +a filtered space with the skeletal filtration. We will therefore sometimes omit the +suffix ‘sk’ and replace |S|sk by |S|, when no confusion should result from this. +Combining geometric realisation with the fundamental crossed complex functor, +we, therefore, have a functor, Π: Simp → Crs, which we will refer to as the +fundamental crossed complex functor, sending a simplicial set |S| to Π(S) := Π(|S|), +where |S| is given its skeletal filtration. +We will use both notations, Π(S) and +Π(|S|), for the fundamental crossed complex of a simplicial set, S, depending on +the requirements of the context. It is worth noting that there are constructions of +82Actually, |f|: |S| → |S′| is furthermore regular, so sending open cells of |S| to open cells of +|S′|. + +A CATEGORIFICATION OF QUINN’S TQFT +182 +Π(S), up to isomorphism, that do not use the geometric realisation, using instead +purely algebraic techniques; see [54], once it is available, Sauvageot’s thesis, [104, +Section 3.2], and [26, Proposition 2.2.]. +7.4.2. A-colourings of simplicial sets. Let A be a crossed complex. By using Re- +mark 194, crossed complex maps, f, from Π(|S|) to A can be specified, uniquely, +by giving the value of f on those elements of Π(|S|) given by the non-degenerate +simplices of S. +We need to develop this a bit further. We start by fixing some terminology and +notation, and by recalling some results on simplicial sets. +Let ∆ be the simplex category. Recall that the objects of ∆ are non-negative +integers, n, or more exactly the finite ordinals, [n] = {0 < 1 < . . . < n}, and the +morphisms from [m] to [n], are the non-decreasing maps, +{0 < · · · < m} → {0 < · · · < n}. +A simplicial set, S, is then a functor, S : ∆op → Set, and, if n is a non-negative +integer, the set, Sn, of n simplices is the image, S(n), of [n] under S. +Let, again, n be a non-negative integer. We let ∆(n) be the n-simplex, here +considered as a simplicial set. This is defined to be the representable functor, +∆(n)(m) := ∆([m], [n]), +that is, +∆(n) : ∆op → Set. +The set of m-simplices of ∆(n) is, thus, the set of non-decreasing maps, σ, from [m] +to [n]. Such an m simplex can be represented by a string, (a0, a1, . . . , am), where +ak = σ(k), and thus we have, 0 ≤ a0 ≤ a1 ≤ · · · ≤ am ≤ n. +An m-simplex, σ, is non-degenerate if the map, σ, is injective, so we have +0 ≤ a0 < a1 < · · · < am ≤ n. +We can express any simplicial set as a coend, i.e., as a colimit, of copies of +standard simplices. This gives, in its simplest form, +S ∼= +ˆ n +Sn × ∆(n), +where, here, Sn×∆(n) is, in fact, just shorthand for the coproduct of copies of ∆(n) +labelled by the n-simplices of S, i.e., the Sn-fold copower of ∆(n). This interprets +as taking lots of labelled copies of the various standard simplices, and then glueing +them along common faces, also taking into account the degeneracies. +Returning to the standard n-simplex, ∆(n), this has a unique non-degenerate +n-simplex, namely that given by the identity map, id[n] : [n] → [n], which is, of +course, non-decreasing. All other simplices in ∆(n), of any dimension, are images +of this n-simplex under the face and degeneracy maps. +It is often useful to note that, given a simplicial set, S, the set, Sn, of n-simplices +of S is in bijective correspondence with the set, Simp(∆(n), S), where, if σ ∈ Sn, +we have a unique map, ˜σ : ∆(n) → S, which, in dimension n, sends id[n] to σ. The +behaviour of that map on any other simplex of ∆(n) is completely determined by +that specified assignment. +The geometric realisation, |∆(n)|, of ∆(n) is the geometric n-simplex, ∆n, with +the obvious CW-decomposition; see, for instance, [55, §4.2]. We will often write ∆n +for |∆(n)|, and will thus think of it as both a CW-complex and as the corresponding + +A CATEGORIFICATION OF QUINN’S TQFT +183 +filtered space, using the skeletal filtration. +We will also need the fundamental +crossed complex, Π([n]) := Π(|∆(n)|). +Following on from the coend description of a simplicial set, S, we note that we +have a similar formula for its geometric realisation, +|S| ∼= +ˆ n +Sn × ∆n. +This follows from the fact that geometric realisation is a left adjoint, so preserves +colimits. We will use this shortly to get a similar coend formula for Π(S). +Consider a crossed complex, +A = . . . +∂−→ A1 +n +∂−→ A1 +n−1 +∂−→ . . . +∂−→ A1 +2 +∂−→ A1 +1 +t +⇒ +s A0. +Definition 208. An A-colouring of a simplicial set, S, is defined to be a map of +crossed complexes, +f : Π(S) → A. +This definition needs taking apart. We will recall how this corresponds to the +intuition of labelling the simplices of S with elements of A. We start with the +basic example, namely the case S = ∆(n). This should reinforce the description of +A-colourings given in [53, section 2.12.1 and, in particular, subsection 2.32], whilst +giving it in a bit more detail. +By using Remark 194, crossed complex maps, f : Π([n]) → A, are determined +by what they do on the non-degenerate simplices of ∆(n). As we have said, the +idea is that this is a labelling of the edges, and higher dimensional faces of ∆(n), +by elements of appropriate dimension within A, and these are to satisfy some com- +patibility rules which are analogues of higher dimensional cocycle rules. +Such a crossed complex map consists of the following information (and note that +we will tacitly be using the homotopy addition lemma from [26, p. 99] here): +• a map, f0 : {0, . . . , n} = ∆(n)nd +0 +→ A0, so picking out n + 1 objects of A; +• a map, f1 : ∆(n)nd +1 +→ A1 +1, such that if (a, b) ∈ ∆(n)nd +1 , so 0 ≤ a < b ≤ n, +d0(a, b) = b, and d1(a, b) = a, then +t(f1(a, b)) = f0(d0(a, b)), and s(f1(a, b)) = f0(d1(a, b)), +and the element f1(a, b) of A goes between the images of the vertices, a +and b, +f0(a) +f1(a,b) +−−−−→ f1(b) ∈ A1; +• a map, f2 : ∆(n)nd +2 +→ A1 +2, such that for each (a, b, c) ∈ ∆(n)2, so with +0 ≤ a < b < c ≤ n, we have: +β(f2(a, b, c)) = f0(a) = f0(d1d2(a, b, c)), +so the image is in the vertex group, A1 +2(a), and, as the edges fit together as +follows83: +f0(b) +f2(b,c) +�● +● +● +● +● +● +● +● +f2(a,b,c) +f0(a) +f1(a,b) +�① +① +① +① +① +① +① +① +f1(b,c) +� f0(c), +the boundary of this element being: ∂(f2(a, b, c)) = f1(a, b) f1(b, c) f1(a, c)−1; +83remember that d0(a, b, c) = (b, c), etc. + +A CATEGORIFICATION OF QUINN’S TQFT +184 +• a map f3 : ∆(3)nd → A1 +3, which is such that, given (a, b, c, d) ∈ ∆(n)3, +β(f3(a, b, c, d)) = f0(a) = f0(d1d2d3(a, b, c, d)), +so once again it is in the appropriate dimension of the vertex crossed com- +plex over f0(a), whilst the boundary84 is +∂(f3(a, b, c, d)) = +� +f2(b, c, d) ⊳ f1(a, b)−1� +f2(a, b, d) f2(a, c, d)−1 f2(a, b, d)−1, +and +• for 3 < i ≤ n, a map, fn : ∆(n)nd +i +→ A0 +i , such that if (a0, . . . , ai) ∈ ∆(n)i, +then β(fi(a0, . . . , ai)) = f0(a0) = f0(d1d2 . . . di(a0, . . . , ai)), whilst the +boundary, +∂(fi(a1, . . . , ai)) = +� +fi−1(di(a0, . . . , ai)) ⊳ f1(a0, a1)−1� +n +� +j=1 +� +fi−1(dj(a0, . . . , ai)) +�� +(−1)j� +, +where we note that (a0, a1) = d2d3 . . . di(a0, . . . , ai). +The following makes the link with [53, 2.32. Definition], which we can take as +an alternative, more algebraic definition of an A-colouring of a simplicial set S. +Proposition 209 (Algebraic definition of A-colouring). An A-colouring of S is +given by a sequence of maps, fn : Snd +n +→ An, for n = 0, 1, . . . , satisfying the same +relations as in the ∆(n) case, above. +This means that, on an n-simplex, σ, the composite, f ◦ ˜σ is an A-colouring of +∆(n). +□ +We may refer to the data for this form of the definition as giving an algebraic +A-colouring of S. +Before continuing, we note the following useful Theorem, which is Proposition +2·2 of [26]. +Theorem 210. Let S be a simplicial set. Let |S| be its geometric realisation, with +the skeletal filtration. We have a natural isomorphism +Π(|S|) ∼= +ˆ n +Sn × Π([n]). +□ +In other words, Π(S) is obtained by glueing copies of the basic Π([n]) together +by the same rules as used for expressing S as a coend as on page 182. We note the +use of Sn × Π([n]) as shorthand for the Sn-fold copower of Π([n]), analogously to +its use in the previous coend formulae. +In [26], this Theorem is proved as a consequence of the higher homotopy van +Kampen theorem, [27, Theorem 8.1.5]. This also follows from Remark 194, using +the explicit cell decomposition of |S|. +It is also a consequence of the fact that +Π : Simp → Crs is a left adjoint, a fact that can be proved directly, without using +the van Kampen theorem, but using that Π has an algebraic description. We will +introduce the right adjoint of Π very shortly. +84The diagram below shows that we can divide the boundary of ∆(3) into two parts, +b +� +d0 +d2 +�❃❃❃❃❃❃❃❃ +c +� +b +� c +� +a +� +� +d +a +� +� +d3 +d1 +��������� +d +This explains where this formula comes from, as the boundary is the difference between the two +parts. + +A CATEGORIFICATION OF QUINN’S TQFT +185 +We have the following, which again follows from Remark 194, and that explicit +cell decomposition of |S|. Some more discussion is to be found in [53, §2.30.1]. +Lemma 211. The two definitions of A-colouring of a simplicial set are equivalent. +In other words, crossed complex maps, f : Π(|S|) → A, are in natural one-to-one +correspondence with algebraic A-colourings of S. +□ +More generally, we have the following. +Lemma 212. Let K and L be disjoint subcomplexes of S. Let i(K,S) : K → S +and i(L,S): L → S be the inclusion maps (in the category of simplicial sets). Let +f : Π(|K|) → A and f ′ : Π(|L|) → A be crossed complex maps. There is a one-to- +one correspondence between crossed complex maps, h: Π(|S|) → A, that make the +diagram below commute, +A +Π(|K|) +Π(i(K,S)) +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +f +�s +s +s +s +s +s +s +s +s +s +Π(|L|) +f ′ +�❏❏❏❏❏❏❏❏❏❏ +Π(i(L,S)) +�✉✉✉✉✉✉✉✉✉ +Π(|S|), +h +� +and (algebraic) A-colourings of S extending those colourings of K and L given by +f and f ′. +□ +7.4.3. Review of nerves and classifying spaces of a crossed complexes. Source ma- +terial for classifying spaces of crossed complexes can be found in [26], and [18], also +in [27, §9.10] and [53], as well as in various other of the sources cited earlier. +The classifying space functor, B : Crs → CGWH, is defined as the composite +of a nerve functor, N : Crs → Simp, and the geometric realisation functor from +Simp to CGWH. +This nerve functor, in fact, goes back to Blakers, [14], so +precedes the publication of much of the foundational sources on crossed complexes +by some time. It is an example of the general procedure for producing analogues +of the singular complex functor. +Clearly we have a functor, Π ◦ ∆: ∆ → Crs, sending [n] ∈ ∆ to Π([n]) = +Π(|∆(n)|), which, on varying n, gives a cosimplicial crossed complex. +Definition 213. The nerve of a crossed complex, A, is the simplicial set, N(A), +whose set of n-simplices is +N(A)n := Crs(Π([n]), A). +The maps between the different dimensions are induced from ∆: ∆ → Crs, so, if +µ : [m] → [n] in ∆, the corresponding map, N(A)µ, from N(A)n to N(A)m is +Crs(Π(µ), A). +This defines the nerve functor N : Crs → Simp. +For us, one of the most useful facts about the nerve functor is the following. +Proposition 214 (Brown-Higgins). The nerve functor, N : Crs → Simp, is right +adjoint to the fundamental crossed complex functor, Π: Simp → Crs. +Proof. See [26, Theorem 2.4]. +□ +Given a simplicial set, S, and crossed complex, A, we, thus, have a bijection, +φA +S : Crs(Π(|S|), A) → Simp(S, N(A)), + +A CATEGORIFICATION OF QUINN’S TQFT +186 +natural in both S and A. +Another fundamental fact that is underpins the use of the nerve is the following. +Proposition 215. Let A be a crossed complex, then N(A) is a Kan simplicial set. +Proof. See [26]. +□ +Remark 216. Kan complexes are often called ∞-groupoids, or, more exactly, weak +∞-groupoids. The Kan complexes of form N(A) for a crossed complex, A, are +special, however, as noted on page 100 of [26], and correspond to strict ∞-groupoids. +The relationship generalises that between bicategories, which are sometimes called +weak 2-categories and (strict) 2-categories. +A good introduction to strict ∞-groupoids is given in [2], §1.3. The equivalence +with crossed complexes is briefly discussed in §2.2 of that paper. The original dis- +cussion ot this equivalence is in [26], but from a slightly different point of view, +namely a globular rather than a simplicial one. The term used in that source is +ω-groupoid. +Another approach, which explores a more simplicial view, is given by Verity, +[120], in his proof of the Street-Roberts conjecture, which originated in some notes +from 1978 by John Roberts, developed from his approach to algebraic quantum field +theory. Verity uses the theory of complicial sets, which closely resembles that of +the T -complexes mentioned in [26]. Complicial sets model strict ∞-categories. An +introduction to that theory can be found in [103]. +As stated above, the classifying space construction that we will be using is ob- +tained from the nerve by taking geometric realisation. +Definition 217 (Classifying space of a crossed complex [26]). The classifying space, +BA, of a crossed complex, A, is defined as the geometric realisation, |N(A)|, of +N(A). +Notation 218. As usual, let A = (An)n∈Z+ +0 be a crossed complex. By construction, +A0 ∼= N(A)0, and so each object a ∈ A0 of A gives rise to a 0-simplex of N(A), +therefore to a vertex of the CW-complex, which will be denoted ˜a ∈ BA. +Given simplicial sets, K and L, their simplicial mapping space, i.e., the function +complex of [86, §6], will be denoted SIMP(K, L). We note that Simp becomes a +cartesian closed category with this function space construction. In particular, we +have SIMP(X, Y )0 = Simp(X, Y ), the set of simplicial set maps from X to Y . +As is well known, if K and L are simplicial sets, with L a Kan complex, then +we have a natural weak homotopy equivalence, |SIMP(K, L)| → TOP(|K|, |L|) = +|L||K|85. Explicitly, this weak homotopy equivalence sends the equivalence class of +(f : K × ∆(n) → L, s), seen as an element of |SIMP(K, L)|, so s ∈ |∆(n)|, to the +function |K| → |L|, such that k �→ |f|(k, s). An explicit proof that this is a weak +homotopy equivalence is in [53, page 131]. This clearly is natural in K. Given a +simplicial map, f : K → L, the weak homotopy equivalence sends the corresponding +vertex of SIMP(K, L) to the geometric realisation, |f|: |K| → |L|, of f. +The technical results collected up, for convenience, in the next theorem are due +to Brown–Higgins, [26], and Tonks, [115, 116]. They are discussed in the cubical, +as opposed to the simplicial, setting by Brown–Higgins–Sivera in [27], and, to some +extent, in a simplicial setting in [18]. +Theorem 219 (Brown–Higgins; Brown–Higgins–Sivera; Tonks). As usual, let A = +(An)n∈Z+ +0 be a crossed complex, and take S to be a simplicial set, then: +85as usual | | : Simp → CGWH denotes geometric realisation, and TOP(|K|, |L|) = |L||K| +denotes the corresponding function space in CGWH. + +A CATEGORIFICATION OF QUINN’S TQFT +187 +(1) there is a natural isomorphism of groupoids, π1(A) := π1(A, A0) ∼= π1(BA, � +A0), +where � +A0 = {˜a | a ∈ A0}86, and hence, +(2) there is a natural bijection, π0(A) ∼= π0(BA); +(3) let a ∈ A0, and let n be a positive integer. We have a natural isomorphism, +πn(A, a) ∼= πn(BA, ˜a), preserving the actions of π1(A, A0) and π1(BA, ˜ +A0). +(4) There is a weak homotopy equivalence of simplicial sets, +ηA +S : N +� +CRS(Π(|S|), A) +� +→ SIMP(S, N(A)), +which, at the level of 0-simplices, coincides with the bijection, +φA +S : Crs(Π(|S|), A) → Simp +� +S, N(A) +� +, +given by the adjunction Π ⊣ N of Proposition 214. This weak homotopy equiva- +lence is natural in S87, and also in A. +(5) There is a weak homotopy equivalence, +ηA +S : +��N +� +CRS(Π(|S|), A) +��� → TOP(|S|, BA)). +This weak homotopy equivalence is natural in both S and A. +□ +We will use the results in the previous theorem without giving a proof here, +rather we note 1) 2) and 3) form parts of [26, Proposition 2.6]; for 4), see [26, +Theorem A] and [18, Proposition 3.1.]. For 5), we refer again to [26, Theorem A] +and [18, Proposition 3.1.], and then proceed by composing with the canonical weak +homotopy equivalence, |SIMP(S, N(A))| → TOP(|S|, |N(A)|). +Let f : Π(|S|) → A be a crossed complex map. The adjunction Π ⊣ N gives a +simplicial set map, φA +S (f): S → N(A). Its geometric realisation gives a continuous +map, +|φA +S (f)|: |S| → BA = |N(A)|, +and then +ηA +S ( ˜f) = |φA +S (f)|, +where ˜f is the vertex of the classifying space, |N(CRS(Π(|S|), A))|, corresponding +to f, following Notation 218. +We note that item (5), above, links the classifying space of the crossed complex +mapping space (i.e. internal hom) with the topological mapping space, from the +realisation of S to the classifying space of A. This is the starting point, in the +setting with B = BA, for computing the Quinn finite total homotopy TQFT, and +its extended versions. +Lemma 220. There is a natural isomorphism of groupoids, +π1 +� +CRS(Π(|S|), A), CRS0(Π(|S|), A) +� +∼= π1 +� +TOP(|S|, BA), +� +|φA +S (f)| : f ∈ CRS0(Π(|S|), A) +�� +. +Proof. This follows from the first point of Theorem 219, together with the weak +homotopy equivalence, ηA +S : |NCRS(Π(|S|), A)| → TOP(|S|, BA)), since ηA +S is in- +jective on the set { ˜f : f ∈ CRS0(Π(|S|), A)}88. +□ +86using Notation 218 +87We note however that ηA +S is not simplicially natural in S, for which fact see [116] and [18]. +88Note that if a weak homotopy equivalence, g : X → Y , between spaces, is injective on +X0 ⊂ X, then g induces an isomorphism of groupoids π1(X, X0) ∼ += π1(Y, g(X0)). + +A CATEGORIFICATION OF QUINN’S TQFT +188 +7.5. Fibrations of crossed complexes and profunctors from fibrant spans. +7.5.1. Fibrations of crossed complexes. We recall the notion of fibrations of groupoids, +which was originally given in [19], and is discussed in [27, B.7]. We then turn to +fibrations of crossed complexes, which is, also, given in that second source, [27, +Definition 12.1.1]. +Definition 221. Let G′ = (s, t: G′ +1 → G′ +0) and G = (s, t: G1 → G0) be groupoids. +A map, (f1, f0): G′ → G, of groupoids, is said to be a fibration of groupoids if, +given any x ∈ G0, and x′ in G′ +0 with f0(x′) = x, and any arrow in G of form +(y +g−→ x) ∈ G1, so ending at x, there exists at least one arrow, +(y′ +g′ +−→ x′) ∈ G′ +1, +with +f1 +� +y′ +g′ +−→ x′� += (y +g−→ x). +This, then, is a ‘path lifting’ or, more precisely, an ‘arrow lifting’ condition. +Definition 222. Let A = (An)n∈Z+ +0 and B = (Bn)n∈Z+ +0 be crossed complexes. A +map, f = (fn)n∈Z+ +0 : A → B, of crossed complexes is called a fibration of crossed +complexes if +(1) f1 : A1 → B1 is a fibration of groupoids, +and +(2) given any integer n ≥ 2, and any x ∈ A0, the group homomorphism, +An(x, x) → Bn(f0(x), f0(x)), +induced by fn : An → Bn, is surjective. +An important link with Kan fibrations of simplicial sets is given in the following +result; see [26, Proposition 6.2]. For a proof in the cubical, as opposed to simplicial +set, setting, see [27, Proposition 12.1.13]. +Lemma 223. Let p: A → B be a crossed complex map. The following are equiva- +lent: +• p: A → B is a fibration, +• the induced map on nerves, N(p): N(A) → N(B), is a Kan fibration. +□ +The next result will play a major role later. It appears to be new, however not +unexpected. Recall Notation 199 for the induced map in this setting. +Let A be a crossed complex. Let X be a CW-complex, and Y be a subcomplex +of X with ι: Y → X denoting the inclusion, which induces a crossed complex map, +Π(ι): Π(Y ) → Π(X). +Lemma 224. The induced crossed complex map between internal homs, +Π(ι)∗ : CRS(Π(X), A) → CRS(Π(Y ), A), +is a fibration of crossed complexes. +Proof. (We point out that this map goes in the opposite direction as we are applying +CRS(Π(−), A).) +When A has a unique object, which is our main case of interest, a proof follows +directly from the second point of Lemma 206. This argument can be easily adapted +for the case when A has more than one object. +□ + +A CATEGORIFICATION OF QUINN’S TQFT +189 +Remark 225. A model category theoretical proof of Lemma 224 follows from the +fact that the category, Crs, of crossed complexes is a monoidal model category, +which was observed by Sauvageot, in [104], and the fact that the crossed complex +map, Π(Y ) → Π(X), induced by the inclusion, is a cofibration89 in that structure; +see [27, Proposition 12.1.4. and Example 7.3.19]. This point of view is explored in +[54]. +By [26, Proposition 6.2, ii], given p: A → B, a fibration of crossed complexes, p +has the right-lifting property with respect to the map, Π({0}) → Π(I), induced by +inclusion, where, as usual, I = [0, 1]. +Let I × I be given the usual product CW-decomposition, and let U be made of +the left, right and bottom sides of I × I, with the obvious skeletal filtration. The +map, Π(U) → Π +� +I × I +� +, induced by the inclusion, is a trivial cofibration of crossed +complexes, and hence has the left-lifting property with respect to all fibrations of +crossed complexes; see [22, Proposition 2.6]. From this, we can easily prove the +following result. +Lemma 226 (The functor derived from a fibration of crossed complexes). Let +A = (An)n∈Z+ +0 and B = (Bn)n∈Z+ +0 be crossed complexes, and let p: A → B be a +fibration between them. There is a functor, +Fp : π1(B, B0) → Set, +in full Fp = F(p: A→B), such that Fp sends b ∈ B0 to π0(p−1(b)), where the crossed +complex p−1(b) is the fibre of p: A → B, at b ∈ B0, as in Definition 184. +Given a morphism, [γ]: b → b′, in π1(B, B0), the map, +Fp([γ]): π0(p−1(b)) → π0(p−1(b′)), +is defined from the right-lifting property of, p: A → B, with respect to the map +Π({0}) → Π(I). +□ +This lemma is a version of Lemma 96 for crossed complexes. Note that a result +as strong as Lemma 96 is not likely to hold, since fibrations of crossed complexes, +as defined above, do not necessarily satisfy the full homotopy lifting condition, i.e. +they are not necessarily Hurewicz fibrations; see [22, Proposition 2.2]. +7.5.2. Fibrant spans of crossed complexes. The notion of fibrant span of spaces, of +course, adapts to other contexts, and can be formulated in terms of fibrant objects +in a category of form, CΛ, where C is any reasonably structured model category. In +particular, we can define the notion of fibrant spans of crossed complexes, which +we will discuss below, for completeness, even though most of what is written below +will not be used directly in the remainder of the paper. The theory of fibrant spans +of crossed complexes very closely parallels that of fibrant spans of spaces, so we will +not give proofs of results if the proofs of the spatial case are easily adapted to this +other context. +Definition 227. A span, +B +p +�❥❥❥❥❥❥ +q �❚ +❚ +❚ +❚ +❚ +❚ +A0 +A1, +of crossed complexes, is said to be fibrant if the induced map, +⟨p, q⟩ : B → A0 × A1, +is a fibration of crossed complexes. +89See [27, section 12.1] and [22] for a more detailed discussion of cofibrations and trivial +cofibrations of crossed complexes. We just need that inclusions, so cofibrations, of CW-complexes +are sent by Π to cofibrations of crossed complexes and similarly for trivial cofibrations. + +A CATEGORIFICATION OF QUINN’S TQFT +190 +We note that, given any crossed complex, A, any projection from a product, +A×A′ → A will be a fibration. It follows that in a fibrant span of crossed complexes, +both p and q will be fibrations. +Remark 228. We have two potential meanings for ‘fibration’, and thus for ‘fibrant +span’. As was mentioned earlier, the category, Crs, has a model category structure, +given by Brown and Golasi´nski, [22], where fibrations are as in Definition 222. We +also have a cylinder-cocylinder based homotopy structure in which ‘fibration’ means +an analogue of a Hurewicz fibration, and in which the relevant analogue of ‘weak +equivalence’ is that of homotopy equivalence with respect to the adjoint cylinder- +cocylinder pair, A ⊗ I and AI := CRS(I, A). We note that, at the time of writing, +it does not seem to be known if this latter theory does actually give a model category +structure on Crs. It is clear, and explicitly proved in both [22] and [54], that any +Hurewicz fibration is a fibration in the sense used here. The two notions agree on +all cofibrant crossed complexes. +We can use the definitions and constructions of the injective model category +structure on CrsΛ, induced from either structure. (Recall also Remark 49 for con- +text.) In our applications of these fibrant spans, we will almost always be handling +homotopy equivalences rather than weak equivalences, so many of these technical +‘difficulties’ will evaporate! +Example 229. Given any morphism, f : A → B, of crossed complexes, we can +use the analogy of the mapping cocylinder construction to replace f by a fibration, +Nf := A ×B BI +ρ(f) +−−−→ B, just as in the spatial case, (see Example 50). Applying +this to an ordinary span, +(f, M, g) := +� +M +f +�✐✐✐✐✐✐ +g +�❯ +❯ +❯ +❯ +❯ +❯ +A +B +� +, +and taking ϕ = ⟨f, g⟩, we can replace this span by a fibrant one, +Nϕ +�❥❥❥❥❥❥ +�❚ +❚ +❚ +❚ +❚ +❚ +A +B, +which is homotopy equivalent to the original one. +We call this fibrant span a +fibrantly resolved span corresponding to (f, M, g). +Example 230. Let A be a crossed complex, then we can apply the above process +to the identity span on A, and note that +AI +sA +�❥❥❥❥❥❥ +tA �❚ +❚ +❚ +❚ +❚ +❚ +A +A +is a fibrant span, since the morphism, AI → A × A, is a (Hurewicz) fibration of +crossed complexes. +Definition 231. If, in a span, (p, M, p′) : B → B′, of crossed complexes, all three +crossed complexes are homotopy finite90, then we say (p, M, p′) is a homotopy finite +span of crossed complexes. +The following is evident. +Lemma 232. If (p, M, p′) is a homotopy finite span, then there is a natural ho- +motopy equivalence between it and a homotopy finite fibrant span. +□ +90The definition of a homotopy finite crossed complex is the obvious one, and will be given +formally later in §7.6.1. + +A CATEGORIFICATION OF QUINN’S TQFT +191 +It should be clear that the usual composition of spans (here ‘of crossed com- +plexes’), extends to fibrant spans. +Lemma 233. Let B, B′ and B′′ be crossed complexes, and (p, M, p′) : B → B′ +and (p′′, M′, p′′′) : B′ → B′′ be fibrant spans, then in the obvious pullback diagram +(analogous to (11)), the composite span is fibrant. +If, moreover, we start with homotopy finite fibrant spans, then their composite is +also a homotopy finite fibrant span of crossed complexes. +□ +This, thus, gives a composite span +B +(p,M,p′)•(p′′,M′,p′′′) +−−−−−−−−−−−−−−→ B′′. +The notion of equivalence of fibrant spans (cf. Definition (57)), is easily adapted +to the context of crossed complexes, and the results on resolved identity, etc. (Def- +inition (59), and the following discussion) adapt to Crs without difficulty, and so +will not be given again here. (We will explore some of this theory of fibrations and +fibrant spans in the hopefully forthcoming [54].) +7.5.3. The profunctor H(X;Y,Z:A). Let X be a special CW-complex, with Y and Z +being two disjoint subcomplexes of X. As usual, let A be a crossed complex. +There is a natural isomorphism of crossed complexes, +Π(Ysk ⊔ Zsk) ∼= Π(Ysk) ⊔ Π(Zsk), +so it follows from the explicit definition of CRS( , A) in §7.3.2, and also from the +closed monoidal structure in Crs, that we have a natural isomorphism of crossed +complexes, +CRS +� +Π(Ysk) ⊔ Π(Zsk), A +� ∼= CRS +� +Π(Ysk), A +� +× CRS +� +Π(Zsk), A +� +. +Note also that given crossed complexes, C = (Cn)n∈Z+ +0 and C′ = (C′ +n)n∈Z+ +0 , there is +a natural isomorphism of groupoids, +π1(C × C′, C0 × C′ +0) ∼= π1(C, C0) × π1(C′, C′ +0). +Applying Lemma 224, the map, Π(ι): Π(Ysk ⊔Zsk) ∼= Π(Ysk)⊔Π(Zsk) → Π(Xsk), +induced by inclusion, gives a fibration of crossed complexes, +p: CRS(Π(Xsk), A) → CRS(Π(Ysk ⊔ Zsk), A) ∼= CRS(Π(Ysk), A) × CRS(Π(Zsk), A), +where p := Π(ι)∗ = CRS(Π(ι), A). Lemma 226 then gives a functor91, +F(p) : π1 +� +CRS(Π(Ysk), A) +� +× π1 +� +CRS(Π(Zsk), A) +� +∼= π1 +� +CRS(Π(Ysk), A) × CRS(Π(Zsk), A) +� +→ Set. +This leads to the following profunctor, whose construction mimics that of the +profunctor, +H(X′ +(p,M,p′) +−−−−−→ Y ′): π1(X′, X′) ↛ π1(Y ′, Y ′), +associated to a fibrant span, X′ +(p,M,p′) +−−−−−→ Y ′, of CGWH spaces; see Subsection 5.2. +Let X, Y , Z and A be as before. +91If B is a crossed complex, we use two notations for the fundamental groupoid of B, namely +π1(B) and π1(B, B0). If W is CW-complex, and A is a crossed complex, we hence abbreviate +π1 +� +CRS(Π(Wsk), A), Crs(Π(Wsk), A) +� +to π1 +� +CRS(Π(Wsk), A) +� +. + +A CATEGORIFICATION OF QUINN’S TQFT +192 +Definition 234 (The profunctor, H(X;Y,Z:A)). The profunctor, +H(X;Y,Z:A) : π1 +� +CRS(Π(Ysk), A) +�op × π1 +� +CRS(Π(Zsk), A) +� +→ Set, +is defined as the composite, +π1 +� +CRS(Π(Ysk), A) +�op × π1 +� +CRS(Π(Zsk), A) +� (( )−1×id) +−−−−−−−→ +π1 +� +CRS(Π(Ysk), A) +� +× π1 +� +CRS(Π(Zsk), A) +� F (p) +−−−→ Set. +Item (2) of Lemma 206 gives a way to understand the fibration, p = Π(ι)∗, +combinatorially, and hence can be used to write down, explicitly, the functor F(p), +and, hence, the profunctor H(X;Y,Z:A). The details are left to the reader. +Example 235. In the previous definition, put X = [0, 1], Y = {0} and Z = {1}. +Let also G = (s, t: G1 → G0) be a groupoid, and take A = ι1(G) to be that groupoid, +considered as a crossed complex. +Let ∗ be the crossed complex with a unique object, and only identity morphisms, +so Π({0}) ∼= ∗ and Π({1}) ∼= ∗. Clearly, CRS0 +� +∗, ι1(G) +� += G0, and moreover the +groupoid, CRS1 +� +∗, ι1(G) +� +, of crossed complex maps from ∗ to ι1(G), and homotopies +between them, is canonically isomorphic to G. The crossed complex CRS +� +∗, ι1(G) +� +is easily seen to only have identity morphisms at levels n ≥ 2. Therefore, as it +would be expected, we have CRS +� +∗, ι1(G) +� ∼= ι1(G). +Let I be the unit interval groupoid, hence, as observed in Example 191, Π([0, 1]) ∼= +ι1(I). By Theorem 200, morphisms ι1(I) → ι1(G) are in one-to-one correspon- +dence with arrows g : x → y, in G. +Unsurprisingly, the arrows in the groupoid +CRS1 +� +ι1(I), ι1(G) +� +, i.e. the homotopies between crossed complex maps, from ι1(I) +to ι1(G), are in bijection with diagrams, in G, as below, +(73) +x +hL +� +g +� y +hR +� +x′ +h−1 +L +g hR +� y′. +Here, the two vertical arrows, hL : x → x′ and hR : y → y′, are taken from the +groupoid G. The groupoid composition in CRS1 +� +ι1(I), ι1(G) +� +is given by the obvious +vertical composition. As before, for n ≥ 2, all morphisms in CRSn +� +ι1(I), ι1(G) +� +are identity morphisms, for all 2-fold homotopies are trivial. +The crossed complex fibration, of crossed complex mapping spaces, obtained from +the inclusion of {0, 1} in the interval [0, 1] (with the obvious CW-decomposition), +denoted +p: CRS +� +Π([0, 1]), ι1(G) +� +→ CRS +� +Π({0, 1}), ι1(G) +� +∼= CRS +� +Π({0}), ι1(G) +� +× CRS +� +Π({1}), ι1(G) +� +, +is, thus, at the level of groupoids, given by the groupoid map from CRS1 +� +ι1(I), ι1(G) +� +→ +G × G that chooses the two vertical arrows in (73). +Finally, we can see that, +π1 +� +CRS(Π({0}), ι1(G)) +� ∼= G +and +π1 +� +CRS(Π({1}), ι1(G)) +� ∼= G, +and moreover, that the profunctor H +� +[0,1];{0},{1}:ι1(G) +� +is given by the profunctor in +Example 32, that is, the identity profunctor on G. + +A CATEGORIFICATION OF QUINN’S TQFT +193 +7.5.4. Fibrations of crossed complexes and fibrations of mapping spaces. The follow- +ing theorem, which generalises the last point of Theorem 219, will be fundamental in +giving explicit calculations of TQFTs from crossed complexes. This result appears +to be new, however is not unexpected. +Let S be a simplicial set and T a subcomplex of S, with i(T,S) : T → S being +the inclusion map. Its geometric realisation, |i(T,S)|: |T | → |S|, is an inclusion of +CW-complexes (by, for instance, [55, Corollary 4.38]), hence it induces a crossed +complex map, Π(i(T,S)): Π(|T |) → Π(|S|), in fact a cofibration. Lemma 224, then, +gives a fibration of crossed complexes between the appropriate internal homs, +Π(i(T,S))∗ : CRS(Π(|S|), A) → CRS(Π(|T |), A). +Let f : Π(|T |) → A be a crossed complex map and, using Definition 184, con- +sider the crossed complex obtained as the fibre over f ∈ CRS0(Π(|T |), A) = +Crs(Π(|T |, A) in this fibration, Π(i(T,S))∗. We will denote this crossed complex +by +CRS(f)(Π(|S|), A) := (Π(i(T,S))∗)−1(f). +This latter crossed complex, of course, fits inside the pull-back diagram below, +where ˆf is the crossed complex with object set {f}, and only identity arrows in all +dimensions, +(74) +CRS(f)(Π(|S|), A) +� +inc +� CRS(Π(|S|), A) +Π(i(T,S))∗ +� +ˆf +inc +� CRS(Π(|T |), A) . +Since |i(T,S)|: |T | → |S| is an inclusion of CW-complexes, hence a cofibration, +we have a mapping space fibration of CGWH topological spaces, +|i(T,S)|∗ : TOP(|S|, BA) → TOP(|T |, BA). +We also have the crossed complex map, f : Π(|T |) → A, and this gives rise, via +the usual adjunction, cf. Proposition 214, followed by geometric realisation, to a +continuous map, +|φA +T (f)|: |T | → |BA|. +The fibre of |φA +T (f)|: |T | → |BA| under the mapping space fibration will be denoted +TOP(|φA +T (f)|)(|S|, BA) := (|i(T,S)|∗)−1(|φA +T (f)|), +and we have a pullback diagram in CGWH, +TOP(|φA +T (f)|)(|S|, BA) +� +inc +� TOP(|S|, BA) +|i(T,S)|∗ +� +{|φA +T (f)|} +inc +� TOP(|T |, BA) . +Theorem 236. The Brown–Higgins/Brown–Higgins–Sivera/Tonks weak homotopy +equivalence, +ηA +S : +��N +� +CRS(Π(|S|), A) +��� → TOP(|S|, BA), +in Theorem 219, restricts to a weak homotopy equivalence, +|N +� +CRS(f)(Π(|S|), A) +� +| → TOP(|φA +T (f)|)(|S|, BA). + +A CATEGORIFICATION OF QUINN’S TQFT +194 +Proof. Since the nerve functor N : Crs → Simp is a right adjoint, it preserves +limits. Applying N to diagram in (74), we have a pull-back diagram in Simp92, +N +� +CRS(f)(Π(|S|), A) +� +� +inc +� N +� +CRS(Π(|S|), A) +� +N(Π(i(T,S))∗ +� +N( ˆf) +inc +� N +� +CRS(Π(|T |), A) +� +. +The geometric realisation functor, Simp → CGWH, preserves finite limits, [55, +Theorem 4.3.16], so applying geometric realisations to the previous diagram, yields +a pull-back diagram in CGWH, +(75) +|N +� +CRS(f)(Π(|S|), A) +� +| +� +inc +� |N +� +CRS(Π(|S|), A) +� +| +|N(Π(i(T,S))|)∗| +� +|N( ˆf)| +inc +� |N +� +CRS(Π(|T |), A) +� +| . +We have another commutative diagram in CGWH, arising from the natural- +ity93, on varying the simplicial set, S, see [18, Proposition 3.1.] and [26, Theorem +A], of the weak homotopy equivalence, +ηA +S : |N +� +CRS(Π(|S|), A) +� +| → TOP(|S|, BA)). +This gives a commutative diagram, +(76) +|N +� +CRS(Π(|S|), A) +� +| +ηA +S +� +���N +� +Π(i(T,S))∗���� +� +TOP(|S|, BA) +|i(T,S)|∗ +� +|N +� +CRS(Π(|T |), A) +� +| +ηA +T +� TOP(|T |, BA) . +We note that diagrams (75) and (76) share one of their vertical arrows. +Next, we note the following. +• Given that Π(i(T,S))∗ : CRS(Π(|S|), A) → CRS(Π(|T |), A) is a fibration of +crossed complexes (by Lemma 224), then its nerve +N +� +Π(i(T,S))∗� +: N +� +CRS(Π(|S|), A) +� +→ N +� +CRS(Π(|T |), A) +� +is a fibration of simplicial sets, and so its geometric realisation is a fibra- +tion of CGWH topological spaces; see [55, Theorem 4.5.25]94. +The left +downwards arrow of diagram (76) is, therefore, a fibration in CGWH. +• Since the inclusion, |iT,S|: |T | → |S|, is a cofibration, the right downwards +arrow of diagram (76) is a fibration in CGWH. +• The two horizontal maps in diagram (76) are weak homotopy equivalences +in CGWH. +Since (76) commutes, the map ηA +S : |N(CRS(Π(|S|), A))| → TOP(|S|, BA) sends +fibres to fibres. As f : Π(|T |) → A is a crossed complex map, we have that ηA +T ( ˜f) = +|φA +T (f)|; cf. the notation in Remark 218. This follows from the fourth point of +92In general, let A = (An)n∈Z+ +0 and B = (Bn)n∈Z+ +0 be crossed complexes. If B is a sub-crossed +complex of A, meaning that each groupoid, Bn, is a subgroupoid of An, then we have a natural +inclusion of N (B) in N (A). +93Recall [116] and [18, Page 177] that the weak homotopy equivalence is only natural with +respect to simplicial maps, but not natural in the enriched sense. +94Note that geometric realisations of Kan fibrations are only sure to have the homotopy lifting +property with respect to homotopies whose domain is a CGWH space, see [55, Page 185]. + +A CATEGORIFICATION OF QUINN’S TQFT +195 +Theorem 219. The map, ηA +S , restricts to a map on the corresponding fibres, which +we denote +g′ : +� +|N(Π(i(T,S))|∗�−1 +( ˜f) → TOP(|φA +T (f)|)(|S|, BA). +Making use of the pull-back diagram (75), and the uniqueness95 of pull-backs, +this map, g′, gives rise to another map, in CGWH, +g : |N(CRS(f)(Π(|S|), A))| → TOP(|φA +T (f)|)(|S|, BA), +arising from the canonical homeomorphism, given by the uniqueness of pull-backs, +|N(CRS(f)(Π(|S|), A))| ∼= +� +|N(Π(i(T,S))|∗�−1 +( ˜f). +(This g is exactly the map we want to prove is a weak homotopy equivalence.) +We will show that g′, and hence g, is a weak homotopy equivalence, which yields +the statement of the theorem. For this we use the homotopy long exact sequences +of the two vertical fibrations of diagram (76). The pair of weak equivalences ηA +S , +of the total spaces, and ηA +T , of the base spaces, together with g′, the map on the +fibre, maps the first long exact sequence to the latter one. Therefore the five-lemma +proves that g induces an isomorphism for all homotopy groups, and hence g is a +weak homotopy equivalence. +□ +Example 237. Consider again the inclusion, ι, of {0, 1} into [0, 1]. Let G be a +group, viewed as a crossed complex, via ι1(G). The fibration of mapping spaces, +arising from the inclusion ι, will be denoted +P : TOP +� +[0, 1], BG +� +→ TOP +� +{0, 1}, BG +� +. +The crossed complex map, +p: CRS +� +Π([0, 1]), G +� +→ CRS +� +Π({0, 1}), G +� +, +induced by Π(ι): Π({0, 1}) → Π([0, 1]) is made explicit in Example 235. +There is only one crossed complex map f : Π({0, 1}) → G. +The fibre of the +projection, p, at f, is the crossed complex CRS(f)(Π([0, 1]), G), whose set of objects +is the underlying set of G, and with only identity morphisms at all orders. From this +fact, we can see that the fibre of the fibration, P, over the map |φG +{0,1}(f)| → BG, +the map that sends both 0 and 1 to the unique vertex of BG, is homotopic to the +classifying space of CRS(f)(Π([0, 1]), G). The fibre, P −1(|φG +{0,1}(f)|), is, thus, the +disjoint union of contractible spaces, one for each element of G, as it one should +expect, since the classifying space BG is an aspherical space, and π1(BG) ∼= G. +7.6. Computing the homotopy content of a finite crossed complex. The +results in this subsection are essentially all in [49], or [53]. +If A = (An)n∈Z+ +0 is a finite connected crossed complex, then, in order to de- +termine the homotopy content of the classifying space, BA, one does not need to +compute the homotopy groups of BA. The computation can be reduced to an alter- +nating product of cardinalities of sets of certain morphisms in An. This fact (and +its proof) is similar to the fact that the Euler characteristic of a finite CW-complex, +X, i.e. the alternating sum of the ranks of its homology groups, Hi(X), can be +computed as �∞ +i=1(−1)ini, where ni is the number of i-cells of X. As we will see +later, for this reason, the formula for the Quinn TQFT, and its once-extended ver- +sions, greatly simplify when B is the classifying space of a finite crossed complex, +with a single object. +Some notation will need be defined to explore this in more detail. +95up to isomorphism + +A CATEGORIFICATION OF QUINN’S TQFT +196 +7.6.1. Finite and homotopy finite crossed complexes. We will start by defining what +it means for a crossed complex to be finite, and more generally homotopy finite, +which follows naturally from the framework already introduced. +Definition 238. Let A = (An)n∈Z+ +0 be a crossed complex. +• We say that A = (An)n∈Z+ +0 is finite if all the groupoids, An, are finite, and +there exists m ∈ N such that the groupoids, An, contain only identity arrows for +n ≥ m. +• We say that A is homotopy finite if A has only a finite number of path compo- +nents, each of which with a finite number of non-trivial homotopy groups, all of +which are finite. +Clearly, if a crossed complex is finite then it is homotopy finite, but not con- +versely. Using the relationship between crossed complexes and simplicially enriched +groupoids, which we will not give here, one can use the results of [46] to show that, +if A is a homotopy finite, then it is weakly homotopy equivalent to a finite one. +The following result will be implicitly used several times. +Lemma 239. Let X be a special finite CW-complex and let A be a finite crossed +complex. The set, Crs +� +Π(Xsk), A +� +, of crossed complex maps from Π(Xsk) to A, is +finite. +Proof. This follows directly from Remark 194. +□ +Definition 240. Let A be a path-connected homotopy finite crossed complex. We +define the homotopy content of A to be +χπ(A) := +∞ +� +i=1 +|πi(A, c)|(−1)i, +where c is any object of A. +More generally, if B is any homotopy finite crossed complex, we define +χπ(B) := +� +A∈� +π0(B) +χπ(A). +Here � +π0(B) is the set of path-components of B; see Definition 185. +Theorem 219 immediately implies the following lemma. +Lemma 241. Let A be a homotopy finite crossed complex, then its classifying +space, BA, is a homotopy finite space, and χπ(A) = χπ(BA). +□ +Notation 242. Let A = (An)n∈Z+ +0 be a finite crossed complex. Given x ∈ A0 and +n ∈ Z+, define +Θx +n(A) ∈ Z+ +to be the cardinality of the set of morphisms, in the groupoid An, with source x. +Remark 243. Note that, fixing n ∈ Z+ +0 , then Θx +n(A) ∈ Z+ depends only on the +path component in A, or equivalently in π1(A), to which x ∈ A0 belongs. +The following result appears in [49, §4.2.2]. +Lemma 244. Let A = (An)n∈Z+ +0 be a finite crossed complex, then +χπ(A) = +� +x∈A0 +� ∞ +� +i=1 +� +Θx +i (A) +�(−1)i +� +. + +A CATEGORIFICATION OF QUINN’S TQFT +197 +Proof. This follows from a telescopic calculation, similar to the proof of Theorem 46. +A crucial point in the proof, allowing us to pass from a sum over path components +of A to a sum over objects of A, is that, if (s, t: G1 → G0) is a finite groupoid, then +given x ∈ G0, the cardinality of the set of morphisms in G, with source x, is equal +to |G(x, x)| |[x]|. Here [x] is the set of objects of G connected to x by a morphism +of G. Full details are in [49, Lemma 4.8]. +□ +Definition 245. A finite crossed complex, A = (An)n∈Z+ +0 , will be called homoge- +neous if, given a non-negative integer n, and an object x of A, the value of Θx +n(A) +depends only on n. This means that there exists, for each non-negative integer n, +a positive integer, Θn(A), such that, for each x ∈ A0, +Θx +n(A) = Θn(A). +If A is homogeneous, define +Θ(A) = +∞ +� +i=1 +(Θn(A)) +� +(−1)i� +∈ Q. +Note that path-connected finite crossed complexes are automatically homogeneous. +Corollary 246. Suppose that A = (An)n∈Z+ +0 is homogeneous, and so, in particular, +finite, then +χπ(A) = Θ(A) |A0|. +7.6.2. The homotopy content of CRS +� +Π(Xsk), A +� +. Let X be a finite CW-complex, +and Y a subcomplex of X. +Notation 247 (K(n, X) and K(n, X, Y )). Let n be a non-negative integer. +• We set K(n, X) to be the number of n-cells of X. +• More generally, let Y be a subcomplex of X. An n-cell, c, of X is said to +be internal to (X, Y ), if it is not in Y . We let K(n, X, Y ) be the number +of n-cells of X that are internal to the pair (X, Y ). +By applying the results in §7.3.4, and, in particular, Lemma 202, together with +the notions and notation introduced earlier in this Subsection 7.6, we have: +Lemma 248. Let A = (An)n∈Z+ +0 be a finite crossed-complex with a single object. +Let X be a finite special CW-complex. +The crossed complex CRS(Π(Xsk), A) is homogeneous, in the sense of Definition +245, and, in particular, finite. Moreover, we have that, for a positive integer j, +Θj +� +CRS(Π(Xsk), A) +� += +∞ +� +i=0 +|Ai+j|K(i,X). +In particular, +χπ� +CRS(Π(Xsk), A) +� += +��Crs(Π(Xsk), A) +�� +∞ +� +j=1 +� ∞ +� +i=0 +|Ai+j|K(i,X) +�(−1)j +. +□ +This is the special case in which Y is empty, of the following more general result, +in which we let (X, Y ) be a finite CW-pair, with both X and Y being special +CW-complexes. + +A CATEGORIFICATION OF QUINN’S TQFT +198 +Let f : Π(Ysk) → A be a crossed complex map and, as before, let Π(i): Π(Ysk) → +Π(Xsk) be induced by the inclusion i: Y → X. Consider also the induced fibration96 +(Lemma 224) between the ‘internal homs’, +Π(i)∗ : CRS +� +Π(Xsk), A +� +→ CRS +� +Π(Ysk), A +� +. +Lemma 249. The fibre of Π(i)∗ at f 97, i.e., the crossed complex, +CRS(f)� +Π(Xsk), A +� +:= (Π(i)∗)−1(f), +is homogeneous, and if j is a positive integer, then +Θj +� +CRS(f)(Π(Xsk), A) +� += +∞ +� +i=0 +|Ai+j|K(i,X,Y ). +In particular +χπ� +CRS(f)(Π(Xsk), A) +� += +��� +g : Π(Xsk) → A | g ◦ Π(i) = f +��� +∞ +� +j=1 +� ∞ +� +i=0 +|Ai+j|K(i,X,Y ) +�(−1)j +. +We note that the above crossed complex was already considered in §7.5.4, in the +case in which (X, Y ) was the geometric realisation of a simplicial pair. +Proof. This follows from Lemmas 202, 206 and 239. +□ +The particular case when we have a CW-triad98, (X; Y, Z), of finite special CW- +complexes will be very useful when we come to write down explicit formulae for +TQFTs derived from crossed complexes. Recall, from page 54, that by this, Quinn +means that Y and Z are disjoint subcomplexes of a special CW-complex, X, which +then implies that Y ⊔ Z is a subcomplex of X. (The example to have in mind is a +triangulated cobordism, M : S → S′, where X = M, Y = S and Z = S′, although +we will shortly consider more general CW-decompositions than triangulations in +this context.) +Let i(Y,X) : Y → X and i(Z,X) : Z → X be the inclusion maps, so we have a +cellular map, +� i(Y,X) +i(Z,X) +� +: Y ⊔ Z ∼= Y ∪ Z → X, +which gives the inclusion. +Given crossed complex maps, f : Π(Ysk) → A and +f ′ : Π(Zsk) → A, we can combine them into a map, +� f +f ′ +� +: Π +� +(Y ∪ Z)sk +� ∼= Π(Ysk) ⊔ Π(Zsk) → A. +The set of objects of the crossed complex, +CRS +�� f +f ′ +�� +(Π(Xsk), A) = (Π( +� i(Y,X) +i(Z,X) +� +)∗)−1⟨f, f ′⟩ +96(of crossed complexes.) +97(See Definition 184.) +98in the sense of Quinn, see page 54, + +A CATEGORIFICATION OF QUINN’S TQFT +199 +is the set of crossed complex maps, h: Π(Xsk) → A, that make the diagram, +A +Π(Ysk) +Π(i(Y,X)) +�❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +❏ +f +�s +s +s +s +s +s +s +s +s +s +Π(Zsk) +f ′ +�❑❑❑❑❑❑❑❑❑❑ +Π(i(Z,X)) +�ttttttttt +Π(Xsk), +h +� +commute. +In particular, we have the following. +Lemma 250. Let X be a finite special CW-complex with Y and Z, two disjoint +subcomplexes. Let A = (An)n∈Z+ +0 be a finite reduced crossed complex, so with a +single object. Let f : Π(Ysk) → A and f ′ : Π(Zsk) → A be crossed complex maps, +then +χπ +� +CRS +�� f +f ′ +�� +(Π(Xsk), A) +� += +��{h: Π(Xsk) → A : h ◦ Π(i(Y,X)) = f and h ◦ Π(i(Z,X)) = f ′} +�� +∞ +� +j=1 +� ∞ +� +i=0 +|Ai+j|K(i,X,Y ∪Z) +�(−1)j +. +Proof. This follows from the discussion just before the lemma. +□ +Finally, let X be a special CW-complex, A = (An)n∈Z+ +0 a finite reduced crossed +complex, and let f : Π(Xsk) → A be a crossed complex map. By passing to the path- +component, PCf(CRS(Π(X)sk, A)), of f in the crossed complex CRS(Π(X), A), we +have, within the same context as before: +Lemma 251. Let f : Π(Xsk) → A be a crossed complex map, then +χπ� +PCf(CRS(Π(Xsk), A) +� += |[f]CRS(Π(Xsk),A)| +∞ +� +j=1 +� ∞ +� +i=0 +|Ai+j|K(i,X) +�(−1)j +. +Here [f]CRS(Π(Xsk),A) denotes the homotopy class of f (the set of all crossed complex +maps, Π(Xsk) → A, that are homotopic to f). +□ +8. TQFTs and once-extended TQFTs derived from homotopy finite +crossed complexes +As before, let n be a non-negative integer and let A be a homotopy finite (often +finite and reduced) crossed complex, so its classifying space, BA, is a homotopy +finite space. We note that, as usual, we are working over a subfield, κ, of C, as +we have to be able to invert non-zero integers when working with the homotopy +content of spaces. Usually we think of using κ as being Q, but that is just the +minimal case. +In this section, we use the techniques of the homotopy theory of crossed com- +plexes, that were recalled and slightly refined in the previous section, to give explicit +formulae for: + +A CATEGORIFICATION OF QUINN’S TQFT +200 +• Quinn’s finite total homotopy TQFT, see Definition 83, where s ∈ C, +Q(s) +BA : Cob(n,n+1) → VectC, +for which we give explicit formulae in Subsection 8.2; +and +• the finitary once-extended Quinn TQFT, in Definition 154, +2QBA : 2Cob +(n,n+1,n+2) +BA +: 2Cob +(n,n+1,n+2) +BA +→ vProfGrpfin, +which we will treat in §8.3.2. +This will lead to explicit formulae for +• the Morita valued once-extended Quinn TQFT from Definition 168, +2Q +Mor +BA : 2Cob +(n,n+1,n+2) +BA +→ Mor, +which is discussed in §8.3.5. +We will consider our manifolds, Σ, to be provided with what we call simplicial +stratifications, fΣ : |XΣ| → Σ, where XΣ is a finite simplicial set (see Definition +255), and, analogously, we can define simplicial stratifications of cobordisms and +of extended cobordisms. All formulae for the Quinn and the once-extended Quinn +TQFTs will be given in terms of such simplicial stratifications. +Picking n-manifolds equipped with simplicial stratifications, leads naturally to +another variant, 2Cob +(n,n+1,n+2) +st +, of the bicategory 2Cob(n,n+1,n+2), whose ob- +jects are pairs, (Σ, iXΣ), where iXΣ : |XΣ| → Σ is a simplicial stratification, of the +(closed and smooth) n-manifold Σ, and with the rest of the bicategory structure +induced, in the obvious way, from that of 2Cob(n,n+1,n+2). (In particular, cobor- +disms and extended cobordisms do not come with chosen simplicial stratifications.) +In §8.3.4 and §8.3.5, we will also address the construction of two closely related +symmetric monoidal bifunctors, where A is a finite crossed complex, +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +and +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +→ Mor. +The symmetric monoidal bifunctors, just mentioned, do not attach a value to +an n-manifold, Σ, unless it is equipped with either a BA-decoration or a simplicial +stratification, even though the associated groupoids and algebras are unique, up to +a canonical invertible profunctor / Morita equivalence. In order to approach the +literature on the subject, such as [106, 10, 11], in §8.3.6, we will address how to +get rid of this latter dependence on simplicial stratification of the n-dimensional +manifolds. This step, however, is non-canonical, and requires the use of the axiom +of choice. This gives rise to once-extended TQFTs, +� +2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, +and +� +2QMor +A +: 2Cob(n,n+1,n+2) −→ Mor. +The well known (1, 2, 3)-extended TQFT sending S1 to the quantum double of the +group algebra of a finite group, [10, 92, 100], is an example of this latter construc- +tion. +As recalled in the beginning of Part 4, homotopy finite crossed complexes do +not model the homotopy types of all homotopy finite spaces B. The constructions +in this section will not, therefore, give formulae for all possible Quinn TQFTs, + +A CATEGORIFICATION OF QUINN’S TQFT +201 +and once-extended Quinn TQFTs. They do, however, provide formulae for those +derived from, for instance, finite 2-types B, most relevant for higher gauge theory; +cf. for instance, Baez and Schreiber, [8, 3], Baez and Huerta, [6] or Faria Martins +and Picken, [52], where the links between 2-groups / crossed modules and higher +gauge theory are summarised. +8.1. Conventions and nomenclature. In this section, given CGWH spaces, M +and N, the space of functions from M to N (with the CGWH topology) will +be denoted both by N M and TOP(M, N), whichever is more convenient for the +formula in question; the overall conventions are otherwise as in Subsection 2.3. +If M is a CGWH space, and x ∈ M, then PCx(M) denotes the path-component +of x in M, with the induced CGWH topology. Also recall, see item (15) on page +16, that � +π0(M) denotes the set of k-ified path-components of M, which we recall is, +as a set, in one-to-one correspondence99 with π0(M). If f : M → N is a map, then +PCf(TOP(M, N)) will be the set of functions from M to N that are homotopic to +f, and PCf(TOP(M, N)) is given the CGWH topology induced from TOP(M, N). +Definition 252 (Finite Simplicial set). A simplicial set, X, is called finite if it +only has a finite number of non-degenerate simplices. +If a simplicial set X is finite, then, as we recalled earlier, its geometric realisation, +|X|, is naturally a finite CW-complex, with one i-cell for each non-degenenate i- +simplex of X. Moreover, |X| will be a special CW-complex, in the sense of Definition +192. We also have a relative notion in which (X, Y ) is a pair of finite simplicial sets, +meaning that Y is a sub-simplicial set of X, and then |Y | is naturally a subcomplex +of the CW-complex, |X|. +Notation 253. Let (X, Y ) be a pair of finite simplicial sets. Extending Notation +247, we note that K(i, |X|) will also be the number of non-degenerate i-simplices of +X, and K(i, |X|, |Y |), the number of non-degenerate i-simplices of X that are not +in Y . We will extend the use of the notation, removing the geometric realisations +signs for convenience, so that: +• K(i, X) will denote be the number of non-degenerate i-simplices of X, +and +• K(i, X, Y ) will denote the number of non-degenerate i-simplices of X, that +are not in Y . +This notation will appear in the formulae for the TQFTs, and the once-extended +TQFTs, derived from finite crossed complexes. +Classically, see, for instance, [64, pp 107], an (abstract) simplicial complex, K, +is defined to be given by a sequence, K = (K0, K1, . . . ), consisting of a set, K0, +the set of vertices of K, together with, for each i ∈ Z+, a subset, Ki, of the set of +subsets of K0 that have cardinality i. The elements of Ki are called the i-faces (or +vertices if i = 0) of K. By definition, these are to have the property that if F is a +subset of cardinality j of some i-face of K, then F will be itself a j-face of K. +Definition 254. A triangulation of a (smooth) manifold, M, is a homeomorphism, +f : |K| → M, where K is a simplicial complex and |K| is its geometric realisation. +Note that our definition of a triangulation of M makes no reference to the smooth +structure in M. This is not required as our construction of TQFTs only makes use +of the underlying topological manifold of M. Otherwise we would need to consider +smooth and regular triangulations of M as, for example, in [94, Chapter II]. +99The framework of this paper makes it natural to distinguish between π0 and � +π0. + +A CATEGORIFICATION OF QUINN’S TQFT +202 +If a simplicial complex, K = (K0, K1, . . . ), is additionally provided with a total +order on the set, K0, of its vertices, then K gives rise to a simplicial set, K′. +Moreover, given a non-negative integer i, we have a one-to-one correspondence +between the i-faces of K and the non-degenerate i-simplices of K′. This well known +construction appears, for example, in [41, Examples 1.3] and [53, §1.3.1]. Note +that we have a canonical homeomorphism, |K| ∼= |K′|, between the geometric +realisations of the simplicial complex and of the simplicial set. +The geometric +realisation of K′ does not depend on the choice of an order on K0, up to canonical +homeomorphism. +In this paper, it will be convenient to consider a more general variant of “tri- +angulations”, defined in the broader context of simplicial sets. We will call these +simplicial stratifications of the manifold, as in the following definition. +Definition 255 (Simplicial stratifications). Let Σ be a compact (smooth) n-manifold. +Let XΣ be a simplicial set. A simplicial stratification of Σ is a homeomorphism, +fΣ : |XΣ| → Σ. We denote such a simplicial stratification of Σ by (XΣ, fΣ). +(Note that since a simplicial set has a compact geometric realisation if, and only +if, it is finite, XΣ must always be a finite simplicial set in the above.) +More generally, consider an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, between the +closed (smooth) n-manifolds, Σ and Σ′. A simplicial stratification (of the cobor- +dism) is given by a (Quinn) triad, (YM; XΣ, X′ +Σ′), of simplicial sets, so that the sim- +plicial sets, XΣ, and X′ +Σ′, are subcomplexes of the simplicial set, YM, and XΣ ∩X′ +Σ′ +is empty, together with a homeomorphism, (fΣ, gM, f ′ +Σ′), of cospans in CGWH, +(77) +|XΣ| +|j| +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +fΣ +� +|X′ +Σ′| +|j′| +�❥❥❥❥❥❥❥❥❥❥❥ +f ′ +Σ′ +� +|YM| +gM +� +Σ +i +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +Σ′. +i′ +�✐✐✐✐✐✐✐✐✐✐✐✐✐ +M +Here j : XΣ → YM and j′ : X′ +Σ′ → YM denote the obvious simplicial inclusions. +Simplicial stratifications of a cobordism, (i, M, i′): Σ → Σ′, arise, for instance, +from triangulations of M that extend given triangulations of Σ and Σ′, and which, +furthermore, are equipped with a total order on the set of vertices of the triangula- +tion of M. Since simplicial stratifications can, in general, be chosen to be smaller +than triangulations, they have advantages over the triangulated form. Any simpli- +cial stratification can be processed to give a finer triangulation, but typically with +more parts to it. +8.2. TQFTs from homotopy finite and finite crossed complexes. In this +subsection, we will work over the field C. For X a simplicial set, and A a crossed +complex, we recall, from (Brown–Higgins) Lemma 214, that there is a natural +bijection, +φA +X : Crs +� +Π(|X|sk), A +� +→ Simp +� +X, N(A) +� +. +Here |X|sk is the realisation of X, with its skeletal filtration. We will also need to +use the Brown–Higgins–Sivera–Tonks Theorem, [18, 26, 27, 115, 116], that is given +here as Theorem 219, and which gives that we have a natural map100 in CGWH, +which is a weak homotopy equivalence, +(78) +ηA +X : +��N +� +CRS(Π(|X|sk), A) +��� → TOP(|X|, BA). +100A map, ηA +Y : |CRS(Π(Ysk), A)| → TOP(Y, BA), can also be defined, [26], when Y is a CW- +complex, with its cellular decomposition, however the map is, in that case, only specified up to +homotopy. + +A CATEGORIFICATION OF QUINN’S TQFT +203 +Moreover, using the notation in Notation 218, and the discussion following Theorem +219, if f : Π(|X|sk) → A is a crossed complex map, +ηA +X( ˜f) = |φA +X(f)|. +If f, f ′ : Π(|X|sk) → A are homotopic crossed complex maps, then the objects +f, f ′ ∈ CRS0(Π(|X|sk, A) are connected by an arrow in CRS(Π(|X|sk), A), and, +thus, ˜f and ˜f ′ belong to the same path-component of +��N +� +CRS(Π(|X|sk), A) +���. +The weak homotopy equivalence, (78), then gives that the two maps, +|φA +X(f)|, |φA +X(f ′)|: |X| → BA, +are homotopic maps between CGWH spaces. +We also note that π0 +� +CRS(Π(|X|sk), A) +� +is just the set of homotopy classes of +crossed complex maps from Π(|X|sk) to A, for which see Subsection 7.3. +As we saw in Lemma 241, if A is homotopy finite, then its classifying space, BA, +is homotopy finite, and we can consider Quinn’s finite total homotopy TQFT, of +[101, Lecture 4], or more generally as explored here in Subsection 4.3, +Q(s) +BA : Cob(n,n+1) → Vect +with base-space B = BA, and in which s is a complex parameter. +Our main case of study is when A is finite and reduced. In this case, the formulae +for Q(s) +BA become particularly simple. Using Proposition 88 and Theorem 90, the +knowledge of that restricted case is enough to give the calculation in general. (Recall +that it is known that any homotopy finite crossed complex is weak equivalent to a +finite crossed complex. This can be derived from Ellis’ theorem, [46].) +In the theorem below, if X is a simplicial set, we put Π(X) = Π(|X|sk). We +also note that | − | is used to denote both the cardinality of a finite set and the +geometric realisation of a simplicial set. Which one it is, should be clear from the +context. +We will assume, now, that A is a homotopy finite crossed complex. This A will +be fixed throughout the discussion. +As usual, let n be a non-negative integer, and Σ be a closed smooth n-manifold. +Consider a simplicial stratification, (XΣ, iXΣ), of Σ, so with XΣ a simplicial set, +and iXΣ : |XΣ| → Σ, a homeomorphism. +Given a crossed complex map, f : Π(XΣ) → A, we define +f := |φA +X(f)| ◦ i−1 +XΣ. +Here, as φA +X(f) is a simplicial map, φA +X(f) : XΣ → N(A), we have that +f : Σ → BA +is a continuous map. +We have s ∈ C, a fixed parameter. We have an isomorphism of vector spaces, +C +� +π0 +� +CRS(Π(XΣ), A) +�� T (iXΣ ,A) +−−−−−−→ Q(s) +B (Σ) = C +� +�π0(BΣ) +� +, +such that, given f : Π(XΣ) → A and on denoting the homotopy class of the crossed +complex map, f, by [f]CRS(Π(XΣ),A), we have +T (iXΣ, A) +� +[f]CRS(Π(XΣ),A) +� += PCf(BΣ). +Suppose given an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, between the closed n- +manifolds, Σ and Σ′, and a simplicial stratification of the cobordism, (i, M, i′), + +A CATEGORIFICATION OF QUINN’S TQFT +204 +obtained from the cospan of simplicial sets, +XΣ +j +�❙ +❙ +❙ +❙ +❙ +❙ +X′ +Σ′ +j′ +�❦❦❦❦❦❦ +YM +together with a homeomorphism of cospans in CGWH, as below, +(79) +|XΣ| +|j| +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +iXΣ +� +|X′ +Σ′| +|j′| +�❥❥❥❥❥❥❥❥❥❥❥ +iX′ +Σ′ +� +|YM| +iYM +� +Σ +i +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +Σ′ . +i′ +�✐✐✐✐✐✐✐✐✐✐✐✐✐ +M +Given crossed complex maps f : Π(XΣ) → A, f ′ : Π(X′ +Σ′) → A, let +f = |φA +XΣ(f)| ◦ i−1 +XΣ : Σ → BA, +and +f ′ = |φA +X′ +Σ′ (f ′)| ◦ i−1 +X′ +Σ′ : Σ′ → BA. +Theorem 256 (Quinn’s finite total homotopy TQFT, Q(s) +B for B = BA). In the +formula for the matrix elements of Quinn’s finite total homotopy TQFT, Q(s) +B , +which is as follows, +� +PCf(BΣ) | Q(s) +B +� +[(i, M, i′)] +� +| PCf ′(BΣ′) +� += χπ�� +f|BM|f ′�� � +χπ� +PCf(BΣ) +��s � +χπ� +PCf ′(BΣ′) +��1−s +, +and where, as before, +� +f|BM|f ′� += + + + + + +H : M → B +������� +B +Σ +f +�❦ +❦ +❦ +❦ +❦ +❦ +i +�❙ +❙ +❙ +❙ +❙ +❙ +Σ′ +f ′ +�❙❙❙❙❙❙ +i′ +�❦❦❦❦❦❦ +M +H +� +commutes + + + + + +⊂ BM, +with the induced CGWH topology, each factor can be calculated as follows101. +First of all, +(80) +χπ�� +f|BM|f ′�� += χπ +� +CRS +�� f +f ′ +�� +(Π(YM), A) +� +. +If A is finite, and reduced (i.e. with a single object), then 102 +(81) +χπ�� +f|BM|f ′�� += +�������� + + + + + + + +h: Π(YM) → A +�������� +A +Π(XΣ) +f +�❤ +❤ +❤ +❤ +❤ +❤ +❤ +❤ +Π(j) �❯ +❯ +❯ +❯ +❯ +❯ +Π(X′ +Σ′) +f ′ +�❱❱❱❱❱❱❱❱ +Π(j′) +�❤❤❤❤❤ +Π(YM) +h +� +commutes + + + + + + + +�������� +∞ +� +k=1 +� ∞ +� +i=0 +|Ai+k|K(i,YM,XΣ∪X′ +Σ′ ) +�(−1)k +. +101For notation see §7.6.2. +102Here recall that K(i, YM, XΣ ∪ X′ +Σ′) is the number of non-degenerate i-simplices of the +simplicial set YM that are neither in XΣ nor in X′ +Σ′. + +A CATEGORIFICATION OF QUINN’S TQFT +205 +Continuing with the evaluation of the other terms in the formula for +� +PCf(BΣ) | Q(s) +B +� +[(i, M, i′)] +� +| PCf ′(BΣ′) +� +, +we have +χπ� +PCf(BΣ) +� += χπ� +PCf(CRS(Π(XΣ), A)) +� +, +(82) +and +χπ� +PCf ′(BΣ′) +� += χπ� +PCf ′(CRS(Π(X′ +Σ′), A)) +� +, +(83) +so (by Lemma 251), if A is finite and reduced, then +χπ� +PCf(BΣ) +� += +��[f]CRS(Π(XΣ),A) +�� +∞ +� +k=1 +� ∞ +� +i=0 +|Ai+k|K(i,XΣ) +�(−1)k +, +χπ� +PCf ′(BΣ′) +� += +��[f]CRS(Π(X′ +Σ′ ),A) +�� +∞ +� +k=1 +� ∞ +� +i=0 +|Ai+k|K(i,X′ +Σ′ ) +�(−1)k +, +(84) +where, as above, [f]CRS(Π(XΣ),A)| denotes the homotopy class of the crossed complex +map, f, and, analogously, [f ′]CRS(Π(X′ +Σ′ ),A)| is that of f ′. +Proof. This result follows from the general discussion in §7.4.3, §7.5.4 and §7.6.2. +In particular, the crucial ingredient is the Brown–Higgins–Sivera–Tonks weak ho- +motopy equivalence, +ηA +S : |N(CRS(Π(S), A))| → TOP(|S|, BA), +in item (5) of Theorem 219, where S is a simplicial set and A is a crossed complex, +and its refinement in Theorem 236, together with the result that the homotopy +groups of a crossed complex coincide with those of its geometric realisation, for +which see again Theorem 219. +For instance, Equation (80) follows from Theorem 236, applied to the simplicial +inclusion +� j +j′ +� +: XΣ ⊔ X′ +Σ′ → YM, and the crossed complex map, +� f +f ′ +� +: Π(XΣ) ⊔ Π(X′ +Σ′) → A. +We then apply Lemma 241. Finally (81) follows from Lemma 250, and Equation +(84) follows from Lemma 251. +□ +Remark 257 (Independence from simplicial stratifications). Let A be a homotopy +finite crossed complex. Note that, by construction, all formulae for the Quinn fi- +nite total homotopy TQFT, Q(s) +BA, in the previous theorem are independent of the +chosen simplicial stratifications of the n-manifolds, Σ and Σ′, and of the (n + 1)- +cobordism, (i, M, i′): Σ → Σ′. There is no need to make use of Alexander moves, +or equivalently of Pachner moves, to prove triangulation-independence103, as done +for instance in [84, 9, 119]. This is because the formulae were directly derived to +give quantities that are, by construction, topologically invariant, and related to the +homotopy content of function spaces. +Remark 258. By using the previous theorem together with Lemma 212, we can +see that the calculations of Quinn’s finite total homotopy TQFT, Q(s) +BA, for A a +finite crossed complex, and for given simplicial stratifications of the manifolds and +cobordisms concerned, could in theory be computed in finite time. +103nor, for this paper, independence from simplicial stratifications. + +A CATEGORIFICATION OF QUINN’S TQFT +206 +We expect that the techniques just shown will also be applicable for computing, +explicitly, TQFTs derived from finite crossed complexes, which are, furthermore, +equipped with a cohomology class valued in U(1). (The existence of these TQFTs, +generalising Dijkgraaf-Witten TQFTs [43], was suggested in Remark 92, and they +were treated in [53], in the particular case of closed manifolds and crossed modules, +using similar techniques to those of this paper.) Our approach here can likely also +be adapted to give concrete formulae for homotopy quantum field theories derived +from (classifying spaces of) crossed complexes, possibly equipped with appropriate +cohomology classes. These homotopy quantum field theories are treated in [108], +and also in [98, 99]. +We also expect that similar techniques as used in this subsection can be used to +give formula for Quinn’s finite total homotopy TQFT, Q(s) +B , in the case when B +is the classifying space of a finite simplicial group, in which case we would obtain +concrete formulae for all types of Quinn’s finite total homotopy TQFT. (Since finite +simplicial groups model all pointed homotopy finite spaces [46].) This would likely +yield formulae similar to those in [97]. +8.3. The once-extended TQFTs derived from finite crossed complexes. +In this section and the next, we will work over the field of rational numbers Q. +We will also fix a finite crossed complex, A, i.e., actually finite, not just homotopy +finite. +Here we will freely use the notation and results from Section 6, particularly +Subsection 6.3, as well as of our review of the homotopy theory of crossed complexes, +and their classifying spaces, BA, in Section 7. +Let B = BA, which we recall is a homotopy finite space, by Theorem 219. Let n +be a non-negative integer. We will give explicit formulae for some instances of the +finitary once-extended Quinn TQFT, +2QB : 2Cob +(n,n+1,n+2) +B +→ vProfGrpfin, +and consequently of its Morita version, +2Q +Mor +B +: 2Cob +(n,n+1,n+2) +B +→ Mor. +We will also treat a few variants of these once-extended TQFTs, as mentioned in +the beginning of this current section, Section 8, namely: +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +→ Mor, +� +2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, +and +� +2QMor +A +: 2Cob(n,n+1,n+2) −→ Mor. +This will be done at the same time as we examine the dependence of the formulae +that we have give on the choice of the simplicial stratification of an n-dimensional +manifold that is being used. +8.3.1. The BA-decoration of a manifold arising from a simplicial stratification. Let +Σ be a closed (and as usual smooth) n-manifold. +Recall, Definition 152, that, given a HF space, B, a B-decoration, fΣ, of Σ, is +given by a finite subset, f Σ, of the function space, BΣ, of functions from Σ to B, +containing at least one function, f : Σ → B, from each path component of BΣ. If + +A CATEGORIFICATION OF QUINN’S TQFT +207 +B = BA, the classifying space of A. Simplicial stratifications of Σ (see Definition +255) naturally give rise to B-decorations of Σ. We can see this as follows. +Let XΣ be a finite simplicial set. We recall that the Brown–Higgins–Siviera– +Tonks Theorem, here Theorem 219, provides a weak homotopy equivalence, +ηA +XΣ : |CRS(Π(XΣ), A)| → TOP(|XΣ|, BA), +so suppose that we have a simplicial stratification of Σ, given by a homeomorphism +iXΣ : |XΣ| → Σ. A crossed complex map, f : Π(XΣ) → A, gives rise to a continuous +map |φA +XΣ(f)|: |XΣ| → BA. Noting the comments after Theorem 219, we have +|φA +XΣ(f)| = ηA +XΣ( ˜f). +The weak homotopy equivalence, ηA +XΣ, hence gives a BA- +decoration of Σ, defined by +(85) +f Σ +� +iXΣ : |XΣ| → Σ, A +� +:= +� +|φA +XΣ(f)| ◦ i−1 +XΣ | f ∈ CRS0(Π(XΣ), A) +� +. +Note that CRS0(Π(XΣ), A), the set of crossed complex maps from Π(XΣ) to A, is +finite, by Lemma 239. +We will frequently abbreviate the notation and put +fΣ +� +iXΣ : |XΣ| → Σ, A +� abbr. += f Σ(iXΣ, A). +8.3.2. Explicit formulae for the finitary once-extended Quinn TQFT for B = BA. +In this subsection we give, explicitly, the various formulae obtained by taking the +identification, up to natural isomorphism, of the various parts of the once-extended +Quinn TQFT, in the case in which B = BA, in terms of crossed complexes and +related groupoids. We give that description first, but, for ease of reference in the +later examples, we will state this more formally in a theorem at the end of the +description. +We note that to get such a description one has to suppose given +simplicial stratifications of the manifolds, cobordisms and 2-cobordisms. We will +discuss the dependence on these choices in a later subsection. This is analogous, +of course, to taking triangulations, so as to get ‘lattice models’ and ‘state sum’ +models, as we mentioned in Remark 257. +The finitary once-extended Quinn TQFT, in Definition 154, +2QBA : 2Cob +(n,n+1,n+2) +BA +→ vProfGrpfin, +can be specified / described, up to isomorphism, as follows. +(i) If Σ is a closed n-manifold, and iXΣ : |XΣ| → Σ is a simplicial stratification +of Σ, then we have a canonical isomorphism of groupoids, +(86) +2Q +0 +BA +� +Σ, fΣ +� +iXΣ, A) +� ∼= π1 +� +CRS(Π(XΣ), A), CRS0(Π(XΣ), A) +� +. +(ii) Given an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, between the closed n- +manifolds, Σ and Σ′, consider a simplicial stratification of the cobordism, +(i, M, i′), derived from a co-span of simplicial sets, +XΣ +j +�❘ +❘ +❘ +❘ +❘ +❘ +X′ +Σ′ , +j′ +�❦❦❦❦❦❦ +YM +together with a homeomorphism of cospans in CGWH, +|XΣ| +|j| +�❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +❚ +iXΣ +� +|X′ +Σ′| +|j′| +�❥❥❥❥❥❥❥❥❥❥❥ +iX′ +Σ′ +� +|YM| +iYM +� +Σ +i +�❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +❯ +Σ′ . +i′ +�✐✐✐✐✐✐✐✐✐✐✐✐✐ +M + +A CATEGORIFICATION OF QUINN’S TQFT +208 +The simplicial stratifications, iXΣ : |XΣ| → Σ, and iX′ +Σ′ : |X′ +Σ′| → Σ, +of Σ and Σ′, respectively, yield BA-decorations, fΣ(iXΣ, A), of Σ, and +fΣ′(iX′ +Σ′ , A), of Σ′, giving the associated 1-morphism in 2Cob +(n,n+1,n+2) +BA +, +� +Σ, fΣ(iXΣ, A) +� (i,M,i′) +−−−−−→ +� +Σ′, fΣ′(iX′ +Σ′ , A) +� +. +Using Definitions 154 and 234, we have a natural isomorphism of profunc- +tors, +(87) +2Q +1 +BA +�� +Σ, fΣ(iXΣ, A) +� (i,M,i′) +−−−−−→ +� +Σ′, fΣ′(iX′ +Σ′ , A) +�� +∼= Lin ◦ H(|YM|sk;|XΣ|sk,|X′ +Σ′|sk:A). +(Recall also that Lin: Set → Vect is the free vector space functor.) +(iii) Finally, at the level of 2-morphisms, consider an (n, n + 1, n + 2)-extended +cobordism, +(88) +K = + + + + + + + + + + + + +Σ +i +� +ι0 +� +M +iN +� +Σ′ +i′ +� +ι′ +0 +� +Σ × I +iE +� K +Σ′ × I +iW +� +Σ +ι1 +� +j +� M ′ +iS +� +Σ′ +j′ +� +ι′ +1 +� + + + + + + + + + + + + +, +and also a diagram, (a ‘co-window’), of finite simplicial sets, WK, as below, +(89) +WK = + + + + + + + + + + + + + +XΣ +i +� +ι0 +� +YM +iN +� +X′ +Σ′ +i′ +� +ι′ +0 +� +XΣ × ∆(1) +iE +� ZK +X′ +Σ′ × ∆(1) +iW +� +XΣ +ι1 +� +j +� Y ′ +M′ +iS +� +X′ +Σ′ +j′ +� +ι′ +1 +� + + + + + + + + + + + + + +, +together with a homeomorphism of diagrams, +(90) +g : |WK| → K. +Here |WK| is obtained by applying geometric realisation to all components +of WK, and, in order to simplify notation, all ‘components’ of g will be +denoted g. +Consider, also, the frame, fr(WK), of WK, a simplicial set, +(defined exactly as was done in (50) for extended cobordisms), together +with the filler, gWK : fr(WK) → ZK, which is a map of simplicial sets. +Note that g : |WK| → K gives simplicial stratifications for Σ and for Σ′, +which extend to simplicial stratifications of Σ× I and Σ′ × I, and moreover +to simplicial stratifications of the (n + 1)-cobordisms, (i, M, i′): Σ → Σ′ +and (j, M ′, j′): Σ → Σ′. The homeomorphism, |fr(WK)| → fr(K) ∼= ∂K, +induced by g : |WK| → K, then also provides a simplicial stratification of +∂K. The simplicial stratification of K that g gives then restricts to the +latter simplicial stratification over ∂K. +Given two crossed complex maps, +H : Π(YM) → A, and H′ : Π(Y ′ +M′) → A, + +A CATEGORIFICATION OF QUINN’S TQFT +209 +suppose that H and H′ agree on Π(XΣ) and on Π(X′ +Σ′), that is to say +that H and H′ coincide when composed with Π(i): Π(XΣ) → Π(YM) +and with Π(j): Π(XΣ) → Π(Y ′ +M′), and also coincide when composed with +Π(i′): Π(X′ +Σ′) → Π(YM) and with Π(j′): Π(X′ +Σ′) → Π(Y ′ +M′). +Let f : Π(XΣ) → A be equal to H◦Π(i) = H′◦Π(j), and f ′ : Π(X′ +Σ′) → A +be H ◦ Π(i′) = H′ ◦ Π(j′). +We define the following continuous maps, (where we recall that all com- +ponents of g : |WK| → K are denoted g), +H = +��φA +YM (H) +�� ◦ g−1 : M → BA, +H′ = +���φA +Y ′ +M′ (H′) +��� ◦ g−1 : M ′ → BA, +f = +��φA +XΣ(f) +�� ◦ g−1 : Σ → BA, +f ′ = +���φA +X′ +Σ′ (f ′) +��� ◦ g−1 : Σ′ → BA. +Let p: XΣ × ∆(1) → XΣ and p′ : X′ +Σ′ × ∆(1) → X′ +Σ′ be the simplicial +projections, inducing crossed complex maps, Π(p): Π(XΣ×∆(1)) → Π(XΣ) +and Π(p′): Π(X′ +Σ′×∆(1)) → Π(X′ +Σ′). By construction, the crossed complex +maps, +H : Π(YM) → A, +H′ : Π(Y ′ +M′) → A, +f ◦ Π(p): Π(XΣ × ∆(1)) → A, +f ′ ◦ Π(p′): Π(X′ +Σ′ × ∆(1)) → A, +combine104 into one, that will be denoted [H, H′]: Π(fr(WK)) → A. +We then have the following description of the matrix entries105, +(91) +� +PCH +� +{f|BM|f ′} +� +| +� +2Q2 +B([K]) +� +(f,f ′) | PCH′� +{f|BM′|f ′} +�� += χπ +� +CRS +� +[H,H′] +�� +Π(WK), A +�� +χπ +� +PCH′ +� +CRS +�� f +f ′ +��� +Π(Y ′ +M′), A +��� +. +We note that to simplify expressions slightly we have, in the final formulae, written +B for BA. +Theorem 259. Let A be a finite crossed complex, and n a non-negative integer. +The structures specified in (i), (ii), and (iii), above, give the finitary once-extended +Quinn TQFT, in Definition 153, +2QBA : 2Cob +(n,n+1,n+2) +BA +→ vProfGrpfin, +if we restrict to the objects of 2Cob +(n,n+1,n+2) +BA +of the form +� +Σ, f Σ(iXΣ, A) +� +, where +iXΣ : |XΣ| → Σ is a simplicial stratification of a closed smooth n-manifold Σ. +104Here we are implicitly using the fact that the fundamental crossed complex functor preserves +certain colimits, including the colimit defining fr(WK), for which see [27, 8.2.i Coproducts with +amalgamation]. +105It may help to look back at the notation developed at the end of Subsection 7.6, so, for in- +stance, CRS +� +[H,H′] +�� +Π(ZK), A +� +denotes the fibre, as in Definition 184, over [H, H′]: Π(fr(WK)) → +A, of the crossed complex map, +CRS�Π(ZK), A� → CRS�Π(fr(WK)), A�, +induced by the inclusion of fr(WK) into ZK. + +A CATEGORIFICATION OF QUINN’S TQFT +210 +Proof. For the most part, the proof is essentially as in Theorem 256. For instance +Equation (86) follows from Lemma 220, and Equation (91) follows from Lemma 236. +Equation (87) follows from the fact that106 we have a weak homotopy equivalence, +ηA +YM : +��N(CRS(Π(YM), A)) +�� → TOP(|YM|, BA), +by (Brown–Higgins–Sivera–Tonks) Theorem 219. +□ +Remark 260. If we assume, furthermore, that A is reduced, then the crossed +complexes appearing in (91) are homogeneous. This follows from the discussion at +the end of Subsection 7.6. In particular (as in Theorem 256), we can obtain explicit +formulae for their homotopy content, similar to (81) and (84), using Lemmas 250 +and 251. +8.3.3. Dependence of the formulae on simplicial stratifications. We need to address +the dependence of the formulae in Theorem 259 on the choice of simplicial strat- +ifications. We freely use Remark 155, where the dependence on decorations was +discussed. +Let Σ be a closed (and, as usual, smooth) n-manifold. If we choose different +simplicial stratifications, iXΣ : |XΣ| → Σ and jYΣ : |YΣ| → Σ, of Σ, then the corre- +sponding BA-decorations of Σ, +f Σ(iXΣ, A) +and +f Σ(jYΣ, A), +as defined in Equation (85), will, in general, be different. Nevertheless, we have a +canonically defined invertible profunctor, +2Q +0 +BA +� +Σ, f Σ(iXΣ, A) +� +↛ 2Q +0 +BA +� +Σ, fΣ +� +jYΣ, A) +� +. +This profunctor is obtained from 2QBA : 2Cob +(n,n+1,n+2) +BA +→ vProfGrpfin, as +2Q +1 +BA +�� +Σ, f Σ(iXΣ, A) +� (ι0,Σ×I,ι1) +−−−−−−−→ +� +Σ, fΣ(jYΣ, A) +�� +, +where ι0(a) = (a, 0) and ι1(a) = (a, 1), for all a ∈ Σ. +By construction, these profunctors, which connect the groupoids associated to +different simplicial stratifications of Σ, are functorial, so compose well with respect +to further changes in the simplicial stratification, and are compatible with the +profunctors associated to cobordisms. A precise statement follows from Theorem +261, below, and is as in Remark 155. +On the other hand, the formula, in Equation (87), for the profunctors associated +to an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, does not depend on the simplicial +stratification, iYM : |YM| → M, of M, extending that of Σ and Σ′, as shown in +Equation (79). (The simplicial stratifications of Σ and Σ′ were part of the given +data and so are themselves fixed.) +Likewise, in Equation (91), the formula for some of the matrix elements associ- +ated to the natural transformation of profunctors provided by an (n + 2)-extended +cobordism, K in (88), required the choice of a co-window of simplicial sets, as in +(89), and a homeomorphism of diagrams, g : |WK| → K, as shown in (90). The +final result depended, however, only on the simplicial stratifications of Σ and of Σ′ +that g : |WK| → K induces. This is to say that the value in (91) depends neither +on the simplicial stratifications of M and M ′, extending those of Σ and Σ′, nor on +the simplicial stratification of the (n + 2)-manifold with border K, extending those +of M, M ′, Σ × I and Σ′ × I, that g : |WK| → K gives. +106We expect to give more discussion for why the profunctors in (87) are equivalent in [54]. + +A CATEGORIFICATION OF QUINN’S TQFT +211 +8.3.4. The bifunctor, 2QA : 2Cob +(n,n+1,n+2) +st +→ vProfGrpfin. Some of discussion +concerning the (in)dependence of the formulae for the finitary once-extended Quinn +TQFT, 2QBA : 2Cob +(n,n+1,n+2) +BA +→ vProfGrpfin, given in §8.3.3, with respect to +the simplicial stratifications, can be repackaged in a new version of the finitary +once-extended Quinn TQFT. As usual, n is a non-negative integer. +We first define a variant, 2Cob +(n,n+1,n+2) +st +, of the bicategory 2Cob(n,n+1,n+2). +• The objects of 2Cob +(n,n+1,n+2) +st +are pairs, (Σ, iXΣ), where iXΣ : |XΣ| → Σ +is a simplicial stratification of the (closed and smooth) n-manifold Σ. +• The 1-morphisms, (Σ, iXΣ) → (Σ′, iX′ +Σ′), are given by (n + 1)-cobordisms, +(i, M, j): Σ → Σ′, without any chosen simplicial stratification on M (ex- +tending the one that already exists on Σ and Σ′). +• The 2-morphisms, in 2Cob +(n,n+1,n+2) +st +, +� +(i, M, j): (Σ, iXΣ) → (Σ′, iX′ +Σ′) +� +=⇒ +� +(i′, M ′, j′): (Σ, iXΣ) → (Σ′, iX′ +Σ′ ) +� +, +are given by equivalence classes of extended cobordisms107, +K: +� +(i, M, j): Σ → Σ′� +=⇒ +� +(i′, M ′, j′): Σ → Σ′� +, +without any chosen simplicial stratification on K, in (88), extending the +one that already exists on Σ and Σ′. +The rest of the bicategory structure for 2Cob +(n,n+1,n+2) +st +is induced from that of +the bicategory 2Cob(n,n+1,n+2), in the obvious way, as in the construction of the +bicategory 2Cob +(n,n+1,n+2) +B +, in Definition 153. +Given a finite crossed complex, A, we therefore have a bifunctor, +VA : 2Cob +(n,n+1,n+2) +st +−→ 2Cob +(n,n+1,n+2) +BA +, +which, on objects, is such that108 +� +Σ, iXΣ +� VA +�−→ +� +Σ, f(iXΣ, A) +� +, +and on 1-morphisms, +VA� +(i, M, j): (Σ, iXΣ) → (Σ′, iX′ +Σ′ ) +� += (i, M, j): +� +Σ, f(iXΣ, A) +� +→ +� +Σ′, f(iX′ +Σ′ ,A) +� +, +and analogously for 2-morphisms. +The symmetric monoidal structure of 2Cob +(n,n+1,n+2) +BA +, which is naturally de- +rived from that of 2Cob(n,n+1,n+2), was briefly explained at the end of Subsection +6.6. In particular, the tensor product of two BA-decorated n-manifolds is +(Σ, fΣ) ⊗ (Σ′, gΣ′) = (Σ ⊔ Σ′, f Σ ⊗ gΣ′), +where +f Σ ⊗ gΣ′ := +� +⟨φ, φ′⟩ | φ ∈ fΣ and φ′ ∈ gΣ′ +� +. +(Here, given φ: Σ → BA and φ′ : Σ′ → BA, ⟨φ, φ′⟩: Σ ⊔ Σ′ → BA is defined from +the universal property of disjoint unions.) +Analogously, we can define a symmetric monoidal structure in the bicategory +2Cob +(n,n+1,n+2) +st +, where the tensor product of two closed, smooth, n-manifolds, Σ +107as for 2Cob(n,n+1,n+2) +108Given a simplicial stratification, iXΣ : |XΣ| → Σ, of Σ, the corresponding BA-decoration, +f(iXΣ, A) ⊂ TOP(Σ, BA) is as defined by Equation (85). + +A CATEGORIFICATION OF QUINN’S TQFT +212 +and Σ′, provided with simplicial stratifications, iX : |XΣ| → Σ and i′ +X′ : |X′ +Σ′| → Σ′, +is given by +(Σ, iX) ⊗ (Σ′, i′ +X′) := +� +Σ ⊔ Σ′, (iX ⊔′ i′ +X′): |XΣ ⊔ X′ +Σ′| → Σ ⊔ Σ′� +, +where, explicitly, the homeomorphism iX ⊔′ i′ +X′ is defined as the composite +|XΣ ⊔ X′ +Σ′| +∼ += +−→ |XΣ| ⊔ |X′ +Σ′| +iX⊔i′ +X′ +−−−−−→ Σ ⊔ Σ′. +From the fact that Π(XΣ ⊔ X′ +Σ′) ∼= Π(XΣ) ⊔ Π(X′ +Σ′), it can moreover be proved +that VA is compatible with the symmetric monoidal structures of 2Cob +(n,n+1,n+2) +st +and 2Cob +(n,n+1,n+2) +BA +. +This discussion implies the following: +Theorem 261 (The bifunctor 2QA). Let A be a finite crossed complex. There is +a (symmetric monoidal) bifunctor, denoted +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +which is defined as the following composite of bifunctors, +2Cob +(n,n+1,n+2) +st +VA +−−−→ 2Cob +(n,n+1,n+2) +BA +2QBA +−−−−−→ vProfGrpfin. +□ +Note that 2QA is now decorated with a crossed complex A, rather than with its +classifying space BA. This is because the step VA depends on the crossed complex +A, and not only on its classifying space. +8.3.5. Morita valued once-extended TQFTs derived from finite crossed complexes. +Explicit formulae for the Morita valued version of the once-extended Quinn TQFT, +2Q +Mor +BA : 2Cob +(n,n+1,n+2) +BA +→ Mor, +in §6.4.5, can be derived from Theorem 259, by applying the general constructions +from Subsection 6.4. +As above, A denotes a fixed finite crossed complex. +Passing from groupoids, Γ, to their groupoid algebras, Lin(2)(Γ), as in §6.4.1, and +with Σ a closed smooth n-manifold, iXΣ : |XΣ| → Σ being a simplicial stratification +of Σ109, we have a canonical isomorphism of finite dimensional algebras, +(92) +2Q +Mor +BA +� +Σ, fΣ(iXΣ, A) +� ∼= Lin(2)� +π1(CRS(Π(XΣ), A) +� +. +These finite dimensional algebras associated to a closed n-manifold, Σ, with a +simplicial stratification, depend, explicitly, on the chosen simplicial stratification of +Σ. This dependence is, however, in a quite ‘mild’ way, exactly as for the case of +2QBA outlined in §8.3.3. We follow quite closely the discussion of this point, both +the ideas and approach, given in [30, 10.3 Morita equivalence]. If we choose two +simplicial stratifications, iXΣ : |XΣ| → Σ and jYΣ : |YΣ| → Σ, of Σ, then there exists +a canonically defined invertible bimodule, +2Q +Mor +BA +� +Σ, f Σ(iXΣ, A) +� +↛ 2Q +Mor +BA +� +Σ, f Σ(jYΣ, A) +� +, +connecting the algebras thus obtained. This bimodule is, itself, obtained by apply- +ing 2Q +Mor +BA +to the identity cobordism, +Σ +ι0 +�❚ +❚ +❚ +❚ +❚ +❚ +Σ +ι1 +�❥❥❥❥❥❥ +Σ × I +, +109so, in particular, XΣ is a finite simplicial set + +A CATEGORIFICATION OF QUINN’S TQFT +213 +where the source and target are provided with the BA-decorations, fΣ(iXΣ, A) and +f Σ(jYΣ, A), respectively. By construction, these bimodules compose well if we make +further changes to the simplicial stratification, or, for that matter, if we reverse the +‘direction’ of the cylinder exchanging ι0 and ι1. +It is then easily seen (the exposition is exactly as in Remark 155 and §6.4.5) that +these bimodules are invertible in the sense of the theory of Morita equivalence. In +particular, all algebras obtained by considering different simplicial stratifications +of Σ will be Morita equivalent, and a canonically defined Morita equivalence ex- +ists connecting each pair of these algebras. Moreover the bimodules associated to +changes of simplicial stratifications are natural with respect to the bimodules as- +sociated to cobordisms, (i, M, j): Σ → Σ′, where both Σ and Σ′ have a simplicial +stratification, and hence a given BA-decoration. +As before, we have, +Theorem 262 (The bifunctor 2Q +Mor +A +). We have a (symmetric, monoidal) bifunc- +tor, +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +→ Mor, +obtained by the following composition of bifunctors, +2Cob +(n,n+1,n+2) +st +VA +−−−→ 2Cob +(n,n+1,n+2) +BA +2QBA +−−−−−→ vProfGrpfin +Lin(2) +−−−−→ Mor. +□ +8.3.6. “Absolute” once-extended TQFTs derived from finite crossed complexes. As +before we take A to be a finite crossed complex. +As was noted earlier in an analogous context, the once-extended TQFTs, +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +and +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +−→ Mor, +do not give a value to a closed smooth n-manifold Σ, unless Σ is given a sim- +plicial stratification, iΣ : |XΣ| → Σ. In order to construct bifunctors whose do- +main is 2Cob(n,n+1,n+2), and whose target, unlike that of the once-extended Quinn +TQFT110, +2QBA : 2Cob(n,n+1,n+2) −→ vProfGrphf, +only outputs finite groupoids and finite dimensional algebras, we must specify a +symmetric monoidal bifunctor, from 2Cob(n,n+1,n+2) to 2Cob +(n,n+1,n+2) +st +. +If n = 0, the latter is quite easy to do, as 0-dimensional manifolds Σ essentially +have only one simplicial stratification. +For instance, we can observe that a 0- +dimensional manifold Σ has a simplicial stratification given by the obvious bijection +|XΣ| → Σ. Here XΣ is a simplicial set whose set of 0-simplices is Σ, and all i- +simplices with i > 0 are degenerate. In particular, this gives a symmetric monoidal +bifunctor 2Cob(0,1,2) → 2Cob(0,1,2) +st +. +For n ≥ 1, we need to use the axiom of choice, for classes, to construct a sym- +metric monoidal bifunctor 2Cob(n,n+1,n+2) → 2Cob +(n,n+1,n+2) +st +. (As we mentioned +at the beginning of Section 8, this step is non-canonical.) For instance, this can be +done by picking a simplicial stratification, iΣ : |XΣ| → Σ, of each connected com- +pact smooth manifold Σ, and then, if Σ′ is a not-necessarily connected manifold, +the decomposition of Σ′ into path-components provides a simplicial stratification +of Σ′. +110of Definition 149 + +A CATEGORIFICATION OF QUINN’S TQFT +214 +Theorem 263. Let A be a finite crossed complex. We have once-extended TQFTs, +� +2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, +and +� +2QMor +A +: 2Cob(n,n+1,n+2) −→ Mor. +These can be ‘normalised’ so that if Σ is any chosen path-connected (and, as usual, +closed and smooth) n-manifold, and iΣ : |XΣ| → Σ is a simplicial stratification of +Σ, then +� +2QA(Σ) ∼= π1 +� +CRS(Π(XΣ), A), CRS0(Π(XΣ), A) +� +, +and +� +2QMor +A +(Σ) ∼= Lin(2)� +π1 +� +CRS(Π(XΣ), A), CRS0(Π(XΣ), A) +�� +. +Proof. We compose the chosen bifunctor 2Cob(n,n+1,n+2) → 2Cob +(n,n+1,n+2) +st +with +either +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +or +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +−→ Mor. +□ +Remark 264. Under the conditions of the previous theorem, we note that we will +always have that the state space of Quinn’s finite total homotopy TQFT, +Q(s) +BA : 2Cob(n,n+1,n+2) → VectC, +on Σ, is canonically isomorphic to the free vector space on the set of components +of the groupoid π1 +� +CRS(Π(XΣ), A), CRS0(Π(XΣ), A) +� +. In other words, +Q(s) +BA(Σ) = C +� +π0 +� +CRS(Π(XΣ), A), CRS0(Π(XΣ), A) +�� +; +see Theorem 256. This makes it again clear in what sense the once-extended Quinn +TQFT is a categorification of Quinn’s finite total homotopy TQFT. +8.4. Some explicit calculations for the once-extended TQFTs derived +from finite groupoids and 2-groups. In the remainder of this paper, we will +give some examples of the algebras that the once-extended TQFTs111, +2QA : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +and +2Q +Mor +A +: 2Cob +(n,n+1,n+2) +st +−→ Mor, +assign to some n-dimensional manifolds. These will be for low dimensions, n = +0, 1, 2, and when A is the crossed complex given by a finite group, a finite groupoid, +or a crossed module of finite groups (equivalently a finite 2-group). The algebras we +assign to loops and surfaces are particular cases of ‘tube algebras’ considered in [32], +[30, Chapters 10 and 13] and [33, Section 3], in the context of models for excitations +of topological phases, derived from discrete higher gauge theory. (Understanding +these algebras was one of the initial motivations for this paper.) Note that this +will also determine the state spaces of the associated Quinn finite total homotopy +TQFTs, as observed in Remark 264. +111in theorems 261, 262 and 263, above + +A CATEGORIFICATION OF QUINN’S TQFT +215 +8.4.1. The simplest example: +the (0, 1, 2)-extended TQFT derived from a finite +groupoid. Recall from Subsection 7.1 that we can think of a groupoid as a 1- +truncated crossed complex, leading to a functor, ι1 : Grp → Crs. We recall that +this sends a groupoid, G = (s, t: G1 → G0), to the crossed complex, +ι1(G) = · · · → +� +a∈G0 +{ida} → · · · → +� +a∈G0 +{ida} → G1 +t +⇒ +s G0. +In particular, if G is a group, we then have +ι1(G) = · · · → {1G} → · · · → {1G} → G → {∗}, +and so the classifying space, Bι1(G), of the crossed complex, ι1(G), is the usual +simplicially defined classifying space, BG, of the group G. Given a non-negative +integer n, a complex number s ∈ C, and a finite group, G, we thus have the Quinn +finite total homotopy TQFT, +Q(s) +BG : Cob(n,n+1) → VectC, +that is obtained from the classifying space of G, and which coincides with the +Dijkgraaf-Witten TQFT with trivial cocycle112; see [43]. +Let us now turn to the extended case, and look at what happens when n = 0. +Each 0-dimensional manifold is trivially diffeomorphic to the disjoint union of +copies of the singleton manifold, S = {∗}. +On unpacking the construction in +Subsection 7.3, we obtain that the groupoid, +π1 +� +CRS +� +Π{∗}, ι1(G) +� +, CRS0 +� +Π{∗}, ι1(G) +�� +, +is isomorphic to the groupoid G. This leads to the following result, essentially in +[75, 76]. (We are, thus, considering the identification 2Cob(0,1,2) ∼= 2Cob +(0,1,2) +st +, +mentioned in §8.3.6.) +Theorem 265 ((0,1,2)-extended TQFTs derived from finite groupoids). For G a +finite groupoid, there are once-extended TQFTs, +� +2Qι1(G) : 2Cob(0,1,2) → vProfGrpfin, +and +� +2QMor +ι1(G) : 2Cob(0,1,2) → Mor, +such that +� +2Qι1(G) +� +{∗} +� ∼= G, +and +� +2QMor +ι1(G) +� +{∗} +� ∼= Lin(2)(G). +Here Lin(2)(G) is the groupoid algebra of G, over Q, so is the usual rational +group algebra of G, if G is a finite group. +The remaining parts of the specification of these (0, 1, 2)-extended TQFTs can +be obtained from Theorem 259 and Example 235. +112Note that, if considered just in the oriented case, so all the manifolds are orientable, Quinn’s +TQFT can naturally be twisted by cohomology classes, as we noted in Remark 92, using the same +method as in [43, 53]. In that case, one would recover Dijkgraaf-Witten TQFT in full generality. + +A CATEGORIFICATION OF QUINN’S TQFT +216 +Proof. One composes the already mentioned equivalence, 2Cob(0,1,2) +∼ += +−→ 2Cob(0,1,2) +st +, +with either +2Qι1(G) : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +or +2Q +Mor +ι1(G) : 2Cob +(n,n+1,n+2) +st +−→ Mor. +□ +Remark 266. The existence of the (0,1,2)-extended TQFT, +� +2QMor +ι1(G) : 2Cob(0,1,2) → Mor, +in the previous theorem, follows from [106, Theorems 3.52 and 3.5.4 in §3.8]. The +key reason is that oriented (0, 1, 2)-extended TQFTs are given by separable symmet- +ric Frobenius algebras, and groupoid algebras of finite groupoids can be given such +a structure; see [75, Examples 5.1 and 5.2.]. +We give some details, following the notation and conventions of [106, Definition +3.61 and Definition 3.62 in §3.8]. +A symmetric Frobenius algebra, (A, λ, e), by +convention here over Q, is given by an associative Q-algebra, A, with 1, together +with: +• a Q-linear map λ: A → Q, satisfying that λ(ab) = λ(ba), for all a, b ∈ A, +• an element e ∈ A ⊗ A, so e = � +i xi ⊗ yi, satisfying that given any w ∈ A, +we have +� +i +(wxi) ⊗ yi = +� +i +xi ⊗ (yiw). +Moreover, λ and e should satisfy the following compatibility condition, +� +i +λ(xi) ⊗ yi = 1A = +� +i +xi ⊗ λ(yi). +We also recall that an algebra A is called separable if there exists an element +e ∈ A ⊗ A, written e = � +i x′ +i ⊗ y′ +i, similarly satisfying that +� +i +(wx′ +i) ⊗ y′ +i = +� +i +x′ +i ⊗ (y′ +iw), +for any w ∈ A, and moreover such that +� +i +x′ +iy′ +i = 1A. +A groupoid algebra, Lin(2)(G), of a finite groupoid, is a separable algebra, for in- +stance via113 +e = +� +( x +g−→ y ) +1 +Nx +(x +g−→ y) ⊗ (y +g−1 +−−→ x) ∈ Lin(2)(G) ⊗ Lin(2)(G), +where the sum is extended to all morphisms (x +g−→ y) in G, and given an object x, +in G, Nx is the number of morphisms in G with source x. +Given a finite groupoid, G, the data that makes Lin(2)(G) a separable symmetric +Frobenius algebra, [75], is as shown below. +• λ: Lin(2)(G) → Q is defined by λ(x +g−→ y) = +� +1, if (x +g−→ y) = (x +idx +−−→ x) +0, otherwise, +whilst, as above, +113Note that an algebra being separable is a properly, and not a structure, unlike that of an +algebra be given the structure of a Frobenius algebra. + +A CATEGORIFICATION OF QUINN’S TQFT +217 +• e = +� +( x +g−→ y ) +(x +g−→ y) ⊗ (y +g−1 +−−→ x) ∈ Lin(2)(G) ⊗ Lin(2)(G). +By [106, Theorem 3.54], since groupoid algebras of finite groupoids are ∗-algebras, +via the inversion of morphisms, and that ∗-structure is compatible with the Frobenius +structure, Lin(2)(G) gives rise to a unoriented (0,1,2)-extended TQFT, therefore to +a (0,1,2)-extended TQFT as we have defined in this paper. +Using the explicit formulae in Theorem 259, we can then see that given a fi- +nite groupoid, G, the (0, 1, 2)-extended TQFT constructed, as in [106, §3.8.5 and +3.8.6], from the separable symmetric Frobenius algebra +� +Lin(2)(G), λ, e +� +coincides +with � +2QMor +ι1(G). The details are left to the reader. +The results in this paper, in particular, provide a homotopy-theoretical definition +for these oriented once-extended TQFT derived from finite groupoids, in terms of +homotopy orders of function spaces, but our construction, here, applies to consid- +erably more general settings than that. +Example 267. For a positive integer k, we let I(k) be the groupoid with set of +objects {1, . . . , k}, and a single morphism k → k′, given objects k and k′. We note +that all these groupoids are homotopy equivalent to I(1) ∼= {∗}, the final groupoid. +A quick calculation shows that +� +2QMor +I(k) ({∗}) ∼= MQ(k), +the algebra of k × k matrices with entries in Q. We note that this is Morita equiv- +alence to MQ(1), that is to Q itself; see also Example 157 +8.4.2. (1, 2, 3)-extended TQFT derived from finite groups. We now consider (1, 2, 3)- +TQFTs derived from finite groups. The same analysis can be modified to handle +finite groupoids, though the formulae become a bit more complicated. +Let us choose a particular simplicial stratification of S1. We consider S1 with a +simplicial stratification, iS1 : |XS1| → S1, where the simplicial set, XS1, has a single +0-simplex and a single non-degenerate one-simplex. Therefore, iS1 : |XS1| → S1 +gives a CW-decomposition of S1, with a unique 0-cell and a unique 1-cell114.. This +naturally gives simplicial stratifications for arbitrary disjoint unions of S1, by using +the obvious disjoint unions of this simplicial stratification. +Looking at Equation 86, we can now determine the groupoid, +π1 +� +CRS(Π(XS1), A), CRS0(Π(XS1), A) +� ∼= π1 +� +CRS(Π(S1 +sk), A), CRS0(Π(S1 +sk), A) +� +. +(We are using the notation of Example 191.) +The following definition is well known and ‘classical’. +Definition 268 (Action groupoid). Let the group G have a left-action, •, on a +non-empty set X. The action groupoid, X � G, or. in full, X �• G, has X as its +set of objects. Given x, y ∈ X, the set of morphisms, from x to y, is given by the +set of all pairs, (x, g), where g ∈ G is such that g • x = y. The composition of the +morphisms in X � G is then such that +� +x +(x,g) +−−−→ g • x +� +composed with +� +g • x +(g•x,h) +−−−−−→ (hg) • x +� +114We could consider CW-decompositions of S1 with more than one 0-cell. +The algebras +thereby obtained would then, in general, be different, however a natural Morita equivalence con- +nects them all; see §8.3.5. The case of S1 decomposed by using multiple 0-cells is reminiscent of +the calculations in [74, II-d] and in [32, 33]. Credit is due here to discussions with Alex Bullivant, +including [30, Theorem 10.3.2] + +A CATEGORIFICATION OF QUINN’S TQFT +218 +is +� +x +(x,hg) +−−−−→ (hg) • x +� +. +By using Remark 194, or directly by definition, we can see that crossed complex +maps from +Π(S1 +sk) ∼= · · · → {0} → {0} → {0} → Z → {∗} +to +ι1(G) = · · · → {1G} → · · · → {1G} → G → {∗}, +are in one-to-one correspondence with elements g ∈ G. Moreover, by unpacking +the construction in Subsection 7.3, we have an isomorphism of groupoids, +π1 +� +CRS(Π(S1 +sk), A), CRS0(Π(S1 +sk), A) +� ∼ += +−→ G � G, +where G � G is the action groupoid of the left-action of G on itself by conjugation, +that we met back in Example 158. +Given the above, for G a finite group and considering S1 with the simplicial +stratification, iS1 : |XS1| → S1, with a single 0-simplex, we have the following. +Theorem 269. The once-extended TQFTs, +2Qι1(G) : 2Cob +(1,2,3) +st +→ vProfGrpfin, +and +2Q +Mor +ι1(G) : 2Cob +(1,2,3) +st +→ Mor, +are such that +2Qι1(G)(S1, iS1) ∼= G � G, +and +2Q +Mor +ι1(G)(S1, iS1) ∼= Lin(2)(G � G), +where the groupoid algebra is taken over Q. +□ +By the previous theorem, it follows that 2Qι1(G) essentially gives the ω = 1 +case of the (1, 2, 3)-extended TQFT constructed by Morton in [92], categorifying +Dijkgraaf-Witten theory. +This (1,2,3)-extended TQFT 2Qι1(G) is also found in +[100]. +Finally note that we can apply Theorem 263 to the simplicial stratification +iS1 : |XS1| → S1 of S1. +This gives that we have (albeit non-canonical) once- +extended TQFTs, +� +2Qι1(G) : 2Cob(1,2,3) → vProfGrpfin, +and +� +2QMor +ι1(G) : 2Cob(1,2,3) → Mor, +such that +� +2Qι1(G)(S1) ∼= G � G, +and +� +2QMor +ι1(G)(S1) ∼= Lin(2)(G � G). +As we recalled in Example 158, the algebra Lin(2)(G � G) coincides with the +quantum double of the group algebra of G; some extra discussion on this is found +in [123] and also in [34]. +In particular, the argument leading to Theorem 269 +gives another proof (and provides a homotopy theoretical underpinning for) the + +A CATEGORIFICATION OF QUINN’S TQFT +219 +fact that, if G is a finite group, then there exists a Morita valued (1,2,3)-extended +TQFT sending S1 to the quantum double of the group algebra of G, see [10, 92]. +8.4.3. Towards (2, 3, 4)-extended TQFT derived from finite groups. We will briefly +discuss (2, 3, 4)-extended TQFTs derived from finite groups. +There is an infinite number of diffeomorphism classes of surfaces, therefore, the +once-extended TQFTs, +2Qι1(G) : 2Cob +(2,3,4) +st +→ vProfGrpfin, +and +2Q +Mor +ι1(G) : 2Cob +(2,3,4) +st +→ Mor, +will a priori require115 an infinite set of data to be specified, even if the algebras +and profunctors associated to surfaces are specified only up to a natural Morita +equivalence. +For this paper, we will focus only on some examples of the algebras assigned +to S2 and T 2 = S1 × S1. We could consider all other surfaces (orientable and +non-orientable), e.g. by choosing the usual CW-decompositions with unique 0 and +2-cells. +We choose a simplicial stratification, iS2 : |XS2| → S2, where XS2 has a single +0-simplex and a single non-degenerate 2-simplex. We let S2 +sk be the induced CW- +decomposition of S2, already considered in Subsection 7.2. In this case, all crossed +complex morphisms from +Π(S2 +sk) ∼= · · · → {0} → · · · → {0} → Z → {1} → {∗} +to +ι1(G) = · · · → {1G} → · · · → {1G} → {1G} → G → {∗}, +are trivial. However, the possible crossed complex homotopies are in bijection with +G. It is then easy to see that +π1 +� +CRS +� +Π(S2 +sk), ι1(G) +� +, CRS0 +� +Π(S2 +sk), ι1(G) +�� +∼= G. +We now determine the groupoid associated to the 2-torus T 2 = S1 × S1. There +is a simplicial stratification of the 2-disk D2, with two non-degenerate 2-simplices, +meeting along a diagonal edge. Identifying boundary edges in the classical way, this +gives a simplicial stratification of the 2-torus, T 2, here denoted iT 2 : |XT 2| → T 2. +We let T 2 +sk be the induced CW-decomposition of T 2. A quick calculation reveals +that the groupoid, CRS1 +� +Π(T 2 +sk), ι1(G)), of crossed complex maps from Π(T 2 +sk) to +ι1(G), together with the homotopies between them, is isomorphic to the action +groupoid, X � G, where +X = +� +(a, b) ∈ G × G | ab = ba +� +, +with the left-action of G given by g •(a, b) := (gag−1, gbg−1). All 2-fold homotopies +of crossed complex maps from Π(T 2 +sk) to ι1(G), are trivial, so we also have that +π1 +� +CRS +� +Π(T 2 +sk), ι1(G) +� +, CRS0 +� +Π(T 2 +sk), ι1(G) +�� +∼= X � G. +From the construction in Subsection 8.3, particularly §8.3.4, we then obtain: +115This issue will likely disappear if one further categorification level is introduced. + +A CATEGORIFICATION OF QUINN’S TQFT +220 +Theorem 270 ((2,3,4)-TQFTs derived from finite groups). Let G be a finite group. +The once-extended TQFTs, +2Qι1(G) : 2Cob +(2,3,4) +st +→ vProfGrpfin, +and +2Q +Mor +ι1(G) : 2Cob +(2,3,4) +st +→ Mor, +are such that (where the simplicial stratifications, iS2, of the 2-sphere and, iT 2, of +the 2-torus are as described above), +2Qι1(G)(S2, iS2) ∼= G, +and +2Q +Mor +ι1(G)(S2, iS2) ∼= Q(G), +where Q(G) = Lin(2)(G) is the group algebra of the group G over Q, whilst +2Qι1(G)(T 2, iT 2) ∼= X � G, +and +2Q +Mor +ι1(G)(T 2, iT 2) ∼= Lin(2)(X � G), +the groupoid algebra over Q, of the action groupoid X � G. +□ +By applying Theorem 263, it therefore follows that: +Theorem 271. Let G be a finite group. There are once-extended TQFTs, +� +2Qι1(G) : 2Cob(2,3,4) → vProfGrpfin, +and +� +2QMor +ι1(G) : 2Cob(2,3,4) → Mor. +These can be normalised so that +� +2Qι1(G)(S2) ∼= G, +and +� +2QMor +ι1(G)(S2) ∼= Q(G), +whilst +� +2Qι1(G)(T 2) ∼= X � G, +and +� +2QMor +ι1(G)(T 2) ∼= Lin(2)(X � G). +□ +8.4.4. Crossed modules of groups. Crossed modules of groupoids (Definition 180) +are well known to model all homotopy 2-types; see e.g. [85] or [13]. They can be +considered as 2-truncated crossed complexes, and so can be used as the algebraic +‘base’ of some once-extended TQFTs. +For convenience, we give the definition explicitly, but will restrict to the reduced +case, hence for crossed modules of groups. Such algebraic structure will, thus, model +connected homotopy 2-types. +Definition 272 (Crossed module of groups). A crossed module, +G = (∂ : E → G, ⊳), +of groups is given by: + +A CATEGORIFICATION OF QUINN’S TQFT +221 +• a group homomorphism ∂ : E → G, +together with +• a right-action, ⊳, of G on E by automorphisms. +This action is such that: +(1) ∂(a ⊳ g) = g−1 ∂(a) g, for all a ∈ E, g ∈ G (which is called the 1st Peiffer +relation), +(2) a⊳ ∂(e) = e−1 a e, for all a, e ∈ E (which is called the 2nd Peiffer relation). +The definition of crossed modules of groups is classical, going back at least to +Whitehead’s original papers on CW-complexes and crossed complexes, [121, 122]. +Recent treatments are in [20, 27, 7, 31]. As is well know, the category of crossed +modules is equivalent to the category of 2-groups; see e.g. [27, §2.5] and [7]. +A crossed module, G = (∂ : E → G, ⊳), of groups gives rise to a reduced crossed +complex, ι2(G). Explicitly ι2(G) has the form, +(93) +ι2(G) = · · · → {1} → · · · → {1} → E +∂−→ G → {∗}. +The classifying space, BG, of a crossed module, G, is, by definition, the same as +the classifying space of the crossed complex ι2(G). As usual, we may sometimes +use the same symbol for a crossed module and the corresponding crossed complex, +thus suppressing the notation for the inclusion functor, ι2. +If the crossed module, G, is finite, meaning that both E and G are finite, then +BG will be homotopy finite, and for n, a non-negative integer, we have an (n, n+1)- +TQFT, as in Definition 83, where s ∈ C is an arbitrary parameter, +Q(s) +BG : Cob(n,n+1) → VectC, +an (n, n + 1, n + 2)-extended TQFT, as in §8.3.4 +2Qι2(G) : 2Cob +(n,n+1,n+2) +st +−→ vProfGrpfin, +and the corresponding Morita valued form of the latter, as in §8.3.5, +2Q +Mor +ι2(G) : 2Cob +(n,n+1,n+2) +st +−→ Mor. +Finally, we have, non-canonical, once-extended TQFTs, as in Theorem 263, +� +2Qι2(G) : 2Cob(n,n+1,n+2) −→ vProfGrpfin, +and +� +2QMor +ι2(G) : 2Cob(n,n+1,n+2) −→ Mor. +The closed-manifold case of Quinn’s finite total homotopy TQFT, Q(s) +BG, where +G is a finite crossed module, was addressed in [53], in the context of homologically +twisted Yetter TQFT, and it coincides with the Yetter homotopy 2-type TQFT of +[124]. Formulae for the entire TQFT, Q(s) +BG, can be obtained as a particular case +of the formulae in Subsection 8.2, which work more generally for homotopy finite +crossed complexes. A more recent paper, [108], addresses closely related homotopy +quantum field theories derived from finite crossed modules. +In the remainder of this paper, we will give some explicit formulae for the once- +extended TQFTs derived from finite crossed modules, in the cases n = 0, n = 1, +and n = 2. Our description of those once-extended TQFTs will be parallel to the +discussion in §8.4.1, §8.4.2 and §8.4.3, of (0, 1, 2), (1, 2, 3)- and (2, 3, 4)-extended +TQFTs derived from finite groups. A crucial difference, however, is that, when +moving to the crossed module case, 2-fold homotopies may become non-trivial. + +A CATEGORIFICATION OF QUINN’S TQFT +222 +Remark 273. Let G be a crossed module. +In discrete higher gauge theory, as +treated in [31, 93], given an (in this case closed and smooth) n-manifold Σ, which +with a CW-decomposition will give the skeletally filtered manifold Σsk, we have, a +2-groupoid, 2Gauge(Σsk, G), of 2-gauge G-configurations, supported on Σsk, gauge +transformations, connecting 2-gauge G-configurations, and 2-gauge transformations +between gauge transformations; again see [31, 93]. This 2-groupoid was addressed +from the point of view of differential-geometric higher gauge theory in [3, 6, 52]. +Passing to the more general language of crossed complexes, 2Gauge(Σsk, G) is the +2-groupoid associated to the underlying crossed module of the crossed complex, +CRS +� +Π(Σsk), ι2(G) +� +. In particular the groupoid +π1 +� +CRS +� +Π(Σsk), ι2(G) +� +, CRS0 +� +Π(Σsk), ι2(G) +�� +, +which the once-extended Quinn TQFT � +2Qι2(G) associates to Σ, with that particu- +lar CW-decomposition, can be interpreted as the groupoid of 2-gauge G-connections +supported in Σsk, and gauge transformations, considered up to 2-gauge transforma- +tions, between them. +Remark 274. It is very likely that there exists an (n, n + 1, n + 2, n + 3)-extended +TQFT hence sending an n-manifold Σ, with a CW-decomposition, to the 2-groupoid +2Gauge(Σsk, G). We expect to address this in a future publication. +Some of the explicit calculations of the groupoids that the once-extended TQFTs +derived from crossed modules assign to the circle, the 2-sphere and the 2-torus can +also be found in [93], in the language of double groupoids, of 2-connections on +a manifold, and in [32, 33], in the language of tube algebras, whose irreducible +representations model excitations of higher gauge theory models for topological +phases of matter. +8.4.5. (0,1,2)-extended TQFTs derived from finite crossed modules. If G = (∂ : E → +G, ⊳) is a crossed module of groups, then ∂(E) is a normal subgroup of G, as is +clear from the first Peiffer relation. Moreover, from the facts that +Π({∗}) ∼= · · · → {0} → {0} → {0} → {1} → {∗}, +and +ι2(G) = · · · → {1} → · · · → {1} → E +∂−→ G → {∗}, +it is easy to see the following. +Lemma 275. If G = (∂ : E → G, ⊳) is a crossed module of groups, then +• the groupoid, CRS1 +� +Π({∗}), ι2(G) +� +, of crossed complex maps from Π({∗}) to +ι2(G), and homotopies between them, is isomorphic to G, so, in particular, +that groupoid has only one object, +and +• the groupoid, π1 +� +CRS +� +Π({∗}), ι2(G) +� +, CRS0 +� +Π({∗}), ι2(G) +�� +, of crossed com- +plex maps from Π({∗}) to ι2(G), and 2-fold homotopy classes of homotopies +between them, is isomorphic to the quotient group G/∂(E). +□ +As a consequence, in this context, we have the following result116. +Theorem 276 ((0,1,2)-TQFTs derived from finite crossed modules). Let G = +(∂ : E → G, ⊳) be a finite crossed module. +The once-extended TQFTs, +� +2Qι2(G) : 2Cob(0,1,2) → vProfGrpfin, +116As in §8.4.1, we are considering the identification 2Cob(0,1,2) ∼ += 2Cob +(0,1,2) +st +, of §8.3.6.) + +A CATEGORIFICATION OF QUINN’S TQFT +223 +and +� +2QMor +ι2(G) : 2Cob(0,1,2) → Mor, +are such that +� +2Qι2(G)({∗}) ∼= G/∂(E), +and +� +2QMor +ι2(G)({∗}) ∼= Q +� +G/∂(E) +� +, +the group algebra of the group, G/∂(E). +□ +The remaining parts of the specification of � +2Qι2(G) and � +2QMor +ι2(G) can be obtained +from Theorem 259. In particular, following on from Remark 266, restricting to the +oriented case, � +2QMor +ι2(G) is obtained from the following symmetric Frobenius algebra +structure on the group algebra Q +� +G/∂(E) +� +of G/∂(E), a separable algebra, +• λ: Lin(2)(G) → Q is defined by λ([g]) = | ker(∂)| δ +� +[g], 1G/∂(E) +� +, +• e = +1 +| ker(∂)| +� +[g]∈G/∂(E) +[g] ⊗ [g]−1 ∈ Q +� +G/∂(E) +� +⊗ Q +� +G/∂(E) +� +. +The details are left to the reader. +In particular, the (0, 1, 2)-extended TQFTs � +2QMor +ι2(G) and � +2QMor +ι1(G/∂(E)) are equiv- +alent. +As we have just seen, the (0, 1, 2)-extended TQFTs derived from finite crossed +modules are not more general than the (0, 1, 2)-extended TQFTs derived from finite +groups. It is an open problem whether (0,1,2)-extended TQFTs derived from more +general homotopy finite spaces, B, can similarly always be reduced to the finite +group case. +8.4.6. (1, 2, 3)-extended TQFTs derived from finite crossed modules. We give some +explicit formulae for the (1,2,3)-extended TQFTs that can be derived from a finite +crossed module. These give rise to (1,2,3)-extended TQFTs therefore associated to +finite 2-group higher gauge theory [6, 31, 34]. +Some preliminaries are needed. Similar constructions are in [31, 34]. The calcu- +lations shown below are a particular case of those of [51] and [57], which address +the more general case of 2-crossed modules, which are models for pointed homotopy +3-types. +First recall that if a group, G, has a right-action on the group E, by automor- +phisms, then we can form the semidirect product, G ⋉⊳ E. Our convention for the +semidirect product will be +(h′, e′)(h, e) := +� +h′ h, e (e′ ⊳ h) +� +, where h, h′ ∈ G, and e, e′ ∈ E. +This slightly non-standard convention for semidirect products arises from the con- +struction, in Subsection 7.3, of the groupoid, CRS1(A, B), of crossed complex maps +from the crossed complex, A, to the crossed complex, B, and homotopies between +them. +Fix a crossed module, G = (∂ : E → G, ⊳). Consider S1, with the same CW- +decomposition as in §8.4.2, so with a single 0-cell and a single 1-cell. By applying +Remark 194, we can see that crossed complex maps from Π(S1 +sk) to ι2(G) are in +one-to-one correspondence with elements of G. This can also be derived from (93) + +A CATEGORIFICATION OF QUINN’S TQFT +224 +and the fact that +Π(S1 +sk) ∼= · · · → {0} → {0} → {0} → Z → ∗ +ι2(G) = · · · → {1} → · · · → {1} → E +∂−→ G → {∗}. +(94) +To describe homotopies between these crossed complex maps, we can use Lemma +202, or a direct calculation, to see that, given a map f : Π(S1 +sk) → ι2(G), homotopies +with source f (i.e. +1-fold f-homotopies) are in one-to-one correspondence with +elements of G × E, seen as a set. To describe how each such crossed complex f- +homotopy modifies f : Π(S1 +sk) → ι2(G), as explained just after Definition 195, we +use the following result. This is motivated by the construction in Subsection 7.3, +and the following diagram, representing a homotopy of maps from Π(S1 +sk) to ι2(G); +cf. also [98, 99, 31]. Here g, h ∈ G and e ∈ E. +h +e +g +h g ∂(e) h−1 +Lemma 277. We have a left-action, •, of G⋉⊳ E on the underlying set of G, such +that, given g ∈ G and (h, e) ∈ G ⋉⊳ E, we have +(h, e) • g := h g ∂(e) h−1. +Proof. This is proved by direct calculation, using the first Peiffer identity, as follows: +(h′, e′) • +� +(h, e) • g +� += (h′, e′) • +� +h g ∂(e) h−1� += h′ h g ∂(e) h−1 ∂(e′) h′−1 += h′ h g ∂ +� +e (e′ ⊳ h) +� +h−1 h′−1 += +� +h′ h, e (e′ ⊳ h) +� +• g += +� +(h′, e′)(h, e) +� +• g, +where (h′, e′), (h, e) ∈ G ⋉⊳ E and g ∈ G. +□ +Unpacking the construction in Subsection 7.3, we can see that, for G = (∂ : E → +G, ⊳) a crossed module of groups, we have the following. +Lemma 278. The groupoid, CRS1 +� +Π(S1 +sk), ι2(G) +� +, of crossed complex maps from +Π(S1 +sk) to ι2(G), and homotopies between them, is isomorphic to the action groupoid, +G � (G ⋉⊳ E), +where we consider the action, •, of G ⋉⊳ E on G, just defined. +□ +In order to describe the groupoid, π1 +� +CRS +� +Π(S1 +sk), ι2(G) +� +, CRS0 +� +Π(S1 +sk), ι2(G) +�� +, +of crossed complex maps from Π(S1 +sk) to G, and 2-fold homotopy classes of homo- +topies between them, we must quotient out the morphisms G � (G ⋉⊳ E) by 2-fold +homotopies. By using Lemma 202, or a direct calculation, again based on (94), +given a crossed complex map, f : Π(S1 +sk) → ι2(G), 2-fold f-homotopies are in one- +to-one correspondence with elements of E. The resulting 2-groupoid is described +in detail in [93]. + +A CATEGORIFICATION OF QUINN’S TQFT +225 +To see what the quotient groupoid looks like, first note that if a ∈ E, (h, e) ∈ +G ⋉⊳ E, and g ∈ G, we have +(h, e) • g = +� +h ∂(a), (a−1 ⊳ g) e a +� +• g, +where we use the first Peiffer identity. +Lemma 279. We have a right-action ⊳′ of E on the underlying set of the group +G × +� +G ⋊⊳ E +� +, such that: +(g, h, e) ⊳′ a = +� +g, h ∂(a), (a−1 ⊳ g) e a +� +. +(Note that ⊳′ is not necessarily an action by automorphisms.) +Proof. This follows because the action, ⊳, of G on E is by automorphisms. +□ +We therefore have an equivalence relation on the set of morphisms of the groupoid +G � (G ⋉⊳ E), which preserves source and target maps, given by, for all a ∈ E, +� +g +(g,h,e) +−−−−→ (h, e) • g +� +∼ +� +g +(g,h,e)⊳′a +−−−−−−→ (h, e) • g +� += +� +g +� +g,h∂(a),(a−1⊳g) e a +� +−−−−−−−−−−−−−−→ (h, e) • h +� +. +Lemma 280. The composition of morphisms in the action groupoid G � (G ⋉⊳ E) +descends to the quotient under the equivalence relation above. +Proof. This follows from the general construction of the internal hom, CRS(−, −), +in the category of crossed complexes, as in [27, §9.3], and [28] for the case of crossed +modules, as was explained here in Subsection 7.3. A direct proof can be given as +follows, showing the importance of the second Peiffer condition for the construction +to work as stated117. +Consider a chain of composable morphisms in G � (G ⋉⊳ E), +g +(g,h,e) +−−−−→ (h, e) • g +� +(h,e)•g,h′,e′� +−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g, +and hence their composite is +g +� +g,h′h,e (e′⊳h) +� +−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g. +Given a, b ∈ E, the composite of the chain of morphisms in G � (G ⋉⊳ E), below, +g +(g,h,e)⊳′a +−−−−−−→ (h, e) • g +� +(h,e)•g,h′,e′� +⊳′b +−−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g, +or more explicitly, +g +� +g,h∂(a),(a−1⊳g) e a +� +−−−−−−−−−−−−−−→ h g ∂(e) h−1 +� +h g ∂(e) h−1,h′ ∂(b), +� +b−1⊳(hg∂(e)h−1) +� +e′ b +� +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +•g, +is +g +� +g,h′∂(b) h∂(a),(a−1⊳g) e a +�� +b−1⊳(hg∂(e)h−1) +� +e′ b +� +⊳(h∂(a)) +� +−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g. +By using the first and second Peiffer identities, and the fact that the action, ⊳, +of G on E is by automorphisms, this latter morphism in G � (G ⋉⊳ E) simplifies to +� +g, h′ h ∂ +� +(b ⊳ h) a +� +,(a−1 ⊳ g) e +�� +b−1 ⊳ (hg∂(e)h−1) +� +e′ b +� +⊳ h a) +� += +� +g, h′ h ∂ +� +(b ⊳ h) a +� +, (a−1 ⊳ g) e +� +b−1 ⊳ (hg∂(e)) +� +(e′ ⊳ h) (b ⊳ h) a +� += +� +g, h′ h ∂ +� +(b ⊳ h) a +� +, (a−1 ⊳ g) +� +b−1 ⊳ (hg) +� +e (e′ ⊳ h) (b ⊳ h) a +� += +� +g, h′ h, e e′ ⊳ h +� +⊳′ � +(b ⊳ h) a +� +. +117See [57] for how to deal with the case of 2-crossed modules, which are models for homo- +topy 3-types. There the second Peiffer condition is categorified and only holds up to a coherent +isomorphism. + +A CATEGORIFICATION OF QUINN’S TQFT +226 +□ +This leads to the following definition. +Definition 281 (The groupoid G �G). Let G = (∂ : E → G, ⊳) be a crossed module. +We define the groupoid, by abuse of notation denoted by G � G, such that +• the objects of G � G are elements of G; +• the morphisms of G �G are equivalence classes of arrows of the form below, +where g, h ∈ G and e ∈ E, +g +[(g,h,e)] +−−−−−→ (h, e) • g. +Here (g, h, e) ∼ (g, h′, e′) if there exists a ∈ E, such that +h′ = h ∂(a), +and +e′ = (a−1 ⊳ g) e a. +The composite of the chain of morphisms, +g +[(g,h,e)] +−−−−−→ (h, e) • g +�� +(h,e)•g,h′,e′�� +−−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g, +is +g +�� +g,h′ h,e (e′⊳h) +�� +−−−−−−−−−−−−→ +� +(h′, e′)(h, e) +� +• g. +The following lemma follows by an explicit calculation, using the construction +in §7.3.2, of the internal hom, CRS(−, −), in the category of crossed complexes. +Lemma 282. Consider a crossed module G = (∂ : E → G, ⊳), of groups. +The +groupoid, +π1 +� +CRS +� +Π(S1 +sk), ι2(G) +� +, CRS0 +� +Π(S1 +sk), ι2(G) +�� +, +whose objects are crossed complex maps, from Π(S1 +sk) to ι2(G), and whose mor- +phisms are 2-fold homotopy classes of homotopies between them, is isomorphic to +G � G. +□ +The proof of the following theorem follows from exactly the same form of dis- +cussion as that in §8.4.2, where we treated the group case. In particular the chosen +simplicial stratification, of S1, has the form iS1 : |XS1| → S1, where XS1 has a +single 0-simplex and a single non-degenerate one-simplex. This yields a simplicial +stratification on all finite disjoint unions of S1. +Theorem 283. Let G = (∂ : E → G, ⊳) be a finite crossed module. The once- +extended TQFTs, +2Qι2(G) : 2Cob +(1,2,3) +tr +→ vProfGrpfin +and +2Q +Mor +ι2(G) : 2Cob +(1,2,3) +tr +→ Mor, +are such that +2Qι2(G)(S1, iS1) ∼= G � G, +and +2Q +Mor +ι2(G)(S1, iS1) ∼= Lin(2)(G � G). +Here Lin(2) gives the groupoid algebra over Q. + +A CATEGORIFICATION OF QUINN’S TQFT +227 +As a consequence, applying Theorem 263, we have (albeit non-canonical) ex- +tended TQFTs, +� +2Qι2(G) : 2Cob(1,2,3) → vProfGrpfin, +and +� +2QMor +ι2(G) : 2Cob(1,2,3) → Mor, +that can be normalised such that their values on S1 are +� +2Qι2(G)(S1) ∼= G � G +and +� +2QMor +ι2(G)(S1) ∼= Lin(2)(G � G). +As before, recall that the rest of the structures of the once-extended TQFTs can +be obtained from the discussion in §8.3.2. +8.4.7. Towards (2, 3, 4)-extended TQFTs derived from finite crossed modules. We +resume the notation from §8.4.3 and that simpler case. +We will briefly look at (2, 3, 4)-extended TQFTs derived from finite crossed mod- +ules. Very similar constructions are in [32] and [33, Section 3], framed in the context +of excitations of strict 2-group topological phases, and also in [34], framed in the +context of invariants of loop braids derived from discrete higher gauge theory. +We use the simplicial stratifications of the 2-sphere S2 and of the 2-torus T 2 +appearing in §8.4.3. Recall that these yield CW-decompositions of S2, with unique +0- and 2-cells, and of T 2 with a 0-cell, three 1-cells and two 2-cells. +If G = (∂ : E → G, ⊳) is a crossed module of groups, then the action, ⊳, of G on +E restricts to an action of G on ker(∂). By using the second Peiffer relation, the +latter action descends to an action of G/∂(E) on ker(∂), where a ⊳ [g] := a ⊳ g. +Note that +Π(S2 +sk) ∼= · · · → {0} → {0} → {0} → Z → {1} → {∗}, +ι2(G) = · · · → {1} → · · · → {1} → E +∂−→ G → {∗}. +(95) +Therefore, an explicit calculation118 gives the following in which G = (∂ : E → G, ⊳) +is, as before, a crossed module of groups. +Lemma 284. We have an isomorphism of groupoids, +CRS1 +� +Π(S2 +sk), ι2(G) +� ∼= ker(∂) � G, +using the left-action of G on ker(∂), given by g • e := e ⊳ g−1. +We have an isomorphism of groupoids, +π1 +� +CRS +� +Π(S2 +sk), ι2(G) +� +, CRS0 +� +Π(S2 +sk), ι2(G) +�� +∼= ker(∂) � (G/∂(E)), +considering the left-action, of G/∂(E) on ker(∂), such that [g] • e := e ⊳ g−1. +□ +To examine the groupoid, +π1 +� +CRS +� +Π(T 2 +sk), ι2(G) +� +, CRS0 +� +Π(T 2 +sk), ι2(G) +�� +, +associated to the 2-torus, T 2, we start by unpacking the construction in Subsection +7.3. We can see that this groupoid is isomorphic to the groupoid, T 2(G), defined +as follows: +118for conventions on action groupoids see Definition 268. + +A CATEGORIFICATION OF QUINN’S TQFT +228 +• the objects of T 2(G) are diagrams of the form below119 +∗ +g +� +� +h +∗� +h +∗ +g +� +v⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +e′ +e +∗ +g, h, v ∈ G +e, e′ ∈ E, +∂(e) = g−1h−1v, +∂(e′) = v−1gh +; +• the 1-morphisms of T 2(G) are equivalence classes of arrows of the form +given below120, where g, h, v ∈ G, and e, e′ ∈ E, and, in addition, x ∈ G +and a, b, c ∈ E, +∗ +g +� +� +h +∗� +h +∗ +g +� +v⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +�⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +⑧ +e′ +e +∗ +(x,a,b,c) +−−−−−→ +∗ +xg∂(a)x−1 +� +� +xh∂(b)x−1 +∗� +x h ∂(b) x−1; +∗ +x g ∂(a) x−1 +� +x v ∂(c) x−1 +t +t +t +t +t +t +t +t +t +t +t +t +t +t +�t +t +t +t +t +t +t +t +t +t +t +t +t +t +� +c−1 e′ (a⊳h) b +� +⊳x−1 +� +a−1 (b−1⊳g) e c +� +⊳x−1 +∗ +• two arrows, given by (x, a, b, c) and (x′, a′, b′, c′) in T 2(G), where x, x′ ∈ G +and a, a′, b, b′, c, c′ ∈ E, with the same source and target, as in the example +above, are said to be equivalent if there exists a p ∈ E such that +(x′, a′, b′, c′) = +� +x ∂(p), (p−1 ⊳ g) a p, (p−1 ⊳ h) b p, (p−1 ⊳ v) c p +� +. +(In terms of the gauge interpretation mentioned in the footnotes, we are +here identifying two gauge transformations between discrete 2-connections +if they differ by a 2-gauge transformation.) +Finally, +• the composition in the groupoid, T 2(G), is induced by the semi-direct prod- +uct, G ⋉⊳ (E × E × E), with the product action of G, namely (a, b, c) ⊳ g = +(a ⊳ g, b ⊳ g, c ⊳ g). +It follows from the general construction in §7.3.2, that indeed T 2(G) is a groupoid. +The direct calculations are as in §8.4.6. Similar calculations appear in [34]. +Analogously to the discussion in §8.4.3, but now for a finite crossed module, +G = (∂ : E → G, ⊳), of groups, we can get: +Theorem 285 ((2,3,4)-TQFTs derived from finite crossed modules). The once- +extended TQFTs, +2Qι2(G) : 2Cob +(2,3,4) +st +→ vProfGrpfin, +and +2Q +Mor +ι2(G) : 2Cob +(2,3,4) +st +→ Mor, +119For how to interpret these diagrams in terms of fake-flat 2-gauge configurations, see [31, +§3.5], +120For how to interpret these arrows in terms of gauge transformations between fake-flat 2- +gauge configurations, see [31, §4.3.1]), + +A CATEGORIFICATION OF QUINN’S TQFT +229 +are such that +2Qι2(G)(S2, iS2) ∼= ker(∂) � +� +G/∂(E) +� +, +and +2Qι2(G)(T 2, iT 2) ∼= T 2(G), +and, therefore, it follows that +2Q +Mor +ι2(G)(S2, iS2) ∼= Lin(2) � +ker(∂) � +� +G/∂(E) +�� +, +whilst +2Q +Mor +ι2(G)(T 2, iT 2) ∼= Lin(2) � +T 2(G) +� +. +By applying Theorem 263, we, therefore, have non-canonical once-extended +TQFTs, +� +2Qι2(G) : 2Cob(2,3,4) → vProfGrpfin, +and +� +2QMor +ι2(G) : 2Cob(2,3,4) → Mor. +These can be normalised such that, for the 2-sphere S2, +� +2Qι2(G)(S2) ∼= ker(∂) � (G/∂(E)), +and therefore +� +2QMor +ι2(G)(S2) ∼= Lin(2) � +ker(∂) � +� +G/∂(E) +�� +, +and, on the 2-torus, T 2, +� +2Qι2(G)(T 2) ∼= T 2(G), +from which we get +� +2QMor +ι2(G)(T 2) ∼= Lin(2) � +T 2(G) +� +. +8.5. Final note: why should we bother with crossed modules? The TQFTs +obtained from finite crossed complexes are strictly more general than the ones that +can be obtained from groupoids, equivalently from disjoint unions of finite groups. +An easy example showing that this is so arises when n = 4, and +Q(s) +BG : Cob(n,n+1) → Vect, +for G a crossed module, with classifying space BG = Bι2(G). +Consider the CW-decomposition of S4 with a single 0-cell and a single 4-cell, +and no other cells. Consider the CW-decomposition of S2 with a single 0-cell and +a single 2-cell, and the product CW-decomposition on S2 × S2. The latter CW- +decomposition hence has a unique 0-cell, no 1-cells, two 2-cells, no 3-cell, and one +4-cell. These CW-decompositions, of S4 and of S2 × S2, are easily seen to arise +from simplicial stratifications. Moreover, we have +Π(S4 +sk) ∼= {0} → {0} → · · · → {0} → Z → {0} → {0} → {1} → {∗} +and +Π +� +(S2 × S2)sk +� ∼= {0} → {0} → · · · → {0} → Z → {0} → Z ⊕ Z → {1} → {∗}. +Let G be a finite groupoid, and recall +ι1(G) = · · · → +� +a∈G0 +{ida} → · · · → +� +a∈G0 +{ida} → G1 +t +⇒ +s G0. +An explicit calculation gives that the crossed complexes, +CRS +� +Π(S4), ι1(G) +� +and +CRS +� +Π(S2 × S2), ι1(G) +� +, + +A CATEGORIFICATION OF QUINN’S TQFT +230 +each are isomorphic to ι1(G). In particular, the state spaces, +Q(s) +BG(S4) +and +Q(s) +BG(S2 × S2), +where BG = Bι1(G), each have dimension given by the cardinality of the set of +components of G. +On the other hand, if G = (∂ : E → G, ⊳) is a finite crossed module of groups, +then +ι2(G) = · · · → {1} → · · · → {1} → E +∂−→ G → {∗}. +An explicit calculation gives that +π1 +� +CRS +� +Π(S4 +sk), ι2(G) +� +, CRS0 +� +Π(S4), ι2(G) +�� +∼= G/∂(E), +a groupoid with a single object. 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Knot Theory Ramifications, 2(1):113– +123, 1993. 2, 5, 56, 67, 164, 221 +Email address, Jo˜ao Faria Martins: j.fariamartins@leeds.ac.uk +Email address, Timothy Porter: t.porter.maths@gmail.com +(Faria Martins) School of Mathematics, University of Leeds, Leeds, LS2 9JT, United +Kingdom +(Porter) Ynys Mˆon / Anglesey, Cymru / Wales, ex-University of Bangor + diff --git a/VtE0T4oBgHgl3EQflwHV/content/tmp_files/load_file.txt b/VtE0T4oBgHgl3EQflwHV/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bef96c90bfd2d8568151e8b3fbdfdf9541c0c60d --- /dev/null +++ b/VtE0T4oBgHgl3EQflwHV/content/tmp_files/load_file.txt @@ -0,0 +1,7399 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf,len=7398 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='02491v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='CT] 6 Jan 2023 A CATEGORIFICATION OF QUINN’S FINITE TOTAL HOMOTOPY TQFT WITH APPLICATION TO TQFTS AND ONCE-EXTENDED TQFTS DERIVED FROM STRICT OMEGA-GROUPOIDS JO˜AO FARIA MARTINS AND TIMOTHY PORTER Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first revisit the construction of Quinn’s Finite Total Homotopy TQFT, which depends on the choice of a homotopy finite space, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We build our construction directly from homotopy theoretical techniques, and hence, as in Quinn’s original notes from 1995, the construction works in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our aim in this is to provide background for giving in detail the construc- tion of a once-extended TQFT categorifying Quinn’s TQFT, in the form of a symmetric monoidal bifunctor from the bicategory of manifolds, cobordisms and extended cobordisms, initially to the symmetric monoidal bicategory of profunctors (enriched over vector spaces), and then to the Morita bicategory of algebras, bimodules and bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These once-extended versions of Quinn’s TQFT likewise are defined for all dimensions, and, as with the original version, depend on the choice of a homotopy finite space B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To show the utility of this approach, we explicitly compute both Quinn’s finite total homotopy TQFT, and its extended version, for the case when B is the classifying space of a homotopically finite omega-groupoid, in this paper taking the form of a crossed complex, following Brown and Higgins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The constructions in this paper include, in particular, the description of once-extended TQFTs derived from the classifying space of a finite strict 2- group, of relevance for modelling discrete higher gauge theory, but the tech- niques involved are considerably more general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Acknowledgement: The greatest part of the research for this paper was financed by the Leverhulme Trust research project grant RPG-2018-029: Emergent Physics from Lattice Models of Higher Gauge Theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' JFM would like to express his gratitude to Alex Bullivant for discussion that framed some of the ideas supporting the construction of the once-extended version of Quinn’s finite total homotopy TQFT, to Fiona Torzewska for discussions on cobordism categories and on cofibrant cospans, to Ben Horton for discussions on bicategories of extended cobordisms, and also to Paul Martin, Vincent Koppen and Jack Rom¨o for useful discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We both thank Ronnie Brown for discussions on classifying spaces of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The final stages of the writing of this paper were partially funded by the project PTDC/MAT-PUR/31089/2017: Higher Struc- tures and Applications, of FCT (Portugal), and then by EPSRC, via the Programme Grant EP/W007509/1: Combinatorial Representation Theory: Discovering the In- terfaces of Algebra with Geometry and Topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Date: January 6, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' TQFTs, Extended TQFTs, Symmetric Monoidal Bicategory, Sym- metric Monoidal Bifunctor, Profunctors / Distributeurs, Morita Bicategory (of algebras, bimod- ules and bimodule maps), Homotopy Finite Spaces, Function Spaces, Crossed Complexes, Strict Omega-Groupoids, 2-groups, Discrete Higher Gauge Theory ©2022: Jo˜ao Faria Martins and Tim Porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1 A CATEGORIFICATION OF QUINN’S TQFT 2 Note: This is a preliminary version for commenting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Comments, suggestions, corrections, etc, are very welcome, and can be sent to either (or both) of the authors, j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='fariamartins@leeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='uk and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='maths@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Introduction In Lecture 4 of his lecture notes, [101, 1995], on axiomatic topological quantum field theory, Quinn described, what he called the finite total homotopy TQFT, following on from a suggestion of Kontsevich, [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This family of TQFTs, whose construction works in any spatial dimension, had, in special cases, been studied by Dijkgraaf and Witten, [43], and also by Segal, [107].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In that lecture, Quinn sketches the construction, starting from a space, B, which has ‘finite total homotopy’ or, as we will say, ‘is homotopy finite’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This finiteness is to ensure that the resulting theory takes values in the category of finite dimensional vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The basic construction used is quite simple in its main idea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let d be any non- negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Cob(d,d+1) be the symmetric monoidal category of closed smooth d-manifolds, and diffeomorphism classes of (d + 1)-cobordisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given B, and a (smooth and closed) d-manifold, Σ, then the TQFT, which we will denote by QB : Cob(d,d+1) → VectQ, assigns Q[Σ, B] to Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vector space corresponding to Σ is thus based on the set, [Σ, B], of homotopy classes of maps from Σ to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Important examples are when B is the classifying space of a finite group, or of a finite (strict) 2-group, where one retrieves well known examples of TQFTs, such as Dijkgraff-Witten’s TQFT, [43], and the Yetter-Porter TQFT, [124, 97, 53], but there are many other possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a cobordism, M, from Σ to another manifold, Σ′, the construction gives a matrix, and hence a linear transformation from QB(Σ) to QB(Σ′), the matrix being with respect to the given bases of the two vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our main purpose in this paper is to categorify this construction of Quinn to get what we call the once-extended Quinn TQFT, which will be formulated in three different, closely related, ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What do we mean by ‘categorification’?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In very general terms, when categori- fying a theory, one wants to try to replace sets by categories or groupoids, cat- egories themselves by 2-categories, or better bicategories, functions between sets by functors between categories, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', and, when all that is done, to add another layer corresponding to natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, for instance, we want to re- place the category, Cob(d,d+1), by a bicategory / weak 2-category, 2Cob(d,d+1,d+2), incorporating a form of 2-cobordism, or cobordism between cobordisms, between manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We want to replace [Σ, B], which is the same as π0(BΣ), the set of path- components of the mapping space1, BΣ, by the fundamental groupoid, π1(BΣ, BΣ), and then do corresponding adaptations of Quinn’s methodology to obtain Vect- valued profunctors between these groupoids, associated to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally we then construct appropriate transformations between profunctors to be associated to extended cobordisms connecting cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All this we want to linearise, forming the free linear categories on the groupoids, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To make all this work, we need to start by taking apart Quinn’s original method, and, noting that in the published version, [101], a lot is merely sketched, we have in- cluded a more detailed rendition of his theory, and, in fact, will give a parametrised family of variants of his theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Partially because of this, we work not only over Q, but over a more general subfield, κ, of C, as on occasion we will need the extra freedom that that gives us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1which may sometimes be more conveniently written as TOP(Σ, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 3 In developing this theory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we hoped that it would allow calculations that will generalise known ones,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we develop methods of explicit calculation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of both Quinn’s finite total homotopy TQFT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and its categorified versions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in a particular family of cases,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' namely when B is the classifying space of a homotopy finite strict ∞-groupoid,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' using the crossed complex model of such algebraic objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as developed by Brown–Higgins–Sivera,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [27],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and Tonks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, our framework includes the case when B is the classifying space of a strict 2-group, which is relevant for understanding TQFTs and extended TQFTs derived from discrete higher gauge theory, [6, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The framework for explicit calculations developed here can likely be extended in order to allow for combinatorial calculation of Quinn’s finite total homotopy TQFT, and its once-extended versions, whenever the homotopy finite space, B, is represented combinatorially, for instance when B is the classifying space of a finite simplicial group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that finite simplicial groups are considerably more general than crossed complexes, and do model all homotopy fnite spaces [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Such a study will be deferred to a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also expect that the categorification constructed here of Quinn’s finite to- tal homotopy TQFT, to a once-extended TQFT, can be further categorified to a doubly-extended, perhaps even fully-extended TQFT, [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This analysis will likewise be deferred to a subsequent paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in a bit more detail 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The ‘classical’ Quinn finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Vectκ = Vect be the category of κ-vector spaces and linear maps, which will usually be considered together with its usual symmetric monoidal category structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout the paper, given a non-negative integer, d, the symmetric monoidal category of closed d-manifolds and equivalence classes of cobordisms between them will be denoted Cob(d,d+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By a d-dimensional TQFT, we will mean a symmetric monoidal functor from Cob(d,d+1) to Vect, as in, for instance, [82, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that there is no assumption made, nor needed, that our manifolds or cobordisms be oriented, or even orientable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, we will need to work over the category, CGWH, of compactly generated, weak Hausdorff spaces, [87, 113].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a space, B, is called homotopy finite if B has only a finite number of path-components, each of which has only a finite number of non-trivial homotopy groups, all of which are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Also recall that given a homotopy finite space, its homotopy content is defined by the formula below (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' also [101, Lecture 4], [4] and [56, §3]): χπ(B) = � [x]∈π0(B) ��π2(B, x) �� ��π4(B, x) �� ��π6(B, x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π1(B, x) �� ��π3(B, x) �� ��π5(B, x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a fixed homotopy finite space B, Quinn defined what he called the finite total homotopy TQFT, denoted here by QB : Cob(d,d+1) → VectQ, defined for all d ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Part 2 of this paper, particularly in Section 4, we provide a thorough description of the construction of Quinn’s finite total homotopy TQFT, giving full mathematical details, in particular defining, more generally, a parametrised version, Q(s) B : Cob(d,d+1) → VectC, of it, where s is a complex parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All TQFTs, Q(s) B , for fixed B, are related by natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter parameter, s, was not present in Quinn’s original construction, and is closely related to the parameter appearing in the similar groupoidification functor in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Part 2 of this paper is subdivided into two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 4, we give full details of the construction of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Before that, in A CATEGORIFICATION OF QUINN’S TQFT 4 Section 3, which is considerably more technical, we formulate the required homotopy theoretical setting supporting its construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we introduce one of the main technical tools used in this paper, which is the idea of a fibrant span, (p, M, p′): B → B′, of homotopy finite spaces, meaning that we have a diagram of homotopy finite spaces, M p �❥❥❥❥❥❥ p′ �❚ ❚ ❚ ❚ ❚ ❚ B B′, and, crucially, the induced map, � p, p′� : M → B×B′, is required to be a fibration2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These fibrant spans of homotopy finite spaces can be composed by perform- ing the obvious pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, we have a category, HFspan, whose objects are homotopy finite spaces, and morphisms are fibred homotopy classes of fibrant spans connecting them3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The identity on a homotopy finite space, B, in HFspan, is given by the fibred homotopy class of the fibrant span of homotopy finite spaces, (sB, B, tB): B → B, obtained from the path-space fibration, that is, BI sB �❥❥❥❥❥❥ tB �❯ ❯ ❯ ❯ ❯ ❯ B B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here BI is the space of all maps from the unit interval, I to B, where I = [0, 1], and sB(γ) = γ(0) and tB(γ) = γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial step for the construction of Quinn’s finite total homotopy TQFT is a family of functors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' R(s) : HFspan → Vect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (working now over C),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' sending each homotopy finite space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to the free vector space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' C(π0(B)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' over the set π0(B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given a fibrant span,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): B → B′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the matrix elements of the linear map,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' R(s)� (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′) � : R(s)(B) → R(s)(B′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' are given by the equation � PCb(B) ��R(s)� (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′) ���PCb′(B′) � = χπ� ⟨p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′⟩−1(b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' b′) � � χπ(PCb(B)) �s � χπ(PCb′(B′)) �1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here b ∈ B, b′ ∈ B′, and we denote the path component of b in B by PCb(B), and the same for b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The existence of the functor, R(s) : HFspan → Vect, is implicit in the construc- tion in [101], and it is also addressed by G´alvez-Carrillo, Kock, and Tonks, [56], albeit in the framework of ∞-groupoids, and using homotopy pullbacks instead of the usual pullbacks (which we can use here, since we are working with fibrant spans only).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This also generalises the “groupoidification” construction in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a fixed homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The final step of the construction of Quinn’s final total homotopy TQFT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Q(s) B : Cob(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='d+1) → Vect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' made explicit in Section 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' relies on the existence of a functor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' FB : Cob(d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='d+1) → HFspan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' sending a d-manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to the space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' BΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of continuous functions from Σ to B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and sending the equivalence class of a (d + 1)-cobordism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' j): Σ → Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' from Σ to Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' seen as a cospan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Σ i �❘ ❘ ❘ ❘ ❘ ❘ Σ′ j �❧❧❧❧❧❧ S ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to the equivalence class of the following fibrant span of homotopy finite spaces,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' BS i∗ �❥❥❥❥❥❥ j∗ �❚ ❚ ❚ ❚ ❚ ❚ BΣ BΣ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2In this paper “fibration” means Hurewicz fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3A ‘dual’ category of cofibrant cospans is treated in [117].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 5 (Here given f : S → B, then i∗(f) = f ◦i and j∗(f) = f ◦j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') This allows us to define Quinn’s finite total homotopy TQFT, Q(s) B : Cob(d,d+1) → Vect, as the composite functor, Cob(d,d+1) FB −−→ HFspan R(s) −−−→ Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This functor, Q(s) B : Cob(d,d+1) → Vect, can be canonically given the structure of a symmetric monoidal functor, and hence of a TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 4, we also show some properties of Q(s) B , as we change B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, we show that Q(s) B depends only on the homotopy type of B, up to natural isomor- phisms, which fact is not immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, given a homotopy finite space, B, we have an action of the group, E(B), of homotopy classes of homotopy equiv- alences of B, on the TQFTs Q(s) B : Cob(d,d+1) → Vect, by natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are Theorems 90 and 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will show, much later on, in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, how to calculate Q(s) B , explicitly, in the case in which B is the classifying space of a strict ∞-groupoid, [2], noting that such a structure is often, more classically, called an ω-groupoid as in [27], and is often considered (as we will do here) in its form as a crossed complex, in the sense of [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The cases for Q(s) B treated here, when B is the classifying space of a crossed complex, include that in which B is the classifying space of a finite 2-group, and are hence relevant for understanding discrete higher gauge theory, [31, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The explicit formulae that we will give for the TQFTs derived from crossed complexes, via the special case of 2-groups / crossed modules, complement and generalise those of [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter reference considered only invariants of closed manifolds derived from crossed complexes, however providing a homotopy interpretation of the Yetter homotopy 2-type TQFT [124, 97] for the case of closed manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended versions of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Part 3 of this paper, which again is subdivided into two sections, is devoted to the construction of once-extended versions of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let d be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let 2Cob(d,d+1,d+2) be the symmetric monoidal bicategory of d-dimensional closed (and smooth) manifolds, (d + 1)- cobordisms between closed d-manifolds, and diffeomorphism classes of (d + 2)- extended cobordisms connecting (d + 1)-cobordisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [106], for instance, for precise definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let Prof κ = vProf be the symmetric monoidal bicategory with objects small linear categories (meaning categories enriched over Vect).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two small linear categories, C and C′, 1-morphisms from C to C′ are Vect-enriched profunctors, H: C ↛ C′, so, according to our conventions, they are enriched functors, H: Cop × C′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 2-morphisms are natural transformations of such enriched functors from Cop × C′ to Vect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [10, 63] for complete definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We review the definition of the bicategory, vProf, in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, a once-extended TQFT, sometimes called here, more briefly and more vaguely, an extended TQFT, will be, by definition, a symmetric monoidal bifunctor, 2Cob(d,d+1,d+2) → vProf, thought of as a categorified version of a classical TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also consider once-extended TQFTs formulated as symmetric monoidal bifunctors, 2Cob(d,d+1,d+2) → Mor, where Mor is the bicategory of algebras, bimodules and bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter constructions however always originate from bifunctors with target vProf by ‘linearisation’, in a similar way to [92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 6 We will work with the subbicategory, vProfGrp, of vProf, whose objects are groupoids, G = (s, t: G1 → G0), each made into a linear category by applying the free vector space functor, Lin: Set → Vect, to the hom-sets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given groupoids, G and G′, 1-morphisms in vProfGrp, from G to G′, are hence, by definition, Vect-profunctors, H: G ↛ G′, in our conventions these being functors, H: Gop × G′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given Vect-profunctors, H, H′ : Gop × G′ → Vect, a 2- morphism, η: H =⇒ H′, in vProfGrp, is a natural transformation of functors, Gop × G′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are two subbicategories of vProfGrp that we will consider here, namely vProfGrphf, the sub-bicategory of vProf, whose objects are the homotopy finite groupoids, and vProfGrpfin, the sub-bicategory of vProfGrp whose objects are the finite groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first version of the construction of the once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our first construction of a once-extended version of Quinn’s finite total homotopy TQFT is a (symmetric monoidal) bifunctor, 2QB : 2Cob(d,d+1,d+2) −→ vProfGrphf, constructed in Section 6, particularly Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As in the construction of Quinn’s finite total homotopy TQFT, we will factor 2QB by an intermediate homotopy theoretical construction, which we describe in Section 5, where we develop most of the homotopy-theoretical underpinning for the once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we consider a bicategory-like object (however not quite a bicategory), denoted 2span(HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The objects of 2span(HF) are homotopy finite spaces, and the 1-cells of 2span(HF), from X to Y , are fibrant spans, (p, M, p′): X → Y , of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The (not strictly associative) composition, •, of 1-cells in 2span(HF) is again given by the obvious pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each homotopy finite space, X, has a ‘horizontal identity’ of X, given by the path- space fibrant span, (sX, XI, tX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two fibrant spans of homotopy finite spaces, (p, M, p′), (q, N, q′): X → Y, 2-cells in 2span(HF), connecting them, are given by homotopy finite fibrant re- solved 2-spans, W: (p, M, p′) =⇒ (q, N, q′), by definition consisting of diagrams as below, W = X M p � p′ � Y XI sX � tX � L lX � rY � P � Q � Y I sY � tY � X N q � q′ � Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here X, Y, M, N, L are homotopy finite spaces, and, crucially for the construction to work, the induced map below, called the filler of W, is a fibration: (1) L PL −→ M × X×Y (XI × Y I) × X×Y N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (On the right-hand-side, we have the obvious pullback arising from the limit along the exterior faces of the diagram defining W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Again, homotopy finite fibrant resolved 2-spans compose horizontally and vertically, though not associatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the crucial cases arising in the once-extended Quinn TQFT, some particular 1- cells, (p, M, p′): X → Y , have ‘vertical units’, id(p,M,p′) : (p, M, p′) =⇒ (p, M, p′), A CATEGORIFICATION OF QUINN’S TQFT 7 and we moreover have ‘unitor’ 2-cells: ρ(p,M,p′) X : (X (p,M,p′) −−−−−→ Y ) • (Y (sY ,Y I,tY ) −−−−−−−→ Y ) =⇒ (X (p,M,p′) −−−−−→ Y ), and λ(p,M,p′) X : (X (sX,XI,tX) −−−−−−−→ X) • (X (p,M,p′) −−−−−→ Y ) =⇒ (X (p,M,p′) −−−−−→ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout Section 5, we construct an ‘assignment’, H = � π1(−, −), H, 2H) � : 2span(HF) → vProfGrphf, that gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Each homotopy finite space, X, is sent to its fundamental groupoid, π1(X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Given a, homotopy finite, fibrant span, X (p,M,p′) −−−−−→ Y , we have a Vect- profunctor, H � X (p,M,p′) −−−−−→ Y � : π1(X, X)op × π1(Y, Y ) → Vect, such that: (a) given x ∈ X and y ∈ Y , the (by construction, finite dimensional) vector space H(X (p,M,p′) −−−−−→ Y )(x, y) is the free vector space over the path components of the fibre ⟨p, p′⟩−1(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (b) Given morphisms in π1(X, X) and π1(Y, Y ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' equivalence classes of paths, in X and Y , x [γX] −−−→ x′ and y [γY ] −−−→ y′, the linear map, H � X (p,M,p′) −−−−−→ Y �� [γX], [γY ] � : H � X (p,M,p′) −−−−−→ Y � (x′, y) → H � X (p,M,p′) −−−−−→ Y � (x, y′), is induced by any of the homotopy equivalences, between fibres, ⟨p, p′⟩−1(x′, y) → ⟨p, p′⟩−1(x, y′), arising by applying the homotopy lifting property of ⟨p, p′⟩: M → X ×Y to γX and γY , together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here γX is the reverse of the path γX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) Finally, given W: (p, M, p′) =⇒ (q, N, q′), as above, we have a natural transformation, of functors π1(X, X)op × π1(Y, Y ) → Vect, 2HW : H � X (p,M,p′) −−−−−→ Y � =⇒ H � X (q,N,q′) −−−−−→ Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, given objects x ∈ X and y ∈ Y , the linear map, 2HW (x,y) : H � X (p,M,p′) −−−−−→ Y � (x, y) → H � X (q,N,q′) −−−−−→ Y � (x, y), has the following matrix elements, where m ∈ ⟨p, p′⟩−1(x, y) and n ∈ ⟨q, q′⟩−1(x, y), and PL is defined in (1), � PCm � ⟨p, p′⟩−1(x, y) � | 2HW (x,y) | PCn � ⟨q, q′⟩−1(x, y) �� = χπ� P −1 L (m, constx, consty, n) � χπ� PCn(⟨q, q′⟩−1(x, y)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here constx and consty are the constant paths at x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof that 2HW, defined this way, is indeed a natural transformation requires a wealth of careful verifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 8 Our main result in Section 5 is that the assignment, H: 2span(HF) → vProfGrphf, preserves all various compositions, and the horizontal identities, up to applying ap- propriate natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, vertical units and unitors are preserved by H, whenever they exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The hardest calculation is that indeed the natural transformations 2HW are well behaved with respect to the horizontal composition of fibrant resolved 2-spans of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is done in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Having developed the homotopy theoretical context supporting the once-extended Quinn TQFT, Section 6 is devoted to its explicit construction, in three different forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly to the case of Quinn’s finite total homotopy TQFT, we have an assignment, B(−) : 2Cob(n,n+1,n+2) → 2span(HF), sending each manifold, cobordism, or extended cobordism to its space of maps to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This preserves all compositions, identities, and unitors, up to natural homeo- morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, the once-extended Quinn TQFT, 2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, is defined by the composite functor, 2Cob(n,n+1,n+2) B(−) −−−→ 2span(HF) H −→ vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is treated in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We check later, in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, that indeed this bifunctor can naturally be given the structure of a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Σ is a d-dimensional closed smooth manifold, then the groupoid that 2QB associates to Σ is 2QB(Σ) = π1(BΣ, BΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a homotopy finite groupoid, however its set of objects is, in general, uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we will explain how the size of the image groupoids under 2QB can be reduced by considering a closely related bifunctor, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, that we call the finitary once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here the objects of the bicategory, denoted 2Cob (n,n+1,n+2) B , are now B-decorated manifolds, (Σ, f Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are, by definition, closed (and smooth) d-manifolds, Σ, equipped with a B-decoration, fΣ, that is, a finite subset, f Σ, of BΣ, con- taining at least one function in each path-component of BΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 1-morphisms, (Σ, f Σ) → (Σ′, f Σ′), are given by cobordisms, Σ → Σ′, with no further struc- ture, and the same for 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On objects, the finitary once-extended Quinn TQFT gives 2QB(Σ, f Σ) = π1(BΣ, fΣ), and on 1-morphisms and 2-morphisms, we make use of the obvious restrictions of the profunctors and natural transformations given by 2QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoids that 2QB associates to a B-decorated manifold, (Σ, fΣ), explic- itly depend on the B-decoration, fΣ, of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However this dependence is only up to a canonically defined invertible profunctor, which is functorial (up to natural isomorphism) with respect to further changes in the B-decoration, and also natural with respect to the profunctors associated to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita-valued once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, we change the target bicategory of our categorification of Quinn’s finite total homotopy TQFT from vProfGrpfin to Mor, the bicategory of (finite dimensional) algebras, bimodules and bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 9 Our starting point will be the discussion of a naturally defined linearisation bifunctor, Lin(2) : vProfGrpfin → Mor, essentially defined in [90], as part of a Morita equivalence between a linear category C and the algebra [C] that is associated to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On objects, Lin(2) sends a groupoid G to its groupoid algebra, [123, 34], here denoted Lin(2)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' At the level of 1- morphisms, a Vect-profunctor, H: G ↛ G′, of groupoids is then easily converted into a � Lin(2)(G), Lin(2)(G′) � bimodule, whose underlying vector space is � x∈ob(G),y∈ob(G′) H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Likewise, natural transformations of profunctors naturally linearise to bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These simple observations allow us to define yet one more version of the once- extended Quinn TQFT, the Morita valued once-extended Quinn TQFT, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor, by considering the following composite of bifunctors: 2Cob (n,n+1,n+2) B 2QB −−−→ vProfGrpfin Lin(2) −−−−→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is done in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebra, 2QB(Σ, f Σ), that is associated to a B-decorated d-manifold de- pends on the decoration, fΣ, of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However, this dependence is up to a canonically defined Morita equivalence, which is functorial with respect to further changes in the decoration, and natural with respect to the bimodules associated to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicit calculations for classifying spaces of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s finite total homotopy TQFT, Q(s) B , and its ‘finitary’ once-extended versions, 2QB and 2Q Mor B , can, in theory, be combinatorially calculated by passing to one of the existing combinatorial models for homotopy theory, for instance, simplicial sets, or simplicial groups, [41, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicit formulae in this general setting will be deferred to a future paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the last part of this paper, Part 4, we will work within a ‘truncation’ of homotopy theory, obtained by passing to the category of strict infinity groupoids, or ω-groupoids in the nomenclature of [27], which we will consider in their equivalent form as crossed complexes, following Brown and Higgins, [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (See also the more recent monograph, [27], by Brown, Higgins and Sivera.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We will give explicit formula for Quinn’s finite total homotopy TQFT, Q(s) B , and the two finitary versions of the once-extended Quinn TQFT, for cases in which B is the classifying space, BA, of a homotopy finite crossed complex, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that the spaces of the form BA, where A is a homotopy finite crossed complex, do not include all possible homotopy classes of homotopy finite spaces, but include, for instance, those that are 2-types (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' whose homotopy groups, πi, vanish for i ≥ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To this end, in Section 7, we review the homotopy theory of crossed complexes, closely following work of Brown, Higgins, Sivera, [27, 26], and Tonks, [116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our main new results are in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, and, given a subsimplicial set, Y , of a simplicial set, X, and a crossed complex, A, they give a crossed complex model for the fibres of the induced fibration, TOP(|X|, BA) → TOP(|Y |, BA), obtained by restricting a function, f : |X| → BA, to |Y |.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This has direct application to giving explicit formulae for Quinn’s finite total homotopy TQFT, and its extended versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 8, we finally give explicit formulae for Q(s) BA, 2QBA, and 2Q Mor BA , where A is a homotopy finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formulae are given in terms of what we A CATEGORIFICATION OF QUINN’S TQFT 10 call simplicial stratifications, iX : |XΣ| → Σ, of manifolds, Σ, and analogously for cobordisms between manifolds, and extended cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here XΣ is a simplicial set and iX is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Simplicial stratifications are more general than triangulations of manifolds, and typically allow us to decompose a manifold utilising a smaller number of simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It will not be necessary to prove that the formulae given do not depend on the chosen simplicial stratifications, since they are instead proved to coincide with quantities that are, by construction, topological invariants, except when it comes to what the once-extended TQFTs assign to d-manifolds, where the dependence on a simplicial stratification, only, up to naturally defined invertible profunctors, or bimodules, is naturally a feature of the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On that token, when treating the finitary once-extended versions of the Quinn TQFT, derived from a homotopy finite crossed complex, we will address, in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 and §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, yet two more versions of the once-extended Quinn TQFT, denoted 2QA : 2Cob (d,d+1,d+2) st −→ vProfGrpfin, and 2Q Mor A : 2Cob (d,d+1,d+2) st −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here the bicategory, 2Cob (d,d+1,d+2) st , has objects pairs, (Σ, iXΣ), where Σ is a closed smooth d-manifold, and iXΣ is a simplicial stratification of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 1- and 2-morphisms of 2Cob (d,d+1,d+2) st are cobordisms, and diffeomorphism classes of ex- tended cobordisms, without any choice of simplicial stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will, in turn, give rise to the construction of (albeit non canonical) once- extended TQFTs, � 2QA : 2Cob(d,d+1,d+2) −→ vProfGrpfin, and � 2QMor A : 2Cob(d,d+1,d+2) −→ Mor, obtained by picking a simplicial stratification of each path-connected closed d- manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This step in general requires using the choice axiom, but its full force is not necessary when the domain bicategory of a once-extended TQFTs is restricted to a ‘finitary’ sub-bicategory of 2Cob(d,d+1,d+2), for instance when considering pre- sentations of the symmetric monoidal bicategory, 2Cob(1,2,3), as done for example in [11, 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' One useful general theorem proved in this paper is the following (see Theorem 263, in Section 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a finite crossed complex, and d a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have once-extended TQFTs, � 2QA : 2Cob(d,d+1,d+2) −→ vProfGrpfin, and � 2QMor A : 2Cob(d,d+1,d+2) −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They can be ‘normalised’ so that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if Σ is any chosen path-connected closed d- manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and iΣ : |XΣ| → Σ is a simplicial stratification of Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then � 2QA(Σ) is the groupoid whose objects are the crossed complex maps from the fundamental crossed complex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Π(XΣ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of the simplicial set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XΣ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 11 and whose morphisms are crossed complex homotopies (considered up to 2-fold homotopy) between those crossed complex maps from Π(XΣ) to A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and � 2QMor A (Σ) is the groupoid algebra of � 2QA(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is quite a general result, of which we will give some representative examples in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category of crossed complexes includes that of groupoids and of strict 2-groups, as full subcategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Taking A to be a finite group, G, or, more generally, a finite groupoid, the theorem above gives a homotopy theoretical interpretation, and a proof of existence, of the (0, 1, 2)-extended TQFTs derived, as in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8], from the fact that the groupoid algebra of a finite groupoid is a “separable symmetric Frobenius algebra”, see [75, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to the (1, 2, 3)-extended TQFTs context, and considering a simplicial stratification of S1, with single 0- and 1-simplices, and with A a finite group, � 2QMor A associates the quantum double of the group algebra of G to S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a new proof of, and a homotopy theoretical interpretation for, the fact that there exists a Morita valued (1, 2, 3)-extended TQFT, sending S1 to the quantum double of the group algebra of G, which is essentially proven in [92, 10, 100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the overall construction is considerably more general, and it works in all dimensions, and for all finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we also develop, at the end of the paper, the case when A = G, a crossed module of finite groups, which is of relevance for higher gauge theory, [6, 3, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Concretely, we write down, in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, some explicit formulae for the (1, 2, 3)- and (2, 3, 4)-extended TQFTs derived from G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to the language of discrete higher gauge theory, as treated in [31, 93], the algebras that � 2QMor G associates to S1 and to the torus coincide with the ‘tube algebras’ considered in [30, Sections 10 and 13], [32] and [33, Section 3], in the context of models for excitations of topological phases, derived from higher gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These algebras were one of the initial motivations for the work in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A general result, directly following from the theorem above, is that if Σ is an n-manifold, with a simplicial stratification, then there exists an (n, n + 1, n + 2)- extended TQFT that sends Σ to the groupoid of discrete G-connections in Σ and gauge transformation (considered up to 2-gauge transformation) between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We expect that, if G is a finite simplicial group – so that G can represent any finite homotopy type by Ellis’ theorem, [46], – then there will similarly exist once- extended TQFTs, 2QG : 2Cob (d,d+1,d+2) st −→ vProfGrpfin, and 2Q Mor G : 2Cob (d,d+1,d+2) st −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' sending (Σ, iX : |XΣ| → Σ), to the groupoid of simplicial maps from XΣ to W(G), the simplicial classifying space of G, and homotopy classes of maps between them, up to 2-fold homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In that case, 2QG(Σ, iΣ) will be the fundamental groupoid of the simplicial function space, W(G)XΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We hope to address this in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we expect that the recent construction of topological invariants of 4-manifolds derived from 3-groups (2-crossed modules), in [102], is a particular case of the Quinn finite total homotopy TQFT, using a 3-type, B, represented by a 2-crossed module of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 12 In a future publication, we also expect to address the construction of homotopy quantum field theories, including extended ones, derived from crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This should be closely related to the construction in [108].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also hope to address whether Quinn finite total homotopy TQFT can be further categorified, and also explore its twisting by cohomology classes of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Introduction 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The ‘classical’ Quinn finite total homotopy TQFT 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended versions of Quinn’s finite total homotopy TQFT 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first version of the construction of the once-extended Quinn TQFT 6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary once-extended Quinn TQFT 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita-valued once-extended Quinn TQFT 8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicit calculations for classifying spaces of crossed complexes 9 Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Preliminaries 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some general conventions and notation 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' General notation 13 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for groupoids 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for topological spaces 15 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Review of fibrations 17 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Review of fibre homotopy equivalence 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Brief crib-sheet on manifolds, cobordisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 20 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions on bicategories 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions on profunctors 35 Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy theoretical underpinning of Quinn’s finite total homotopy TQFT 43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopically finite spaces and the category HFspan 43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopically finite (HF) spaces 43 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrant spans of HF spaces and their composition 47 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A family of functors, R(s) : HFspan → Vect, derived from the homotopy content 54 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A more detailed review of Quinn’s finite total homotopy TQFT 60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cobordism categories 61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s results on HF function spaces 63 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s finite total homotopy TQFT 64 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A discussion of the monoidality of Quinn’s finite total homotopy TQFT 66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some examples and properties of Quinn’s finite total homotopy TQFT 67 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Changing B 69 Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Once-extended versions of Quinn’s finite total homotopy TQFT 72 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy-theoretical underpinning of the once-extended Quinn TQFT 73 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation and some more basic results about fibrations 73 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor construction 80 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF fibrant resolved 2-spans connecting fibrant spans 85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF resolved fibrant 2-spans and natural transformations of profunctors 89 A CATEGORIFICATION OF QUINN’S TQFT 13 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The horizontal composition of HF resolved 2-spans 98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertical composition of HF resolved 2-spans 108 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Towards horizontal and vertical identities 112 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Comment and Summary 115 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Once-extended versions of Quinn’s TQFT 117 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for the bicategory 2Cob(n,n+1,n+2) 117 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A once-extended version of Quinn’s TQFT 121 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A finitary version of the once-extended Quinn TQFT 125 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita valued extended Quinn TQFT 128 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structure in 2Cob(n,n+1,n+2) 139 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structure of the bifunctor 2QB 157 Part 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Calculations for classifying spaces of ω-groupoids 163 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed complexes: their homotopy theory and their classifying spaces 165 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition of crossed complexes, and related notions 165 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fundamental crossed complexes of filtered spaces 169 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopy of crossed complexes 173 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The classifying space of a crossed complex 180 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrations of crossed complexes and profunctors from fibrant spans 188 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Computing the homotopy content of a finite crossed complex 195 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' TQFTs and once-extended TQFTs derived from homotopy finite crossed complexes 199 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions and nomenclature 201 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' TQFTs from homotopy finite and finite crossed complexes 202 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended TQFTs derived from finite crossed complexes 206 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some explicit calculations for the once-extended TQFTs derived from finite groupoids and 2-groups 214 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Final note: why should we bother with crossed modules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 229 References 230 Part 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Preliminaries We will review some of the background theory in the various areas that will feed into this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Many readers will not need to read these short sections and need only refer to them when the ideas and results, mentioned here, are needed in later sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we also set up some of the necessary notation, a quick skim through to see what is here is probably necessary however.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some general conventions and notation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' General notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let V and W be vector spaces, over a field κ, with given bases, X and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will specify a linear map, f : V → W, by giving its matrix elements, denoted ⟨x | f | y⟩ ∈ κ, where x ∈ X and y ∈ Y , hence f(x) = � y∈Y ⟨x | f | y⟩y for each x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X is a finite set, then its cardinality is denoted |X| 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 4We note that the same notation is used for the geometric realisation of a simplicial set, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For most of the time, the risk of confusion is slight, and on the few occasions that both notations occur in the same expression, we expect that the context allows the meaning to be gleaned unambiguously with little pain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 14 The category of sets is denoted Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category of κ-vector spaces will be denoted Vect, or Vectκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category of topological spaces will be denoted Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If C is a monoidal category, the tensor product functor is denoted by ⊗C : C×C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the cases where the tensor product arises from a coproduct or a product in C, we will also use the notation (respectively), ×C : C × C → C and ⊔C : C × C → C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid, G, will be denoted G = (s, t: G1 → G0), where G1 and G0 are, respectively, the set of morphisms and the set of objects of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Morphisms of G are frequently denoted as (s(g) g−→ t(g)), so s(g) is the source of g and t(g) its target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The identity of x ∈ G0 is denoted 1x = (x 1x −→ x), or 1G x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our convention for notation for composition in this context is that the composition of (x g−→ y) and (y h−→ z) is (x gh −→ z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set of arrows from x to y is denoted G(x, y), or homG(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertex group at x ∈ G is G(x) := homG(x, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A totally disconnected groupoid is a groupoid for which the source and target maps coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Totally disconnected groupoids are frequently denoted in the style A = (β : A1 → A0), where β := s = t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will identify a groupoid having just a single object with its group of mor- phisms of that single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid, G, is said to be discrete if it has no non-identity arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, it is more or less indistinguishable from its set, G0, of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a groupoid may also be called trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We often identify a set, X, with the corresponding discrete groupoid having X as its set of objects, and, of course, just the identity arrows as the arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives an inclusion of the category of sets into that of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This inclusion functor has a left adjoint, sending a groupoid, G, to the set of connected components, π0(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For basic information on the theory of groupoids, see [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A simple, but useful, example of a groupoid is the ‘unit interval groupoid’, often denoted I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this groupoid, we have objects 0 and 1, and only two non-identity morphisms, 0 (0,1) −−−→ 1 and 1 (1,0) −−−→ 0, with the evident compositions and identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will often think of groupoids as modelling very simple homotopy types (1- types).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also recall the notion of homotopically finite space;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Combining the two notions, we will have a notion of homotopy finite, or ho- motopically finite groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is just one of several related finiteness conditions on groupoids, so we list some of the main ones that may be used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid, G, will be said to be finite if both G0 and G1 are finite sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' G may be called locally finite if each ‘hom-set’ G(x, y) is a finite set, (so in particular its vertex groups are all finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' G will be called homotopically finite (or briefly to be a HF-groupoid) if it has finitely many connected components and each vertex group, G(x), is a finite group, (so both π0(G) and π1(G, x) ∼= G(x) are finite for each possible base point, x ∈ G0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We introduce some notation that may be used in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Grp will be the category of groupoids, finGrp will be that of finite groupoids, locfinGrp that of locally finite groupoids, A CATEGORIFICATION OF QUINN’S TQFT 15 and hfGrp that of HF-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will require a certain background of concepts and notation when handling topological spaces, not all of which is considered in many sources on topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Recall that a space X is called weak Hausdorff, see [87, 111] and [114, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='9], if given any continuous map, f : K → X, where K is compact Hausdorff, then f(K) is closed in X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) We will denote by CG the full subcategory of the category Top of topological spaces with objects the compactly generated spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for definitions see, for in- stance, [111], [66, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4], [114, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='9] or [55, page 242] – note that these are called k-spaces in [87, 55, 66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) We have a k-ification functor, denoted k: Top → CG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definitions are in [55, page 242] and [114, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is a right adjoint to the inclusion functor CG → Top;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [55, page 243].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X is a space, then the map k(X) → X given by the identity function, which we will sometimes denote ǫX : k(X) → X, is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives the counit of the adjunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If f : X → Y is a continuous map between topological spaces, then k(f): k(X) → k(Y ) is f itself, at the level of maps between sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (4) If K is a compact Hausdorff space, then a set map, f : K → k(X), is continuous if, and only if the same map f : K → X is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The same holds if K is compactly generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In particular, by noting that all disks, Dn, are compact Hausdorff, it follows that the map ǫX : k(X) → X is a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (5) It follows by the discussion above that if X is weak Hausdorff, then so is k(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (6) As in [87, 55, 113], we will work in the category CGWH, the full subcategory of Top with objects the compactly generated and weak Hausdorff topological spaces, (which we will refer to as CGWH spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These include all compact Hausdorff spaces and all metric spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that CGWH has all small limits and colimits, [80, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note further, see [66, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='22], the limits in CGWH are computed by computing the limits in Top, and then applying the k-ification functor, so, for example, given a pair of CGWH spaces, their product is X × Y = k(X ×0 Y ), where ×0 is the product in Top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Unless otherwise specified, all limits and colimits of CGWH spaces will be calculated in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (7) We note, moreover, that CGWH is a cartesian closed monoidal category, [80, 111].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given CGWH spaces, X and Y , the space of maps from X to Y , will be denoted both by Y X and by TOP(X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X is compact Hausdorff, the topology on Y X is the k-ification of the compact-open topology on the set of maps from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (8) As in [111, 80], a subset, A, of a CGWH space, X, will be always be given the k-ification of the topology induced by X, called the CGWH induced topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that if F is closed in X, then F with the induced topology from X is already CGWH, so k will not modify the topology, hence the CGWH induced topol- ogy on F is the usual induced topology on F as a subspace of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if A ⊂ B ⊂ X, then the k-ification of the topology that X induces on A coincides with the k-ification of the topology that B, with the k-ification of the induced topology from X, induces on A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the inclusion, A → X, is continuous, and A CATEGORIFICATION OF QUINN’S TQFT 16 if A ⊂ X, and f : Y → A, with both X and Y being CGWH spaces, then f is continuous (where A has the k-ification of the induced topology) if, and only if, f is continuous when considered as a map from Y to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (9) Given a CGWH space, X, then XI denotes the space of maps from I = [0, 1] to X with the k-ification of the compact-open topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have continuous maps, which we will often denote, s := sX, t := tX : XI → X with s(γ) = γ(0) and t(γ) = γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notation emphasises that these pick out the source and target of a path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Alternative notation will sometimes be used, for instance, e0(X) and e1(X), standing for evaluation at 0 or 1, as these are more natural if considering XI as being a ‘cocylinder’ on X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' dual to the usual cylinder X × I, when dealing with homotopies between maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In a particular circumstance, in which such a notation might lead to clashes or confusion, we may substitute other notation, but will note the substitution nearer the place it is used5 (10) Given a CGWH space X, and x ∈ X, we write Fx(X) for the space of paths, γ : I → X, starting at x, and Ωx(X) for the space of all paths in D starting and ending in x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are given the CGWH induced topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (11) Given a CGWH space X, and j ∈ {0, 1}, also define the inclusions, ιX j := ιj : X → X × I, by ιj(x) = (x, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We may occasionally use simplified notation for these end inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (12) Given a CGWH space X, we define π0(X), as usual, as the coequaliser, π0(X) := Coeq(XI s −→ −→ t X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note, however, that π0(X) is a CGWH space, and not just a set, and need not have the discrete topology here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is, of course, a natural projection map, p : X → π0(X), and the fibres of that map (in the category CGWH) are obtained by pulling back along maps from singletons to the space π0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The safe way to handle this (so as not to end up with spaces that are not CGWH) is given in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (13) Given a CGWH space, X, and an element, x ∈ X, the path-component that x belongs to will be denoted by PCx(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each path component of X is given the k-ification of the topology induced by X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This CGWH interpretation of path component corresponds to the formula PCx(X) = p−1(p(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (14) The category with objects the CGHW spaces and morphisms from X to Y , being the maps, X → Y , considered up to homotopy will be denoted CGWH/ ≃.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (15) We will consider a functor, �π0 : CGWH/ ≃ → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This sends a CGWH space, X, to the set, �π0(X), of (k-ified) path components in X, in other words to the set of PCx(X), for x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, we note that two or more different elements of X may correspond to, and hence may label, the same path component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We do not want to invoke the axiom of choice and, for instance, choose a representative point in each path component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Let X and Y be CGWH spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a homotopy class, [f]: X → Y , of maps from X to Y , we put �π0(f) � PCx(X) � := PCf(x)(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is an obvious one-to-one correspondence, π0(X) ↔ �π0(X), if we forget the topology in π0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout the paper, it will be useful to distinguish between π0(X) and �π0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will write �π0(X) = {PCx(X) | x ∈ X}, but note again that different x in X may correspond to the same element PCx(X) ∈ �π0(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5An instant of this is when handling simplicial sets, where s is used, almost universally in the literature, for the degeneracy mappings, so conflicts with s as ‘source’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 17 (16) Another very useful way to view �π0(X) is as actually being the set consisting of the fibres of the continuous function, p: X → π0(X), mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As fibres naturally are k-ified, this provides a quick categorical way of handling the index- ation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The same double interpretation of the set of connnected components will be useful later on when we look at connected components of crossed complexes, see page 168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Review of fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us recall, for instance from [87, Chapter 7] or [113, Chapter 5] the definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 1 (Fibration).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let E and B be CGWH spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We say that p: E → B is a Hurewicz fibration (abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' fibration) if the following homotopy lifting property holds: given any CGWH space X, any homotopy, H : X × I → B, and any map, ˆf : X → E, making the diagram with solid arrows, below, commutative, then there exists a map, H′ : X × I → E, making the full diagram commutative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X ι0 � ˆ f � E p � X × I H′ �♥ ♥ ♥ ♥ ♥ ♥ ♥ H � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This H′ : X × I → E is called a lifting of H starting at ˆf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Differently from the conventions in [87, Chapter 7], we do not impose that fibrations are surjective6, hence, given any space B, we have a fibration, ∅ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall that: Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' • The composite of fibrations is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X and Y are CGWH-spaces, then pX : X × Y → X and pY : X × Y → Y are both fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Pullbacks of fibrations are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that if p: E → B is a fibration, and f : X → B is any map (of CGWH spaces), then the map, q: X ×B E → X, appearing in the pullback diagram below is a fibration ([87, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 Lemma]), X ×B E q � � E p � X f � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If p: E → X is a fibration, then, given x ∈ X, the fibre of p at x is Ex := p−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Following our conventions in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, the fibre Ex is given the induced CGWH-topology from E, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the k-ification of the topology that E induces on Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that p is continuous, p−1(x) is closed in E, so p−1(x) is compactly generated already, with the induced topology, so the k-ification step will not modify the topology in Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that we have the following pullback diagram in CGWH, p−1(x) � inc � E p � {x} inc � X, 6This surjectivity condition was dropped in the subsequent [88];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see footnote on page 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 18 where the inc denote the obvious inclusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will make extensive use of the fact that if p: E → B is a fibration, and x, y ∈ B are in the same path-component, then the fibres, p−1(x) and p−1(y), are homotopy equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [87, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need that if E is path- connected and x ∈ X, it follows that all path-components of Ex are homotopically equivalent, [48, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will review some of these results in more detail, later, starting with Lemma 94, page 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A is a subset of X, being considered with the k-ification of the induced topology, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider EA = p−1(A), with the induced CGWH-topology, and the induced map, pA : EA → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to see that we have a pullback diagram in CGWH: EA pA � inc � E p � A inc � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Indeed given a CGWH space, Z, and continuous maps, f : Z → E and g : Z → A, such that p ◦ f = inc ◦ g, there is a unique set map, f ′ : Z → EA, obtained by restricting the codomain of f to EA, which arises from the fact that the diagram above gives a pullback in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The map, f ′, is continuous as a map f ′ : Z → E, hence it is continuous as a map Z → EA, since Z is CGWH and EA has the k- ification of the induced topology, so the diagram above is a pullback in CGWH as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since pullbacks of fibrations are fibrations, we have that pA : EA → A is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, suppose that p: E → A is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let e ∈ E and set x = p(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to see that p(PCe(E)) = PCx(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover the induced map pe : PCe(E) → PCx(X) is a surjective fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is [109, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1], (which can easily be adapted to the CGWH setting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Looking at the dual setting, recall that a map, f : A → X, of CGWH spaces is a cofibration, [87, Chapter 6], or [113, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1], if it satisfies the homotopy extension property: For any CGWH space, B, any map, g : X → B, and any homotopy, H : A × I → B, as in the diagram, A ιA 0 � f � X ιX 0 � g � B A × I H � f×idI � X × I H′ �① ① ① ① there is a homotopy, H′, making the diagram commute, so extend- ing the homotopy H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following two well known results will be used without further comment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : A → X be a cofibration and let B be a CGWH space, then the induced map on mapping spaces, f ∗ : BX → BA, sending φ: X → B to φ ◦ f : A → B, is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (For example, see [87, Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') If (X, Y ) is a CW-complex pair, meaning that Y is a subcomplex of the CW- complex, X, then the inclusion map i: Y → X is a cofibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see, for instance, [55, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Review of fibre homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f, g : X → Y be maps of CHWH spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A homotopy, H : X × I → Y , connecting f to g, will frequently be denoted by f H =⇒ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two fibrations, p: X → B and q: Y → B, over the same space, a fibre map, or fibred map, f : X → Y , is a map such that the diagram below commutes, (2) X p �❅ ❅ ❅ ❅ ❅ ❅ ❅ ❅ f � Y q �⑦⑦⑦⑦⑦⑦⑦⑦ B Two fibre maps, f, g : X → Y , are fibre homotopic if there exists a homotopy, H : X × I → Y , called a fibre or fibred homotopy, connecting f and g, and such that for each (x, t) ∈ X × I, p(x) = q(H(x, t)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We write f H =⇒ B g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We say that a pair of fibre maps, f : X → Y and f ′ : Y → X, realises a fibre homotopy equivalence if we have fibre homotopies, f ◦ f ′ H =⇒ B idX and f ′ ◦ f H′ ==⇒ B idY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that H : X × I → X satisfies p(H(x, t)) = p(x), for each (x, t) ∈ X × I, and similarly for H′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following non-immediate, but well-known, result will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is the dual version of Dold’s Theorem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see the discussion and references in [69, Chapter I, section 6, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 4 (Fibre homotopy equivalence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that in (2), f : X → Y is a homotopy equivalence, then there exists a homotopy inverse, f ′ : Y → X, of f, which is a fibre map, and such that f and f ′ realise a fibre homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a proof, see [87, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5], [23, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Very thorough discussions in the dual case of cofibrations appear in [21, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2: Addendum] and in [69], as mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Almost by definition, we have: Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that f : X → Y , as in (2), is a fibre homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any b ∈ B, the map, f, restricts to a homotopy equivalence, p−1(b) → q−1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We will also need a ‘relative’ version of the result from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is also to be found in [87, Chapter 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First we need two definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first is simply the restriction of the notion of morphism between maps in a category, applied to CGWH and, in particular, to fibrations there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two fibrations, p : D → A and q : E → B, a map from p to q is a pair, (g, f), of maps as in the square, D g � p � E q � A f � B, making that square commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We write (g, f) : p → q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two fibrations, p: D → A and q: E → B, a map, (g, f): p → q, as above, is a homotopy equivalence of fibrations if there are homotopy inverses, f ′ of f and g′ of g, such that p ◦ g′ = f ′ ◦ q, and, in addition, there are homotopies, A CATEGORIFICATION OF QUINN’S TQFT 20 H : g′ ◦ g ≃ idD, and K : g ◦ g′ ≃ idE, that cover homotopies, h: f ′ ◦ f ≃ idA and k: f ◦ f ′ ≃ idB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The relative version of Lemma 4 is then: Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If (g, f): p → q is a map of fibrations in which both g and f are homotopy equivalences, then (g, f): p → q is a homotopy equivalence of fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A proof can be given by pulling q back along g, and then applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is also a relative version of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose p: D → A and q: E → B are fibrations, and (g, f): p → q is a homotopy equivalence between them, then, for any a ∈ A, the induced map on fibres, f : p−1(a) → q−1(f(a)) is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We note that, as this assumes that f is a homotopy equivalence, π0(f) is a bijection and so, if b ∈ B, there is an a ∈ A with f(a) ∈ PCb(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that q−1(b) is homotopy equivalent to p−1(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The actual homotopy equivalence will, of course, depend on the path used to join f(a) and b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Brief crib-sheet on manifolds, cobordisms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A topological manifold of dimension n is a Hausdorff and second countable topological space, S, such that each point of S has a neighbourhood homeomorphic to an open subset of the upper half plane of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A smooth manifold, (S, smtS), is a pair, consisting of a topological manifold, S, and a smooth structure, smtS, on S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see, for instance, [65], or [89, §1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We call S the underlying topological manifold of (S, smtS), and will usually abbreviate (S, smtS) to S, when the context makes this unambiguous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that a topological manifold being smooth is a structure, not a property, and some topological manifolds do not have a smooth structure at all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If M is a compact smooth manifold, then it can be given a finite triangulation, and, in particular, it can be given the structure of a finite CW-complex, see [94].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that if M is a smooth manifold with border, then we can find, again see [94], a triangulation of the pair, (M, ∂M), making ∂M a subcomplex of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the inclusion, ι: ∂M → M, is a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next give a fairly standard definition of cobordism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will need to shift the viewpoint slightly before generalising this to ‘extended cobordisms’, but we will do that in two stages, rewording the definition and notation, before giving that generalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' General references include [72], for the standard idea of cobordism, and, for the higher dimensional extended case, see [106, 91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two n-manifolds, Σ1 and Σ2, are said to be cobordant, if they ‘jointly’ or ‘together’ (‘co’) form the ‘bord’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the boundary, of an (n + 1)- manifold, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that there are embeddings, ij : Σj → ∂S → S, for j = 1, 2, so that the induced map, Σ1 ⊔Σ2 → ∂S, is a homeomorphism (or a diffeomorphism if we are working with smooth manifolds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a cospan in CGWH, Σ1 i1 �❏ ❏ ❏ ❏ ❏ ❏ Σ2 i2 �tttttt S , and we note that � i1 i2 � : Σ1 ⊔ Σ2 → S is an embedding, and is thus a cofibration, as it is, essentially, given by the inclusion of the boundary of a smooth manifold A CATEGORIFICATION OF QUINN’S TQFT 21 into the manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here � i1 i2 � is the obvious induced morphism from the disjoint union, Σ1 ⊔ Σ2, to S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We say that the cospan is cofibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such cofibrant cospans are treated in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the above setting, we say: Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The (n + 1)-manifold, S, or, more precisely, the 5-tuple, (S, Σ1, Σ2, i1, i2), is called a cobordism from Σ1 to Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If the dimension is not evident from the context, we may say that S is an (n+1)- cobordism, or even, occasionally, an (n, n + 1)-cobordism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The manifold, Σ1, is often referred to as the inward boundary, whilst Σ2 is then the outward boundary of the cobordism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For any n-manifold, Σ, the trivial or identity cobordism, is given by S = Σ × I, for I = [0, 1], with the two ends being copies of Σ, via the obvious embeddings, ei(Σ) : Σ ∼= Σ × {i} ֒−→ Σ × I, i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cobordisms compose in the well-known way by gluing the outward boundary of the first to the inward boundary of the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is only defined up to isomorphism, so it is usual to pass to diffeomorphism classes of cobordisms, relative to the boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With smooth cobordisms, a collar on the boundaries has to be chosen so as to ensure that the resulting composite cobordism also has a smooth structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [89, §1 and §3] or [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A collar also comes in in the proof that the trivial cobordism acts as an identity arrow for the composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We refer the reader to the various sources mentioned earlier, but will be considering related issues later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will need to consider certain types of cobordisms between cobordisms, which are thought of as being certain ‘2-cospans’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To introduce these, we will need to have the notion of a manifold with corners.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The study of manifolds with corners was originally developed by Cerf, [39], and Douady, [44], in the early 1960s, as a natural generalisation of the concept of manifolds with boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A (differential) manifold with corners is a generalisation of the standard defini- tion above obtained by replacing the upper half plane of Rn by Rn +, that is [0, ∞)n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have maximal atlases that consist of families of compatible charts, where the charts, (ϕ, U), are of the form ϕ : U → [0, ∞)n, and where two charts, (ϕi, Ui), for i = 1, 2, are said to be compatible if, and only if, ϕ2 ◦ ϕ−1 1 : ϕ1(U1 ∩ U2) → ϕ2(U1 ∩ U2) is a diffeomorphism;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see, for instance, the original sources and Laures, [77], for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For each x ∈ X, with x ∈ U, the number, c(x), of zeros in the coordinates of ϕ(x) does not depend on the choice of chart, (U, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That number is called the index of x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A manifold with corners is a topological manifold with boundary equipped with a maximal atlas of charts, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A face of a manifold with corners is the closure of some connected component of the set of points, {x ∈ X | c(x) = 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 22 We often need to know how the faces fit together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this, J¨anich, [67], in- troduced the notion of an ⟨n⟩-manifold, which is reviewed by Laures, [77], in the context that we need.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A manifold with corners is called a manifold with faces if each x ∈ X belongs to c(x) different connected faces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is an obvious set of simple examples of such objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Take X = Rm +.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a manifold with faces in an obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The faces are the m-different coordinate hyperplanes, say Hi = {x ∈ Rm + | xi+1 = 0}, i = 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , m − 1, and adopting the obvious order (H0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , Hm−1), and taking ∂i = Hi we note that each Hk is a manifold with faces using the faces Hi ∩ Hk, for i ̸= k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives us the basic examples of the following: Definition 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An ⟨n⟩-manifold is a manifold, X, with faces together with a spe- cific ordered n-tuple, (∂0X, ∂1X, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ∂n−1X), of faces of X which satisfy the con- ditions: 1) ∂0X ∪ ∂1X ∪ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∪ ∂n−1X = ∂X, the boundary of X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2) ∂iX ∩ ∂jX is a (possibly empty) face of both ∂iX and ∂jX if i ̸= j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any ⟨n⟩-manifold gives rise to an n-dimensional cubical diagram of topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More precisely let 2 be the ordered set, {0 < 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a category with two objects plus a single non-identity arrow, 0 ← 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set ⟨n⟩ to be the n-fold product category, 2n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given an ⟨n⟩-manifold, X, we can form a functor, X : ⟨n⟩ → CGWH, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If a = (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , an−1) is an object of ⟨n⟩, set X(a) = � ai=1 ∂iX, with X(0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , 0) = X, the morphisms in ⟨n⟩ are then sent to the evident inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that a ⟨0⟩-manifold is just a manifold without boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A ⟨1⟩-manifold is a manifold with boundary and a ⟨2⟩-manifold is a manifold with corners with ∂X = ∂0X ∪ ∂1X, and ∂0X ∩ ∂1X consists of ‘corners’ of index 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For us, the key case will be that with n = 2 and a ⟨2⟩-manifold is a manifold with corners, with the specified faces being (∂0X, ∂1X) and it gives ∂0X ∩ ∂1X � � ∂0X � ∂1X � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Laures, in [77], introduced cobordisms of ⟨n⟩-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will adopt and adapt the form of his definition that is given in [106].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will shift our viewpoint, and notation, on cobordisms, ready for the extended higher dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Y1 and Y2 be smooth closed n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If W is a cobordism from Y1 to Y2, then it is an (n + 1)-manifold with boundary, and hence a ⟨1⟩-manifold, but, in addition, there is a specified decomposition of ∂W, traditionally written something like ∂inW ⊔ ∂outW, together with embeddings, as before, i1 : Y1 → ∂W, i2 : Y2 → ∂W, so that i1(Y1) = ∂inW and i2(Y2) = ∂outW, making i := � i1 i2 � : Y1 ⊔ Y2 → ∂inW ⊔ ∂outW = ∂W an isomorphism / diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For later use, we note a cobordism, as above, corresponds to a cospan, Y1 i1 −→ W i2 ←− Y2, A CATEGORIFICATION OF QUINN’S TQFT 23 but also note that although this process preserves the composition (up to isomor- phism), it does not preserve identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The identity cobordism on a manifold, Y, gives the cospan Y e0(Y ) −−−−→ Y × I e1(Y ) ←−−−− Y, whilst the identity cospan on Y is Y idY −−→ Y idY ←−− Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let now W0, W1 be two (n, n + 1)-cobordisms from Y1 to Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will write ∂Wi = ∂inWi ⊔ ∂outWi ∼= Y0 ⊔ Y1, for i = 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An (extended) (d − 2, d − 1, d)-cobordism from W0 to W1 will be a ⟨2⟩-manifold, S, equipped with a decomposition and isomorphism, W0 ⊔ W1 g−→ ∂0,inS ⊔ ∂0,outS = ∂0S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a decomposition and isomorphism, (Y1 × I) ⊔ (Y2 × I) f−→ ∂1,inS ⊔ ∂1,outS = ∂1S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so that g−1 ◦ f : Y1 × {0} ⊔ Y2 × {0} → ∂inW0 ⊔ ∂outW0, and g−1 ◦ f : Y1 × {1} ⊔ Y2 × {1} → ∂inW1 ⊔ ∂outW1, which are to be compatible with the structural isomorphisms of W0 and W1 and the decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A good way of getting a picture of such a structure is via a double cospan diagram of a special kind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (It is in fact a cofibrantly resolved double cospan, but to justify that name here would prolong the discussion too much for a mere ‘crib-sheet’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') This looks like: Y1 i1,0 � e0(Y1) � W0 gin � Y2 i2,0 � e0(Y2) � Y1 × I fin � S Y2 × I fout � Y1 i1,1 � e1(Y1) � W1 gout � Y2 i2,1 � e1(Y2) � where the top and bottom rows give the two cobordisms, W0 and W1, but the left and right ‘vertical’ cobordisms are trivial / identity ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a double cobordism is not the most general form one could imagine, but corresponds to the fact that such ‘2-cospans’ form something more like a bicategory with these as the 2-cells, rather than a double bicategory of 2-cospans in some sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This point is explored in Jeff Morton’s paper, [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That being said, a second point to note is that the left and right cospans are not trivial cospans, as such would have form Y idY −−→ Y idY ←−− Y , but that could not correspond to a cobordism as the middle term would need to be of higher dimension than the ends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will be applying mapping space constructions, BY , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', to these various types of cobordisms and cospans, and will then take up some of these ideas again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions on bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will have frequent need to use the termi- nology of the theory of bicategories7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The basics of bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A basic introduction to this theory can be found in Leinster’s [78], with a more thorough and complete description given in Borceux’s [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We may also use the summary to be found in [106], and the relevant parts of the draft book, [68], by Johnson and Yau.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also use that last reference as one of the sources for definitions relating to monoidal categories and monoidal bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A bicategory, B, is specified by the following: a collection of objects, denoted Ob(B), or sometimes B0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each pair of objects, a, b in B, a locally small category, B(a, b), whose ob- jects are 1-morphisms from a to b, whose morphisms are called 2-morphisms and whose composition is sometimes referred to as vertical composition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for objects, a, b, c in B, there are composition functors, ca,b,c : B(a, b) × B(b, c) → B(a, c), and, for each object a in B, a functor, Ia : [0] → B(a, a), (where [0] is the ‘singleton’ category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The functors, c, are called horizontal compositions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and natural isomorphisms, α : ca,b,d ◦ (cb,c,d × id) ⇒ ca,c,d ◦ (id × ca,b,c), λ : ca,b,b ◦ (Ib × id) ⇒ id, and ρ : ca,a,b ◦ (id × Ia) ⇒ id, called, respectively, the associator and the left and right unitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are required to satisfy the pentagon and triangle identities, which we omit here, referring the reader to Borceux, [17], and the many other sources in the liter- ature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Although when discussing specific bicategories, we will more often than not use generic composition symbols such as • or ◦, but when it is clear whether we intend horizontal or vertical composition, it can be useful to have available some specific notation that distinguishes them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In such cases, we may use #0 for hori- zontal composition and #1 for the vertical one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The motivation for the symbolism is that in horizontal composites the ‘intersection’ of the two 2-morphisms is the object which is the codomain of the first 2-morphism and the domain of the second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Objects live in dimension 0 of the data specifying the bicategory, hence #0 would be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For vertical composition of suitable 2-morphisms, the ‘intersection’ is a 1-morphism, so lives in dimension 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two classes of examples of bicategories usually given are the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Any (strict) 2-category is a bicategory in which the natural isomorphisms, α, λ and ρ, are all identity arrows, (in particular, the composition is associa- tive, and the identities are identities ‘on the nose’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have that any category gives a ‘locally discrete’ bicategory in which each hom-category, B(a, b), is ‘just a set’, or more exactly is a discrete category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Any monoidal category, (M, ⊗, I), gives a bicategory, M[1], in which there is just one object, which we will denote by ∗, and where M[1](∗, ∗) = M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition, c∗∗, is the tensor, ⊗, and the identity, I∗, is the monoidal identity, I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7We recall that these are weak 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 25 We will see several other examples of bicategories later on coming from spans, profunctors, both set and vector space valued ones, from bimodules, and, of course given the context, from cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Much of this paper will involve checking that various constructions are compatible with the bicategorical properties of their do- main and codomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A and B be bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A bifunctor, sometimes also called a homomorphism, or a pseudo-functor8, F : A → B, or more completely (F, ϕ), consists of (1) a function, F : A0 → B0, mapping objects to objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) for each pair of objects, a, a′ in A, a functor, Fa,a′ : A(a, a′) → B(F(a), F(a′));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) natural isomorphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ϕa0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each triple,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of objects in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as shown in the diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A(a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a1) × A(a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a2) cA � Fa0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a1 ×Fa1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a2 � ✖✖✖✖� ϕa0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a2 A(a0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a2) Fa0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a2 � B(F(a0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(a1)) × B(F(a1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(a2)) cB � B(F(a0),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(a2)) and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [0] IA a � IB F (a) �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ☞☞☞☞� ϕa A(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a) Fa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a � B(F(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(a)) such that certain diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' expressing compatibility with the corresponding associators and unitors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' commute,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' again,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we will not give them here as they can easily be found in the literature and we will give them in a simplified case slightly later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If the ϕa0,a1,a2 and ϕa are all identities, F is said to be a strict homo- morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It may sometimes be useful to weaken the conditions on the ϕ requiring that they just be natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, the term ‘morphism’ may be used, as in Leinster’s notes, [78].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As any 2-functor, F : A → B, between two 2-categories, yields a homomorphism between the corresponding bicategories, the above should be seen as a natural gen- eralisation of a 2-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is, thus, natural to consider transformations between homomorphisms between bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation: Given a 1-morphism, b1 h−→ b2, in a bicategory B, we denote by h∗ and h∗, the natural transformations / induced morphisms, h∗ : B(b, b1) → B(b, b2), and h∗ : B(b2, b) → B(b1, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8We will tend to use this last term in a particular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 26 Definition 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let (F, ϕ), (G, ψ) : A → B be two homomorphisms between bicate- gories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A transformation, also called a pseudo-natural transformation, σ : F → G, is given by 1-morphisms, σa : F(a) → G(a), for each object a in A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' natural isomorphism, as in the diagram, A(a, a′) Fa,a′ � Ga,a′ � ✒✒✒✒� σa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='a′ B(F(a), F(a′)) (σa)∗ � B(G(a), G(a′)) (σa′ )∗ � B(F(a), G(a′), and, thus, for each f : a → a′, a 2-morphism, σf : G(f)σa ⇒ σa′F(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before we omit the conditions for compatibility with the other struc- ture, referring to the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given the fact that bicategories have an extra layer of structure than do cate- gories, it is not surprising that not only does one consider bifunctors / homomor- phisms, and transformations between them, but also some sort of 2-transformation between transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are called modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given F, G : A → B, as before, and σ, θ : F → G, two transforma- tions, a modification, Γ : σ → θ, consists of a 2-morphism, Γ1 : σa ⇒ θa, for every object a in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are required to make the following square commute, G(f)σa id#Γa � σf � G(f)θa θf � σa′F(f) � θa′F(f), for every 1-morphism f : a → a′ in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We refer the reader to [106] for how to compose transformations, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', so that one gets a bicategory Bicat(A, B) with the resulting structure, provided the bicategories are small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not be using this that much, so will refer to the sources if we need to use it, rather than introduce more conventions here It will be useful, from time to time, to be able to use the following analogues of some basic categorical notions adapted to work internally within a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We suppose that A is a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) Given a pair of 1-morphisms, f : A → B and u : B → A, in A, we say f is left adjoint to u9, and written f ⊣ u, if there are two 2-morphisms, η : 1A ⇒ uf, and ε : fu ⇒ 1B, such that the following equations hold: (u ∼ = ←→ 1Au η·u −−→ uf · u ∼ = ←→ u · fu u·ε −−→ u · 1B ∼ = ←→ u) = idu, and (f ∼ = ←→ f1A fη −→ f · uf ∼ = ←→ fu · f εf −→ 1B · f ∼ = ←→ f) = idf, where we have written ∼= to label the evident unitors and associators, or their in- verses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 9and u is right adjoint to f, A CATEGORIFICATION OF QUINN’S TQFT 27 (ii) A 1-morphism, f : A → B, in A is an equivalence if there is a 1-morphism, g : B → A, and two 2-isomorphisms, gf ∼ = =⇒ 1A and fg ∼ = =⇒ 1B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is said to be an adjoint equivalence if f ⊣ g and both η and ε are isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 6, we will consider several examples of bifunctors / pseudo-functors, F : A → B, in which the domain, A, is ‘locally discrete’ in the sense mentioned on page 24, so each A(x, y) is a discrete category, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will often just say that A is a category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, in most of the cases that we will need, A will be a groupoid as all its 1-morphisms are invertible;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' but the codomain, B, is a bicategory, usually that of 2Cob(n,n+1,n+2), any of the span or cospan bicategories, vProf, one of its variants, or Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, F allows one to specify structure very easily in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We repeat the specification of a pseudo-functor, but in the simplified form that this context allows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have for each object, a in A, an object, F(a), in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each pair, a0, a1, of objects in A, a functor, Fa0,a1 : A(a0, a1) → B(F(a0), F(a1)), and, as A(a0, a1) is discrete, this just means a family of 1-morphisms, F(f) : F(a0) → F(a1), where f : a0 → a1 ∈ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each composable pair of morphisms, a0 f−→ a1, a1 g−→ a2, in A, an invertible 2-morphism, ϕg,f : F(g)F(f) ⇒ F(gf) in B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each object, a, in A, an invertible 2-morphism, ϕa : idF (a) ⇒ F(ida), in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These must satisfy the following conditions: compatibility with the associator in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so given,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in addition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' h : a2 → a3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the following diagram commutes: (F(h)F(g))F(f) � aB � F(hg)F(f) �❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ F((hg)f) = F(h(gf)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(h)(F(g)F(f)) � F(h)F(gf) �❧ where the unlabelled arrow are derived from the various ϕ 2-cells10,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' compatibility with the right and left unitors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (which are ‘equalities’ in A),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each f : a0 → a1 in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the diagrams: F(f) · idF (a0) � ρB F (a0) �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ F(f) · F(ida0) � F(f · ida0) = F(f) 10Note that as A is a category (hg)f = h(gf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 28 and idF (a1) · F(f) � λB F (a1) �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ F(ida0) · F(f) � F(ida1 · f) = F(f), where the unlabelled arrows are the evident ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' When formalising or analysing a structure in a category or bicategory, the struc- ture is often expressed in terms of the commutativity of certain diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose we have a commutative diagram in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can think of this as a functor D : I → A, where I is some ‘template’ for the commutative diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can then think of D as a (trivially structured) pseudo-functor and compose it with our given F : A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The result will be a pseudo-functor from I to B, so a ‘pseudo-commutative’ diagram in B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note all the 2-cells in this diagram will be invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As a more-or-less trivial example, we can take I = [2], the small category corre- sponding to the ordered set, 0 → 1 → 2, and the D will correspond to a commuta- tive diagram of form a1 a12 �❈ ❈ ❈ ❈ ❈ ❈ ❈ ❈ a0 a01 �⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ ⑤ a02 � a2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The corresponding 2-diagram, FD, in B will be the pseudo-commutative one having an invertible 2-arrow from F(a12)F(a01) to F(a02), together with three invertible 2-arrows, idF (ai) ⇒ F(ida1), for i = 0, 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (In some contexts, these latter 2- arrows may, themselves, be identity 2-arrows, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the pseudo-functor, F, may be ‘normalised’, but the general case is important to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We will need such constructions especially when looking at ‘inducing’ a descrip- tion of structure on certain bicategories, D, in the context of monoidal bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Monoidal bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need the bicategorical analogues of monoidal and symmetric monoidal categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' One of the motivating examples for the notion of a symmetric monoidal bicate- gory is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let R be a commutative ring, and let Alg(R) be the bicategory such that the objects are R-algebras, denoted A, B, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the morphisms from A to B are the left-right (A, B)-bimodules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the 2-morphisms are the bimodule homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The monoidal product is the tensor product over R, so the unit is R itself, considered as an (R, R)-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bicategory is also often denoted Alg2(R) or, as later in this paper, by MorR, or simply by Mor if the commutative ring, R, is ‘understood’ or ‘fixed’ for the relevant section;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Definition 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Later on, around that definition, we will introduce this more formally as we need some more explicit detail as to its construction and its structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It will be one of the main codomain bicategories for the once-extended Quinn theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A monoidal bicategory is a bicategory that also has a monoidal structure, up to the equivalence inherent in the bicategorical context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They can be defined in various ways, for instance, as a tricategory having just one object, [58, 61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Other definitions mention Gray categories11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each of these is fairly complex to give, and 11For which see, for example, [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 29 needs a few more definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following is one of the simpler ones in as much as it seems fairly clearly motivated by the definition of monoidal category suitably weakened with equality replaced by equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It does use some bicategorical language that we have not given earlier, but is, perhaps, fairly self explanatory12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A monoidal bicategory, A, consists of a bicategory, A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a pseudofunctor/homomorphism, ⊗ : A × A → A, so ⊗ is ‘functorial up to isomorphism’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a pseudofunctor/homomorphism, I : 1 → A, where 1 is the unit bicategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' an adjoint equivalence, sometimes called the monoidal associator, diagram- matically denoted A3 ⊗×A � A×⊗ � ✂✂✂✂� α A2 ⊗ � A2 ⊗ � A, in Bicat(A3, A), corresponding to associativity of ⊗ in a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This adjoint equivalence consists of α, its adjoint, α∗, with unit, ηα, and counit, εα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To see what these do, we take a triple, A, B, C, of objects in A, so (A, B, C) is in A3, and then we have that αCBA : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A), whilst α∗ CBA : C ⊗ (B ⊗ A) → (C ⊗ B) ⊗ A, with unit and counit, ηα CBA : Id ⇒ α∗ CBA ◦ αCBA and εα CBA: αCBA ◦ α∗ CBA ⇒ Id, being isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, given 1-morphisms, f : C → C′, g : B → B′ and h: A → A′, we have natural 2-morphisms, for instance, (C ⊗ B) ⊗ A αCBA � ✖✖✖✖� α(f,g,h) (f⊗g)⊗h � (C′ ⊗ B′) ⊗ A′ αC′B′A′ � C ⊗ (B ⊗ A) f⊗(g⊗h) � C′ ⊗ (B′ ⊗ A′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' adjoint equivalences, sometimes called the monoidal unitors, A2 ⊗ �❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ A I×A �⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ A � ✤✤ ✤✤� ℓ A 12The structure and laws are well illustrated in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Corner’s thesis, [40] in §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, in the draft book by Johnson and Yau, [68] and in Mike Stay’s article, [110], which has some excellent diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 30 and A2 ⊗ �❆ ❆ ❆ ❆ ❆ ❆ ❆ ❆ A A×I �⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ ⑥ A � ✤✤ ✤✤� r A in Bicat(A2, A), corresponding to left and right unitors13;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and an invertible modification giving the analogue of the pentagon rule for monoidal product in a monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is called the pentagonator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its com- ponent 2-morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for objects A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' D in A looks like,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (D ⊗ (C ⊗ B)) ⊗ A � D ⊗ ((C ⊗ B) ⊗ A) �❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ((D ⊗ C) ⊗ B) ⊗ A �♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ ♦ �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ⇓πDCBA D ⊗ (C ⊗ (B ⊗ A)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (D ⊗ C) ⊗ (B ⊗ A) �❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ where each of the unlabelled arrows corresponds to a use of an associator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' possibly combined with an identity on an object,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as in the usual pentagon rule for monoidal categories14;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and invertible modifications, µ � , λ � and ρ � , called the middle, left and right 2- unitors, respectively, with component 2-morphisms, for objects A and B in A, (B ⊗ I) ⊗ A � ✏✏✏✏� µB,A B ⊗ (I ⊗ A) B⊗ℓA � B ⊗ A r∗ B⊗A � = � B ⊗ A, (I ⊗ B) ⊗ A ℓB⊗A � α �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ⇓λ � BA B ⊗ A, I ⊗ (B ⊗ A) ℓB⊗A �✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ B ⊗ A B⊗r∗ A � r∗ B⊗A �● ⇓ρ � BA B ⊗ (A ⊗ I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (B ⊗ A) ⊗ I α �r r r r r r r r r r This data is required to satisfy three pasting diagrams, which we omit15, but which are well presented in Johnson and Yau, [68], and in [58, 61], from the point of view of the more general tricategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In a string diagram form, they are also to be found in Corner, [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are easier to draw, but still quite complex to read16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To ease our way towards a sketch of the definition of symmetric monoidal bi- category, we will briefly recall the corresponding definition of symmetric monoidal category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Although originally introduced directly by specifying that there was a natural isomorphism, X ⊗Y ∼= Y ⊗X, satisfying certain axioms, for our purposes it is slightly better to go via the definition of braided monoidal category, so we briefly recall that first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 13Leaving the reader to expand that condition on the lines above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 14see Johnson and Yau, [68], for a fully labelled version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 15as being far too large and complicated to include, and to type set, for the limited use we will have of them, 16and we have not used this string diagram notation elsewhere in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 31 (We have adapted the definition given in Etingof, Gelaki, Nikshych and Ostrik, [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Definition 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A braided monoidal category is a monoidal category,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ⊗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' I),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' equipped with a natural isomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' RX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y : X ⊗ Y ∼= Y ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' called the braiding,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' such that the diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X ⊗ (Y ⊗ Z) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='� (Y ⊗ Z) ⊗ X ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='�P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='P ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='P ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='♥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='♥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='♥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='♥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='♥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='commute for all choices of objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in C,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and where each arrow is an evident application of the associator,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' its inverse or of the braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A braided monoidal category, C, is said to be symmetric if, for all X, Y in C, RY,X ◦ RX,Y = idX⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A symmetric monoidal bicategory categorifies the above, so replacing equalities by structural morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A complete description of symmetric monoidal bicate- gories can be found in [60, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Loosely speaking, we have a bicategory, B, which is monoidal as above (Definition 26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The monoidal structure is assumed to be braided,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have a pseudo-natural transformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' R: ⊗ → ⊗ ◦ τ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of bifunctors from B × B to B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where τ arises from swapping coordinates,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [60,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' page 4234],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in particular,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for every X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Y in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' there is an equivalence (within B),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' RX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y : X ⊗ Y ≃ −→ Y ⊗ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and also invertible 2-cells between the two obvious composites from (X ⊗Y )⊗Z to Y ⊗(Z ⊗X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and similarly from X ⊗(Y ⊗Z) to (Z ⊗X)⊗Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so replacing equality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the diagrams of Definition 27,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by invertible 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We will denote these 2-cells by RX|Y Z and RXY |Z as seems to be the fairly standard notation currently in use;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [68] and [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A full definition of braided monoidal bicategories can be found in [60, Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is another intermediate step before getting to the final form for ‘sym- metric’, rather than merely ‘braided’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Following [62, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definitions], a sylleptic monoidal bicategory is a braided monoidal bicategory with a syllepsis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a A CATEGORIFICATION OF QUINN’S TQFT 32 structure is a natural isomorphism, given by, for each pair X, Y of objects, an isomorphism, νXY : RY XRXY ∼ = −→ IdX⊗Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The structure of a symmetric monoidal bicategory is, then, to satisfy one additional axiom which says that the two ways of rewriting RXY RY XRXY to RXY , one using νXY , the other using νY X, agree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An excellent list of examples of symmetric monoidal bicategories can be found on page 2 of Mike Stay’s paper, [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will select a few of most relevance to this work, adapting some to fit the context here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also add a few others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Yet others will be included later on, once the necessary terminology has been introduced, and, for those here, we will simply mention them briefly, with a reference to where they are discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' V −Cat: If V is a symmetric monoidal category, then the 2-category, V − Cat, of V-categories forms a symmetric monoidal 2-category, and thus a symmetric monoidal bicategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Kelly, [70], page 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, this applies when V is the symmetric monoidal category of small categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Bicategories with finite products: One has that any bicategory, A, with binary product, − × −, and terminal object, 1, underlies a symmetric monoidal bicategory with −×− as its tensor product and 1 as its unit object;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='15 of Carboni, Kelly, Walters and Wood, [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Essentially the same arguments work for bicategories with finite coproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Span(C): As is well known, a span from A to B in a category, C, is a diagram C f �❥❥❥❥❥❥ g �❯ ❯ ❯ ❯ ❯ ❯ A B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For each pair A, B, there is a category, Span(C)(A, B), of spans, as above, and, if C has pullbacks, then we can compose spans in a well known way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a bicategory, Span(C);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Borceux, [17], Examples 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If C is a category with finite products, then the bicategory, Span(C), is a symmetric monoidal bicategory17, in which the tensor product on both objects and spans is given by the product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This needs taking apart a little as there are subtleties that are important later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, the objects of Span(C) are just the objects of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any two objects, A1 and A2, in Span(C), and thus in C, their tensor product will be A1 ⊗ A2 := A1 × A2, whilst the tensor product of two spans is (A1 ← C1 → B1) ⊗ (A2 ← C2 → B2) := (A1 × A2 ← C1 × C2 → B1 × B2), where the maps are as one would expect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To define the associator on objects, we suppose that we have three objects, A1, A2 and A3, and we need a ‘morphism’ (in Span(C)), αA1A2A3 : (A1 ⊗ A2) ⊗ A3 → A1 ⊗ (A2 ⊗ A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Within the base category, C, we have an associator (iso)morphism, (the usual one coming from the universal property of products), aA1A2A3 : (A1 × A2) × A3 → A1 × (A2 × A3), which satisfies the requirements that the pentagon diagrams commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Re- member C is a monoidal category with the product as tensor, so it is in a simpler setting than Span(C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') 17In fact, Span(C) is a compact closed bicategory in the sense of Stay’s paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 33 In Span(C), as we said, the associator transformation is to be made of spans, and the one that works is (A1 × A2) × A3 id ←− (A1 × A2) × A3 aA1A2A3 −−−−−−→ A1 × (A2 × A3), in other words, using the way that C can be thought if as being ‘embedded’ in Span(C), using the second legs of the spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The associator is not just αA1A2A3, but has to be part of an adjoint equivalence, so we need a α∗ A1A2A3 going the other way, which is given by the reverse span, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' using the ‘first leg’, and we also need η and ε as in Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That ‘second leg’ then quickly shows how to specify η and ε for Span(C) in terms of the corresponding ones in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The unitors and the braidings are similarly handled giving them first in C before transferring them to Span(C) using the second leg process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More general results are described in much more detail in [110].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A special case of this is when C = Set and this gives, after a bit of adap- tation and verification, a symmetric monoidal bicategory structure to the category of relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Prof: The bicategory of profunctors, or distributeurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be revisited after we have recalled the basic theory in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProf: The bicategory of Vect-enriched categories, enriched profunctors and enriched natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again discussion of this is postponed to the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cospan(C): If we replace C by the opposite category then we have that, if C has finite colimits, Cospan(C) will be a symmetric monoidal bicategory, having coproduct, ⊔, as its tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2Cob(d,d+1,d+2): Let d be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is well known that Cob(d,d+1), the category of closed smooth d-manifolds and diffeomorphism classes of cobordisms between them, [89, 37], forms a symmetric monoidal category with coproduct / disjoint union, ⊔, as the tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As proved in [106], 2Cob(d,d+1,d+2), the bicategory of closed smooth d-manifolds, their cobordisms, and diffeomorphism classes of extended cobordisms be- tween cobordisms, is a symmetric monoidal bicategory, again having ⊔ as its tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will sketch the construction of the symmetric monoidal structure of 2Cob(d,d+1,d+2) in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Alg(R), also denoted Mor: We mentioned this important example earlier in Example 25.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here R is a commutative ring, the objects are R-algebras, the 1-morphisms are bimodules and the 2-morphisms are homomorphisms of bimodules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The tensor product is the tensor product over R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This example is sometimes denoted Alg2(R), to indicate that it is the 2- categorical version of the category of algebras and bimodules between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We, in fact, will use yet another name namely the Morita bicategory of R and consequently denote it MorR, as it is a good context for emphasising Morita equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will introduce it in more detail in Definition 164, where R will be a subfield, κ, of C, as that will be the only case that we will be needing later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' BimodR: In the previous example, there is no reason to restrict the ‘al- gebras’ to having a single object18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can form a bicategory, which we will denote by BimodR for the moment, having R-linear categories as ob- jects, with bimodules over them as its 1-morphisms and (R-linear) natural transformations as its 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 18in the sense of Mitchell’s paper, [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 34 This is nearly the same as the bicategory, vProf, of R-linear profunc- tors, which we will be using in the case that R = κ, and for which see the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is considered, for instance, in [10] as Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will treat a related example later, but note that this is another example of a symmetric monoidal bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The tensor product of two R-linear categories, C and D, is defined to have as its objects, the pairs (C, D), with C ∈ C0 and D ∈ D0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', objects from the relevant categories, and, following Mitchell, [90], (C ⊗ D)((C, D), (C′, D′)) := C(C, C′) ⊗R D(D, D′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For bimodules CMC′, and DND′, their tensor product is the tensor prod- uct over R of the various parts of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the structural isomorphisms, braidings, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', are defined in an evident natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Many of the above examples are linked by symmetric monoidal bifunctors, but we will not give a full formal detailed definition of such here, and will merely sketch one, directing the reader to Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5 of [106], and [61, 60, 62], for a more detailed version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A and B be two symmetric monoidal bicategories, in which the structures involved, such as the tensor, will be provided with a suffix to show to which of A and B they relate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Sketch only) A symmetric monoidal bifunctor, F : A → B, con- sists of the following data: a homomorphism, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a bifunctor), F : A → B, between the underlying bicategories;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a transformation, (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a pseudo-natural transformation),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' χ: ⊗B ◦(F × F) ⇒ F ◦ ⊗A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of bifunctors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' from A × A to B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' so we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given objects,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A0 and A1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a 1-morphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' χA0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A1 : F(A0) ⊗B F(A1) → F(A0 ⊗A A1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and given 1-morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f0 : A0 → A′ 0 and f1 : A1 → A′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have a natural 2-cell in B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' F(A0) ⊗B F(A1) F (f0)⊗BF (f1) � χ(A0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A1) � ✖✖✖✖� χ(f0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f1) F(A′ 0) ⊗B F(A′ 1) χ(A′ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A′ 1) � F(A0 ⊗A A1) F (f0⊗Af1) � F(A′ 0 ⊗A A′ 1),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' which is compatible with compositions and horizontal identities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and we also have a transformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i: IB ⇒ F ◦ IA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' together with corresponding adjoint equivalence transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' χ∗ and i∗,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (and with the relevant adjunction data);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' invertible modifications, ω, γ and δ, measuring compatibility with the rele- vant associators and unitors (see Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of [106]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and an invertible modification, u, giving compatibility with the braiding (see Fig 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6 of [106], or [60, page 4239], for the relevant diagram, so that u : F(RA) ◦ χ ⇒ χ ◦ RB : F(B) ⊗B F(A) → F(A ⊗A B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 35 This data is to satisfy certain axioms, which we omit, referring to the discussion in [106], Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', for further details including further references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In par- ticular the compatibility conditions in [61, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3] / [58, page 17] hold, dealing with the preservation of the monoidal structure by F, and those of [60, page 4239] hold, similarly describing the symmetric monoidal structure of the bifunctor F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will sketch the construction of the entire structure of a symmetric monoidal bifunctor when we prove, in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, that the once-extended Quinn TQFT, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions on profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The detailed theory of profunctors can be found in many texts and on-line sources, for instance, [17, Chapter 7], [81, Section 5] and the nLab [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' When searching for such theory, it is important to note that the terms ‘distributor’ and ‘bimodule’ are often used alternative names for profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will give a very minimal sketch here and include a bit of a ‘crib- sheet’, whilst we are doing that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some background and basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In a general context, given two small categories, A and B, we have: Definition 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A (Set-valued) profunctor from A to B is a functor, F : Aop × B → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be sometimes written F : A ↛ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With a suitable notion of composition, these profunctors, for varying domain and codomain categories, together with their natural transformations, form a bicategory Prof, whose construction will be briefly recalled below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Warning: There are several different conventions used in the literature as to the ‘direction’ of the profunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' One of the most current, but not the one that we will use, is to say a profunctor, F : A ↛ B, is a functor, Bop × A → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This can be confusing, but makes no essential difference to the resulting theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The choice between them is influenced by the context of their use, but this means that, when referring to a source on profunctors, the reader is advised to check the convention being used in that source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In our case, with a geometry-inspired concatenation composition rule for cobor- disms, and a compatible algebra-inspired rule for composition (tensor product) of bimodules, our conventions will not be the same as many of the sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This may cause some slight difficulty when looking at the classical source material on pro- functors / distributors, but is, we think, a good and consistent mix of conventions for our use of that theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any such profunctor, F : Aop × B → Set, determines a functor, F: Aop × B → Vect, by composing F with the free vector space functor, Lin: Set → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The resulting linear, or Vect-valued, profunctor will quite often also be denoted, by abuse of notation, by the same symbol, F : A ↛ B, or sometimes (as above) by the symbol for the functor, converted to a boldface type, so F, if both F and its ‘linearisation’ are needed in a discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will usually work with the Vect-enriched analogue, vProf, [10, 63], of the bicategory Prof, and, within that, the sub-bicategory, vProfGrp, whose objects are derived from groupoids, G = (s, t: G1 → G0), each made into a linear category, A CATEGORIFICATION OF QUINN’S TQFT 36 which we may occasionally write as Lin(G), by applying the free vector space functor, Lin: Set → Vect, to the hom-sets of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given groupoids, G and G′, 1-morphisms from G to G′, in vProf, will be con- sidered to be functors, H: Gop × G′ → Vect, from now on called Vect-profunctors from G to G′, although sometimes, no doubt, we will abbreviate that term, just saying ‘profunctors’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a (small) category, then we have the bivariant hom-functor, A(−, −) : Aop×A → Set, which is a Set-valued profunctor from A to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A is a Vect-enriched category, the natural analogue of the above is A(−, −) : Aop×A → Vect, and so is a Vect-valued profunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We denote the latter by IdA, and call it the identity profunctor on the (linear) category, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We may shorten this to IdG, if A is the linearisation, Lin(G), of a groupoid, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a functor, F : A → B, we can define two profunctors, ϕF : A ↛ B and ϕF : B ↛ A, by ϕF (A, B) = B(F(A), B), whilst ϕF (B, A) = B(B, F(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In case F is the identity functor on A, the two profunctors coincide and are the same as that in the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A profunctor that is isomorphic to one of these two forms is said to be representable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These profunctors are, in fact, adjoint 1-cells in the bicategory, Prof, whose structure we are sketching here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose, now, that η : F ⇒ G is a natural transformation of functors from A to B, then, for each A in A, we have a morphism, η(A) : F(A) → G(A), and there are induced 2-arrows, ϕη : ϕG ⇒ ϕF , given by ϕη(A) : ϕG(A) ⇒ ϕF (A), is ϕη(A) := (η(A))∗ : B(G(A), B) → B(F(A), B), if B is an object of B, and ϕη : ϕF ⇒ ϕG, given, dually, by ϕη(A) := (η(A))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That these do give 2-morphisms / natural transformations is easy to check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This process also respects composition of natural transformations, so if η′ : G ⇒ G′ is another natural transformation, then ϕη′η and ϕη′ϕη are equal as, for an object A of A, they both induce the composite, (η′η)(A) = η′(A)η(A) and so give the same induced morphism from B(G′(A), B) to B(F(A), B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that the 2-category of small categories, considered as a bicategory ‘bifunctorially embeds19’ in the bicategory, Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be important when we consider the monoidal bicategory structure on Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, if F is an equiva- lence of categories, then ϕF is an equivalence in the bicategorical sense within Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Other similar statements hold for adjointness, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', but we will not be needing them as much.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 19or ‘pseudo-functorially embeds’ A CATEGORIFICATION OF QUINN’S TQFT 37 Given Vect-profunctors, H, H′ : Gop × G′ → Vect, a 2-morphism, or 2-cell, η: H =⇒ H′, between them, is a natural transformation of functors, Gop × G′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hence, we have, given x ∈ G0 and y ∈ G′ 0, a linear map, ηx,y : H(x, y) → H′(x, y), which is natural in both x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G, H, K be groupoids, or more generally small categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given Vect- profunctors, H: G ↛ H, and H′ : H ↛ K, their composite, H • H′ : G ↛ K, will be the Vect-profunctor such that, if x ∈ G0 and z ∈ K0, then20 (3) (H • H′)(x, z) := ˆ y∈H0 H(x, y) ⊗ H′(y, z) = � � y∈H0 H(x, y) ⊗ H′(y, z) � / ≃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, fixing x ∈ G and z ∈ K, the equivalence relation, ≃, is generated (as a linear equivalence relation21) by for y, y′ ∈ Y , vx,y ∈ H(x, y) and v′ y′,z ∈ H(y′, z) and an arrow, y h−→ y′, in H, vx,y ⊗ H′(y h−→ y′, z 1z −→ z)(v′ y′,z) ≃ H(x 1x −→ x, y h−→ y′)(vx,y) ⊗ v′ y′,z, or, more informally, (4) vx,y ⊗ h · v′ y′,z ≃ vx,y · h ⊗ v′ y′,z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note the convention on the order of composition that we are using, and would remind the reader of the Warning that we placed a short while back.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This convention is used because it reflects the geometric intuition, being a concatenation order of composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It also reflects a useful convention for the bicategory, Mor, of algebras, bimodules and bimodule maps, to which Prof is closely related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we just have Set-valued profunctors, this formula for composition still make sense by interpreting ⊗ as ×, and we note that Lin preserves that composition in the evident way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, there is a projection, (5) proj : � y∈H0 H(x, y) ⊗ H′(y, z) → (H • H′)(x, z), which we will need later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any element in (H • H′)(x, z), we can represent it by an element in H(x, y) ⊗ H′(y, z), for some y, but, working with that, just as in the setting of bimodules, any resulting calculation has to be shown to be independent of the y chosen, that is, it must be invariant under the action of the arrows of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose we have functors, A F−→ B G −→ C, then we have corresponding profunctors, ϕF and ϕG, so can form their composite, ϕF • ϕG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is not hard to check that this composite is isomorphic to ϕGF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More precisely we have a natural isomorphism, ϕF • ϕG =⇒ ϕGF , and this is part of the data that says that ϕ(−) is a pseudo-functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use this later in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, especially on page 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly we have ϕG : C ↛ B and ϕF : B ↛ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Composing these to form ϕG • ϕF , we find that this is isomorphic to ϕGF : C ↛ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of general profunctors and so, in particular, of Vect-profunctors has left and right (lax) identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose we have H: G ↛ H, then we can compose it with the (enriched) hom-functor, G(−, −) : Gop × G → Set, or to 20Note that this coend is, a priori, defined only up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, we always implicitly choose a natural realisation for all limits and colimits appearing, as we have done below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 21i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as an equivalence relation whose quotient is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 38 Vect after applying Lin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This acts like a left identity on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a natural isomorphism, (6) λH G : G(−, −) • H =⇒ H, often called the ‘left unitor’, and similarly another natural isomorphism, ρH H : H • H(−, −) =⇒ H, the ‘right unitor’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are easy to write down, for example, using equation (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given natural transformations, η: H1 =⇒ H2, between Vect-profunctors, G ↛ H, and η′ : H′ 1 =⇒ H′ 2, between Vect-profunctors, H ↛ K, we have a natural transformation, (η • η′): H1 • H′ 1 → H2 • H′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, given x ∈ G0 and z ∈ K0, then (η•η′)(x,z) sends the equivalence class of vx,y ⊗v′ y,z to the equivalence class of η(x,y)(vx,y)⊗η′ (y,z)(v′ y,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here y ∈ H0, vx,y ∈ H1(x, y) and v′ y,z ∈ H2(y, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, given x ∈ G0 and z ∈ K0, the linear map, (η •η′)(x,z), is the unique map that makes the diagram below commute: (7) � y∈H0 H1(x, y) ⊗ H′ 1(y, z) � y∈H0 η(x,y) ⊗ η′ (y,z) � proj � ˆ y∈H0 H1(x, y) ⊗ H′ 1(y, z) (η • η′)(x,z) � � y∈H0 H2(x, y) ⊗ H′ 2(y, z) proj � ˆ y∈H0 H2(x, y) ⊗ H′ 2(y, z) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Matrix elements for natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G and H be groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In most examples appearing in this paper, Vect-profunctors, Gop × H → Vect, arise from set-valued profunctors, F : Gop × H → Set, by applying the free vector space functor Lin: Set → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However the natural transformations between profunc- tors will be full fledged natural transformations between Vect-valued profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a profunctor, F : Gop × H → Set, we thus have that its linearisation, F = Lin ◦ F : Gop × H → Vect, comes with given bases on each of its constituent vector spaces, which means that for the purposes of the presentation of the encoded data, or for calculation, we can use matrices and other tools and insights from classical representation theory22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An object of Gop × H is, of course, a pair, (x, y) ∈ G0 × H0, and if we denote a typical element of F(x, y) by f, we can consider f as an element of the natural, given basis for F(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For g : x → x′ in G1, we then have, for each y ∈ H0, a linear map, F(g, y): F(x′, y) → F(x, y), and so a matrix with matrix elements, ⟨f ′ | F(g, y) | f⟩, and F(g, y)(f ′) = � f∈F (x,y) ⟨f ′ | F(g, y) | f⟩f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly, if h: y → y′ ∈ H1, we have F(x, h): F(x, y) → F(x, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This mean that the evident actions of G and H on the family of vector spaces, F(x, y), for x ∈ G0 and y ∈ H0, come with a given matrix representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This looks like a ‘many object’ bimodule, and we will recall the relationship with the theory of bimodules over algebras more fully in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 22In what follows, we will make the assumption that all the F (x, y) are finite sets, as that is true in the situations that will be of interest to use later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 39 For the moment, we need to examine the way of describing the natural transfor- mations between such profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose F and F′ : Gop × H → Vect are two Vect-valued profunctors, linearised from some Set-valued ones, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fur- ther suppose ϕ: F ⇒ F′ is a natural transformation from F to F′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For each (x, y) ∈ G0 × H0, we then have a linear mapping, ϕ(x, y): F(x, y) → F′(x, y), and hence, for each f ∈ F(x, y) and f ′ ∈ F ′(x, y), once again a matrix element, ⟨f | ϕ(x, y) | f ′⟩, so that we have a ‘state sum’ description, ϕ(x, y)(f) = � f ′∈F ′(x,y) ⟨f | ϕ(x, y) | f ′⟩f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fact that ϕ is a natural transformation means that it must be compatible with changes along any g : x′ → x and h: y → y′, and so must satisfy equations involving the various F(g, h): F(x, y) → F(x′, y′), and the corresponding matrix representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' When we deal with the categorified ‘once-extended’ version of Quinn TQFT, in Section 5, and in particular §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, our methods, will give naturality from general constructions, and then a description of the natural transformations in terms of these matrices and state sums, rather than starting with the families of matrices and doing quite complex manipulations to show that they define natural transfor- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some lemmas on coends of functors from groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the convenience of the reader, we collect a few elementary lemmas, whose explicit formulation can be difficult to find in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They will be useful when giving some explicit details in the proof that the once-extended Quinn TQFT is indeed a bifunctor, especially when it comes to the preservation of horizontal compositions, see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2 and Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G = (s, t: G1 → G0) be a groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a functor F : Gop × G → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The coend of F is a universal wedge, making the diagram, F(y, x) F (g,idx) � F (idy,g) � F(x, x) px � F(y, y) py � ˆ z∈G0 F(z, z), commute, for all choices of morphisms g : x → y in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore, as written above, ˆ z∈G0 F(z, z) = � � z∈G0 F(z, z) � � ∼, where ∼ is the smallest equivalence relation that makes the diagram above commute for all choices of g : x → y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As G is a groupoid, given any g : x → y and g′ : x′ → y′, the map, F(g, g′): F(y, x′) → F(x, y′), is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives: Lemma 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have ˆ z∈G0 F(z, z) = � � z∈G0 F(z, z) � � ≃, where ≃ is the equivalence relation in which ux ∈ F(x, x) ≃ uy ∈ F(y, y) A CATEGORIFICATION OF QUINN’S TQFT 40 if there exists g : x → y such that: F � g−1 : y → x, g : x → y � (ux) = uy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In fact, this shows that the equivalence relations ∼ and ≃ are really the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that any groupoid G comes with a contravariant functor (−)−1 : G → G, that is the identity on objects and sends g : x → y to g−1 : y → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is also the diagonal functor, ∆: G → G × G, sending x ∈ G0 to (x, x) ∈ G0 × G0, and with ∆ � x g−→ y � = � (x, x) (g,g) −−−→ (y, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hence we have a functor, F ◦ � (−)−1 × id � ∆: G → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The previous lemma gives: Lemma 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a canonical bijection ˆ z∈G0 F(z, z) ∼= colim � F ◦ � (−)−1 × id � ∆ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Let A and B be sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a linear map, f : κ(A) → κ(B), between free vector spaces and equivalence relations, ∼A and ∼B, on A and B, such that f descends to a map f ′ : κ(A/ ∼A) → κ(B/ ∼B), then, given a ∈ A and b ∈ B, the matrix elements of f ′ satisfy � [a]|f ′|[b] � = � b′∈[b] � a|f|b′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Combined with the previous discussion, this gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let F, F ′ : Gop × G → Set be functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider, for each (x, y) ∈ G0 × G0, a linear map, η(x,y) : κ � F(x, y) � → κ � F ′(x, y) � , such that putting all of the η(x,y) together gives a natural transformation, η: Lin ◦ F =⇒ Lin ◦ F ′, where Lin: Set → Vect is the free vector space functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Further consider the induced map (see [81, Notation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='15]) between coends, ˆ x∈G0 η(x,x) : ˆ x∈G0 κ(F(x, x)) → ˆ x∈G0 κ(F ′(x, x)), that is (since the free vector space functor preserves colimits, and with a minor abuse of notation), ˆ x∈G0 η(x,x): κ � ˆ x∈G0 F(x, x) � → κ � ˆ x∈G0 F ′(x, x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its matrix elements satisfy, for each z ∈ G0, uz ∈ F(z, z) and wz ∈ F ′(z, z), � [uz] ����� ˆ x∈G0 η(x,x) ����� [wz] � = � w′ z∈Orb(wz) � uz|η(z,z) |w′ z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here Orb(wz) is the orbit of wz, under the action of the group homG(z, z), on F ′(z, z), defined as , vz ⊳ (g−1 : z → z) := F ′(g−1 : z → z, g : z → z)(vz), where vz ∈ F ′(z, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 41 On the other hand, if z′ ∈ G0 belongs to a different connected component from z in G, and if tz′ ∈ F ′(z′, z′), then � [uz] ����� ˆ x∈G0 η(x,x) ����� [tz′] � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the previous discussion, and the fact that the diagram below commutes, where the vertical arrows are the canonical projections, � x∈G0 κ(F(x, x)) � � x∈G0 η(x,x) � � x∈G0 κ(F ′(x, x)) � κ � ˆ x∈G0 F(x, x) � ˆ x∈G0 η(x,x) � κ � ˆ x∈G0 F ′(x, x) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Prof and its relatives as symmetric monoidal bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next need to link up these bicategories of profunctors with considerations of our previous discussions on symmetric monoidal bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To start with, there is a natural monoidal bicategory structure on Prof itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is explicitly given by Cattani and Winskel, [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Define ⊗ on objects to be simply the product, ×, of categories, and then on 1-morphisms, F : A0 ↛ B0 and G : A1 ↛ B1, take F ⊗ G to be the composite (A0 × A1)op × (B0 × B1) ∼ = −→ (Aop 0 × B0) × (Aop 1 × B1) F ×G −−−→ Set × Set ×Set −−−→ Set, so on an object, ((A0, A1), (B0, B1)), it takes the value F(A0, B0)×G(A1, B1) given by the product in Set of the two image sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On 2-morphisms, if α : F ⇒ F ′ and β : G ⇒ G′, then (α ⊗ β)(A0,A1),(B0,B1) = α(A0,B0) × β(A1,B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Together with this, one can prove that Prof supports a symmetric monoidal bicategory structure in which the associativity and braiding profunctors are in- herited from the associativity and braiding morphisms of the 2-category of small categories, functors and natural transformation, by applying the ϕ(−)-construction of Examples 33 and 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the braiding in Prof is given explicitly as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose G and H are small categories, (which for us will usually be groupoids), then there is an isomorphism of categories, R : G × H ∼ = −→ H × G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We therefore have a profunctor, ϕR : G × H ↛ H × G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is an equivalence, and gives the required braiding, RGH := ϕR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a symmetric monoidal bicategory structure to Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to the κ-linear setting and Vect-enriched profunctors, then the above constructions adapt well, and vProf is a symmetric monoidal bicategory, whose objects are the κ-linear categories, [63, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On objects, we take A ⊗ B to be the tensor product (over κ) of such things, as was discussed in the previous section (page 34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The tensor product of two profunctors is obtained by almost the same composite as above except that, of course, we replace Set by Vect and A CATEGORIFICATION OF QUINN’S TQFT 42 ×Set = − × − by ⊗Vect = − ⊗κ −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 2-morphisms look after themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The braiding structure is as in the Prof case, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The sub-bicategories, vProfGrp, vProfGrphf, and vProfGrpfin, are easily seen all to inherit symmetric monoidal bicategory structures from the larger bicat- egory, where we use the terminology on groupoids from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In partic- ular if we think of the tensor product of two groupoids, G and H, in vProfGrp as being the usual cartesian product, G × H, of groupoids, then, after linearising the groupoids, that product is sent to the tensor product of the two linearised categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All these bicategories are variants of the basic bicategory of bimodules, Bimodκ, approached from a different direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The linearisation functor induces a symmetric monoidal bifunctor from Prof to vProf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we restrict attention to profunctors defined on groupoids, this functor restricts to a symmetric monoidal bifunctor from ProfGrp, the sub-bicategory of Prof defined on groupoids, to vProfGrp, and, clearly, this further restricts to the homotopy finite, and finite subcases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Summary of conventions and notation for bicategories of profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To summarise the terminology and notation for some of the instances that we will be needing later on in one place, we have (for terminology, see also Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Prof: the (symmetric monoidal) bicategory of categories, set-valued pro- functors, and their natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProf: the (symmetric monoidal) bicategory of Vect-enriched categories, enriched profunctors and enriched natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is also a symmetric monoidal bicategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProfGrp: the sub-bicategory of vProf, whose objects are groupoids, each made into a Vect-enriched category by applying the free vector space functor to the sets of morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here 1-morphisms, G ↛ H, are functors, Gop × H → Vect, and are called Vect-profunctors, and 2-morphisms are natural transformations of such functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The horizontal compositions of 1- and 2-morphisms are explained in equations (3) and (7), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This inherits a symmetric monoidal structure from vProf, in which the tensor product of the groupoids, G and H, is the usual cartesian product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProfGrphf: the sub-bicategory of vProf, whose objects are the homo- topy finite groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProfGrpfin: the sub-bicategory of vProfGrp whose objects are the finite groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Comments: (i) As is well known, any group can be considered as a groupoid having just one object, so let G be a (finite) group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' When we consider G in vProf, the resulting linear category, as it has just one object, is effectively just a κ-algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is precisely the group κ-algebra of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From this viewpoint, for G and H (finite) groups, a Vect-profunctor, G ↛ H, is the same as a G-H bimodule over κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of such ‘bimodules’ is exactly that used in more classical treatments, and is given by a tensor product in the usual way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus take the viewpoint that the profunctors that we are considering here, are many object versions of bimodules and that the κ-linear categories obtained from the groupoids, are many object groupoid κ-algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) The relationship between these Vect-enriched categories and the κ-algebras that are sometimes called ‘groupoid algebras’ is classical, being one of the themes of Mitchell’s 1972 paper, [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will expand on this later on in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 43 Part 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy theoretical underpinning of Quinn’s finite total homotopy TQFT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopically finite spaces and the category HFspan 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopically finite (HF) spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 38 (Homotopically finite (HF) space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A space, B, is called homotopi- cally finite (abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF) if B is CGWH23 and, moreover, B has only a finite set of path-components, each of which has only a finite set of non-trivial homotopy groups, all of which are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly finite disjoint unions and finite products of HF-spaces are HF24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each path-component of a HF-space is also HF (after possibly applying the k-ification functor in order to make it a CGWH space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the literature, one finds some alternative terminology used for homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lurie, [83, Appendix E], calls them ‘π-finite spaces’ whilst Anel, [1], uses the term ‘truncated coherent space’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following is essentially in [56, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4], albeit stated in the context of ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: E → B be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given b ∈ B, we let Eb := p−1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Suppose that B is path-connected, and that p is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If any two of B, E and Eb are homotopy finite, then so is the third.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Let B be any space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B and E are homotopically finite, then so is Eb for each b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B and each Eb are homotopically finite (for each b ∈ B), then so is E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, if p: E → B is a fibration, and E and B are HF, then each fibre of p is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Follows from the homotopy long exact sequence of p: E → B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Equation (8) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The second point of the following result will be crucial for what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is stated in [56, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='13] for ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a pullback diagram of spaces, where p: E → B is a fibration, X ×B E q � � E p � X f � B, then q is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X, E and B are HF, then X ×B E is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That q is a fibration follows from the standard fact that pullbacks of fi- brations are fibrations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [87, §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 Lemma].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us prove that X ×B E is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use the previous lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By assumption, X is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We only need to prove that the fibres of q are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By the lemma below, given x ∈ X, the fibre of q at x is homeomorphic to p−1(f(x)), which is HF since B and E are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using Lemma 40 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') □ 23Recall that this is an abbreviation for “compactly generated and weak Hausdorff”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Sub- section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 24Equally clearly infinite disjoint unions and products may not be!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 44 Note that the following result is not immediate, given that k-ification was applied to both product and induced topologies, see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ X, the fibre of q: X ×B E → X at x is homeomorphic to p−1(f(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a pullback diagram in CGWH, where the inc denote the obvious inclusion maps, q−1(x) � inc � X ×B E q � {x} inc � X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By applying the pullback pasting lemma, the outer rectangle of the diagram below on the left is a pullback in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The diagram on the right below is a pullback in CGWH as well, q−1(x) � inc � X ×B E q � � E p � {x} inc � X f � B, p−1(f(x)) � inc � E p � {x} f◦inc � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we get that p−1(f(x)) is homeomorphic to q−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of a HF-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 43 (Homotopy content).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a path connected HF-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of B is defined as , χπ(B) = ��π2(B, x) �� ��π4(B, x) �� ��π6(B, x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π1(B, x) �� ��π3(B, x) �� ��π5(B, x) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∈ Q, where x ∈ B is any point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In general, if X is a HF space, and using the notation in item (15) on page 16, define χπ(X) = � B∈�π0(X) χπ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also put χπ(∅) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that for all other HF spaces F, we have χπ(F) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We note that what we have called ‘homotopy content’ is called ‘homotopy order’ in [101, Lecture 4], ‘homotopy cardinality’ in [4], and also, more recently, in [56, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of a space also appeared in [49], without being given a name, and, there, was also considered for crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will consider that form separately a bit later on here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The case of ∞-groupoids is treated in [56], which proves similar results to the one below, in that context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that homotopic HF spaces, and, more generally, weakly homotopic HF spaces, have the same homotopy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The customary examples are (i) when X is a finite set, thought of as a discrete space, then χπ(X) is the usual cardinality of X, and (ii) when X is the classifying space of a finite groupoid, G, then χπ(X) = � [x]∈π0(G) 1 |G(x)|, which is the groupoid cardinality of G, in the sense of Baez and Dolan, [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will generalise the previous formula for the case of crossed complexes of groupoids in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of a HF-space: properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B and B′ are HF-spaces, then χπ(B ⊔ B′) = χπ(B) + χπ(B′) and χπ(B × B′) = χπ(B)χπ(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first equation is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second follows from the fact that �π0(B × B′) ∼= {A × A′|A ∈ �π0(B), A′ ∈ �π0(B′)}, and that, if x ∈ B and x′ ∈ B′, then πn � B × B′, (x, x′) � ∼= πn(B, x) × πn(B′, x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, we have: χπ(B × B′) = � A∈�π0(B), A′∈�π0(B′) ��π2 � A × A′��� ��π4 � A × A′��� ��π6 � A × A′��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π1 � A × A′��� ��π3 � A × A′��� ��π5 � A × A′��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = � A∈�π0(B), A′∈�π0(B′) ��π2 � A ��� ��π4 � A ��� ��π6 � A ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π1 � A ��� ��π3 � A ��� ��π5 � A ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π2 � A′��� ��π4 � A′��� ��π6 � A′��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ��π1 � A′��� ��π3 � A′��� ��π5 � A′��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = χπ(B) χπ(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ More generally, Lemma 46 (Quinn, [101], Baez–Dolan, [4], and Galv´ez-Carillo–Kock–Tonks, [56]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that p: E → B is a fibration of HF-spaces and that B is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B be arbitrary, then, recalling Eb = p−1(b) is the fibre at b, χπ(E) = χπ(B) χπ(Eb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof we give below is as hinted at in the above references, with some crucial technical details added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If E is empty, then so is Eb, and in this case there is nothing to prove.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B is empty, then so are E and Eb, and there is nothing to prove either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We are left with the case that E, B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, it follows that p: E → B is surjective, as B is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, if E′ is a path-component of E, the restriction p′ : E′ → B of p is also surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose, firstly, that E is path-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ E and b = p(x), then, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [87, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 52] or [64, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 376], the homotopy long exact sequence of p: E → B, at b and x reads (8) → πi(Eb, x) ι→ πi(E, x) ∂→ πi(B, b) δ→ πi−1(Eb, x) → .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ι→ π1(E, x) ∂→ π1(B, b) δx → π0(Eb) ι−→ π0(E) = {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, for the last stages of the sequence, the exactness means the following: we have a left-action, ⊲, of π1(B, b) on π0(Eb) (reviewed in Lemma 97), whose stabiliser subgroup at the path-component, PCx(Eb), of x ∈ Eb, is ∂(π1(E, x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the map δx: π1(B, b) → π0(Ex), which is defined as δx(g) = g ⊲ PCx(Eb), is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Also note that, by the orbit-stabiliser theorem, |π0(Eb)| = |π1(B, b)|/|∂(π1(E, x))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The exactness of the sequence (8) yields that: |πi(E, x)| = |∂(πi(E, x))| |ι(πi(Eb, x))|, if i ≥ 1, |πi(B, b)| = |∂(πi(E, x))| |δ(πi(B, b))|, if i ≥ 2, |π1(B, b)| = |∂(π1(E, x))| |π0(Eb)|, |πi(Eb, x)| = |δ(πi+1(B, b))||ι(πi(Eb, x))|, if i ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 46 Therefore, noting that B and E are by assumption path-connected, χπ(B) = 1 |∂(π1(E, x))| |π0(Eb)| +∞ � k=2 ���∂(πk(E, x)) �� ��δ(πk(B, b)) �� �((−1)−k) , χπ(E) = +∞ � k=1 ���∂(πk(E, x)) �� ��ι(πk(Eb, x)) �� �((−1)−k) , and also, χπ(Eb) = ��π0(Eb) �� +∞ � k=1 ���i(πk(Eb, x)) �� ��δ(πk+1(B, b)) �� �((−1)−k) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crucially, in the last equation, we also used the fact that given that p: E → B is a fibration, and E is path-connected, all path-components of Eb = p−1(b) are homotopy equivalent, [48, Proposition 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is reviewed in Lemma and Definition 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have χπ(Eb)χπ(B) = +∞ � k=1 ���i(πk(Eb, x)) �� ��∂(πk(E, x)) �� �� (−1)k� = χπ(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose now that E may have more that one path-component (but recall that we still take B to be path-connected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let E1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , En be the path-components of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let pk : Ek → B be the restriction of p to Ek, for each k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each pk is itself a fibration, and is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Fk = p−1 k (b) = Eb ∩ Ek.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that we have an obvious continuous bijection ⊔n k=1Fk → Eb, which is always a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We therefore have: χπ(E) = n � k=1 χπ(Ek) = n � k=1 χπ(Fk) χπ(B) = χπ� n � k=1 Fk � χπ(B) = χπ(Eb) χπ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We have the following, which is very useful later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: E → B be a fibration, where B and E are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If b ∈ B, and Eb = p−1(b), then25 χπ(E) = � [b]∈π0(B) χπ(Eb) χπ(PCb(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here we have chosen a representative of each path-component of B, noting that if b and b′ are in the same path-component then Eb is homotopic to Eb′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B is empty, then so is E, so the result follows trivially, so we suppose that B ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That if b ∈ B, then Eb is HF follows from Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given [b] ∈ π0(B), put E[b] = p−1(PCb(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The restriction, pb : E[b] → PCb(B), of p: E → B, is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have weak homotopy equivalences, � [b]∈π0(B) PCb(B) → B and � [b]∈π0(B) E[b] → E, 25Recall that, if b ∈ B, then the path-component of b, in B, with the induced CGWH topology, is denoted PCb(B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 47 therefore χπ(E) = χπ� � [b]∈π0(B) E[b] � = � [b]∈π0(B) χπ� E[b] � = � [b]∈π0(B) χπ(Eb) χπ(PCb(B)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrant spans of HF spaces and their composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Before we intro- duce fibrant spans in detail, we should briefly motivate why we are going to use them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The objects considered in basic TQFTs are manifolds of some type, and the cobordisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a set-up gives a cospan of CGWH spaces, Σ i �❘ ❘ ❘ ❘ ❘ ❘ Σ′ j �❧❧❧❧❧❧ S and we have that the induced map, Σ ⊔ Σ′ → S, is an inclusion, and furthermore a cofibration;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see later in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, starting on page 61, for a more detailed discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that such cofibrant cospans of spaces are studied in detail in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To study the state spaces associated to the manifolds, we form the space of maps from such manifolds to a ‘classifying space’ B, which we will take in Section 4 to be homotopy finite, but, in so doing, we convert a cospan, as above, to a span, BS i∗ �❦❦❦❦❦❦ j∗ �❚ ❚ ❚ ❚ ❚ ❚ BΣ BΣ′, where i∗ and j∗ denote the obvious restriction maps, and we note that the induced map from BS to BΣ × BΣ′ is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To study that type of situation, we need to understand fibrant spans and we will examine them in some generality, not just in this particular function space set-up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the point of view of the dual language of cofibrations, the duals of most results in this subsection can be found in [117, 118] The techniques used there are very similar, but, of course, dual to those used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: A → B be a continuous map of CGWH spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If b ∈ B, recall (Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3) that we topologise p−1(b) with the k-ification of the topology that B induces on A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since p−1(b) is closed, p−1(b) is CGWH, so the k-ification will not, in fact, alter the topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 48 (Fibrant span).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B, B′ and M be CGWH-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A fibrant span, B (p,M,p′) −−−−−→ B′, from B to B′, is a diagram in CGWH of form, (9) M p �❥❥❥❥❥❥ p′ �❚ ❚ ❚ ❚ ❚ ❚ B B′, where the induced map � p, p′� : M → B × B′ is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If all three spaces are HF, we will say this is a HF fibrant span or a fibrant span of HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the Hurewicz / Strøm model structure on CGWH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [112].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Λ be the category {−1 ← 0 → 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is an inverse category in the sense used in, for instance, [66, §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we give the injective model structure to CGWHΛ, then weak equivalences and cofibrations are given objectwise, see [66, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3], whilst the fibrant spans are precisely the fibrant objects in that category, CGWHΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 48 Example 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To obtain lots of examples of such fibrant spans, we can use the classical fibrant replacement process for turning an arbitrary map into a fibration, and then apply that in a suitable way to an arbitrary span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first recall that fibrant replacement construction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [87, Chapter 7, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This replaces an arbitrary map, f : A → B, by a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first form the path space, BI, with e0(B) : BI → B being ‘evaluation at 0’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a fibration, and we can form the pullback along f, Nf πf � pr1 � BI e0(B) � A f � B, so Nf ∼= {(a, γ) | a ∈ A, γ : I → B, f(a) = γ(0)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set ρ(f) : Nf → B to be e1(B) ◦ πf, so given by ρ(a, γ) = γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This map is a fibration, and the natural maps between Nf and A, induced by the constant path map from B to BI, make them homotopy equivalent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [87, Chapter 7, §3], or [64, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, page 407].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to apply this to a span, X f �❦❦❦❦❦❦ g �❚ ❚ ❚ ❚ ❚ ❚ A B, where no assumption is made about the induced map, ϕ := ⟨f, g⟩ : X → A×B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We want to replace it by a fibration, ρ(ϕ) : Nϕ → A × B, and will thus get a fibrant span, Nϕ p �❥❥❥❥❥❥ p′ �❚ ❚ ❚ ❚ ❚ ❚ A B, where Nϕ ∼= {(x, γ1, γ2) | f(x) = γ1(0), g(x) = γ2(0)}, p(x, γ1, γ2) = γ1(1), whilst p′(x, γ1, γ2) = γ2(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The trivial or identity span on X is then X idX �❥❥❥❥❥❥ idX �❯ ❯ ❯ ❯ ❯ ❯ X X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is clearly not a fibrant span if X is non-empty, so we want to replace it by a fibrant replacement, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', by using the process sketched above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Once that is done, we note that, in fact, that fibrant replacement has a simpler formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a fibrant span of spaces given by (10) XI sX �❥❥❥❥❥❥ tX �❚ ❚ ❚ ❚ ❚ ❚ X X, where, if γ : I → X, then sX(γ) = γ(0), corresponds to e0(X), and tX(γ) = γ(1), corresponding to e1(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can see directly that we have a fibration, ⟨sX, tX⟩: XI → X × X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows, for instance, from the fact that the inclusion, ι: {0, 1} → I, is a cofibration, and hence the induced map, ι∗ : XI → X{0,1}, is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that X×X ∼= X{0,1}, as the category CGWH is cartesian closed and {0, 1} ∼= {0}⊔{1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We think of the span in Equation (10) as being a fibrantly ‘resolved’ replacement for the identity span on X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X is HF, then so is XI, as it is homotopy equivalent to X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 49 Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a fibrant span, B (p,M,p′) −−−−−→ B′, as in Equation (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Both maps, p: M → B and p′ : M → B′, are fibrations, and moreover, given b ∈ B and b′ ∈ B′, both of the induced maps, p−1(b) → B′ and p′−1(b′) → B, are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the first point, given the fact that both projections, B × B′ → B and B ×B′ → B′, are fibrations, and also that the composite of fibrations is a fibration, it follows that both p and p′ are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second point follows from the fact that pullbacks of fibrations are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Specifically, consider a map f : X → B′ and a lifting, ˆf : X → p−1(b), of f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', p′ ◦ ˆf = f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a homotopy, h: X × I → B′, with h(x, 0) = f(x), where x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and the homotopy, h′ : X × I → B × B′, given by h′(x, t) = (b, h(x, t)), where x ∈ X and t ∈ I, then ⟨p, p′⟩( ˆf(x)) = h′(x, 0), for x ∈ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A lifting of h′ to M (given from the fact that ⟨p, p′⟩: M → B × B′ is a fibration) gives the desired lifting of h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Spans can be composed in a well known way, but we will only need this when working with HF fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The calculations related to Quinn’s TQFT are, from the perspective of this paper, calculations on homotopy invariants of intersections of fibres of fibrant spans and how they react to composition, but this only in the case in which the spaces are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We start the study of such in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the moment we will mostly concentrate on the prop- erties of fibrant spans that depend on the spaces being HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B (p,M,p′) −−−−−→ B′ be a fibrant span of HF-spaces from B to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any (b, b′) ∈ B × B′, then the fibre, ⟨p, p′⟩−1(b, b′) ⊂ M, is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemma 40, given that M and B × B′ are both HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that the fibrant span, B (p,M,p′) −−−−−→ B′, is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B and b′ ∈ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The spaces p−1(b), and p′−1(b′) and also the fibre of ⟨p, p′⟩: M → B × B′, over (b, b′), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', � p, p′�−1(b, b′), are all HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Lemma 52, we have a fibration p′ : p−1(b) → B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibres have the form � p, p′�−1(b, b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are also fibres for the fibration, ⟨p, p′⟩: M → B × B′, so they must be HF, as M and B × B′ both are, by Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B, B′ and B′′ be HF-spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider HF fibrant spans, B (p,M,p′) −−−−−→ B′, and B′ (p′′,M′,p′′′) −−−−−−−→ B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We form the obvious pullback, as in the diagram, (11) M ×B′ M ′ P � q �❣❣❣❣❣ q′ �❲ ❲ ❲ ❲ ❲ M p �❦❦❦❦❦❦ p′ �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ M ′ p′′ �❣❣❣❣❣❣❣❣❣ p′′′ �❯ ❯ ❯ ❯ ❯ ❯ B B′ B′′, where P = p′ ◦ q = p′′ ◦ q′, then the span, denoted B (p,M,p′)•(p′′,M′,p′′′) −−−−−−−−−−−−−−→ B′′, defined to be B (p◦q,M×B′ M′,p′′′◦q′) −−−−−−−−−−−−−−→ B′′, is a fibrant span of HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have that � p ◦ q, P, p′′′ ◦ q′� : M ×B′ M ′ → B × B′ × B′′ is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 50 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That � p ◦ q, P, p′′′ ◦ q′� is a fibration is clear from the fact that � p, p′� and � p′′, p′′′� are fibrations, and from the universal property of pullbacks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It follows that � p ◦ q, p′′′ ◦ q′� is also a fibration, for the projection, B × B′ × B′′ → B × B′′, is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To prove that M ×B′ M ′ is homotopically finite, it suffices (by Lemma 40) to observe that B×B′×B′′ is HF and that the fibres of the fibration, � p◦q, P, p′′′◦q′� , have the form ( � p, p′�−1(b, b′)) × ( � p′′, p′′′�−1(b′, b′′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each fibre is thus HF, since both components of the product are, by Lemma 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Definition 56 (Composition of HF fibrant spans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The HF fibrant span, B (p,M,p′)•(p′′,M′,p′′′) −−−−−−−−−−−−−−→ B′′, is called the composite of (p, M, p′): B → B′ and (p′′, M ′, p′′′): B′ → B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category HFspan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that this will be a non-locally small category, as we have a class of maps between objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will, however, not cause any difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The class of objects of HFspan is the class of all HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given HF-spaces, B and B′, the class of morphisms from B to B′ is given by equivalence classes of HF fibrant spans, (p, M, p′): B → B′, as we now explain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will make use of the materials on fibre homotopy equivalence recalled in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 57 (Equivalent HF fibrant spans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B and B′ be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two HF fibrant spans, (p, M, p′): B → B′ and (q, N, q′): B → B′, are said to be equivalent if there exist fibred maps, Ψ: M → N and Ψ′ : N → M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' maps making the diagrams below commute, (12) M Ψ � p �♥♥♥♥♥♥ p′ �◗ ◗ ◗ ◗ ◗ ◗ B B′ N q �PPPPPP q′ �♠ ♠ ♠ ♠ ♠ ♠ and M p �♦♦♦♦♦♦ p′ �◗ ◗ ◗ ◗ ◗ ◗ B B′, N Ψ′ � q �❖❖❖❖❖❖ q′ �♥ ♥ ♥ ♥ ♥ ♥ realising a fibre homotopy equivalence, with respect to the fibrations, ⟨p, p′⟩: M → B × B′ and ⟨q, q′⟩: N → B × B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that homotopies, H : M × I → M and H′ : N × I → N, exist such that: (1) H(m, 1) = Ψ′(Ψ(m)) and H(m, 0) = m for each m ∈ M;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) p(H(m, t)) = p(m) and p′(H(m, t)) = p′(m), for each m ∈ M and t ∈ I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) H′(n, 1) = Ψ(Ψ′(n)) and H′(n, 0) = n, for each n ∈ N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (4) q(H(n, t)) = q(n) and p′(H(n, t)) = q′(n), for each n ∈ N and t ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Ψ and Ψ′ are inverses of each other, then the HF fibrant spans are said to be isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Standard arguments prove that indeed this defines an equivalence relation on the class of all HF fibrant spans, from B to B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An equivalence class of HF fibrant spans, from B to B′, will usually be denoted [(p, M, p′)]: B → B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the context and notation of Definition 57, we recall that pullbacks along fibrations are homotopy limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that ⟨p, p′⟩: M → B × B′ and ⟨q, q′⟩: N → B × B′ are fibrations, several conditions in the definition of equivalence between HF fibrant spans are, in fact, redundant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Lemma 4, it follows that: Lemma 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two HF fibrant spans, (p, M, p′): B → B′ and (q, N, q′): B → B′, are equivalent if there exists a map, Ψ: M → N, making the left-most diagram of (12) commute and such that Ψ: M → N is a homotopy equivalence of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A CATEGORIFICATION OF QUINN’S TQFT 51 Definition 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a HF space, B, put, idHFspan B = [(sB, BI, tB)], to be the equivalence class of (13) BI sB �❦❦❦❦❦❦ tB �❚ ❚ ❚ ❚ ❚ ❚ B B, (see Example 51), under the equivalence relation in Definition 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 60 (The category HFspan).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of HF fibrant spans in Defi- nition 55 descends to the quotient under the equivalence relation in Definition 57, and, with this, the identities satisfy the evident rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, we thus have a non-locally small category, HFspan, whose objects are the HF-spaces, and in which the morphisms from B to B′ are the equivalence classes of HF fibrant spans, connecting B and B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a HF space, B, the identity in B is given by [(s, BI, t)]: B → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That the composition descends to the quotient follows from the univer- sal property of pullbacks26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More precisely, suppose that Ψ1 : M1 → N1 and Ψ′ 1 : N1 → M1 realise a fibred homotopy equivalence between the HF fibrant spans, (p1, M1, q1): B1 → B and (p′ 1, N1, q′ 1): B1 → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that Ψ2 : M2 → N2 and Ψ′ 2 : N2 → M2 realise a fibred homotopy equivalence between the HF fibrant spans (p2, M2, q2): B → B2 and (p′ 2, N2, q′ 2): B → B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is as in the diagram, (14) M1 p1 �④④④④④④④④ q1 �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ Ψ1 � � Ψ′ 1 M1×BM2 � � M2 Ψ2 � � Ψ′ 2 p2 �tttttttttt q2 �❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ B1 B B2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' N1 p′ 1 �❈❈❈❈❈❈❈❈ q′ 1 �t t t t t t t t t t t N1×BN2 � � N2 p′ 2 �❏❏❏❏❏❏❏❏❏❏❏ q′ 2 �③ ③ ③ ③ ③ ③ ③ ③ The universal property of pullbacks gives maps, (Ψ1×BΨ2): M1×BM2 → N1×BN2, co-gluing Ψ1 and Ψ2, and (Ψ′ 1×BΨ′ 2): N1×BN2 → M1×BM2 doing the same for the other pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose fibred homotopies (using the notation in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5): Ψ′ 1 ◦ Ψ1 H1 ====⇒ B1×B idM1, Ψ1 ◦ Ψ′ 1 H′ 1 ====⇒ B1×B idN1, Ψ′ 2 ◦ Ψ2 H2 ====⇒ B×B2 idM2, Ψ2 ◦ Ψ′ 2 H′ 2 ====⇒ B×B2 idN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conditions 1 to 4 of Definition 57 imply that these homotopies can be (co)glued to homotopies, J : (M1×BM2) × I → M1×BM2 and J′ : (N1×BN2) × I → N1×BN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, they are such that (Ψ′ 1×BΨ′ 2) ◦ (Ψ1×BΨ2) J =====⇒ B1×B2 idM1×BM2, (Ψ1×BΨ2) ◦ (Ψ′ 1×BΨ′ 2) J′ =====⇒ B1×B2 idN1×BN2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 26The arguments are essentially identical to those proving that cospans of spaces and maps between them can be arranged into a bicategory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 52 To handle the point about identities, let B and B′ be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a HF fibrant span, (p, M, q): B → B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us prove that we have maps Ψ and Ψ′, as below, realising an equivalence of HF fibrant spans, (15) M Ψ � p �❦❦❦❦❦❦❦❦❦ q �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ B B′, BI×BM p′ �❙❙❙❙❙❙ q′ �❦ ❦ ❦ ❦ ❦ ❦ and M� Ψ′ p �❥❥❥❥❥❥❥❥ q �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ B B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' BI×BM p′ �❙❙❙❙❙❙ q′ �❥ Here we consider the obvious pull-back, appearing as the diamond in the diagram: BI×BM �❥❥❥❥❥❥ �❚ ❚ ❚ ❚ ❚ ❚ q′ � p′ � BI sB �❧❧❧❧❧❧ tB �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ M p �✐✐✐✐✐✐✐✐✐ q �❘ ❘ ❘ ❘ ❘ ❘ B B B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We put Ψ(m) = (� p(m), m), where � p(m) is the constant path at p(m) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly Ψ is fibred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Lemma 4, in the context of Lemma 58, we only need to prove that Ψ: M → BI×BM is a homotopy equivalence of spaces (as opposed to a homotopy equivalence of fibred spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A homotopy inverse of Ψ: M → BI×BM is given by the map Φ: BI×BM → M such that Φ(γ, m) = m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that this is not a fibred map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We have that Φ ◦ Ψ = idM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand Ψ(Φ(γ, m)) = ( � γ(1), m), for each (γ, m) ∈ BI×BM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here � γ(1)) is the constant path at γ(1) ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The following homotopy connects Ψ ◦ Φ and idBI×BM: (γ, m, t) ∈ (BI×BM) × I �→ � s �→ γ � t + (1 − t)s � , m � ∈ BI×BM, for s ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That the resolved identity spans given by the mapping spaces are also identities on the right is dealt with similarly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A ‘dual’ category to HFspan, whose objects are spaces, and morphisms are cofibred homotopy equivalence classes of cofibrant cospans was constructed in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our methods of proofs here are very similar, but, of course, needed switching from cofibred to fibred homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let HFiso be the subcategory of CGWH with objects the HF spaces, and homeomorphisms of HF spaces as morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a functor, I : HFiso → HFspan, given by, if X is a HF space, then I(X) = X, and if f : X → Y is a homeomorphism of HF spaces, then I(f) is the equivalence class of the span, XI sX �❥❥❥❥❥❥ f◦tX�❚ ❚ ❚ ❚ ❚ ❚ X Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (It is likely that a similar functor will map the category with objects the HF spaces, and morphisms the homotopy classes of homotopy equivalences of HF spaces, to HFspan, but we will not consider this, nor do we need it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is clear that if f : X → Y is a homeomorphism of HF spaces, then I(f) = (sX, XI, f ◦ tY ): X → Y is a HF fibrant span, since (sX, XI, tX): X → X is a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, I sends the identities in HFiso to the identities in HFspan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 53 Let f : X → Y and g : Y → Z be homeomorphisms of HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We check that I(g ◦ f) = I(f) • I(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To see this, look at the diagram below, where the top diamond is a pullback, defining the composite I(f) • I(g): XI ×Y Y I proj1 �tttttttttt proj2 �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ XI sX �⑤⑤⑤⑤⑤⑤⑤⑤ f◦tX �❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ Y I sY �sssssssssss g◦tY �❇ ❇ ❇ ❇ ❇ ❇ ❇ ❇ X Y Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XI sX �❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙❙ g◦f◦tX �❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ ❦ The map, Ψ: XI ×Y Y I → XI, such that, Ψ(γ, γ′)(t) = � γ(2t), if t ∈ [0, 1/2], f −1(γ′(2t − 1)), if t ∈ [1/2, 1], is a homeomorphism that makes the obvious diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This shows that I(g ◦ f) = I(f) • I(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a more general version of the notion of equivalence of (HF) spans, as given in Definition 57, that will be useful slightly later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that spans form a category, CGWHΛ, as noted in Remark 49, in which a morphism is simply a natural transformation, (p, M, p′) (f−1,f0,f1) � (q, N, q′) B f−1 � M p � f0 � p′ � B′ f1 � C N q � q′ � C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, we take the Hurewicz / Strøm model structure on CGWH, and the injective model structure on CGWHΛ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A morphism, such as (f−1, f0, f1), is thus a cofibration, in that model structure, if each of f−1, f0, and f1 is a cofibration in CGWH, and is a weak equivalence if each of these maps is a weak equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As, in the Hurewicz / Strøm model category structure, the weak equivalences are, in fact, ‘strong’ homotopy equivalences, we make the following definition: Definition 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two fibrant spans, (p, M, p′): B → B′ and (q, N, q′): C → C′, are said to be homotopy equivalent if there is a morphism, (f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′), in which each fi is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, in this case, (f−1, f0, f1) is a homotopy equivalence27, and, in the setting in which f−1 and f1 are the respective identities, we retrieve the notion of equivalence given in Definition 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These conditions28 do not depend on the 27which would be called a weak equivalence in the injective model structure on the category of spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 28We note that much of this depends on using homotopy equivalences and not just weak equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Although these coincide on the spaces that we are handling, it is, in practice, the stronger form of the definition that is being used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 54 spaces involved being homotopy finite, but it is only in that case that we will be using them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that fibrant spans are cofibrant-fibrant objects in the injective model structure in CGWHΛ, so a homotopy equivalence of fibrant spans will actually be a strong homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a map, (f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′), of fibrant spans, we get an induced map, (f0, ⟨f−1, f1⟩) : ⟨p, p′⟩ → ⟨q, q′⟩, of fibrations (in the sense of Definition 6), so, if (f−1, f0, f1) is a homotopy equiva- lence, then (f0, ⟨f−1, f1⟩) will be a homotopy equivalence of fibrations, (Definition 7), and by Proposition 8, there will be a homotopy inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, if b ∈ B and b′ ∈ B′, then by Corollary 9, or by using the fact that (f−1, f0, f1) is a homotopy equivalence, we have Proposition 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If (f−1, f0, f1) : (p, M, p′) ⇒ (q, N, q′) is a homotopy equivalence of fibrant spans, then, for any (b, b′) ∈ B × B′, the induced map on fibres, ⟨p, p′⟩−1(b, b′) → ⟨q, q′⟩−1(f−1(b), f1(b′)), is a homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We will return to this result slightly later on, both in the particular case of an equivalence of fibrant spans going between B and B′ and in this more general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A family of functors, R(s) : HFspan → Vect, derived from the homo- topy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The results in this subsection are closely related to some given in [56], where they are stated in the language of ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They were, in fact, essentially implicit in [101, Section 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The indexation of the family of functors is, however, a generalisation of the α-degroupoidification set-up introduced by Baez, Hoffnung and Walker in [5], Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, which we will very briefly recall shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The setting here is particularly suited to constructing Quinn’s finite total homo- topy TQFT, and explicitly to compute it in a number of cases, as well as moving towards extended versions of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The point about the parameter, s, is then that, for s = 0, one has Quinn’s theory, as we will see shortly, but, for other values of s, one also gets functors linked to other TQFTs, in the normalisations in which they were initially constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s finite total homotopy TQFT and degroupoidification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We pause in the development of the background theory to set the scene for the more detailed treatment of Quinn’s theory, which is needed if we are to categorify that theory to an extended form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will revisit Quinn’s Lecture 4, [101], ‘once over lightly’, and then look at parts of the paper, [5], on the ‘groupoidification’ and consequent ‘degroupoidification’ of linear algebra, mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn considers a fixed space, B, which is often a classifying space of a finite group, but in any case has to be HF, in our terminology29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The TQFT, Z, has to assign a ‘state space’ to each space, Y , and Quinn takes Z(Y ) abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = ZB(Y ) := Q[Y, B], that is the rational vector space with basis the set, [Y, B], of homotopy classes of maps from Y to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rˆole of cobordisms in his theory is, then, played by what he calls CW- triads30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This he takes to mean a CW-complex, X, together with two (disjoint) 29which is, of course, adapted from Quinn’s 30The term ‘CW-triad’ is used more generally in the literature to mean a triple of CW- complexes as with Quinn, except that the disjointness of the two subcomplexes is not required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 55 subcomplexes, Y1 and Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given such a triple, he wants to define a linear map, ZX : Z(Y1) → Z(Y2), so, if [f1] ∈ [Y1, B], then we will have ZX([f1]) = � [f2] µX,f1,f2[f2], (see [101, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 340] for more details and interpretation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The condition that Z is to be a monoidal functor imposes conditions on how the assignment µ must behave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' He says that these conditions essentially determine µ, but does not elaborate on this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formula he gives, in his notation31, is µX,f1,f2 = #πMapf1(X, B)[f2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is the homotopy content of the space of maps from X to B that restrict to f1 on Y1, and which are homotopic to f2 on Y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note the asymmetric treatment of the two boundaries, which is mentioned [101, page 340].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, we will treat the two boundaries symmetrically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') This definition works, but, in [101], many details are only sketched or left as an exercise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Those ‘details’, although fairly ‘clear’ are, in fact, quite tricky to check in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Quinn makes reference to a preprint from 1991, but that does not seem to be readily available now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') From our viewpoint, the CW-triad defines a cofibrant cospan and so, on passing to mapping spaces, yields a fibrant span of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In that context, Mapf1(X, B)[f2] is an example of a ‘spatial slice’ in the terminology that we will introduce below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to ‘groupoidification’ and ‘homotopy linear algebra’, in the paper by Baez, Hoffnung and Walker, [5], the setting is now the linear algebra corresponding to spans of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again we give a brief ‘recall’ and, again, adopt or adapt the terminology of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a groupoid, X, they assign to it the real vector space with the set of connected components of X as basis, (so that is R[X], in their notation or, for us, R[π0(X)]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They also consider RX, defined to be the vector space {ψ : X → R}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the space of maps from π0(X) to the reals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, RX ∼= R[X]∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now, given a span (of groupoids), S q �❦❦❦❦❦❦ p �❚ ❚ ❚ ❚ ❚ ❚ Y X, which is required to satisfy a ‘tameness’ condition32, in their Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7, they show how such a (q, S, p) yields a linear operator, S � : RX → RY , (and note the contravariance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This operator is defined by (S � ψ)([y]) = � [x]∈X � [s] |Aut(x)| |Aut(s)| ψ([x]), (and again we refer the reader to [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 501] for the full explanation of the notation and the limits on the ‘index variable’ [s]33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We again note the lack of symmetry in the formula and, in fact, quoting from [5, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 513]: “one might wonder why this formula uses information about Aut(x), but not Aut(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The answer is that we made an arbitrary choice of conventions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is another equally nice choice and, in fact, an entire family of choices interpolating between these two.” 31so his #π is our χπ, 32which we will not be needing so we omit here, 33as this involves the ‘tameness’ criterion that we mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 56 They then show how, for each α ∈ R, one gets a linear operator, Sα � : RX → RY , on replacing |Aut(x)| by |Aut(x)|1−α|Aut(y)α, and that the case α = 1 2 has nicer symmetry properties than that of α = 0, which they had introduced earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, in the context of TQFTs, we can similarly define a family of linear operators associated to a cobordism, and indexed by a parameter, for us, s in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be done shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the symmetric case, in which s = 1 2, the resulting TQFT is essentially that of Yetter in [124], and for, in that paper, B being the classifying space of a crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also see that these TQFTs, thus obtained for different choices of s, are all naturally isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now return to our context of HF-spans, generalising the tame spans of groupoids of [5] and adapting the α-groupoidification to that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Spatial slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some terminology and a bit of notation will be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It will be widely used later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The basic idea of what we will call a ‘spatial slice’ is that of the fibre, ⟨p, p′⟩−1(b, b′), for an HF fibrant span, as before, and with b ∈ B, and b′ ∈ B′, but given the way that we will have to use these, we need a somewhat simpler notation, as it will be used in formulae which, in any case, are quite complicated enough!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 67 (Spatial slice).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B (p,M,p′) −−−−−→ B′ be a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B, b′ ∈ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Define the following (possibly empty) space, {b ��(p, M, p′) ��b′} abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = {b ��M ��b′} := {m ∈ M : p(m) = b and p′(m) = b′}, which will be called here the spatial slice of B (p,M,p′) −−−−−→ B′, at b ∈ B and b′ ∈ B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the abbreviated notation, {b|M|b′}, does not show the dependence on p: M → B and p′ : M → B′, but we will use it more often than the more complete {b ��(p, M, p′) ��b′}, so as not to overload the various formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also define the following spaces, also called spatial slices, but over the various subsets of B, or B′, as indicated: {b ��M ��PCb′(B′)} := {m ∈ M : p(m) = b and p′(m) ∈ PCb′(B′)}, {PCb(B) ��M ��b′} = {m ∈ M : p(m) ∈ PCb(B) and p′(m) = b′}, and {PCb(B) ��M ��PCb′(B′)} = {m ∈ M : p(m) ∈ PCb(B) and p′(m) ∈ PCb′(B′)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B (p,M,p′) −−−−−→ B′ be a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We collect some useful facts about its spatial slices that will play important rˆoles later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since ⟨p, p′⟩: M → B × B′ restricts to a fibration, {PCb(B) ��M ��PCb′(B′)} → B × B′, and, by Lemma 52, p: M → B and p′ : M → B′ restrict to fibrations {PCb(B) ��M ��PCb′(B′)} → B and {PCb(B) ��M ��PCb′(B′)} → B′, the fibres, or more generally, the inverse images are, respectively, the spaces, {b ��M ��b′}, {b ��M ��PCb′(B′)} and {PCb(B) ��M ��b′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the homotopy type of the spaces, {b ��M ��b′}, {b ��M ��PCb′(B′)} and {PCb(B) ��M ��b′}, depends only on the path-components of b in B and b′ in B′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 57 Lemma 52 also gives that p and p′ restrict to fibrations, {b|M|PCb′(B′)} → PCb′(B′) and {PCb(B)|M|b′} → PCb(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibres, again, have the form {b|M|b′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All the spaces, {b ��M ��b′}, {b ��M ��PCb′(B′)}, {PCb(B) ��M ��b′} and {PCb(B) ��M ��PCb′(B′)}, are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemma 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can therefore take their homotopy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let (p, M, p′) be a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have χπ({PCb(B) ��M ��PCb′(B′)}) = χπ({b ��M ��PCb′(B)}) χπ(PCb(B)) = χπ({PCb(B) ��M ��b′}) χπ(PCb′(B′)) = χπ({b ��M ��b′}) χπ(PCb(B)) χπ(PCb′(B′)), in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemma 46 applied to the fibrations in Remark 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First recall Definition 67 and Lemma 69, and, as we want to define some linear maps, recall also the General Notation and terminology relating to matrix elements that we mentioned near the start of the paper on page 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B (p,M,p′) −−−−−→ B′ be a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We introduce a matrix over C, parametrised by a complex valued index, s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given (non-empty) path-components, PCb(B), PCb′(B′), of B and B′, define the following complex valued ‘matrix-elements’, � PCb(B) ��R (s)(p, M, p′) ��PCb′(B′) � = χπ� {b ��M ��b′} � � χπ(PCb(B)) �s � χπ(PCb′(B′)) �1−s = χπ({PCb(B) ��M ��PCb′(B′)})) � χπ(PCb(B)) �s−1 � χπ(PCb′(B′)) �−s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As usual, we have written {b|M|b′} for {b|(p, M, p′)|b′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We note, as well, that the homotopy type of {b|(p, M, p′)|b′} depends only on the path-components, in B and B′, that b and b′, respectively, belong to, so χπ({b ��M ��b′}) is indeed a function of PCb(B) and PCb′(B′), only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following result is essentially in [101].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A version of this result for groupoids and spans appears in [5, Theorem 41], whilst a version for ∞-groupoids is in [56, Proposition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The matrix elements corresponding to R (s) are multi- plicative with respect to composition of HF fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' consider HF fibrant spans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): B → B′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (p′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′′′): B′ → B′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and their composition,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' connecting B to B′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M ×B′ M ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' PR) = (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′) • (p′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′′′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' defined from the diagram below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where the middle diamond is a pullback,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (16) M ×B′ M ′ PL � PR � P � q �❤❤❤❤❤❤ q′ �❱ ❱ ❱ ❱ ❱ ❱ M p �♠♠♠♠♠♠ p′ �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ M ′ p′′ �❤❤❤❤❤❤❤❤❤ p′′′ �❙ ❙ ❙ ❙ ❙ ❙ B B′ B′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 58 then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given b ∈ B and b′′ ∈ B′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have � PCb(B) ��R (s)(PL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M ×B′ M ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' PR) ��PCb′′(B′′) � = � [b′]∈π0(B′) � PCb(B) ��R (s)(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′) ��PCb′(B′) � � PCb′(B′) ��R (s)(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′′) ��PCb′′(B′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We apply Theorem 47 to the fibration, Pb,b′′ : {b ��M ×B′ M ′��b′′} → B′, obtained by restricting P : M ×B′ M ′ → B′ to {b ��M ×B′M ′��b′′} = ⟨PL, PR⟩−1(b, b′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This map, Pb,b′′, is a fibration, since, by Lemma 55, � PL, P, PR � : M ×B′ M ′ → B × B′ × B′′ is a fibration34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We, then, have � PCb(B) ��R (s)(PL, M ×B′ M ′, PR) ��PCb′′(B′′) � = χπ(PCb(B))s χπ(PCb′′(B′′))1−s χπ({b ��M ×B′ M ′��b′′}) = χπ(PCb(B))s χπ(PCb′′(B′′))1−s � [b′]∈π0(B′) χπ(P −1 b,b′′(b′)) χπ(PCb′(B′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now note that, as spaces, P −1 b,b′′(b′) = {b ��(p, M, p′) ��b′} × {b′��(p′, M ′, p′′) ��b′′}, so χπ(P −1 b,b′′(b′)) = χπ({b ��(p, M, p′) ��b′}) χπ({b′��(p′, M ′, p′′) ��b′′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This yields the main formula in the statement of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a path-connected HF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose a base-point ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Ω∗(B) denote the pointed loop space of (B, ∗), that is, the space of all loops in B starting and ending in ∗, then Ω∗(B) is HF, and χπ(Ω∗(B)) = 1/χπ(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is the analogue of [56, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='10], which is for ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let F∗(B) denote the space of all paths, α : [0, 1] → B, starting at ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is, of course, a fibration, F∗(B) → B, sending a path to its second end- point, and we note that F∗(B) is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since both F∗(B) and B are HF, it follows that the fibre at ∗, which is Ω∗(B), is HF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since F∗(B) is contractible, we have that χπ(F∗(B)) = 1, and Lemma 46 gives that χπ(Ω∗(B)) χπ(B) = χπ(F∗(B)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose B is a HF space and let b, b′ ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let s ∈ C, then the matrix element, � PCb(B) ��R (s)(sX, B[0,1], tY ) ��PCb′(B) � = δ � PCb(B), PCb′(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, let f : B → B′ be homeomorphism of HF spaces, then, � PCb(B) ��R (s)(sX, B[0,1], f ◦ tY ) ��PCb′(B′) � = δ � PCb(B), PCf −1(b′)(B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, if X is a set, which we will need to be the set of path-components of B, we take δ: X × X → {0, 1}, to be such that δ(x, y) is 0, if x ̸= y, and δ(x, x)) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 34cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the proof of the second part of Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 59 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We prove the most general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First of all note that if PCb(B) ̸= PCf −1(b′)(B), then � PCb(B) ��R (s)(sX, B[0,1], f ◦ tY ) ��PCb′(B′) � = 0, as in this case {b ��(sB, B[0,1], f ◦ tB) ��b′} is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, � PCb(B) ��R (s)(sX, B[0,1], f ◦ tY ) ��PCf(b)(B′) � = χπ� {b ��(sB, B[0,1], f ◦ tB) ��f(b)} � χπ(PCb(B))s χπ(PCf(b)(B′))1−s = χπ� {b ��(sB, B[0,1], tB) ��b} � χπ(PCb(B))s χπ(PCb(B))1−s = χπ� Ωb(PCb(B)) � χπ(PCb(B)) = 1, where we have used Lemma 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We thus have35 Theorem 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a functor, R(s) : HFspan → VectC, such that: if B is a HF space, then R(s)(B) = C(�π0(B)), the free vector space over the set of all path-components of B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for [(p, M, p′)]: B → B′, a morphism in HFspan, the matrix elements for the linear map, R(s)� [(p, M, p′)] � : C(�π0(B)) → C(�π0(B′)), are given by � PCb(B) ��R(s)� [(p, M, p′)] ���PCb′(B′) � := � PCb(B) ��R (s)(p, M, p′) ��PCb′(B′) � , for given path-components PCb(B) ∈ �π0(B) and PCb′(B′) ∈ �π0(B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Compatibility of R(s) with the composition and identities in HFspan follows from Lemmas 71 and 73, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What we have not yet shown is that the matrix elements R (s) are invariant under equivalence of spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows by using lemma 5, combined with Definition 57, as made explicit in Proposition 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a functor, ×: HFspan × HFspan → HFspan, sending the equivalence class of, � M p �♠♠♠♠♠♠ q �◗ ◗ ◗ ◗ ◗ ◗ X Y , M ′ p′ �♠♠♠♠♠♠ q′ �◗ ◗ ◗ ◗ ◗ ◗ X′ Y ′ � , to the equivalence class of, M × M ′ p×p′ �❤❤❤❤❤ q×q′ �❱ ❱ ❱ ❱ ❱ X × X′ Y × Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that the latter span is fibrant, since the product of two fibrations is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from straightforward calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ It can furthermore be proved that HFspan is a monoidal category, with this tensor product, but we will not need this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For details in the dual case of cofibrant cospans, see [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If ⊗: VectC × VectC → VectC denotes the tensor product functor in VectC, then we have a natural isomorphism of functors from HFspan × HFspan to VectC, η: ⊗ ◦ � R(s) × R(s)� =⇒ R(s) ◦ ×.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 35The notation �π0(B) is explained at the end of Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 A CATEGORIFICATION OF QUINN’S TQFT 60 It is such that, given HF spaces X and X′, and x ∈ X, x′ ∈ X′, then ηX,X′� PCx(X)⊗PCx′(X′) � = PC(x,x′)(X × X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If X and X′ are spaces, then we have a natural bijection from �π0(X)×�π0(X′) to �π0(X × X′), sending � PCx(X), PCx′(X′) � to PC(x,x′)(X × X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The naturality of η then follows from the calculation below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider two HF fibrant spans, M p �♥♥♥♥♥♥ q �P P P P P P X Y, and M ′ p′ �♠♠♠♠♠♠ q′ �❘ ❘ ❘ ❘ ❘ ❘ X′ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If x ∈ X, x′ ∈ X′, y ∈ Y and y′ ∈ Y ′, we have: � PC(x,x′)(X × X′) |R (s)� (p, p′), M × M ′, (q, q′) � |PC(y,y′)(Y × Y ′) � = χπ �� (x, x′) | � (p, p′), M × M ′, (q, q′) � | (y, y′) �� χπ(PC(x,x′)(X × X′) �s χπ� PC(y,y′)(Y × Y ′) �1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that we have homeomorphisms, � (x, x′) | � (p, p′), M × M ′, (q, q′) � | (y, y′) � ∼= {x | (p, M, q) | y} × {x′ | (p′, M ′, q′) | y′}, PC(x,x′)(X × X′) ∼= PCx(X) × PCx′(X′), and PC(y,y′)(Y × Y ′) ∼= PCy(Y ) × PCy(Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It therefore follows that, � PC(x,x′)(X × X′) | R (s)� (p, p′), M × M ′, (q, q′) � | PC(y,y′)(Y × Y ′) � = � PCx(X) | R (s)� (p, M, q) � |PCy(Y ) � � PCx′(X′) | R (s)� (p′, M ′, q′) � |PCy′(Y ′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The last equation follows from Lemma 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Finally, note the following result, that follows from a straightforward calculation using the conventions in Definition 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let s, t be complex numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a natural isomorphism, ηs,t : R(s) =⇒ R(t), of functors, from HFspan to VectC, which is such that, if X is a space, and x ∈ X, then, ηs,t X (PCx(X)) = χπ (PCx(X))s−t PCx(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A more detailed review of Quinn’s finite total homotopy TQFT First let us recall some basic definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Most of the time, we will be able to do the necessary constructions without this level of detail, just as Quinn does in the primary source, [101], where he discusses a version of the theory using just CW complexes, but very occasionally, these results, or their consequences, are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They also help to tie Quinn’s theory into the general theory of TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 61 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cobordism categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a positive integer n, let Cob(n,n+1) be the monoidal category of compact smooth manifolds and cobordisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Details are discussed in many places in the literature, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [37, 89].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note, once again, that we make no assumption that orientations on manifolds and cobordisms are given, or even that they exist36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The class of objects of Cob(n,n+1) is given by all compact smooth-manifolds of dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given compact smooth n-manifolds, Σ and Σ′, morphisms in Cob(n,n+1) from Σ to Σ′, are equivalence classes of cobordisms, (i, S, j): Σ → Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here a cobordism is a cospan of compact smooth manifolds and smooth maps, Σ i �◗ ◗ ◗ ◗ ◗ ◗ Σ′ , j �❧❧❧❧❧❧ S where i and j are smooth maps inducing a diffeomorphism, ⟨i, j⟩: Σ ⊔ Σ′ → ∂(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two cobordisms, (i, S, j), (i′, S′, j′): Σ → Σ′, are considered equivalent if a smooth diffeomorphism, f : S → S′, exists, making the diagram, S f � Σ i �s s s s s s i′ �❑ ❑ ❑ ❑ ❑ ❑ Σ′ , j �▼▼▼▼▼▼ j′ �qqqqqq S′ commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use a hopefully evident notation for the equivalence classes of cobordisms, but may later forget to use it if no ambiguity should arise from its omission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of morphisms, [(i, S, j)]: Σ → Σ′ and [(i′, S′, j′)]: Σ′ → Σ′′, is done as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first consider the pushout, S ⊔Σ′ S′, in CGWH, as in the diagram below (the nodes in the first and second rows contain the underlying topological manifolds of the corresponding smooth manifolds): Σ i �❄ ❄ ❄ ❄ ❄ ❄ ❄ ❄ Σ′ j �✈✈✈✈✈✈✈✈✈✈ i′ �❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ Σ′′ j′ �⑤⑤⑤⑤⑤⑤⑤⑤ S k �● S′ k′ �✈✈✈✈✈✈✈✈✈ S ⊔Σ′ S′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The topological space, S ⊔Σ′ S′, is a topological manifold, see [89, §1] or [65, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2], and ⟨k ◦i, k′ ◦j′⟩: Σ⊔Σ′′ → ∂(S ⊔Σ′ S′) is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This yields the cospan below in CGWH, where again the nodes, as yet, only denote topological manifolds, Σ k◦i �▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Σ′′ k′◦j′ �♣♣♣♣♣♣♣ S ⊔Σ′ S′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As is well known, S ⊔Σ′ S′ can be given a smooth structure, which ‘restricts’ to the smooth structures in S and in S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This smooth structure, despite not being unique, as it depends on the choice of a collar of Σ′ in S and in S′, is unique up to a diffeomorphism, which is the identity on ∂(S ⊔Σ′ S′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for discussion see [89, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 36Quinn’s finite total homotopy TQFT makes no assumption of orientability of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 62 The composition, •, of morphisms, [(i, S, j)]: Σ → Σ′ and [(i′, S′, j′)]: Σ′ → Σ′′, in Cob(n,n+1) is then given as � [(i, S, j)]: Σ → Σ′� � [(i′, S′, j′)]: Σ′ → Σ′′� := � [(k ◦i, S ⊔Σ′ S′, k′ ◦j′)]: Σ → Σ′′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given an n-manifold, Σ, the identity cobordism, idCob(n,n+1) Σ : Σ → Σ, is the equivalence class of the cospan, Σ ιΣ 0 �❑ ❑ ❑ ❑ ❑ ❑ ❑ Σ ιΣ 1 �sssssss Σ × I .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here ιΣ i (x) = (x, i), for all i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structures in Diffn and Cob(n,n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Diff n denote the category of closed n-manifolds and diffeomorphism between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a functor, I′ : Diffn → Cob(n,n+1), which is the identity on objects, and such that I′(f : Σ → Σ′) is the equivalence class of the cospan, Σ ιΣ 0 �❑ ❑ ❑ ❑ ❑ ❑ ❑ Σ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιΣ′ 1 ◦f −1 �qqqqqqq Σ × I The proof of this fact is dual to that of Lemma 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This, of course, implies that each I′(f) will be an invertible morphism in Cob(n,n+1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', that the cobordism is invertible up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is well known that this functor descends to the category with objects the closed n-manifolds and morphisms isotopy classes of diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will, however, not be used in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is another way of obtaining a cobordism from a diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Instead of using f −1 in the right (co)leg, we could have used f in the left one, that is, Σ ιΣ′◦f 0 �❑ ❑ ❑ ❑ ❑ ❑ ❑ Σ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιΣ′ 1 �qqqqqqq Σ′ × I This gives an equivalent cobordism as f × I : Σ × I → Σ′ × I fits in a commutative diagram Σ × I f×I ∼ = � Σ ιΣ′ 0 ◦f �◆ ◆ ◆ ◆ ◆ ◆ ιΣ 0 �♦ ♦ ♦ ♦ ♦ ♦ ♦ Σ′ , ιΣ′ 1 ◦f −1 �PPPPPPP ιΣ′ 1 �♥♥♥♥♥♥ Σ′ × I so gives the same functor as the previous construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will later on ‘categorify’ this second I′-construction, when we are considering the symmetric monoidal bicategory structure on 2Cob(n,n+1,n+2) in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that both Cob(n,n+1) and Diff n are symmetric monoidal categories, where the tensor product on objects is given by the disjoint union, Σ⊔Σ′, of closed n-manifolds, Σ and Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For both symmetric monoidal bicategories, the unit object is the empty manifold, ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Diffn, the tensor product of morphisms is achieved as in (CGWH, ⊔), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by performing the disjoint union of diffeomorphisms, namely (f1 : Σ1 → Σ′ 1) ⊔ (f2 : Σ2 → Σ′ 2) = (f1 ⊔ f2): Σ1 ⊔ Σ2 → Σ′ 1 ⊔ Σ′ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The associativity constraints, braiding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', in Diffn are also as those in (CGWH, ⊔).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 63 In Cob(n,n+1), the monoidal structure is based on the functor, ⊔: Cob(n,n+1) × Cob(n,n+1) → Cob(n,n+1), so is obtained from the disjoint union of cobordisms, which descends to their equiv- alence classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is dual to the construction in Lemma 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The associativity and unit constraints, and the braiding in Cob(n,n+1), are obtained from those of Diffn by applying I′ : Diff n → Cob(n,n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s results on HF function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As mentioned earlier, in the ‘once over lightly’ description of Quinn’s finite total homotopy TQFT, on page 55, Quinn uses various results on mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a HF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 79 (Quinn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a finite CW-complex, then BX is a HF-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof follows from an induction on the number of cells of X, by making use of the following lemma in each induction step;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [101, Chapter 4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let i be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let a space, Y , be obtained from the CW-complex X by attaching an i-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose BX is HF, then so is BY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Y be obtained from X by attaching an i-cell along f : Si−1 → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We hence have a pushout diagram, Si−1 � f � X � Di � Y , where both vertical arrows are induced by inclusion of subcomplexes, hence they are cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to function spaces, and using the fact that CGWH is monoidal closed, we have a pullback diagram, where the vertical arrows are moreover fibrations, given that they are ‘dual’ to cofibrations, BY � � BDi � BX f ∗ � BSi−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Concretely, each vertical arrow is obtained by restricting a function defined on a CW-complex to a subcomplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now apply Lemma 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since BX and BDi are HF, the first by assumption, the second since BDi is contractible, the proof is reduced to proving that BSi−1 is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is proved in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any positive integer n, the space, BSn, is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again the proof is by induction in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The base case follows from the fact that BS0 = B{0,1} ∼= B × B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The induction step follows by observing that we have the following pullback diagram, where the vertical arrows are fibrations: BSn � � BDn � BDn � BSn−1 and noting, once again, that BDn is HF, since it is contractible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A CATEGORIFICATION OF QUINN’S TQFT 64 Lemma 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer, and B be a HF-space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) There is a functor, F0 B : (Diffn)op → HFiso37, that sends a closed and smooth n-manifold, Σ, to BΣ and a diffeomorphism, f : Σ → Σ′, to the induced map38, f ∗ : BΣ′ → BΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) There is a functor, FB : Cob(n,n+1) → HFspan, that sends the equivalence class of a cobordism, Σ i �◗ ◗ ◗ ◗ ◗ ◗ Σ′, j �❧❧❧❧❧❧ S to the equivalence class of the HF fibrant span, (17) BΣ � i∗ ❘ ❘ ❘ ❘ ❘ ❘ BΣ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � j∗ ❦❦❦❦❦❦ BS Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Σ is a compact smooth manifold, then Σ has a finite triangulation, and, in particular, it can be given the structure of a finite CW-complex, [94], so, by Lemma 79, BΣ is HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rest of (1) follows from the fact that CGWH is a cartesian closed category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the second point, also note that if S is a smooth manifold with boundary, then we can find, again by, for instance, [94], a triangulation of the pair (S, ∂(S)) making ∂(S) a subcomplex of S, so the inclusion ι: ∂(S) → S is a cofibration39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As a consequence, the induced map, ι∗ : BS → B∂(S), is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To see that the HF span in (17) is fibrant, note that B∂(S) ∼= BΣ⊔Σ′ ∼= BΣ × BΣ′, where we used the fact that CGWH is cartesian closed in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hence FB sends (equivalence classes of) cobordisms of manifolds to (equivalence classes of) HF fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That FB preserves the compositions of Cob(n,n+1) and HFspan follows again from the fact that CGWH is cartesian closed, and, in particular, that the contravariant functor, B( ) : CGWH → CGWH, sends colimits to limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally units are preserved by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the following, we let n be a non-negative integer, B be a HF space, and fix some s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will work over the complex number field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we mentioned in the introduction, in [101, Lecture 4], Quinn defined what he called the finite total homotopy TQFT, which we will denote by QB : Cob(n,n+1) → Vect, depending on an arbitrary, but fixed, homotopy finite space, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we will see, this is one of a family of such constructions, one for each s ∈ C, albeit all related by natural isomorphisms, as in Proposition 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A particular case of the general s case appeared in [5, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 Proposition, page 513] in the context of TQFT-like functors for spans of groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 83 ((The s-indexed form of) Quinn’s finite total homotopy TQFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quinn’s finite total homotopy TQFT, Q(s) B : Cob(n,n+1) → Vect, is defined to be the composite of the functors, FB : Cob(n,n+1) → HFspan and R(s) : HFspan → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 37See Definition 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 38I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the map sending φ: Σ′ → B to φ ◦ f : Σ → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 39An alternative proof of this fact is in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 65 We will write QB for Q(0) B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This was Quinn’s original normalisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') For the second of these functors, see Theorem 74, and use Lemma 82 to allow its application here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Taking this apart, and in more detail, given a closed n-manifold, Σ, then Q(s) B (Σ) = C(�π0(BΣ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, between the closed n- manifolds, Σ1 and Σ2, we have that the matrix elements of the resulting linear operator are given by the equation below, for continuous functions f : Σ1 → B and f ′ : Σ2 → B, � PCf(BΣ1) | Q(s) B ([(i1, S, i2)]) | PCf ′(BΣ2) � = χπ� {f|(i∗ 1, BS, i∗ 2)|f ′} � � χπ(PCf(BΣ1)) �s� χπ(PCf ′(BΣ2)) �1−s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We are using the notation of Definition 67, so {f|(i∗ 1, BS, i∗ 2)|f ′} abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = {f|BS|f ′} = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 H : S → B | B Σ1 f �❧ i1 �❙ ❙ ❙ ❙ ❙ ❙ Σ2 f ′ �❙❙❙❙❙❙ i2 �❦❦❦❦❦❦ S H � commutes \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following elementary lemma will be used later in the proof that the functor Q(s) B : Cob(n,n+1) → Vect, is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use the notation in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B be a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) There is a symmetric monoidal functor, TB : Diff n → Vect, such that TB(Σ) = Q(s) B (Σ) = C � �π0(BΣ) � , and given f : Σ → B and f ′ : Σ′ → B, and a diffeomorphism, φ: Σ → Σ′, then the matrix elements satisfy � PCf(BΣ) | TB(φ: Σ → Σ′) | PCf ′(BΣ′) � = � 1, if PCf(BΣ) = PCf ′◦φ(BΣ), 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) If φ: Σ → Σ′ is a diffeomorphism, then Q(s) B (I′(φ)) = TB(φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The existence of TB, and that it can be upgraded to be a symmetric monoidal functor, follows from standard results from algebraic topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The crucial point is that the functor π0 : CGWH → Set is a symmetric monoidal functor, since, given CGWH spaces X and Y , we have a natural bijection, η′′ X,Y : π0(X) × π0(Y ) → π0(X ×Y ), such that, for x ∈ X and y ∈ Y , (PCx(X), PCy(Y )) �→ PC(x,y)(X ×Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, the natural isomorphism, η′′ : ×◦(π0×π0) =⇒ π0◦×, is associative, which means that the diagrams below commutes, for all CGWH spaces X, Y and A CATEGORIFICATION OF QUINN’S TQFT 66 Z, � π0(X) × π0(Y ) � × π0(Z) η′′ X,Y ⊗π0(Z) � αSet π0(X),π0(Y ),π0(Z) � π0(X) × � π0(Y ) × π0(Z) � π0(X)⊗η′′ Y,Z � � π0(X × Y ) � ⊗ π0(Z) η′′ X×Y,Z � π0(X) ⊗ � π0(Y × Z) � η′′ X,Y ×Z � π0 � (X × Y ) × Z) � π0 � αCGWH X,Y,Z � � π0 � X × (Y × Z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we consider the bijection, ǫ: {∗} → π0({∗}), we can see that, given a CGHW space X, the two diagrams pertaining to the unitality of η′′, commute, for instance, π0(X) × {∗} π0(X)×ǫ � ρSet π0(X) � � π0(X) π0(X) × π0({∗}) η′′ X,{∗} � π0(X × {∗}), π0(ρCGWH X ) � so the triple (TB, η′′, ǫ) is a (strong) monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that the diagram, π0(X) × π0(Y ) η′′ X,Y � τ Set π0(X),π0(Y ) � π0(Y ) × π0(X) η′′ Y,X � π0(X × Y ) π0(τ CGWH X,Y ) � π0(Y × X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' clearly commutes, (TB, η′′, ǫ) is also a symmetric monoidal functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The free vector space functor, Lin: Set → Vect, is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By us- ing the fact that CGWH is a monoidal closed category, the functor F0 B : (Diffn)op → HFiso (for notation see Lemma 84) is also symmetric monoidal, in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now note that TB is given by the following composition of functors: Diffn (−)−1 −−−−→ (Diffn)op F 0 B −−→ HFiso inc −−→ CGWH π0 −→ Set Lin −−→ Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second point of the lemma follows from the second part of Lemma 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Note that a part of the monoidal structure of TB is a natural isomorphism, ⊗ ◦ (TB × TB) =⇒ TB ◦ ⊔, which, by tracking the sequence of compositions above, explicitly is such that, given closed smooth n-manifolds, Σ and Σ′, and maps, f : Σ → B and f ′ : Σ′ → B, PCf(BΣ) ⊗ PCf ′(BΣ′) �→ PC⟨f,f ′⟩ � BΣ⊔Σ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A discussion of the monoidality of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let, once again, B be a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The functor, Q(s) B : Cob(n,n+1) → Vect, can be upgraded to be a symmetric monoidal functor, which we will also denote by Q(s) B : Cob(n,n+1) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 67 The key part of the construction is the natural isomorphism, η′ : ⊗ ◦ � Q(s) B × Q(s) B � =⇒ Q(s) B ◦ ⊔, of functors from Cob(n,n+1) × Cob(n,n+1) to Vect defined as the composite, ⊗ ◦ � Q(s) B × Q(s) B � = ⊗ ◦ � R(s) ◦ FB × R(s) ◦ FB) η◦(FB×FB) ========⇒ R(s) ◦ × ◦ (FB × FB) ∼ = =⇒ Q(s) B ◦ FB ◦ ⊔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here η is defined in Lemma 76, and the last natural isomorphism follows from the fact that CGWH is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, given compact smooth n-manifolds Σ and Σ′, η′ Σ,Σ′ : Q(s) B (Σ) ⊗ Q(s) B (Σ′) → Q(s) B (Σ ⊔ Σ′) is such that, given f : Σ → B and f ′ : Σ′ → B, PCf(BΣ) ⊗ PCf ′(BΣ′) �→ PC⟨f,f ′⟩ � BΣ⊔Σ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural isomorphism can easily be proved to be ‘associative’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' meaning that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given closed smooth n-manifolds Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Σ′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the following diagram commutes (where we omitted the labels in the associativity constraints in Vect),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � Q(s) B (Σ) ⊗ Q(s) B (Σ′) � ⊗ Q(s) B (Σ′′) η′ Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′⊗Q(s) B (Σ′′) � αVect � Q(s) B (Σ) ⊗ � Q(s) B (Σ′) ⊗ Q(s) B (Σ′′) � Q(s) B (Σ)⊗η′ Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′′ � � Q(s) B (Σ ⊔ Σ′) � ⊗ Q(s) B (Σ′′) η′ Σ⊔Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′′ � Q(s) B (Σ) ⊗ � Q(s) B (Σ′ ⊔ Σ′′) � η′ Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′⊔Σ′′ � Q(s) B � (Σ ⊔ Σ′) ⊔ Σ′′) � Q(s) B � αCob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1) Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Σ′′ � � Q(s) B � Σ ⊔ (Σ′ ⊔ Σ′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That this diagram commutes, follows from the fact that, using the definition of the monoidal structure of Cob(n,n+1) sketched in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, we have Q(s) B � αCob(n,n+1) Σ,Σ′,Σ′′ � = Q(s) B � I′� αDiff n Σ,Σ′,Σ′′ �� = TB � αDiff n Σ,Σ′,Σ′′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we have used the second point of Lemma 84 in the last step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have that the diagram above commutes, since the functor TB is monoidal, by the first point of Lemma 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that TB and Q(s) B coincide on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The remaining bits of the proof of the fact that Q(s) B : Cob(n,n+1) → Vect can be turned into a symmetric monoidal functor follow exactly the same pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some examples and properties of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The ground field for Quinn’s finite total homotopy TQFT can be taken to be Q, for s = 1 or s = 0, or the Galois closure of Q, for s = 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The case s = 0 was the one developed in [101], whilst the case s = 1/2 coincides with the conventions in [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that we have monoidal natural isomorphisms connecting all normalisations of Quinn’s finite total homotopy TQFT, obtained by applying Proposition 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again the remaining details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Methods for concrete calculations of Quinn’s TQFT will be addressed in Part 4, where B will be the classifying space of a homotopy finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There we will also discuss the methods of calculation in the extended version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the A CATEGORIFICATION OF QUINN’S TQFT 68 moment, we will restrict our attention to some simple examples and observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are essentially ‘well known’, and easy to calculate, but are included to lay the ground for the corresponding examples and results for the extended case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A trivial, but sometimes useful, example is from the case in which B = {∗}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', a singleton space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For any Σ, BΣ is also a singleton space, as there is a unique map from Σ to {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have π0(BΣ) is a singleton set, and Q(s) B (Σ) ∼= C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that (i1, S, i2): Σ1 → Σ2, then BS is also a singleton, FB(S) is the terminal span and, not surprisingly, the 1 × 1 matrix Q(s) B ([(i1, S, i2)] is just the identity 1 × 1 ‘matrix’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have that Q(s) B is the trivial TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This trivial example is related to a simple construction for when B is a product space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that B = B1 × B2, then BΣ ∼= BΣ 1 × BΣ 2 , so we have Q(s) B (Σ) ∼= Q(s) B1(Σ) ⊗ Q(s) B2(Σ), and similarly for the operation on the cobordisms40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fact that {∗} is the unit for the product, ×, is reflected here by the fact that, as C is the unit for the ⊗ in Vect, the operation of tensoring a general Quinn type TQFT with Q(s) ∗ will be trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Quite often, it is assumed that B is a path-connected space, but it is interesting to see what happens if it is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We start in a very simple case, namely that in which B = {0, 1}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', a two point discrete space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can think of a map Σ → B as ‘labelling’ or ‘colouring’ the path components of the space, Σ, by elements from B, so we may refer to a B-colouring of a space in more generality41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first look at Q(s) B (Σ) for Σ a connected manifold, then BΣ ∼= BΣ 0 ⊔ BΣ 1 , where B0 = {0}, and B1 = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, Q(s) B0(Σ) ∼= C, and the same holds for Q(s) B1(Σ), and so Q(s) B (Σ) ∼= Q(s) B0(Σ) ⊕ Q(s) B1(Σ) ∼= C ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, now, Σ is not connected, then we can write it as Σ ∼= Σ′ ⊔ Σ′′, with both parts non-empty, and, as Q(s) B is a monoidal functor, Q(s) B (Σ) ∼= Q(s) B (Σ′) ⊗ Q(s) B (Σ′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By induction on the size of π0(Σ), suppose Σ is a disjoint union of connected n-manifolds, ⊔k i=1Σi, then Q(s) B (Σ) ∼= (C ⊕ C)⊗k, the tensor product of k-copies of C2 = C ⊕ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Next, suppose that S is a cobordism, (i, S, j): Σ1 → Σ2, and that the (n + 1)- manifold, S, is connected, then BS = BS 0 ⊔BS 1 , and the linear map, Q(s) B (i, S, j), is simply Q(s) B0(i, S, j) ⊕ Q(s) B1(i, S, j), the direct sum of the corresponding linear maps, so giving a block decomposition of the corresponding matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If S is disconnected, then Q(s) B (i, S, j) will be the tensor product of the values on the connected components in the evident way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The argument used in this second example adapts well to handle the general case of a not necessarily path-connected B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To make this explicit, we recall the idea of direct sum as defined initially in [45] in the case of 2-D TQFTs, but later applied, in [105], to general TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will repeat the definition in the form used by Sawin in the above cited [105].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that Z1 and Z2 are two TQFTS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 40The link with the example is that the singleton space is the unit space to the (cartesian) monoidal structure on spaces, and C is the unit for the monoidal structure on Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 41Later on, in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, and in a more general situation, we will look at the case in which B = BA, the classifying space of a crossed complex, and then we will talk of A-colourings of (regular) CW-complexes and related structures rather than BA-colourings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 69 Definition 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The direct sum, Z1 ⊕ Z2, of Z1 and Z2 is the theory which: associates, to each connected Σ, the vector space Z1(Σ) ⊕ Z2(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' associates, to each disconnected Σ, the tensor product of the vector spaces associated to its components;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' associates, to each connected cobordism, (i, S, j): Σ1 → Σ2, the linear map, Z1(i, S, j)⊕Z2(i, S, j), interpreted as an operator on the appropriate vector spaces, and associates to each disconnected cobordism, the tensor product of the values on the components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On generalising the argument used in the above example, we clearly have: Proposition 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) If B = B1 ⊔ B2, then Q(s) B ∼= Q(s) B1 ⊕ Q(s) B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) In general, for B a HF space, Q(s) B has a direct sum decomposition as � x∈π0(B) Q(s) Bx, where Bx is the connected component labelled by x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Changing B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given these constructions and their properties, one might think that there was some possible functoriality of Q(s) B with B itself, but recall, for instance from [37, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5 and Appendix A2], that if ϕ : Z1 =⇒ Z2 is a (monoidal) natural transformation between TQFTs, then it is a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If f : B1 → B2 is a general continuous map, we therefore should not expect that there would be some sort of induced ‘morphism’ between Q(s) B1 and Q(s) B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can explore the failure of such na¨ıve functoriality with a simple example42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We put ourselves in Cob(1,2), and denote by Σi, i = 1, 2, two 1- manifolds with S : Σ1 → Σ2, a 2-manifold with boundary, given as a cobordism between the two 1-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For B1, we take {0, 1}, for B2, the singleton space, {∗}, and f : B1 → B2, the unique such map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The particular case that we will examine will have Σ1 = S1 a ⊔ S1 b , the disjoint union of two circles, where the suffixes are the labels by which we will refer to the components of Σ1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', π0(Σ1) = {a, b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly, we want Σ2 = S1 c and π0(Σ2) = {c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Taking the previous discussion in Examples 85 and 86, but applying it to our simple case here, we see that π0(BΣ1 1 ) consists of 4 elements, which we will denote ( a b 0 0 ), ( a b 1 1 ), ( a b 0 1 ) and ( a b 1 0 ), where, for instance, in the last of these, S1 a is sent to 1 in {0, 1}, whilst S1 b is sent to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly, π0(BΣ2 1 ) = {( c 0 ) , ( c 1 )}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next look at the ‘pair of trousers’ cobordism, T : Σ1 → Σ2, going from the two circles ‘near the ankles’ to the circle at the ‘waist’, and we note that BT 1 , also consists of just two points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since T is connected, any continuous map, β : T → B1, must either be constant with value 0, or constant with value 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We write BT 1 = {( t 0 ) , ( t 1 )}, and note that, in the induced fibrant span, BT 1 i∗ �❦❦❦❦❦❦ j∗ �❙ ❙ ❙ ❙ ❙ ❙ BΣ1 1 BΣ2 1 , whilst j∗ is an isomorphism, i∗ sends ( t 0 ) to ( a b 0 0 ), and ( t 1 ) to ( a b 1 1 ), so the fibres of i∗ over the other two points of BΣ1 1 are empty43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we now replace B1 by B2 in the above, then BΣ1 2 = {( a b ∗ ∗ )}, BΣ2 2 = {( c ∗ )}, and BT 2 = {( t ∗ )}, with the corresponding fibrant span being the terminal span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 42This example is very simple, but it shows what the problem is in general, so we will give the calculation in a lot of detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 43Recall that fibrations are not assumed to be surjective, so can have empty fibres as mentioned in Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 70 The map, f : B1 → B2, of course, sends anything with a 0 or 1 to the analogous one with ∗ in the same place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Turning to the corresponding vector spaces, the diagram, Q(s) B1(Σ1) f∗ � Q(s) B1(T ) � Q(s) B2(Σ1) Q(s) B2(T ) � Q(s) B1(Σ2) f∗ � Q(s) B2(Σ2) is, in fact, just C4 + � proj1,2 � C id � C2 + � C, where proj1,2(x1, x2, x3, x4) = (x1, x2), and the two maps labelled + just sum the coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This clearly does not commute!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The problem, here, arises with the colourings of the disconnected Σ1, which do not extend to colourings of the cobordism, T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have, in general, that there is no functoriality in the B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Under certain circumstances, however, a map, f : B1 → B2, does induce a natural transformation between Q(s) B1 and Q(s) B2, which as we noted must, then, be a natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If f : B1 → B2 is a homotopy equivalence, then f induces a monoidal natural isomorphism, f∗ : Q(s) B1 =⇒ Q(s) B2, between Q(s) B1 and Q(s) B2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural isomorphism is defined in the following way: if Σ is a closed n-manifold, then the linear map, (f∗)Σ : Q(s) B1(Σ) → Q(s) B2(Σ), is such that, given g : Σ → B1, then (f∗)Σ � PCg � BΣ 1 �� = PCf◦g(BΣ 2 � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', is post-composition with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We always have such a family of mappings, f∗ : Q(s) B1 → Q(s) B2, induced by post-composition with f, but, as we saw, in general, this need not define a natural transformation, due to possible incompatibility with the cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose (i, S, j) : Σ1 → Σ2 is a cobordism, thus giving us fibrant spans, BS i i∗ �❦❦❦❦❦❦ j∗ �❙ ❙ ❙ ❙ ❙ ❙ BΣ1 i BΣ2 i , for i = 1, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The function f : B1 → B2 gives us a commutative diagram, BΣ1 1 f Σ1 � BS 1 i∗ � f S � j∗ � BΣ2 1 f Σ2 � BΣ1 2 BS 2 i∗ � j∗ � BΣ2 2 , in which the vertical maps are all homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that this is not a morphism in the category HFspan, but does relate to a higher category structure A CATEGORIFICATION OF QUINN’S TQFT 71 on the class of HF spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The commutative diagram induces a map of fibrations, (18) BS 1 ⟨i∗,j∗⟩ � f S � BΣ1 1 × BΣ2 1 f Σ1×f Σ2 � BS 2 ⟨i∗,j∗⟩ � BΣ1 2 × BΣ2 2 , where the vertical arrows are homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have a diagram of vector spaces and linear maps, Q(s) B1(Σ1) (f∗)Σ1 � Q(s) B1 (S) � Q(s) B2(Σ1) Q(s) B2(S) � Q(s) B1(Σ2) (f∗)Σ2 � Q(s) B2(Σ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To check that this diagram commutes, in this context, we pick basis elements in Q(s) B1(Σ1) and Q(s) B1(Σ2), and compare the matrices corresponding to the left-hand side, with those on the right-hand side, with respect to the image basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First we note that �π0(BΣ 1 ) and �π0(BΣ 2 ) are related by the bijection induced from f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider arbitrary maps g : Σ1 → B1 and g′ : Σ2 → B1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To prove that the diagram commutes, it suffices to prove that � PCg(BΣ1 1 ) | Q(s) B1([(i, S, j)]) | PCg′(BΣ2 1 ) � = � PCf◦g(BΣ1 2 ) | Q(s) B2([(i, S, j)]) | PCf◦g′(BΣ2 2 ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Unpacking the notation, this amounts to comparing the corresponding fibres of the horizontal fibrations in diagram in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These fibres are homotopy equivalent by Corollary 66, applied to the map of fibrations given in (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that these isomorphisms respect the monoidal structure and also the composition, which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ This result is very useful as it, for instance, implies that any contractible B gives a TQFT isomorphic to the trivial one Q(s) {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That is easy to see directly of course, but equally well we can sometimes simplify the calculations of some Q(s) B , by replacing a given B by a homotopy equivalent one, for instance by using a presentation of its homotopy type that is smaller than that given initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More interestingly, perhaps, we note, from the proof, how the isomorphism f∗ depends only on the homotopy class of the homotopy equivalence f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They are also compatible with composition of such homotopy equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives: Theorem 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any HF-space, B, there is an action of the group, E(B), of homotopy classes of self homotopy equivalences of B on the TQFT Q(s) B , by natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We leave aside, for the moment, the application of this result within specific calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally in this section, we should mention the possibility of twisting the TQFT by a cohomology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 92 (Cohomology twisting of Quinn’s TQFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that if we restrict to oriented n-manifolds and oriented cobordisms Quinn’s TQFT, Q(s) B , can be also A CATEGORIFICATION OF QUINN’S TQFT 72 be twisted by a cohomology class in Hn(B, U(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again this is as explained in [43], for B the classifying space of a finite group, or, more generally, as done in [53], in the closed case, where B is the classifying space of a finite crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We hope to give some explicit formulae in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Part 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Once-extended versions of Quinn’s finite total homotopy TQFT Part 3 of this paper consists of two sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first, section 5, looks at the homotopy-theoretical underpinning of the once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second, section 6, gives the detailed construction of that extended TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout Part 3, we will work with an arbitrary, but fixed, subfield, κ, of the complex field44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again we let B be a homotopy finite space, and n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this part of the paper, we will see how the s = 0 case, QB : Cob(n,n+1) → Vect = Vectκ, of the Quinn finite total homotopy TQFT (abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the Quinn TQFT), given in Definition 83, can be ‘categorified’ to a once-extended Quinn TQFT, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, in Definition 149.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, as introduced in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8, and, in particular, in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the bicategory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' vProfGrphf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' has objects the homotopy finite groupoids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the 1-morphisms being Vect-valued profunctors between groupoids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and the 2- morphisms natural transformations of profunctors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2) is the bi- category with objects the closed (and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by convention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' smooth) n-manifolds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the 1-morphisms being the (n + 1)-cobordisms between closed n-manifolds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and the 2- morphisms the equivalence classes of (n+2)-cobordisms between (n+1)-cobordisms (by convention,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' with corners);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [106, 91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Σ be a closed and smooth n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Typically, the groupoid 2QB(Σ), despite being homotopy finite, is still uncountable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to reduced the size of the target groupoids, we will consider a bicategory, 2Cob (n,n+1,n+2) B , with ob- jects B-decorated, closed, and smooth n-manifolds, with the rest of the bicategory structure induced by that of 2Cob(n,n+1,n+2), as discussed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will then consider another once-extended TQFT, called the finitary once-extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Definition 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bicategory, vProfGrpfin, is the full sub-bicategory of vProfGrphf, with objects the finite groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This, in turn, gives rise to another once-extended TQFT, called the Morita valued once-extended Quinn TQFT, in Definition 168, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor, where Mor is the bicategory of κ-algebras, bimodules and bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebraic construction showing how to go from the bicategory vProfGrpfin to the bicategory Mor, starting from groupoid algebras, may be of independent interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is laid out in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Depending on which setting is chosen, the groupoids, or algebras that 2QB and 2Q Mor B assign to a closed manifold, Σ, with a B-decoration, explicitly depend on the B-decoration of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However this dependence is up to a canonically defined, 44For instance, we can take κ = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 73 and invertible, profunctor or bimodule, which is functorial with respect to further changes in the decoration (up to natural isomorphism), and natural with respect to the profunctors, or bimodules, assigned to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is discussed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 and §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5 We will show explicit examples of calculations of these once-extended TQFTs later on in Part 4, Section 8, for the case in which B is the classifying space of a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This includes the case of classifying spaces of finite 2-groups, as appear in higher gauge theory, see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [3, 6, 52, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we will further see in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, when n = 0, and for the case where B is the classifying space of a finite groupoid G, the extended Quinn TQFT gives a homotopy theoretical proof of the existence of the Morita valued once-extended (0,1,2)-TQFT, derived, elsewhere, from the fact that the groupoid algebra of G is a ‘separable symmetric Frobenius algebra’;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8] and [75, Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark: It is possible to define a one-parameter categorification of the Quinn TQFT QB, but we will not deal with that here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This would considerably increase the complexity of our formulae, without adding much more generality to our con- struction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly to our exposition of Quinn’s finite total homotopy TQFT, we will factor its once-extended version through a homotopy theoretical bicategorical object, later denoted 2span(HF), whose objects are HF spaces, 1-morphisms are fibrant HF spans, and 2-morphisms consist of HF-fibrant resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be done in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended versions of the Quinn finite total homotopy TQFT will then be treated in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, we will sketch the symmetric monoidal structure of the once-extended versions of Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy-theoretical underpinning of the once-extended Quinn TQFT 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation and some more basic results about fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will need some additional results and notation about fibrations, as defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Holonomy maps and the functor, FM : π1(B, B) → CGWH/ ≃, associated to a fibration p: M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 that given a CGWH space, B, we defined the maps, sB = s, tB = t: BI → B, such that s(γ) = γ(0) and t(γ) = γ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a fibration, ⟨s, t⟩: BI → B × B, from which we constructed the identities in HFspan;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Lemma 59 and Definition 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given γ ∈ BI, the reverse path to γ will be denoted γ, so γ(u) = γ(1 − u), for each u ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also consider the map, const: B → BI, sending x ∈ B to the constant path, constx, at x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that the fibre of x ∈ B is denoted Mx := p−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let M ×BBI denote the pullback of the maps, p: M → B and s: BI → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider also the canonical projections, proj1 : M ×B BI → M and proj2 : M ×B BI → BI, A CATEGORIFICATION OF QUINN’S TQFT 74 so we have a pullback diagram, M ×B BI proj1 �✉✉✉✉✉✉✉✉✉✉ proj2 �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ M p �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ BI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' sB �sssssssssss B We have a continuous function, λM : I × (M ×B BI) → M, arising from the diagram below and the homotopy lifting property of p: M → B, M ×B BI proj1 � {0}×( ) � M p � I × (M ×B BI) λM �❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ ❢ idI×proj2 � I × BI (u,γ)�→γ(u) � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here � {0} × ( ) � (m, γ) = (0, m, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Definition 93 (Holonomy Map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A function, λM, making the diagram above com- mute will be called a holonomy map on the fibration p: M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The nomenclature “holonomy map” is borrowed from differential geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will frequently write “holonomy” rather than “holonomy map”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The holonomy maps considered here are equivalent to the “path-lifting functions” in [87, Chapter 7], and the “lifting functions” in [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following string of classical results are to be found, essentially, in [48] or [87, Chapter 7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They follow from simple application of the appropriate homotopy lifting property, and are ‘well known’, but we give a reference for each one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a fixed holonomy map, λM, on p: M → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let γ be a path, in B, from x ∈ B to y ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the map ΓM γ : Mx → My, defined by ΓM γ (m) := λM(1, m, γ), for all m ∈ Mx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Up to homotopy of maps from Mx to My, the map, ΓM γ : Mx → My, then, depends only on the homotopy class of γ (and, in particular, not on the chosen holonomy map, λM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, ΓM γ : Mx → My is a homotopy equivalence between the fibres Mx and My, with a homotopy inverse to ΓM γ : Mx → My given by ΓM γ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' See [87, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6 (Change of fiber)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let φ be a path in M, and γ = p ◦ φ, its image path in B, then ΓM γ (φ(0)) and φ(1) are in the same path-component of Mγ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, if M is path-connected and x, y ∈ B, then if A ∈ �π0(Mx) and A′ ∈ �π0(My) are path components of the chosen fibres, it follows that A and A′ are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Concretely, choose m ∈ A and m′ ∈ A′, and a path, φ, in M connecting m to m′, then ΓM γ : Mx → My restricts to a map, A → A′, giving the desired homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here γ = p ◦ φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' See [48, page 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A CATEGORIFICATION OF QUINN’S TQFT 75 Suppose that we have paths, x γ−→ y and y γ′ −→ z, then ΓM γγ′ is homotopic to ΓM γ′ ◦ ΓM γ , as maps from Mx to Mz;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [87, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, given x ∈ B, the map, ΓM constx : Mx → Mx, is homotopic to the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that CGWH/ ≃ denotes the category with objects the CGHW spaces, with morphisms being homotopy classes of maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a functor, FM : π1(B, B) → CGWH/ ≃, (where π1(B, B) is the fundamental groupoid of B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ B, FM(x) := Mx, and given a path, x γ−→ y, in B, then, FM(x [γ] −→ y) := [ΓM γ ]: Mx → My.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here [ΓM γ ] is the homotopy class of ΓM γ : Mx → My.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This functor, FM, depends only on the fibration, p: M → B, and not on the chosen holonomy map, λM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' See [87, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We, thus, have a functor � π0 ◦ FM : π1(B, B) → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It sends x ∈ B to �π0(Mx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (For notation see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Given a path, x γ−→ y, in B, the functor is such that, if m ∈ Mx, � (�π0 ◦ FM)(x [γ] −→ y) � (PCm(Mx)) = PCλM(1,m,γ)(My) = PCΓM γ (m)(My), where we recall that PCm(Mx) denotes the path-component of m in Mx;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Sub- section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This functor depends only on the fibration, p: M → B, and not on the chosen holonomy map, λM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally in this string of lemmas, we have Lemma 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are left and right actions of π1(B, x) on �π0(Mx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are such that, if m ∈ Mx, γ ∈ Ωx, the loop space of B based at x, and [γ] is the associated element of π1(B, x), then: [γ]⊲PCm(Mx) = PCΓM γ (m)(Mx), and PCm(Mx) ⊳ [γ] = PCΓM γ (m)(Mx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The following result will be needed when addressing why the once-extended Quinn TQFT can be given the structure of a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: E → X and p′ : E′ → X′ be fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have a fibration, (p × p′): E × E′ → X × X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The functor, FE×E′ : π1(X × X′, X × X′) → CGWH/ ≃, provided by (p × p′): E × E′ → X × X′, is given by the composition of the functors below, π1(X × X′, X × X′) ∼= π1(X, X) × π1(X′, X′) F E×F E′ −−−−−−→ (CGWH/ ≃) × (CGWH/ ≃) ×CGWH −−−−−−→ CGWH/ ≃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here ×CGWH is the product monoidal structure on CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will revisit this in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 76 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On objects, this follows from the fact that, if x ∈ X and x′ ∈ X′, then (p×p′)−1(x, x′) = p−1(x)×p′−1(x′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On morphisms, this follows from the fact that a holonomy map for the fibration, (p × p′): E × E′ → X × X′, can be obtained by doing the “product” of those of p: E → X and p: E′ → X′, namely, I×(E×X XI)×(E′×X′X′I) ∋ (t, e, γ, e′, γ′) �→ � λE(t, e, γ), λE′(t, e′, γ′) � ∈ E×E′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Definition 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose a subset, xB, of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The functor, FM xB : π1(B, xB) → CGWH/ ≃, is defined by restricting the functor, FM : π1(B, B) → CGWH/ ≃, to π1(B, xB), the full subgroupoid of π1(B, B), with set of objects xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have, thus, also defined a functor, �π0 ◦ FM xB : π1(B, xB) → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter functor, �π0 ◦ FM xB : π1(B, xB) → Set, is one of the basic ingredients for the reduction of the once-extended Quinn TQFT to give something more amenable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set, xB, is a choice of a set of base-points, typically, but not exclusively, for the components of B, or, at the other extreme, we could take xB to be the set of all the elements of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can choose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we pick xB finite, this can be used to reduce the groupoids, π1(B, B), and their action (interpreted as a functor) to the more classical setting of a set of fundamental groups and their action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This allows one to reduce a Vect-enriched category to something nearer to a finite group algebra together with categories of bimodules over them, in fact to the Morita bicategory that we will be recalling later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will need such many pointed extensions of quite a few otherwise classical results, which are not that easy to find given in an explicit form in the literature, and so will give them in a bit of detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that, Definition 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A pair, (X, xX), of topological spaces is said to be 0-connected if the set, xX, has at least one point in each path-component of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 102.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose a subset, xB, of B such that (B, xB) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a natural bijection, F : colim(�π0 ◦ FM xB) = � � x∈xB �π0(Mx) � � ∼ → �π0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ xB and m ∈ Mx, this sends the equivalence class of PCm(Mx) to PCm(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, the map, F ′ : � x∈xB �π0(Mx) → �π0(M), such that, if x ∈ xB and m ∈ Mx, then PCm(Mx) �→ PCm(M), descends to colim(�π0 ◦ FM xB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us explain this a bit more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x [γ] −→ y in π1(B, xB), if m ∈ Mx and n ∈ Ny, and we have that � �π0 ◦ FM xB([γ]) � (PCm(Mx)) = PCn(My), then this means that λM(1, m, γ) is in PCn(My).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From this, it follows that m and n are in the same path-component in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that the 0-connectedness of (B, xB) was not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A CATEGORIFICATION OF QUINN’S TQFT 77 Now to prove that F is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that, given x, x′ ∈ xB, and m ∈ Mx and m′ ∈ Mx′, we have PCm(M) = PCm′(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose any path, φ, in M, starting in m and ending in m′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let γ = p ◦ φ, then, by using Lemma 95, it follows that PCΓM γ (m)(M) = PCm′(M), so � (�π0 ◦ FM xB)(x [γ] −→ y) � (PCm(Mx)) = PCm′(My).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Again note that the 0-connectedness of (B, xB) was not used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') That F is surjective follows analogously, but here we use the fact that (B, xB) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we are given m ∈ M, there is a path, γ, in B connecting p(m) ∈ B to some x ∈ xB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Put m′ = ΓM γ (m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We then have F([PCm′(Mx)]) = PCm(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The path components of pullbacks along fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B and q: N → B be fibrations and consider the pullback diagram in CGWH given by the diamond in the diagram below, where we put P = q ◦ proj2 = p ◦ proj1, (19) M×BN proj2 �❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ proj1 �✈✈✈✈✈✈✈✈✈ P � M p �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' q �✉✉✉✉✉✉✉✉✉✉ B It is clear, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', by the universal property of pullbacks, that P is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let λM and λN be holonomy maps for the fibrations, p: M → B and q: N → B, then a holonomy map, λM×BN, for the fibration, P : M ×B N → B, can be given such that, if we have a path x γ−→ y in B, and a point, (m, n) ∈ Mx×Nx ⊂ M ×B N, (20) λM×BN(t, m, n, γ) = � λM(t, m, γ), λN(t, n, γ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given b ∈ B, recall that we put Mb = p−1(b), and Nb = q−1(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fi- bre, P −1(b), of P at b, is homeomorphic to Mb × Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a bijection, from �π0(Mb) × �π0(Nb) to �π0(P −1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bijection sends � PCm(Mb), PCn(Nb) � to PC(m,n)(P −1(b)), where m ∈ Mb and n ∈ Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have functors, FM, FN : π1(B, B) → CGWH/ ≃, given by the fibrations p: M → B and q: N → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, and using (20), the functor, FM×BN : π1(B, B) → CGWH/ ≃, given by the fibration, P : M ×B N → B, is naturally isomorphic to the composition of the functors below, where � FM, FN� is given by the universal property of a product, π1(B, B) ⟨F M,F N⟩ −−−−−−→ (CGWH/ ≃) × (CGWH/ ≃) ×CGWH −−−−−−→ CGWH/ ≃, and where ×CGWH : CGWH × CGWH → CGWH denotes the product functor in CGWH, which descends to a functor, also denoted ×CGWH : (CGWH/ ≃) × (CGWH/ ≃) → CGWH/ ≃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Explicitly, ×CGWH sends a pair, (X, Y ), of CGWH spaces to their product, X×Y , and analogously for maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The functor, �π0 ◦ FM×BN : π1(B, B) → Set, is, thus, naturally isomorphic to ×Set ◦ � �π0 ◦ FM, �π0 ◦ FN� : π1(B, B) → Set, where ×Set : Set × Set → Set is the product functor in Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given xB ⊂ B, it also follows that the functor, �π0 ◦ FM×BN xB : π1(B, xB) → Set, A CATEGORIFICATION OF QUINN’S TQFT 78 is naturally isomorphic to ×Set ◦ � �π0 ◦ FM xB, �π0 ◦ FN xB � : π1(B, xB) → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that the pair (B, xB) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a bijection, colim � ×Set ◦ � �π0 ◦ FM xB, �π0 ◦ FN xB � � −→ �π0(M ×B N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bijection is such that, if x ∈ xB, m ∈ Mx and n ∈ Nx, then, [(PCm(Mx), PCn(Nx))] �−→ PC(m,n)(M ×B N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemma 102 combined with the previous discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Any groupoid, G, comes with a contravariant functor ( )−1 : G → G, that is the identity on objects and sends each morphism to its inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we have a functor, ×Set ◦ � �π0 ◦ FM xB ◦ ( )−1 × �π0 ◦ FN xB � : π1(B, xB)op × π1(B, xB) → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives us the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that a generalisation of this lemma, written in the context of ∞-groupoids, is in [56, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Lemma 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: M → B and q: N → B be fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose xB ⊂ B such that the pair (B, xB) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a bijection, ˆ x∈xB � (�π0 ◦ FM) ◦ ( )−1(x) � × � (�π0 ◦ FN)(x) � → �π0(M ×B N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Noting that, ˆ x∈xB � (�π0 ◦ FM) ◦ ( )−1(x) � × � (�π0 ◦ FN)(x) � = � � x∈xB (�π0 ◦ FM)(x) × (�π0 ◦ FN)(x) � � ∼, given x ∈ xB, the bijection sends the equivalence class of (PCm(Mx), PCn(Nx)) to PC(m,n)(M ×B N), where m ∈ Mx and n ∈ Nx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the previous lemma, since any arrow in π1(B, xB) is in- vertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that, here, Lemma 36, in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, is useful to translate between languages, that of coends and the more classical form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of path-components of pullbacks along fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This lemma will be implicitly used below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : E → X be a fibration, with E ̸= ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let e ∈ E and x = f(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) The induced map, fe : PCe(E) → PCx(X), is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) If k ∈ f −1 e (x), then PCk(f −1 e (x)) = PCk(f −1(x)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first point is immediate from the homotopy lifting property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the second point, note that clearly PCk(f −1 e (x)) ⊆ PCk(f −1(x)) as sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The reverse inclusion also holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is because a path in f −1(x), starting in k ∈ f −1 e (x), cannot leave PCe(E), so it is a path in f −1 e (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That the topologies in the path components coincide follows from item (8) on page 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Consider two fibrations, p: M → B, and q: N → B, and the resulting fibration, P : M ×B N → B, of diagram (19), and suppose that M, N and B are homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The space, M ×B N, is homotopy finite, so P : M ×B N → B is a fibration of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 79 This lemma is a particular case of Lemma 55, and of [56, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8], but here is a direct proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre of the fibration, P : M ×B N → B, at b ∈ B, is homeomorphic to Mb × Nb, a product of homotopy finite spaces (by Lemma 40), so each fibre of P is homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since B is homotopy finite, it follows that the total space of P : M ×B N → B is homotopy finite, again by Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Let b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Lemma 97, we have a right action of π1(B, b) on �π0(Mb)×�π0(Nb) ∼= �π0(P −1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This action is such that, if γ : I → B connects b to b, then, given m′ ∈ Mb and n′ ∈ Nb, so (m′, n′) ∈ M ×B N, we have, � PCm′(B) × PCn′(B) � ⊳ [γ] = � PCΓM γ (m′)(Mb), PCΓN γ (n′)(Nb) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that, given the form of the induced holonomy map, (20), for P : M ×B N → B, it follows that � ΓM γ (m′), ΓN γ (n′) � is in the same path component of (m′, n′) in M ×B N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now fix b ∈ B, and elements, m ∈ Mb, n ∈ Nb, in the fibres of the two fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibration, P : M ×B N → B, restricts to a map, P(m,n) : PC(m,n)(M ×B N) → PCb(B), which is a fibration by Lemma 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Assuming that M, N and B are homotopy finite, then P(m,n) : PC(m,n)(M ×B N) → PCb(B) is a fibration of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the fibre of P(m,n) : PC(m,n)(M ×B N) → PCb(B) has only a finite number of path-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Until the end of this subsection45, consider two fibrations, p: M → B and q: N → B, of homotopy finite spaces, and also the induced fibration, P : M ×BN → B, therefore, again of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 107 (T M×BN (m,n) ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B, and also m ∈ Mb, n ∈ Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We write T M×BN (m,n) for the number of path-components of the fibre of P(m,n) : PC(m,n)(M ×B N) → PCb(B) at b ∈ B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Almost by definition it follows that: Lemma 108.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B, m ∈ Mb and n ∈ Nb, then T M×BN (m,n) equals the cardinality of the orbit of PC(m,n)(P −1(b)) under the right-action of π1(B, b) on �π0(P −1(b)), derived from the fibration P : M ×B N → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We prove that �π0(P −1 (m,n)(b)) ⊂ �π0(P −1(b)) coincides with the π1(B, b)-orbit, of PC(m,n)(P −1(b)), inside �π0(P −1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let m′ ∈ Mb and n′ ∈ Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If PC(m′,n′)(P −1(b)) ∈ �π0(P −1 (m,n)(b)), then, in particular, (m′, n′) ∈ Mb × Nb ⊂ M ×B N is in the same path-component as (m, n) in M ×B N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose a path, φ, in M ×B N connecting (m, n) and (m′, n′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying Lemma 95, it follows that PC(m,n)(P −1(b))⊳ [p(φ))] = PC(m′,n′)(P −1(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rest follows by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Lemma 109.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let b ∈ B, m ∈ Mb, and n ∈ Nb, then χπ� PC(m,n)(M ×B N) � = T M×BN (m,n) χπ(PCb(B)) χπ(PCm(Mb)) χπ(PCn(Nb)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 45We will not always repeat this for each result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 80 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The decomposition of P −1(b) ∼= Mb × Nb, into path-components, gives a weak homotopy equivalence, � (A,A′)∈�π0(Mb)×�π0(Nb) A × A′ → Mb × Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each path component of P −1 (m,n) is also a path-component of P −1(b) ∼=Mb×Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using Lemma 105.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A priori, however, there may be fewer path-components in P −1 (m,n)(b) than there are in Mb × Nb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we use Lemma 46, applied to the HF fibration, P(m,n) : PC(m,n)(M ×B N) → PCb(B), we obtain χπ� PC(m,n)(M ×B N) � = χπ(PCb(B)) χπ� P −1 (m,n)(b) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now, by Lemma 95, all path-components of P −1 (m,n)(b) are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The result follows from the fact that, by definition, we have T M×BN (m,n) such path components, each of which is homotopic to PC(m,n) � P −1 (m,n)(b) � = PC(m,n) � P −1(b) � ∼= PC(m,n)(Mb × Nb) ∼= PCm(Mb) × PCn(Nb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This, of course, uses that the homotopy content of a product of HF spaces is the product of the homotopy contents, and that homotopy content is additive with respect to disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the results of the previous sections, we will show that each fibrant span gives us a profunctor with Vect-values, and that the composition of fibrant spans in Definition 56 translates under this to composition of profunctors, as described in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that our results are related to those of [56, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Cardinality as a functor], which were written in the language of ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor associated to a fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a fibrant span from X to Y , then we have a diagram in CGWH of form, M p �❥❥❥❥❥❥ p′ �❚ ❚ ❚ ❚ ❚ ❚ X Y, where the induced map, � p, p′� : M → X × Y , is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ X and y ∈ Y , recall, from Definition 67, that we defined the spatial slice at x and y as being {x|(p, M, p′)|y} := ⟨p, p′⟩−1(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Also recall that we will frequently abbreviate {x|(p, M, p′)|y} to {x|M|y}, when no ambiguity arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a holonomy map, λM, for the fibration, � p, p′� : M → X × Y , of form, λM : I × (M ×X×Y (X × Y )I) → M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using the notation of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Let x, x′ ∈ X and y, y′ ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given paths, x γX −−→ x′ in X, and y γY −−→ y′ in Y , the holonomy map, λM, induces a homotopy equivalence, ΓM ⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′}, m �→ λM(1, m, γX, γY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 81 Here ⟨γX, γY ⟩ is the path in X × Y such that I ∋ u �→ � γX(u), γY (u) � ∈ X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy class of ΓM ⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′} depends only on the fibration, ⟨p, p′⟩: M → X × Y , and not on the chosen holonomy map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This yields, by Lemma 96, a functor, FM : π1 � X × Y, X × Y )) ∼= π1(X, X) × π1(Y, Y ) → CGWH/ ≃, where FM(x, y) := {x|M|y} and FM� x [γX] −−−→ x′, y [γY ] −−−→ y′� := [ΓM (γX,γY )].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For convenience, we will repeat our conventions, from Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8, for pro- functors between groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fix a subfield, κ, of the complex field, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that Vectk = Vect denotes the category of κ-vector spaces and linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 110 (Set-profunctor and Vect-profunctor).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G and G′ be groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A Set-profunctor, H : G ↛ G′, is a functor, H : Gop × G′ → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A Vect-profunctor, or Vect-valued profunctor, H: G ↛ G′, is a functor, H: Gop × G′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The free vector space functor, from Set to Vect, is denoted Lin = Linκ : Set → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each Set-profunctor, H : Gop × G′ → Set, gives rise to a Vect-valued profunctor, H := Lin ◦ H : Gop × G′ → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that Lin preserves colimits, this operation preserves the composition of profunctors (in Set and in Vect, see Equation (3)), up to canonical natural isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 111 (The Vect-profunctor associated to a fibrant span).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a fibrant span, of GCWH spaces X (p,M,p′) −−−−−→ Y, that is, � M p �✐✐✐✐✐✐ p′ �❯ ❯ ❯ ❯ ❯ ❯ X Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its associated Vect-profunctor, denoted, H(X (p,M,p′) −−−−−→ Y ): π1(X, X) ↛ π1(Y, Y ), which we will frequently abbreviate to HM : π1(X, X) ↛ π1(Y, Y ), is, by definition, HM = Lin ◦ �π0 ◦ FM ◦ � ( )−1 × id � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Taking this apart, given x ∈ X and y ∈ Y , HM(x, y) is the free vector space over �π0({x|M|y}), the set of path-components of the fibre of ⟨p, p′⟩: M → X × Y , at (x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given morphisms, in π1(X, X) and π1(Y, Y ), say x [γX] −−−→ x′ and y [γY ] −−−→ y′, the linear map, HM� [γX], [γY ] � : HM(x′, y) → HM(x, y′), is induced by the homotopy equivalence, between fibres, ΓM ⟨γX,γY ⟩ : {x′|M|y} → {x|M|y′}, by applying �π0 : CGWH → Set, and then Lin: Set → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here ⟨γX, γY ⟩ is the path in X × Y , such that I ∋ u �→ � γX(1 − u), γY (u) � ∈ X × Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 82 The following result will implicitly be used a number of times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that X (p,M,p′) −−−−−→ Y is homotopy finite, so that X, Y and M are homotopy finite spacesm then the profunctor, HM : π1(X, X) ↛ π1(Y, Y ), is a 1-morphism in the bicategory, vProfGrphf, of homotopy finite groupoids and Vect-profunctors between them46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ X and y ∈ Y , the vector space, HM(x, y), is finite dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since X and Y are homotopy finite, it follows that the groupoids, π1(X, X) and π1(Y, Y ), each are homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ X and y ∈ Y , then by Lemma 40, the fibre {x|M|y} = ⟨p, p′⟩−1(x, y) is homotopy finite, and thus it only has a finite number of path-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Notation 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, choose subsets, xX and yY , of X and Y , respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The restriction, of HM : π1(X, X)op × π1(Y, Y ) → Vect, to π1(X, xX) × π1(Y, yY ), is a Vect-profunctor, π1(X, xX) ↛ π1(Y, yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use three different notations for it: H(xX,yY )(p, M, p′): π1(X, xX)op × π1(Y, yY ) → Vect, H(xX,yY ) � X (p,M,p′) −−−−−→ Y � : π1(X, xX)op × π1(Y, yY ) → Vect, and finally, H M (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, depending on the context and the amount of detail needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will be mainly interested in the case when X (p,M,p′) −−−−−→ Y is homotopy fi- nite, and furthermore both sets, xX and yY , are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, we therefore have that H M (xX,yY ) : π1(X, xX) ↛ π1(Y, yY ) is a 1-morphism in the bicategory vProfGrpfin, of finite groupoids and Vect-profunctors between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal-like structure of H M (−,−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following result will be used when proving that the constructions of once-extended Quinn TQFTs, given here, do indeed give bifunctors, which are symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider two fibrant spans of homotopy finite spaces, (p, M, q): X → Y and (p′, M ′, q′): X′ → Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX ⊂ X, x′ X′ ⊂ X′, yY ⊂ Y and y′ Y ′ ⊂ Y ′, and form the product HF fibrant span, (p × p′, M × M ′, q × q′): X × X′ → Y × Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a natural isomorphism from the profunctor, H(xX×x′ X′ ,yY ×y′ Y ′ ) � p × p′, M × M ′, q × q′� : π1(X × X′, xX × x′ X′)op × π1(Y × Y ′, yY × y′ Y ′) → Vect, 46See §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5 A CATEGORIFICATION OF QUINN’S TQFT 83 to the profunctor obtained from the following composition of functors, π1(X × X′, xX × x′ X′)op × π1(Y × Y ′, yY × y′ Y ′) ∼ = −→ � π1(X, xX)op × π1(Y, yY ) � × � π1(X′, x′ X′)op × π1(Y, y′ Y ′) � H(xX ,yY )(p,M,q)×H(x′ X′ ,y′ Y ′ )(p′,M′,q′) −−−−−−−−−−−−−−−−−−−−−−−−−−→ Vect × Vect ⊗Vect −−−−→ Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ xX, x′ ∈ x′ X′, y ∈ yY and y′ ∈ y′ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural isomorphism is such that, if m ∈ {x|M|y} and m′ ∈ {x′|M ′|y′}, then PCm � {x|M|y} � ⊗ PCm′� {x′|M ′|y′} � ←→PC(m,m′) � {(x, x′)|M × M ′|(y, y′)} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemma 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In order to prove that the once-extended Quinn TQFT, in its various forms, gives a symmetric monoidal bifunctor, it is convenient to change slightly the language of the previous result, approximating that of Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This point will be made concrete later, in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Consider the canonical natural isomorphisms, of groupoids, m(X,X′) : π1(X, xX) × π1(X′, x′ X′) → π1(X × X′, xX × x′ X′), m(Y,Y ′) : π1(Y, yY ) × π1(Y ′, y′ Y ′) → π1(Y × Y ′, yY × y′ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using the notation of Example 33, they yield profunctors, ϕm(X,X′) : π1(X, xX) × π1(X′, x′ X′) ↛ π1(X × X′, xX × x′ X′), ϕm(Y,Y ′) : π1(Y, yY ) × π1(Y ′, y′ Y ′) ↛ π1(Y × Y ′, yY × y′ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Continuing the notation of Lemma 114, we have Lemma 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a diagram of Vect-profunctors, and a natural isomorphism, χ(M,M′), or in full χ� (p,M,q),(p′,M′,q′) �, of Vect-profunctors, as shown below, π1(X, xX) × π1(X′, x′ X′) H(xX ,yY )(p,M,q)⊗H(x′ X′ ,y′ Y ′ )(p′,M′,q′) � ϕ m(X,X′) � ✚✚✚✚� χ(M,M′) π1(Y, yY ) × π1(Y ′, y′ Y ′) ϕ m(Y,Y ′) � π1(X × X′, xX × x′ X′) H(xX ×x′ X′ ,yY ×y′ Y ′ )(p×p′,M×M′,q×q′) � π1(Y × Y ′, yY × y′ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ xX, x′ ∈ x′ X′, (y1 γ−→ y2) ∈ π1(Y, yY ) and (y′ 1 γ′ −→ y′ 2) ∈ π1(Y, yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural isomorphism is such that, referring to the notation in Equation (3), if m ∈ {x|M|y1} and m′ ∈ {x′|M ′|y′ 1}, then the equivalence class of � PCm � {x|M|y1} � ⊗ PCm′� {x′|M ′|y′ 1} �� ⊗ m(Y,Y ′)(γ, γ′), is sent to the equivalence class of id(x,x′)⊗ H M×M′ (xX×x′ X′ ,yY ×y′ Y ′ ) � id(x,x′), m(Y,Y ′)(γ, γ′) �� PC(m,m′) � {(x, x′)|M ×M ′|(y2, y′ 2)} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the previous lemma, and elementary properties of pro- functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A CATEGORIFICATION OF QUINN’S TQFT 84 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Is H a functor, or perhaps a bifunctor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now we know how to construct profunctors and Vect-profunctors from fibrant spans, we should ask how that con- struction behaves with respect to composition of fibrant spans, and also what does it do to identity spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will examine preservation of composition here, whilst preservation of identities will be discussed later, being a consequence of Lemma 145, as it is more convenient to package it with similar results later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider HF fibrant spans, (p1, M1, p′ 1): X → Y and (p2, M2, p′ 2): Y → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall the definition of the composition47, (p1, M1, p′ 1) • (p2, M2, p′ 2) = (p1, M1 ×Y M2, p′ 2): X → Z, which is, itself, a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To recall and extend notation from earlier, we repeat the relevant commutative diagram, in Equation (11), M1×Y M2 P � �❧❧❧❧❧❧❧ �❘ ❘ ❘ ❘ ❘ ❘ ❘ p1 � p′ 2 � M1 p1 �rrrrrr p′ 1 �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ M2 p2 �❧❧❧❧❧❧❧❧❧ p′ 2 �▲ ▲ ▲ ▲ ▲ ▲ X Y Z, in which the middle diamond is a pullback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall, from Lemma 55, that ⟨p1, P, p′ 2⟩: M1 ×Y M2 → X × Y × Z is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given holonomies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for ⟨p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′ 1⟩: M1 → X × Y and ⟨p2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′ 2⟩: M2 → Y × Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' denoted48,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λM1 : I × � M1 ×X×Y (XI × Y I) � → M1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and λM2 : I × � M2 ×Y ×Z (Y I × ZI) � → M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (respectively),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then a holonomy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λM1×Y M2 : I × � (M1 ×Y M2) ×X×Y ×Z (XI × Y I × ZI) � → M1 ×Y M2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for ⟨p1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′ 2⟩: M1 ×Y M2 → X × Y × Z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is obtained from the holonomies,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λM1 and λM2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the obvious way,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' namely,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' m2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (γX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' γY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' γZ) � �→ � λM1� t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' m1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (γX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' γY ) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λM2� t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' m2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (γY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' γZ) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following result shows that the profunctor construction is compatible with composition of fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose subsets, xX, yY and zZ, of X, Y and Z, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that (Y, yY ) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a canonical isomorphism of Vect- profunctors from π1(X, xX) to π1(Z, zZ), ηM1,M2 (xX,yY ,zZ) : H M1 (xX,yY ) • H M2 (yY ,zZ) = ˆ y∈yY H M1 (xX,yY )(−, y) ⊗ H M2 (yY ,zZ)(y, −) =⇒ H M1×Y M2 (xX,zZ) , 47see Lemma 55 and Definition 56, 48Note that (X × Y )I ∼ = XI × Y I and (Y × Z)I ∼ = Y I × ZI, canonically, since CGWH is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 85 such that, if x ∈ xX and z ∈ zZ, then, given any y ∈ yY , m1 ∈ {x|M1|y} and m2 ∈ {y|M2|z}, we have that the linear map49, (ηM1,M2 (xX,yY ,zZ))(x,z) : ˆ y∈yY H M1 (xX,yY )(x, y) ⊗ H M2 (yY ,zZ)(y, z)→H M1×Y M2 (xX,zZ) (x, z), sends the equivalence class of PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ∈ � y∈yY H M1 (xX,yY )(x, y) ⊗ H M2 (yY ,zZ)(y, z) to PC(m1,m2)({x|M1 ×Y M2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows by combining the previous discussion with Lemma 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here it may help to note the comments at the beginning of the proof of Lemma 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Note that, in a situation in which we have two pairs of composable fibrant spans, say, (p1, M1, p′ 1): X → Y and (p2, M2, p′ 2): Y → Z, and also, ( ˆp1, ˆ M1, ˆp′ 1): ˆX → ˆY and ( ˆp2, ˆ M2, ˆp′ 2): ˆY → ˆZ, then the natural isomorphisms in Lemma 116 are compatible with those of Lemma 115.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We leave it to the reader to write down the corresponding commutative dia- gram of natural transformations between profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We still have to handle what H does to (resolved) identities, for which see Lemma 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Even with the preservation of identities, we have only been addressing the prop- erties of the ‘category’ of fibrant spans, but the category of fibrant spans, say of HF spaces, should have some more bicategorical aspect, and without that in evidence the full question asked in the title of this section cannot be answered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF fibrant resolved 2-spans connecting fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' So far we have, in the main, been handling only the 1-categorical structure related to fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now introduce ‘spans between spans’, that is ‘2-spans’, or, as we will call them ‘windows’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will later be investigating how this second level of structure on the spatial side is reflected, via the profunctor construction, in the ‘linear algebra’, and, of course, this is the beginning of the extension of Quinn’s theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF fibrant windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 117 (Window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By a window, W, we will mean a diagram of, as usual, CGWH spaces of the form below, (21) W = X M p � p′ � Y Z Pl � Ql � L l � r � P � Q � W Pr � Qr � X′ N q � q′ � Y ′, so it is a ‘span of spans’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 49We are using the notation in Equation (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 86 The boundary, bd(W), of the above window, is the following diagram, (22) bd(W) = X M p � p′ � Y Z Pl � Ql � W Pr � Qr � X′ N q � q′ � Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By the frame, fr(W), of the window, W, above, we will mean the following limit, (23) fr(W) = M × X×Y (Z × W) × X′×Y ′ N ∼= lim(bd(W)), and the filler, PL, of the window, W, is given by the naturally defined map, (24) PL : L → fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The restrictions of the diagram, (21), to each of its four boundary spans, are called the top, bottom, left and right boundary spans of W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Respectively, these are: (25) M p �⑤⑤⑤⑤ p′ �❇❇❇❇ X Y, N q �⑤⑤⑤⑤ q′ �❈❈❈❈ X′ Y ′, Z Pl�⑧⑧⑧⑧ Ql�❇❇❇❇ X X′, W Pr �⑤⑤⑤⑤ Qr�❊❊❊❊ Y Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are also the middle horizontal and middle vertical spans of the window W, defined to be L P�③③③③ Q �❈❈❈❈ M N and L l �⑤⑤⑤⑤ r�❉❉❉❉ Z W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, we will see windows as being “oriented” from top to bottom and from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 118 (HF fibrant window).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A fibrant window is a window, W, as in (21), such that: (1) the filler, PL : L → fr(W), is a fibration, and, (2) the four boundary spans (top, bottom, left and right) of W are all fibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, in addition, all the spaces appearing in diagram (21) are HF, then the window, W, will be called a HF fibrant window.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following are some immediate consequences of the definition of HF fibrant windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Suppose that a fibrant window, W, as in (21), is HF, then its frame fr(W) = M × X×Y (Z × W) × X′×Y ′ N is a HF space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows by applying Lemma 41 to the pullback diagram, (26) M × X×Y (Z × W) × X′×Y ′ N � � M × N ⟨p,p′⟩×⟨q,q′⟩ � Z × W τ◦(⟨Pl,Ql⟩×⟨Pr,Qr⟩) � (X × Y ) × (X′ × Y ′), where τ is the obvious transposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This uses that the top and bottom boundary spans, of W, are fibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 87 (2) There are two naturally defined maps, fr(W) → Z × W and fr(W) → M × N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Both are fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the first map, fr(W) → Z × W, this follows from the pullback diagram (26) above together with Lemma 41, and similarly for the map fr(W) → M × N, by symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) Composing with the filler, PL : L → fr(W), of W, which by definition is a fibration, this gives: Lemma 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If a window, W, is fibrant, then so are the middle horizontal and middle vertical spans, namely, L P�④④④④ Q �❈ ❈❈❈ M N, and L l �⑤⑤⑤⑤ r�❉❉❉❉ Z W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Isomorphic windows and equivalent fibrant windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 120 (Isomorphic windows and equivalent fibrant windows).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two win- dows, W1 and W2, as below, so with the same boundary, (27) W1 = X M p � p′ � Y Z Pl � Ql � L1 l1 � r1 � P1 � Q1 � W Pr � Qr � X′ N q � q′ � Y ′ and W2 = X M p � p′ � Y Z Pl � Ql � L2 l2 � r2 � P2 � Q2 � W Pr � Qr � X′ N q � q′ � Y ′, and thus, in particular, fr(W1) = fr(W2), are said to be isomorphic if there exists a homeomorphism, F : L1 → L2, making the obvious three dimensional diagram commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is equivalent to saying that there exist maps, F : L1 → L2 and F ′ : L2 → L1, making the diagrams below commute, L1 F � PL1 �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ L2, PL2 �✇✇✇✇✇✇✇✇ fr(W2) and L1 � F ′ PL1 �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ L2, PL2 �✇✇✇✇✇✇✇✇ fr(W2) such that F ◦ F ′ = idL2 and F ′ ◦ F = idL1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, if W1 and W2 are fibrant, then W1 and W2 are called equiv- alent if there exist F : L1 → L2 and F ′ : L2 → L1 making the diagrams above commute, together with fibred homotopies (see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5), F ′ ◦ F H1 =====⇒ fr(W2) idL1 and F ◦ F ′ H2 =====⇒ fr(W2) idL2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to see that equivalence between fibrant windows is an equivalence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will, of course, be mostly interested in the situation in which all the spaces involved are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF fibrant resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 88 Definition 121 (HF fibrant resolved 2-span).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given HF fibrant spans,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' from X to Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' which we picture as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) � (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q′) � Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and also in full,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M p �③③③③ p′ �❉ ❉ ❉ ❉ X Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' N q �④④④④ q′ �❈ ❈ ❈ ❈ X Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a HF fibrant resolved 2-span,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' W = (lX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' rY ): (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′) =⇒ (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' q′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' also written W = (lX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' rY ): � (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): X → Y � =⇒ � (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' q′): X → Y � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' or,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' simply,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) � (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q′) � ⇓ W Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is a HF fibrant window of form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as below50,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (28) W = X M p � p′ � Y XI sX � tX � L lX � rY � P � Q � Y I sY � tY � X N q � q′ � Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that X, Y, M, N, and L are HF spaces, (and, thus, so are XI and Y I), and the filler of W (see Definition 117), below, is a fibration, (29) L PL −→ fr(W) = M × X×Y (XI × Y I) × X×Y N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that the top and bottom boundary spans in (28) are already fibrant, by assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The left and right boundary spans are also fibrant, by construction, see Example 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We will frequently identity a HF-fibrant resolved 2-span with its filler, L PL −→ fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crucially for our constructions later on, the middle horizontal and middle vertical spans in (28) are fibrant, by Lemma 119.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The ‘resolved’ terminology arises from the fact that we allow the left and right boundaries of a 2-span to take values in the respective path spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For 2-spans in a usual setting, the left and right vertical edges would be identity spans, but to ensure the result is fibrant, we must ‘resolve’ those vertical edges, replacing them with their ‘fibrant replacements’, as in Example 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is in line with the definition of extended cobordisms between cobordisms of manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our definition was also designed so that the horizontal composition of resolved 2-spans, defined in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, is a resolved 2-span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 50note the notation in item (9) in page 16, and also Definition 59 A CATEGORIFICATION OF QUINN’S TQFT 89 Definition 122 (Equivalent and isomorphic HF fibrant resolved 2-spans).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X and Y be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two HF resolved 2-spans, W1, W2 : � (p, M, p′): X → Y ) =⇒ � (q, N, q′): X → Y ), will be said to be equivalent if they are equivalent as HF fibrant windows, and, similarly, W1 and W2 are isomorphic if they are isomorphic as HF fibrant windows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using the nomenclature of Definition 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') As usual, the above definition is given in the case that we will be using, but we note that it nowhere uses that fact that the spaces are homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF resolved fibrant 2-spans and natural transformations of profunc- tors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The next step is to study these HF resolved fibrant 2-spans before turning to how their properties are reflected by the profunctor construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The spatial 2-slices of a HF resolved fibrant 2-span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X and Y be HF- spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider HF fibrant spans, (p, M, p′), (q, N, q′): X → Y, and a HF fibrant resolved 2-span, W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′), connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its underlying HF fibrant window, W, is the commutative di- agram of solid arrows in Equation (30), just below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The dashed arrows showing the inclusion, x �→ constx, of a space, X, into the corresponding path space, XI, via constant paths, do not necessarily commute with the rest of the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They will, however, be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (30) W = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X const � ♣ ① ✤ ❋ ◆ M p � p′ � Y const � ◆ ❋ ✤ ① ♣ XI sX � tX � L lX � rY � P � Q � Y I sY � tY � X const � ◆ ❋ ✤ ① ♣ N q � q′ � Y const � ♣ ① ✤ ❋ ◆ \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We repeat that, X, Y, M, N, L are HF spaces, hence XI and Y I are HF-spaces, and that the filler, PL : L → fr(W), of W, below, is a fibration, (31) L PL −→ fr(W) = M × X×Y (XI × Y I) × X×Y N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall, from Definition 67, that the spatial slices of the fibrant spans, (p, M, p′) and (q, N, q′), are defined as {x|M|y} = ⟨p, p′⟩−1(x, y) = {m ∈ M : p(m) = x and p′(m) = y}, {x|N|y} = ⟨q, q′⟩−1(x, y) = {n ∈ N : q(n) = x and q′(n) = y}, and these spatial slices will be homotopy finite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Lemma 54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 123 (Spatial 2-slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′) be a HF resolved 2-span, as in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X, y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define51 the following space, which we call the spatial 2-slice, of W, at (x, y), [x|L|y] := ⟨lX, rY ⟩−1(constx, consty) = � l ∈ L : lX(l) = constx, rY (l) = consty � ⊂ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 51and note the square brackets rather than braces here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That distinction will be needed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 90 Given m ∈ {x|M|y} and n ∈ {x|N|y}, also consider the following space, which we call the spatial 2-slice, of W, at (m, x, y, n), \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb := P −1 L (m, constx, consty, n) = � l ∈ L : P(l) = m, lX(l) = constx, rY (l) = consty, Q(l) = n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, given paths, x γX −−→ x′ and y γY −−→ y′, in X and Y , m ∈ {x|M|y} and n ∈ {x′|N|y′}, we define the following space, also called a spatial 2-slice of W, but at (m, γX, γY , n), \uf8ee \uf8f0 m γX L γY n \uf8f9 \uf8fb := P −1 L (m, γX, γY , n) = � l ∈ L : P(l) = m, lX(l) = γX, rY (l) = γY , Q(l) = n � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the context of this definition, note that, given x ∈ X and y ∈ Y , the maps, P : L → M and Q: L → N, canonically restrict to maps, denoted Px,y : [x |L| y] → {x|M|y} and Qx,y : [x |L| y] → {x|N|y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, given m ∈ {x|M|y} and n ∈ {x|N|y}, we have the following, \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb = ⟨Px,y, Qx,y⟩−1(m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 124.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X, and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) The induced map, ⟨Px,y, Qx,y⟩: [x |L| y] → {x|M|y} × {x|N|y}, is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) If m ∈ {x|M|y} and n ∈ {x|N|y}, the homotopy type of the space, \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb , depends only on the path-components, in {x|M|y}, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' {x|N|y}, contain- ing m, resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first item follows by direct application of the homotopy lifting property of the fibration, PL : L → fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second follows from the fact that all fibres of a fibration over a path-connected space are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Recalling that we are assuming that X, Y, L, M, N are HF, we have: Lemma 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All spaces appearing in Definition 123 are HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the spatial 2-slice, \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb = P −1 L (m, constx, consty, n), this follows from the fact that both L and fr(W) are HF, (for the latter fact, see the discussion just after Definition 118), and Lemma 40, applied to the fibration PL : L → fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 91 The same sort of argument holds for the general spatial 2-slices, namely, \uf8ee \uf8f0 m γX L γY n \uf8f9 \uf8fb = P −1 L (m, γX, γY , n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a fibration, ⟨Px,y, Qx,y⟩: [x |L| y] → {x|M|y} × {x|N|y}, in which the spaces, {x|M|y} and {x|N|y} are both HF (see Lemma 54), and all of whose fibres, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', all of the P −1 L (m, constx, consty, n), are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have [x |L| y] is also HF by Lemma 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We define: Definition 126 (Vertical span of slices).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let W = (lX, P, L, Q, rY ): (p, M, p′) =⇒ (q, N, q′) be a HF resolved 2-span, as in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibrant span, [x|W|y] := \uf8eb \uf8ec \uf8ed [x|L|y] Px,y �♦♦♦♦♦ Qx,y �◆ ◆ ◆ ◆ ◆ {x|M|y} {x|N|y} \uf8f6 \uf8f7 \uf8f8 , will be called the vertical span of slices, of W, at x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By the discussion just given, [x|W|y] is a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly we have \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb = ⟨Px,y, Qx,y⟩−1 (m, n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We continue to fix a HF resolved 2-span, W, as in (30) and prove that, in several important cases, the spatial 2-slices of W are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Firstly consider holonomy maps for the fibrations, ⟨p, p′⟩: M → X × Y and ⟨q, q′⟩: N → X × Y , which will be denoted λM : I × (M ×X×Y (X × Y )I) → M and λN : I × (N ×X×Y (X × Y )I) → N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given paths x γX −−→ x′ in X, and y γY −−→ y′, in Y , these holonomy maps induce homotopy equivalences (where we are using the notation of Lemma 94), ΓM ⟨γX,γY ⟩ : {x|M|y} → {x′|M|y′} and ΓN ⟨γX,γY ⟩ : {x|N|y} → {x′|N|y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here ⟨γX, γY ⟩ is the path, in X × Y , such that, for u ∈ I, u �→ � γX(u), γY (u) � ∈ X × Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 127.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a HF resolved 2-span as in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x, x′ ∈ X and y, y′ ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that γX : I → X is a path in X, connecting x to x′, and that γY : I → Y is one in Y , connecting y to y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let m ∈ {x|M|y}, n ∈ {x|N|y} and n′ ∈ {x′|N|y′}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) The two spaces, \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb and \uf8ee \uf8f0 ΓM ⟨γX,γY ⟩(m) x′ L y′ ΓN ⟨γX,γY ⟩(n) \uf8f9 \uf8fb , are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 92 (2) The three spaces, \uf8ee \uf8f0 m γX L γY n′ \uf8f9 \uf8fb , \uf8ee \uf8f0 m x L y ΓN ⟨γX,γY ⟩(n′) \uf8f9 \uf8fb , and \uf8ee \uf8f0 ΓN ⟨γX,γY ⟩(m) x′ L y′ n′ \uf8f9 \uf8fb , are homotopically equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We recall that given a path γ, then γ denotes its reverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first point follows from the fact that the two spaces are fibres of the fibration, PL : L → fr(W), over points in the same path component of fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Indeed, the two spaces are P −1 L (m, constx, consty, n), and P −1 L � ΓM ⟨γX,γY ⟩(m), constx′, consty′, ΓN ⟨γX,γY ⟩(n) � , respectively, for points in fr(W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A path, in fr(W), connecting (m, constx, consty, n), and � ΓM ⟨γX,γY ⟩(m), constx′, consty′, ΓN ⟨γX,γY ⟩(n) � , is given by t �→ � λM� t, m, ⟨γX, γY ⟩ � , constγX(t), constγY (t), λN� t, n, ⟨γX, γY ⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second point follows again by identifying the spaces with fibres of the fibra- tion PL : L → fr(W), and examining different points of the same path component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, a path, in fr(W), connecting (m, γX, γY , n′) and � m, constx, consty, ΓN ⟨γX,γY ⟩(n′) � , is s �→ � m, γX s , γY s , λN� s, n′, ⟨γX, γY ⟩ �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, given a map γ : I → B, where B is a space, and s ∈ [0, 1], we have written γs : I → B for the path t �→ γ � (1 − s)t � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The natural transformation of profunctors associated to a HF resolved fibrant 2-span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let κ be a subfield of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to recap a little on the notation, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let W = (lX, P, L, Q, rY ): � (p, M, p′): X → Y � =⇒ � (q, N, q′): X → Y � be a HF fibrant resolved 2-span, as in (30), connecting the HF fibrant spans, (p, M, p′) and (q, N, q′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We assume given, or chosen, subsets xX ⊂ X and yY ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have Vect-profunctors, as defined in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, in particular in Notation 113, H M (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, and H N (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are the restrictions, to π1(X, xX)op × π1(Y, yY ), of the profunctors, HM : π1(X, X)op × π1(Y, Y ) → Vect, and HN : π1(X, X)op × π1(Y, Y ) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 93 We want to define a natural transformation of profunctors, which will be denoted 2H W (xX,yY ) : H M (xX,yY ) =⇒ H N (xX,yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural transformation will, itself, be the restriction of a natural transforma- tion, denoted 2HW : HM =⇒ HN, which we will define first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, given x ∈ X and y ∈ Y , HM(x, y) and HN(x, y) are both κ-vector spaces, so to specify a natural transformation, as required, we have to specify a linear transformation, 2HW (x,y), from HM(x, y) to HN(x, y), depending on x and y in a ‘natural’ way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that, by Lemma 112, both vector spaces are finite dimensional, we may specify this linear map by giving its matrix elements with respect to the evident bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a HF resolved 2-span, W: (p, M, p′) =⇒ (q, N, q′), as in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ X and y ∈ Y , we define the linear map, 2HW (x,y) : HM(x, y) → HN(x, y), where HM(x, y) = Lin(�π0({x|M|y})) and HN(x, y) = Lin(�π0({x|N|y})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is to have the following matrix elements, with respect to the usual bases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given m ∈ {x|M|y} and n ∈ {x|N|y}, (32) � PCm({x|M|y}) | 2HW (x,y) | PCn({x|N|y}) � := χπ \uf8eb \uf8ed \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ� PCn({x|N|y}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We refer to Definition 123 for notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that, by Lemma 125, all the spatial 2-slices met here are HF spaces, so we can consider their homotopy content.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 129.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a HF resolved 2-span, W: � (p, M, p′): X → Y � =⇒ � (q, N, q′): X → Y � , as in (30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X and y ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given m ∈ {x|M|y} and n ∈ {x|N|y}, the value of the right-hand-side of Equation (32) depends only on the path-components, in {x|M|y} and in {x|N|y}, respectively, to which m and n belong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The family of linear maps, 2HW (x,y): HM(x, y) → HN(x, y), for all x ∈ X and y ∈ Y , together defines a natural transformation of Vect-profunctors, 2HW : � HM : π1(X, X) ↛ π1(Y, Y ) � =⇒ � HN : π1(X, X) ↛ π1(Y, Y ) � , and, therefore (by Lemma 112) a 2-morphism in the bicategory vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first statement follows directly from item (2) of Lemma 124, and the fact that the homotopy content of a homotopy finite space is a homotopy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second statement follows from point (1) of Lemma 127, given the explicit forms of HM and HN in Definition 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also use the fact that, given paths γX : x → x′ in X, and γY : y → y′ in Y , the holonomy map, λN, for ⟨q, q′⟩: N → X × Y , gives rise to a homotopy equivalence, between fibres, ΓN ⟨γX,γY ⟩ : {x|N|y} → {x′|N|y′}, A CATEGORIFICATION OF QUINN’S TQFT 94 and, in particular, induces a bijection between the sets of path components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More- over, given n ∈ {x|N|y}, ΓN ⟨γX,γY ⟩ therefore restricts to a homotopy equivalence, PCn � {x|N|y} � ∼= PCΓN ⟨γX ,γY ⟩(n) � {x′|N|y′} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Definition 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Choose xX ⊂ X and yY ⊂ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The natural transformation, 2H W (xX,yY ) : H M (xX,yY ) =⇒ H N (xX,yY ), is defined by restricting 2HW : HM =⇒ HN to π1(X, xX)op × π1(Y, yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, given x ∈ xX and y ∈ yY , the linear map, (2H W (xX,yY ))(x,y) : H M (xX,yY )(x, y) → H N (xX,yY )(x, y), is, therefore, 2HW (x,y): HM(x, y) → HN(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 131.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' When κ = C, there is a 1-parameter version of 2HW : HM =⇒ HN, which is denoted 2H(W,s) : HM =⇒ HN, where s ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This has as matrix elements � PCm({x|M|y}) | 2H(W,s) (x,y) | PCn({x|N|y}) � := χπ \uf8eb \uf8ed \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ� PCn({x|N|y}) � χπ(PCx(X))1−sχπ(PCy(Y ))s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All results go through with this extra generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a special case of a more general 2-parameter version, also involving the vertical direction of the spatial 2- slices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is left to the reader to explore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal like structure of 2H W (−,−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following string of results will be used later on (Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5) to prove that the constructions of the once-extended Quinn TQFTs, defined in this paper, give bifunctors which, furthermore, can be given symmetric monoidal structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our starting point is Lemmas 114 and 115 of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 132.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider HF fibrant resolved 2-spans, denoted W: � (p, M, q): X → Y � =⇒ � (f, N, g): X → Y � , and W′ : � (p′, M ′, q′): X′ → Y ′� =⇒ � (f ′, N ′, g′): X′ → Y ′� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The diagrams for W and W′ will be (33) X M p � q � Y XI sX � tX � L l � r � P � Q � Y I sY � tY � X N f � g � Y and X′ M ′ p′ � q′ � Y ′ X′I sX′ � tX′ � L′ l′ � r′ � P ′ � Q′ � Y ′I sY ′ � tY ′ � X′ N ′ f ′ � g′ � Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 95 (1) The window, W × W′, below, is a HF fibrant resolved 2-span, � (p × p′, M × M ′, q × q′): X × X′ → Y × Y ′� =⇒ � (f × f ′, N × N ′, g × g′): X × X′ → Y × Y ′� , X × X′ M × M ′ p×p′ � q×q′ � Y × Y ′ (X × X′)I ∼= XI × X′I sX×sX′ � tX×tX′ � L × L′ l×l′ � r×r′ � P ×P ′ � Q×Q′ � Y I × Y ′I ∼= (Y × Y ′)I sY ×sY ′ � tY ×tY ′ � X × X′ N × N ′ f×f ′ � g×g′ � Y × Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Let x ∈ X, x′ ∈ X′, y ∈ Y and y′ ∈ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given m ∈ {x|M|y}, n ∈ {x|N|y}, m′ ∈ {x′|M ′|y′} and n′ ∈ {x′|N ′|y′}, we have that, � PC(m,m′)({(x, x′)|M × M ′|(y, y′)}) | 2HW×W′ ((x,x′),(y,y′)) | PC(n,n′)({(x, x′)|N × N ′|(y, y′)}) � equals � PCm({x|M|y}) | 2HW (x,y) | PCn({x|N|y}) � � PCm′({x′|M ′|y′}) | 2HW′ (x′,y′) | PCn′({x′|N ′|y′}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first point follows from the fact that the product of fibration is a fibra- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second follows from the fact that, clearly, \uf8ee \uf8f0 (m, m′) (x, x′) L × L′ (y, y′) (n, n′) \uf8f9 \uf8fb ∼= \uf8ee \uf8f0 m x L y n \uf8f9 \uf8fb × \uf8ee \uf8f0 m′ x′ L′ y′ n′ \uf8f9 \uf8fb , and PC(n,n′) � {(x, x′)|N × N ′|(y, y′)} � ∼= PCn � {x|N|y} � × PCn′� {x′|N ′|y′} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formula therefore follows from the fact that the homotopy content of HF spaces is multiplicative with respect to their product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In order to prove that the once-extended Quinn TQFT is a symmetric monoidal bifunctor, it is convenient to change the language of the previous result, to a lan- guage closer to that of Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, combining Lemma 132 with Lemma 115, whose notation we follow, gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 133.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX ⊂ X, x′ X′ ⊂ X′, yY ⊂ Y and y′ Y ′ ⊂ Y ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two natural transformations of Vect-profunctors, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the two 2-morphisms in vProfGrphf, obtained by pasting the diagrams in vProfGrphf, below, coincide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' π1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' xX) × π1(X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' x′ X′) H(xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )(f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g)⊗H(x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y′ Y ′ )(f ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g′) � H(xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q)⊗H(x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y′ Y ′ )(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q′) � ⇓ � 2H W (xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )⊗2H W′ (x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y′ Y ′ ) � ϕ m(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′) � ✘✘✘✘� χ(N,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N′) π1(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' yY ) × π1(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y′ Y ′) ϕ m(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′) � π1(X × X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' xX × x′ X′) H(xX ×x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ×y′ Y ′ )(f×f ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N×N ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g×g′) � π1(Y × Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' yY × y′ Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 96 and π1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' xX) × π1(X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' x′ X′) H(xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q)⊗H(x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y′ Y ′ )(p′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q′) � ϕ m(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′) � ✘✘✘✘� χ(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′) π1(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' yY ) × π1(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y′ Y ′) ϕ m(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′) � π1(X × X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' xX × x′ X′) H(xX ×x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ×y′ Y ′ )(p×q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M×M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q×q′) � H(xX ×x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ×y′ Y ′ )(f×f ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N×N ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g×g′) � ⇓ � 2H W×W′ (xX ×x′ X′ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ×y′ Y ′ ) � π1(Y × Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' yY × y′ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We continue to follow the notation in Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 134.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given pairs of spaces, (X, xX), (X′, x′ X′) and (X′′, x′′X′′), with X, X′ and X′′ homotopy finite, we have an obvious invertible 2-morphism in vProfGrphf, as shown in the diagram below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We have condensed the notation, so π(X) means π1(X, xX), X × X′ means (X × X′, xX × x′X′), and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') � π(X) × π(X′) � × π(X′′) ϕ αGrp (π(X),π(X′),π(X′′ )) �✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ϕ m(X,X′)×idπ(X′′) �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ π(X) × � π(X′) × π(X′′) � idπ(X)×ϕ m(X′,X′′) � ❴❴❴❴� ω(X,X′,X′′) π(X × X′) × π(X′′) ϕ m(X×X′,X′′) � π(X) × � π(X′ × X′′) � ϕ m(X,X′×X′′) �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ π � (X × X′) × X′′� ϕ π � αCGWH (X,X′,X′′) � �✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ π � X × (X′ × X′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This 2-morphism of profunctors arises from the fact that, if we switch back the profunctors in the arrows of the diagram above, to the functors that gave rise to them, then, applying Example 33, gives rise to a commutative diagram of functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Indeed, note that, in general, if F : C → C′ and G: C′ → C′′ are functors, then we have a canonical natural isomorphism from the profunctor, ϕG◦F : C ↛ C′′, to the composition of the profunctors, C ϕF ↛ C′ ϕG ↛ C′′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for more details see Example 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 2-morphisms, ω(X,X′,X′′) in vProfGrphf satisfy an obvious cocycle con- dition, given pairs of spaces, (X, xX), (X′, x′X′), (X′′, x′′X′′), and (X′′′, x′′′X′′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The equations satisfied are those in [61, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3] / [58, page 17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from an explicit calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These cocycles, ω(X,X′,X′′), are furthermore compatible with the 2H W (−,−) and the χ(−,−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that we are given three HF spans, (p, M, q): X → Y, (p′, M ′, q′): X′ → Y ′, and (p′′, M ′′, q′′): X′′ → Y ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 97 The 2-morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' χ(−,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='−),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in vProfGrphf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' arising from Lemma 115,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' can be pasted in two different ways,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as written below (where we have again condensed the nota- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the obvious way),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � π(X) × π(X′) � × π(X′′) ϕ m(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′)×idπ(X′′) � ✘✘✘✘� χ(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′)⊗HM′′ (HM⊗HM′ )⊗HM′′ � � π(Y ) × π(Y ′) � × π(Y ′′) ϕ m(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′)×idπ(X′′) � π(X × X′) × π(X′′) HM×M′ ⊗HM′′ � ϕ m(X×X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′′) � ✘✘✘✘� χ(M×M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′′) π(Y × Y ′) × π(Y ′′) ϕ m(Y ×Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′′) � π � (X × X′) × X′′� ϕ π � αCGWH (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′′) � � ✘✘✘✘� ∼ = H(M×M′)×M′′ � π � (Y × Y ′) × Y ′′� ϕ π � αCGWH (Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′′) � � π � X × (X′ × X′′) � HM×(M′×M′′) � π � Y × (Y ′ × Y ′′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and � π(X) × π(X′) � × π(X′′) ϕ αGrp (π(X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='π(X′ ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='π(X′′)) � ✘✘✘✘� ∼ = (H M⊗H M′ )⊗H M′′ � � π(Y ) × π(Y ′) � × π(Y ′′) ϕ αGrp (π(Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='π(Y ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='π(Y ′′)) � π(X) × � π(X′) × π(X′′) � HM⊗(HM′ ⊗HM′′ ) � π(X)×ϕ m(X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′′) � ✘✘✘✘� H M⊗χ(M′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′′) π(Y ) × � π(Y ′) × π(Y ′′) � π(Y )×ϕ m(Y ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′′) � π(X) × π(X′ × X′′) ϕ m(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′×X′′) � ✘✘✘✘� χ(M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M′×M′′) HM⊗HM′×M′′ � π(Y ) × π(Y ′ × Y ′′) ϕ m(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y ′×Y ′′) � π � X × (X′ × X′′) � H M×(M′×M′′) � π � Y × (Y ′ × Y ′′) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the first of these diagrams fits together with the diagram for ω(X,X′,X′′) on the left, and the various 2-morphisms compose well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second lower dia- gram likewise composes, this time with ω(Y,Y ′,Y ′′) and on the right, again giving a second composite 2-morphism, which has the same source and target composite 1-morphisms as the first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two composite 2-morphisms obtained as above by pasting, re- spectively, ω(X,X′,X′′) or ω(Y,Y ′,Y ′′) to the two diagrams, are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctors, H(xX×x′ X′ ,yY ×y′ Y ′ ) � X × X′ (p×q,M×M′,q×q′) −−−−−−−−−−−−→ Y × Y ′ � : π1(X × X′, xX × x′ X′)op × π1(Y × Y ′, yY × y′ Y ′) → Vect, as well as the natural transformation between them that are obtained from fibrant resolved 2-spans, are similarly well behaved with respect to swapping the order of coordinates, and with respect to products with trivial spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We leave it to the reader to unpack what this means in terms of diagrams similar to those that were just presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 98 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The horizontal composition of HF resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' HF resolved 2- spans can be composed both horizontally and vertically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we first look at the horizontal composition before considering the effect of that composition on the corresponding natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The horizontal composition of HF resolved 2-spans in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider HF fibrant spans, (p1, M1, p′ 1): X → Y and (p2, M2, p′ 2): Y → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recalling the definition of their composition from Lemma 55 and Definition 56, we have that (p1, M1, p′ 1) • (p2, M2, p′ 2) = (p1, M1 ×Y M2, p′ 2): X → Z is itself a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is defined by the pullback diagram, appearing as the diamond in the commutative diagram below, which we repeat from earlier for the notation being used, M1×Y M2 �❣❣❣❣❣ �❲ ❲ ❲ ❲ ❲ p1 � p′ 2 � M1 p1 �❥❥❥❥❥❥ p′ 1 �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ M2 p2 �❣❣❣❣❣❣❣❣❣ p′ 2 �❚ ❚ ❚ ❚ ❚ ❚ X Y Z, and from which we extract the HF fibrant span, M1×Y M2 p1 �qqqqqq p′ 2 �▼ ▼ ▼ ▼ ▼ ▼ X Z, which is the composite of the given pair of fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a diagram of spaces, HF spans, and HF resolved 2-spans, as below, X (p1,M1,p′ 1) � (q1,N1,q′ 1) � ⇓ W1 Y (p2,M2,p′ 2) � (q2,N2,q′ 2) � ⇓ W2 Z , where the diagrams for W1 and W2 are as shown below, (34) X M1 p1 � p′ 1 � Y XI sX � tX � L1 l1 � r1 � P1 � Q1 � Y I sY � tY � X N1 q1 � q′ 1 � Y and Y M2 p2 � p′ 2 � Z Y I sY � tY � L2 l2 � r2 � P2 � Q2 � ZI sZ � tZ � Y N2 q2 � q′ 2 � Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need to consider the fillers, Definition 117, of W1 and W2, as usual denoted by PL1 : L1 → fr(W1) and PL2 : L2 → fr(W2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 99 We will define the horizontal composite, W1#0W2, of W1 and W2, in such a way that W1#0W2 is a HF resolved 2-span, which fits inside the diagram below, (35) X (p1,M1×Y M2,p′ 2) � (q1,N1×Y N2,q′ 2) � ⇓ W1#0W2 Z = X (p1,M1,p′ 1)•(p2,M2,p′ 2) � (q1,N1,q′ 1)•(q2,N2,q′ 2) � ⇓ W1#0W2 Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is done by considering the obvious pullback along the common vertical HF fibrant span in (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly the horizontal composition of W1 and W2 will be given by the window (36) W1#0W2 := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M1 ×Y M2 p1 � p′ 2 � Z XI sX � tX � L1 ×Y I L2 lX � rZ � P � Q � ZI sZ � tZ � X N1 ×Y N2 q1 � q′ 2 � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we will need to consider the pullback diagram included (as the middle dia- mond) in the commutative diagram below, in which, (i) ΨL1,L2 : L1 ×Y I L2 → Y I is ΨL1,L2 = l2 ◦ proj2 = r1 ◦ proj1, (ii) lX = l1 ◦ proj1, and, (iii) rZ = r2 ◦ proj2, (37) L1 ×Y I L2 lX � rZ � ΨL1,L2 � proj1 �tttttttttt proj2 �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ L1 r1 �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ l1 �⑤⑤⑤⑤⑤⑤⑤ L2 l2 �tttttttttt r2 �❈ ❈ ❈ ❈ ❈ ❈ ❈ XI Y I ZI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have obvious maps, ¶: L1 ×Y I L2 → M1 ×Y M2, induced by P1 : L1 → M1 and P2 : L2 → M2, and Q: L1 ×Y I L2 → N1 ×Y N2, induced by Q1 : L1 → M1 and Q2 : L2 → N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will briefly explain why W1#0W2, in Equation (36), is a HF fibrant resolved 2-span, fitting inside diagram (35).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows by a sequence of observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Lemma 55, together with the fact that the top, bottom and middle hori- zontal (Lemma 119) spans of the diagrams in (34) are HF fibrant, implies that all spaces appearing in diagram (36) are HF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 100 (2) Secondly, we note that the naturally defined map, below, which is the filler of W1#0W2, is a fibration, PL1,L2 : L1 ×Y I L2 → lim \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M1 ×Y M2 p1 � p′ 2 � Z XI sX � tX � ZI sZ � tZ � X N1 ×Y N2 q1 � q′ 2 � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The argument is very similar to that in the proof of Lemma 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (a) First note that we can put the fillers, PL1 : L1 → fr(W1) and PL2 : L2 → fr(W2), together, to obtain a map, as below, (38) PL1,L2 : L1 ×Y I L2 → lim \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M1 p1 � p′ 1 � Y M2 p2 � p′ 2 � Z XI sX � tX � Y I sY � tY � ZI sZ � tZ � X N1 q1 � q′ 1 � Y N2 q2 � q′ 2 � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fact that both the fillers, PL1 and PL2, are fibrations, together with the universal property of pullbacks, then gives that PL1,L2 is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (b) We also have a naturally defined map, (39) Pout : lim \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M1 p1 � p′ 1 � Y M2 p2 � p′ 2 � Z XI sX � tX � Y I sY � tY � ZI sZ � tZ � X N1 q1 � q′ 1 � Y N2 q2 � q′ 2 � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 −→ lim \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M1 ×Y M2 p1 � p′ 2 � Z XI sX � tX � ZI sZ � tZ � X N1 ×Y N2 q1 � q′ 2 � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the fact that ⟨sY , tY ⟩: Y I → Y × Y is a fibration, it follows that Pout is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (c) We thus have that PL1,L2 = Pout ◦ PL1,L2 is a fibration, for it is given by the composition of fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 101 We now need to analyse the composite window, W1#0W2, via its spatial 2-slices, \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb , with, of course, x ∈ X, z ∈ Z, (m1, m2) ∈ M1 ×Y M2 and (n1, n2) ∈ N1 ×Y N2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Taking this apart a bit, just listing elementary properties, we get, with the conventions as in (34): x = p1(m1), z = p′ 2(m2), and there is some y = p′ 1(m1) = p2(m2) ∈ Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' x = q1(n1), z = q′ 2(n2), and there is some y′ = q′ 1(n1) = q2(n2) ∈ Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the two elements, y and y′, in Y , need not be the same here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However note that m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′} and n2 ∈ {y′|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, if ℓ = (ℓ1, ℓ2) ∈ \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb, then it is required to satisfy: – P(ℓ) = (m1, m2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' – Q(ℓ) = (n1, n2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' – ℓ = (ℓ1, ℓ2) ∈ L1 ×Y I L2, so r1(ℓ1) = l2(ℓ2), which is in Y I, of course, and so it is some path, γ : [0, 1] → Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see diagram (34).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' – Referring again to that diagram, the path γ = r1(ℓ1) starts at sY � r1(ℓ1) � = p′ 1(P1(ℓ1)) = p′ 1(m1) = p2(m2) = y, and, similarly, it ends at q2(n2) = y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Summarising this for future reference, we have: Lemma 137.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we have ℓ = (ℓ1, ℓ2) ∈ \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb , then r1(ℓ1) is a path in Y , from y := p2(m2) to y′ := q2(n2), and so y and y′ will be in the same path component of Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The discussion and results above have several important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a) If m2 ∈ M2 and n2 ∈ N2 are such that p2(m2) and q2(n2) are in different path components of Y , then the spatial 2-slice, \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb , is empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' b) If the spatial 2-slice, \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb , is non-empty, then it has the same homotopy type as a spatial 2-slice, \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n′ 1, n′ 2) \uf8f9 \uf8fb , having m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n′ 1 ∈ {x|N1|y} and n′ 2 ∈ {y|N2|z}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', with the same y in both expressions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More precisely, A CATEGORIFICATION OF QUINN’S TQFT 102 c) using the notation of Lemma 137, if y and y′ are in the same path-component of Y , we can always find a representative, (n′ 1, n′ 2), of the path-component, PC(n1,n2)({x|N1 ×Y N2|z}), with n′ 1 ∈ {x|N1|y} and n′ 2 ∈ {y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This uses the last point of Lemma 55, and applying the homotopy lifting prop- erty of the fibration, {x|N1 ×Y N2|z} → Y, to a path connecting y′ to y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Since PC(n1,n2)({x|N1 ×Y N2|z}) = PC(n′ 1,n′ 2)({x|N1 ×Y N2|z}), and using Lemma 124, we then have \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb ∼= \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n′ 1, n′ 2) \uf8f9 \uf8fb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With that choice, namely that we take ‘the same y’, as in the above, there is an action of the space, Ωy(Y ), of loops of the pair, (Y, y), on the corresponding spatial 2-slice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have: Lemma 138.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ X, y ∈ Y and z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The map, ΨL1,L2 : L1 ×Y I L2 → Y I in diagram (37), induces a fibration, ΨL1,L2 : \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb −→ Ωy(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the fact that the map, PL1,L2, in (38) is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We will use this fact shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The natural transformation associated to the horizontal composition of HF resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We assume given a diagram of spaces, HF spans, and HF resolved 2-spans, and their horizontal composite, as shown52, X (p1,M1,p′ 1) � (q1,N1,q′ 1) � ⇓ W1 Y (p2,M2,p′ 2) � (q2,N2,q′ 2) � ⇓ W2 Z, and X (p1,M1×Y M2,p′ 2) � (q1,N1×Y N2,q′ 2) � ⇓ W1#0W2 Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX ⊂ X, yY ⊂ Y and zZ ⊂ Z be subsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 130 gives natural transformations of profunctors, therefore 2-morphisms in vProfGrphf, 2H W1 (xX,yY ) : H M1 (xX,yY ) =⇒ H N1 (xX,yY ), 2H W2 (yY ,zY ) : H M2 (yY ,zY ) =⇒ H N2 (yY ,zZ), and 2H W1#W2 (xX,zZ) : H M1×Y M2 (xX,zZ) =⇒ H N1×Y N2 (xX,zZ) , where, in more notational detail, H M1 (xX,yY ) : π1(X, xX) ↛ π1(Y, yY ), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 52The underlying windows, W1 and W2, are as in (34), and their horizontal composite is as in (36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 103 We will now prove the important fact that, with the assumption that (Y, yY ) is 0-connected, and noting Proposition 116, the natural transformation arising from W1#0W2 is obtained by horizontally composing the natural transformations given by W1 and W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We leave the reader to explore what might happen when the condition on (Y, yY ) does not hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial fact that we use is the conclusion of Lemma 138, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', that ΨL1,L2 : \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb → Ωy(Y ), is a fibration of HF spaces, (of course, subject to the conditions required there).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be used together with Theorem 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notation and results in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 will also play a key role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our conventions on profunctors are laid out in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX ⊂ X, yY ⊂ Y and zZ ⊂ Z be subsets with (Y, yY ) 0-connected, then 2H W1#0W2 (xX,zZ) = 2H W1 (xX,yY ) • 2H W2 (yY ,zZ), as natural transformations of Vect-profunctors, H M1 (xX,yY ) • H M2 (yY ,zY ) =⇒ H N1 (xX,yY ) • H N2 (yX,zY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we have abused notation, and noted that, by Proposition 116, we have canonical natural isomorphisms, of Vect-profunctors, H M1 (xX,yY )•H M2 (yY ,zZ) = ˆ y∈yY H M1 (xX,yY )(−, y)⊗H M2 (yY ,zZ)(y, −) ηM1,M2 (xX ,yY ,zZ ) ========⇒ H M1×Y M2 (xX,zZ) , and H N1 (xX,yY )•H N2 (yY ,zZ) = ˆ y∈yY H N1 (xX,yY )(−, y)⊗H N2 (yY ,zZ)(y, −) ηN1,N2 (xX ,yY ,zZ ) ========⇒ H N1×Y N2 (xX,zZ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, the natural isomorphism, ηM1,M2 (xX,yY ,zZ), is such that, if x ∈ xX and z ∈ zZ, and given any y ∈ yY , m1 ∈ {x|M1|y} and m2 ∈ {y|M2|z}, then the linear map, � ηM1,M2 (xX,yY ,zZ) � (x,z), sends the equivalence class of PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) to PC(m1,m2)({x|M1 ×Y M2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will prove that the following diagram of natural transformations com- mutes53, ˆ y∈yY H M1 (xX,yY )(−, y) ⊗ H M2 (yY ,zZ)(y, −) 2HW1 (xX ,yY )•2HW2 (yY ,zZ ) � ηM1,M2 (xX ,yY ,zZ ) � H M1×Y M2 (xX,zZ) 2HW1#0W2 (xX ,zZ ) � ˆ y∈yY H N1 (xX,yY )(−, y) ⊗ H N2 (yY ,zZ)(y, −) ηN1,N2 (xX ,yY ,zZ ) � H N1×Y N2 (xX,zZ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 53The argument in the proof gives that the diagram commutes even if (Y, yY ) is not 0- connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However in this case the horizontal natural transformations are not necessarily natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 104 To this end,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we prove that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given x ∈ xX and z ∈ zZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the following diagram of linear maps commutes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (40) � y∈yY HM1(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y) ⊗ HM2(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' z) proj � ˆ y∈yY H M1 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y) ⊗ H M2 (yY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ)(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' z) � 2H W1 (xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )•2H W2 (yY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z) � � ηM1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M2 (xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z) � H M1×Y M2 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ) (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' z) � 2H W1#0W2 (xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z) � ˆ y∈yY H N1 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY )(x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y) ⊗ H N2 (yY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ)(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' z) � ηN1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='N2 (xX ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z) � H N1×Y N2 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ) (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' z),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where proj is the projection mentioned earlier,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in Equation (5) on page 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let y, y′ ∈ yY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let also m1 ∈ {x|M1|y}, and m2 ∈ {y|M2|z}, and then n1 ∈ {x|N1|y′}, and n2 ∈ {y′|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first consider the linear map, F tr : � y∈yY HM1(x,y) ⊗ HM2(y, z) → H N1×Y N2 (xX,zZ) (x, z), obtained from the path in diagram (40) passing through the top right corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The corresponding matrix elements, � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ��F tr�� PC(n1,n1)(N1 ×Y N2) � , are defined, by Equation (32), to be (41) χπ \uf8eb \uf8ed \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ� PC(n1,n2)({x|N1 ×Y N2|z}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we noted in part a) of the discussion after Lemma 137, the spatial 2-slices, \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb are empty if, in the notation from that lemma, y and y′ are not in the same path component of Y , so if y and y′ are in different components of Y , the matrix elements of F tr in (41) have value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, using part c) of that same discussion, if y and y′ are in the same path-component of Y , we can always find a representative, (n′ 1, n′ 2), of the path-component, PC(n1,n2)({x|N1 ×Y N2|z}), with n′ 1 ∈ {x|N1|y} and n′ 2 ∈ {y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consequently, when the matrix elements in (41) are non zero, we can suppose that y′ = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We therefore let y ∈ yY , m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}, and compute the value of (41) in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use Lemma 138 and apply Theorem 47 to the fibration, ΨL1,L2 : \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb → Ωy(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 105 Let γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , γr, where r = |π1(Y, y))| = |π0(Ωy(Y ))|, be representatives of the different path-components of Ωy(Y ), which we recall is homotopy finite (Lemma 72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will suppose, with no loss of generality, that γ1 = consty, the constant path at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the notation in Definition 123, we have χπ \uf8eb \uf8ed \uf8ee \uf8f0 (m1, m2) x L1 ×Y I L2 z (n1, n2) \uf8f9 \uf8fb \uf8f6 \uf8f8 = r � i=1 χπ� Ψ −1 L1,L2(γi) � χπ(PCγi(Ωy(Y ))) (42) = r � i=1 χπ \uf8eb \uf8ed \uf8ee \uf8f0 m1 constx L1 γi n1 \uf8f9 \uf8fb × \uf8ee \uf8f0 m2 γi L2 constz n2 \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ(PCγi(Ωy(Y ))) = r � i=1 χπ \uf8eb \uf8ed \uf8ee \uf8f0 m1 constx L1 γi n1 \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ \uf8eb \uf8ed \uf8ee \uf8f0 m2 γi L2 constz n2 \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ(PCγi(Ωy(Y ))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using point (2) of Lemma 127, each term in this sum has form χπ �� m1 x L1 y ΓN1 ⟨constx,γi⟩(n1) �� χπ �� m2 y L2 z ΓN2 ⟨γi,constz⟩(n2) �� χπ(PCγi(Ωy(Y ))), which we will note for future use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Next we apply Lemma 109 to calculate the next term, that is, χπ� PC(n1,n2)({x|N1 ×Y N2|z}) � , in expression (41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the commutative diagram, below, where the middle diamond is a pullback, N1×Y N2 P � �♠♠♠♠♠♠♠ �◗ ◗ ◗ ◗ ◗ ◗ ◗ q1 � q′ 2 � N1 q1 �rrrrrr q′ 1 �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ N2 q2 �❧❧❧❧❧❧❧❧❧ q′ 2 �▼ ▼ ▼ ▼ ▼ ▼ X Y Z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Put {x|N1} = q−1 1 (x) and {N2|z} = q′ 2 −1(z), then q′ 1 : N1 → Y and q2 : N2 → Y induce fibrations, qr : {N2|z} → Y and ql : {x|N1} → Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Lemma 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover we have a pullback diagram, as in the diagram below, where Px,z is the unique map making the diagram commute, {x|N1 ×Y N2|z} Px,z � proj1 �♦♦♦♦♦♦♦♦♦♦♦ proj2 �❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ {x|N1} ql �P P P P P P P P P P P P P {N2|z} qr �♥♥♥♥♥♥♥♥♥♥♥♥♥ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have that Px,z is a fibration (see the proof of Lemma 71), and its fibre over y ∈ Y satisfies P −1 x,z(y) ∼= {x|N1|y} × {y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Lemma 109, we have, for n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}, an expres- sion of form (43) χπ� PC(n1,n2)({x|N1 ×Y N2|z} � = T {x|N1×Y N2|z} (n1,n2) χπ(PCy(Y )) χπ(PCn1({x|N1|y})) χπ(PCn2({y|N2|z})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 106 Here T {x|N1×Y N2|z} (n1,n2) is the number of path-components of the fibre over y of the fibration, P n1,n2 x,y : PC(n1,n2)({x|N1 ×Y N2|z}) → PCy(Y ), obtained by restricting Px,z to PC(n1,n2)({x|N1 ×Y N2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, by Lemma 108, T {x|N1×Y N2|z} (n1,n2) is the cardinality of the orbit of the path-component, PC(n1,n2)({x|N1|y} × {y|N2|z}), under the right action, ⊳, of π1(Y, y) on the set �π0 � {x|N1|y} × {y|N2|z} � ∼= �π0({x|N1|y}) × �π0({y|N2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter is the set of path-components of the fibre of the fibration Px,z : {x|N1 ×Y N2|z} → Y at y, and the action is as in Lemma 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This action is derived in the obvious way from the product action of π1(Y, y) on �π0({x|N1|y}) × �π0({y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Going back to (41), we now put (42) and (43) together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two more observations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall r = |π0(Ωy(Y ))| = |π1(Y, y)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) We have homotopy equivalences, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , r}, PCn1({x|N1|y}) ∼= PCΓN1 ⟨constx,γi⟩(n1)({x|N1|y}), (44) and PCn2({y|N2|z}) ∼= PCΓN2 ⟨γi,constz⟩(n2)({y|N2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (45) These are induced by the homotopy equivalences, ΓN1 ⟨constx,γi⟩ : {x|N1|y} → {x|N1|y} and ΓN2 ⟨γi,constz⟩ : {y|N2|z} → {y|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using Lemma 95 here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (2) By Lemma 140 below, all path-components, PCγi(Ωy(Y )), of the loop space, of Y at y, are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We therefore have χπ(PCγi(Ωy(Y ))) = χπ(Ωy(Y ))/|π1(Y, y)|, for each i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Putting everything together, we have: � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,n1)(N1 ×Y N2) � = T {x|N1×Y N2|z} (n1,n2) χπ(PCy(Y ))χπ(Ωy(Y )) |π1(Y, y)| � |π1(Y,y)| � i=1 � PCm1({x|M1|y})|(2H W1 (xX,yY ))(x,y)|PCΓN1 (constx,γi)(n1)({x|N1|y}) � � PCm2({y|M2|z})| � 2H W2 (yY ,zZ) � (y,z)|PCΓN2 (γi,constz)(n2)({y|N2|z}) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By the definition of the right action, ⊳, of π1(Y, y) on the set, �π0 � {x|N1|y} × {y|N2|z} � ∼= �π0({x|N1|y}) × �π0({y|N2|z}), A CATEGORIFICATION OF QUINN’S TQFT 107 the set of path-components of the fibre of the fibration, Px,z : {x|N1×Y N2|z} → Y , at y, and from the discussion in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, we have � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,n1)(N1 ×Y N2) � = T {x|N1×Y N2|z} (n1,n2) |π1(Y, y)| � � g∈π1(Y,y) � PCm1({x|M1|y})|(2H W1 (xX,yY ))(x,y)|PCn1({x|N1|y})⊳g � � PCm2({y|M2|z})| � 2H W2 (yY ,zZ) � (y,z)|PCn2({y|N2|z}) ⊳ g �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that we also have used that, by Lemma 72, χπ(PCy(Y ))χπ(Ωy(Y )) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now let T {x|N1×Y N2|z} (n1,n2) denote the orbit of the element below, � PCn1({x|N1|y}, PCn2({y|N2|z}) � ∈ �π0({x|N1|y}) × �π0({y|N2|z}), under the right action of π1(Y, y) on the set, �π0({x|N1|y} × {y|N2|z}), and thus on �π0({x|N1|y}) × �π0({y|N2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Recall again that this is the set of path-components of the fibre of the fibration, Px,z : {x|N1 ×Y N2|z} → Y at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') By Lemma 108, |T {x|N1×Y N2|z} (n1,n2) | = T {x|N1×Y N2|z} (n1,n2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let S{x|N1×Y N2|z} (n1,n2) denote the cardinality of the stabiliser of the same element, � PCn1({x|N1|y}, PCn2({y|N2|z}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the elementary fact that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if a finite group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' acts on a set,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then given any pairs of elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' k and l,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the same orbit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the cardinality of {g ∈ G : k ⊳ g = l} is that of the stabiliser subgroup of k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' on applying this to the case of G being π1(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' invoking the orbit-stabiliser theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' firstly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' that S{x|N1×Y N2|z} (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n2) T {x|N1×Y N2|z} (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n2) = |π1(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y)|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and thus � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) |F tr| PC(n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n1)(N1 ×Y N2) � = S{x|N1×Y N2|z} (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n2) T {x|N1×Y N2|z} (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n2) |π1(Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' y)| × � � PCm1({x|M1|y}) | � 2H W1 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y) | PCn′ 1({x|N1|y}) � � PCm2({y|M2|z}) | � 2H W2 (yZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ) � (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z) | PCn′ 2({y|N2|z}) � = � � PCm1({x|M1|y})| � 2H W1 (xX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='yY ) � (x,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='y)|PCn′ 1({x|N1|y}) � � PCm2({y|M2|z})| � 2H W2 (yZ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='zZ) � (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='z)|PCn′ 2({y|N2|z}) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where each sum is indexed by the set of elements of form � PCn′ 1({x|N1|y}),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' PCn′ 2({y|N2|z}) � in T {x|N1×Y N2|z} (n1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that x ∈ xX, z ∈ zZ, and that we took y, y′ ∈ yY , where y and y′ are in the same path-component in Y (so without loss of generality we take y = y′), and also m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′}, and n2 ∈ {y′|N2|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter formula gives exactly the corresponding matrix element, � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ��F bl�� PC(n1,n1)(N1 ×Y N2) � , A CATEGORIFICATION OF QUINN’S TQFT 108 of the linear map associated to the path in (40) passing through the bottom left corner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from (7), because if x ∈ xX and z ∈ zZ, the linear bijection, ˆ y∈yY H N1 (xX,yY )(x, y) ⊗ H N2 (yY ,zZ)(y, z) � ηN1,N2 (xX ,yY ,zZ ) � (x,z) � H N1×Y N2 (xX,zZ) (x, z), is such that given y ∈ yY , n1 ∈ {x|N1|y} and n2 ∈ {y|N2|z}, it sends the equivalence class of PCn1({x|N1|y}) ⊗ PCn2({y|N2|z}) ∈ � y∈yY H N1 (xX,yY )(x, y) ⊗ H N2 (yY ,zZ)(y, z) to PC(n1,n2)({x|N1 ×Y N2|z}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 37 in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 is useful to translate between the categorical and the combinatorial languages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now suppose that y, y′ ∈ yY are not in the same path-component in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If m1 ∈ {x|M1|y}, m2 ∈ {y|M2|z}, n1 ∈ {x|N1|y′}, and n2 ∈ {y′|N2|z}, we already saw that � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ��F tr�� PC(n1,n1)(N1 ×Y N2) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying the second point of Lemma 37, it also follows that, � PCm1({x|M1|y}) ⊗ PCm2({y|M2|z}) ��F bl�� PC(n1,n1)(N1 ×Y N2) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore, diagram (40) commutes as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In the proof, we used the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Y be any CGWH space54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All path components of Ωy(Y ) = {γ ∈ Y I : sY (γ) = tY (γ) = y} are homotopic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Py = {γ ∈ Y I : sY (γ) = y}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a path-connected space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a fibration, ty : Py → Y , induced by tY : Y I → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly Ωy(Y ) is the fibre of ty at y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hence all path-components of Ωy(Y ) are homotopy equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This uses Lemma 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertical composition of HF resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Checking that hori- zontal composition translates via the profunctor construction to horizontal compo- sition of the corresponding natural transformations required some delicate count- ing arguments, the corresponding checks for the vertical composition require other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Preliminaries for the vertical composition of HF resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a CGWH space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Example 51, there is a fibrant span, (sX, XI, tX): X → X, from which we constructed the identity of X in the category HFspan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composite, (see Definition 56), (sX, XI, tX) • (sX, XI, tX): X → X is the fibrant span, (sX, XI ×X XI, tY ): X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, XI ×X XI = {(γ, γ′) ∈ XI × XI | γ(1) = γ′(0)}, and we recall that sX(γ, γ′) = γ(0) and tX(γ, γ′) = γ′(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We consider the homeomorphism, FX : XI ×X XI → XI, defined as (46) FX(γ, γ′)(t) = � γ(2t), t ∈ [0, 1/2], γ′(2t − 1), t ∈ [1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 54which need not be HF A CATEGORIFICATION OF QUINN’S TQFT 109 Clearly FX makes the diagram below commute, XI×XXI FX � sX �✉✉✉✉✉✉✉✉✉✉ tX �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ X X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XI sX �■■■■■■■■■■ tX �t t t t t t t t t t We thus have that FX is an isomorphism (of fibrations) over X×X, or, equivalently, of fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertical composition of HF resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X and Y be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a diagram of fibrant HF resolved 2-spans as shown below, ⇓ W2 X (p2,M2,p′ 2) � (p1,M1,p′ 1) � (q1,N1,q′ 1) �Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ⇓ W1 Here we have HF fibrant resolved 2-spans of the form: W2 : � (p2, M2, p′ 2): X → Y � =⇒ � (p1, M1, p′ 1): X → Y � , and W1 : � (p1, M1, p′ 1): X → Y � =⇒ � (q1, N1, q′ 1): X → Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly the windows, W1 and W2, have the form: (47) W1 = X M1 p1 � p′ 1 � Y XI sX � tX � L1 l1 � r1 � P1 � Q1 � Y I sY � tY � X N1 q1 � q′ 1 � Y and W2 = X M2 p2 � p′ 2 � Y XI sX � tX � L2 l2 � r2 � P2 � Q2 � Y I sY � tY � X M1 p1 � p′ 1 � Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We want to construct a vertical composite, W2#1W1, which should be a HF resolved 2-span, such that W2#1W1 : (p2, M2, p′ 2) =⇒ (q1, N1, q′ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We do this in two steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The first step is exactly as when we constructed the horizontal composition of HF fibrant resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Namely, we perform the obvious pullback along the common horizontal spans of W1 and W2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This yields the following HF fibrant A CATEGORIFICATION OF QUINN’S TQFT 110 window, W2#′ 1W1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M2 p2 � p′ 2 � Y XI ×X XI sX � tX � L2 ×M1 L1 l1 � r2 � P2 � Q1 � Y I ×Y Y I sY � tY � X N1 q1 � q′ 1 � Y \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, given (ℓ2, ℓ1) ∈ L2 ×M1 L2, we have written l1 � (ℓ2, ℓ1) � = � l2(ℓ2), l1(ℓ1) � , r2 � (ℓ2, ℓ1) � = � r2(ℓ2), r1(ℓ1) � , and also, P2 � (ℓ2, ℓ1) � = P2(ℓ2), Q1 � (ℓ2, ℓ1) � = Q1(ℓ1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To prove that W2#′ 1W1 is a HF fibrant window, we can use the same argument that we used for the horizontal composition of HF resolved 2-spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now need to ‘adjust’ the left and right vertical spans of W2#′ 1W1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To this end, we use the homeomorphisms, below, of fibrant spans;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XI×XXI FX � sX �✉✉✉✉✉✉✉✉✉✉ tX �■ ■ ■ ■ ■ ■ ■ ■ ■ ■ X X XI sX �■■■■■■■■■■ tX �✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ ✉ and Y I×Y Y I FY � sY �✈✈✈✈✈✈✈✈✈✈ tY �❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ Y Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Y I sY �❍❍❍❍❍❍❍❍❍❍ tY �✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ and the commutative diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X M2 p2 � p′ 2 � Y XI sX �r r r r r r r r r r r r tY �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ XI ×X XI sX � tX � FX � L2 ×M1 L1 l1 � r2 � P2 � Q1 � Y I ×Y Y I sY � tY � FY � Y I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' sY �▲▲▲▲▲▲▲▲▲▲▲▲ tY �rrrrrrrrrrrr X N1 q1 � q′ 1 � Y This yields what will be called the vertical composite of the fibrant resolved 2-spans, W2 and W1, as displayed below, W2#1W1 := X M2 p2 � p′ 2 � Y XI sX � tX � L2 ×M1 L1 FX◦l1 � FY ◦r2 � P2 � Q1 � Y I sY � tY � X N1 q1 � q′ 1 � Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, the window, W2#1W1, is a HF fibrant resolved 2-span such that W2#1W1 : � (p2, M2, p′ 2): X → Y � =⇒ � (q1, N1, q′ 1): X → Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 111 Given a HF resolved 2-span, W = (lX, P, L, Q, rY ): � (p, M, p′): X → Y � =⇒ � (q, N, q′): X → Y � , and (x, y) ∈ X × Y, we have its vertical HF span of slices at x and y, as defined in Definition 126, which idenoted [x|W|y] = � P W x,y, [x|L|y], QW x,y � : {x|M|y} → {x|N|y} = \uf8eb \uf8ec \uf8ed [x|L|y] P W x,y �♦♦♦♦♦ QW x,y �◆ ◆ ◆ ◆ ◆ {x|M|y} {x|N|y} \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following simple lemma will be essential later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We continue to use the notation introduced earlier in this section and are using the composition of HF fibrant spans as in Definition 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There exists an isomorphism55 of HF spans, from {x|M2|y} to {x|N1|y} [x|W2#1W1|y] ∼= [x|W2|y] • [x|W1|y].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the fact that the concatenation of two paths, a γ−→ a and a γ′ −→ a, in any space, see (46), is a constant path at a if, and only if, γ, γ′ = consta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ As before, suppose that we have HF spaces, X and Y , and vertically composable HF fibrant resolved 2-spans, W2 : � (p2, M2, p′ 2): X → Y � =⇒ � (p1, M1, p′ 1): X → Y � , W1 : � (p1, M1, p′ 1): X → Y � =⇒ � (q1, N1, q′ 1): X → Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX and yY be subsets of X and Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the corresponding Vect-profunctors, H M2 (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, H M1 (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect, and H N1 (xX,yY ) : π1(X, xX)op × π1(Y, yY ) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The HF fibrant resolved 2-spans W2 and W1 give rise to natural transformations of Vect-profunctors, as in Definition 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composite of the natural transformations, H M2 (xX,yY ) 2HW2 (xX ,yY ) � H M1 (xX,yY ) 2HW1 (xX ,yY ) � H N1 (xX,yY ) , is H M2 (xX,yY ) 2H W2#1W1 (xX ,yY ) � H N1 (xX,yY ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 55See Definition 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 112 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ xX and y ∈ yY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We claim that the composite of the linear maps, H M2 (xX,yY )(x, y) � 2HW2 (xX ,yY ) � (x,y) � H M1 (xX,yY ) � 2HW1 (xX ,yY ) � (x,y) � H N1 (xX,yY ) , is H M2 (xX,yY ) � 2H W2#1W1 (xX ,yY ) � (x,y) � H N1 (xX,yY ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows by combining Lemma 141 with the s = 0 case of Lemma 71, and using Definition 130 and Equation (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Towards horizontal and vertical identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We still have to examine if the suggested compositions, both horizontal and vertical, have identities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is needed, also, to partially answer the query left over from §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertical identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let (p, M, q): X → Y be an HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define the following window56, id2 (p,M,q) := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M p � q � Y XI sX � tX � M I lX � rY � sM � tM � Y I sY � tY � X M p � q � Y \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, given γ : I → M, we put lX(γ) = p ◦ γ and rX(γ) = q ◦ γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This definition is motivated by the construction of the bicategory 2Cob(n,n+1,n+2), below in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the diagram above is a function space coun- terpart of the vertical identity of a cobordism, as given in item (7) on page 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 143.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We do not know whether id2 (p,M,q) is, in general, a fibrant window or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Whenever id2 (p,M,q) is a fibrant window (which holds in all cases required in the construction of the once-extended Quinn TQFT, in Section 6), we note that it will be a HF fibrant resolved 2-span, connecting (p, M, q) to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is because M I, XI and Y I are all HF, as they are homotopic to M, X and Y , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let (p, M, q): X → Y be a HF fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that id2 (p,M,q) is a HF fibrant resolved 2-span, therefore connecting (p, M, q) to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this situation, given any subsets xX ⊆ X and yY ⊆ Y , the natural transformation, 2H id2 (p,M,q) (xX,yY ) : H M (xX,yY ) =⇒ H M (xX,yY ), is the identity natural transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ xX and y ∈ yY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If m, n ∈ {x|M|y}, then � PCm({x|M|y}) | (2H id2 (p,M,q) (xX,yY ))(x,y) | PCn({x|M|y}) � = χπ \uf8eb \uf8ed \uf8ee \uf8f0 m x M I y n \uf8f9 \uf8fb \uf8f6 \uf8f8 χπ� PCn({x|M|y}) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 56As usual, given a path, γ, in a space, X, sX(γ) = γ(0) and tX(γ) = γ(1), and the similarly for Y , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 113 Now note that, by Example 51 and Remark 68, we have an HF fibrant span, {x|M|y}I s{x|M|y} �❣❣❣❣❣ t{x|M|y}�❲ ❲ ❲ ❲ ❲ {x|M|y} {x|M|y}, and that \uf8ee \uf8f0 m x M I y n \uf8f9 \uf8fb = � s{x|M|y}, t{x|M|y} �−1(m, n), so we only need to apply the first part of Lemma 73, for the case s = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Horizontal identities and unitors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a HF space and consider the HF fibrant span, (sX, XI, tX): X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xX ⊆ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The associated Vect-profunctor, H XI (xX,xX) : π1(X, xX)op × π1(X, xX) → Vect, is such that, given x ∈ xX and y ∈ xX, (x, y) H XI (xX ,xX ) �−−−−−−−−−−→ Lin � �π0({x|XI|y}) � ∼= Lin � homπ1(X,xX)(x, y) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A holonomy map, λXI, see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, for the fibration, ⟨sX, tX⟩: XI → X × X, can be constructed so that, given paths in X, γ : x → y, γl : x′ → x and γr : y → y′, ΓXI (γl,γr)(γ) = γl ∗ γ ∗ γr, the concatenation of three paths, each fitting into a third of [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, this means that, as a morphism, from x′ to y′, in π1(X, xX), (applying the comments just after Definition 111), HXI (xX,xX)([γl], [γr])([γ]) = [γl][γ][γr].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor associated with the identity span, (sX, XI, tX): X → X, is, there- fore, canonically isomorphic to the horizontal identity, Idπ1(X,xX), of π1(X, xX), in the bicategory vProfGrp;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Example 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Continuing this approach, we now discuss a type of “would be” unitor for HF fibrant spans, given by certain HF fibrant resolved 2-spans, and also how the bona fide unitors in the bicategory vProfGrp can be obtained from the former by computing the associated natural transformations of profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be crucial for constructing the once-extended Quinn TQFT in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will only discuss left unitors, as the case of right unitors is analogous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X and Z be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a HF fibrant span, (p, M, q): X → Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We suppose, and this will be the case in all settings that we need for constructing the once-extended Quinn TQFT in Section 6, that the following conditions, (1) – (3), are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (These conditions may seem a bit mysterious at this stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However, as we will see later, they arise naturally from the construction of the unitors of a cobordism in the bicategory 2Cob(n,n+1,n+2), when looking at their function space counterparts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, especially item (9), starting on page 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (1) We have a homeomorphism, Φ: XI ×X M → M, making the diagram (48) M p �✐✐✐✐✐✐✐✐✐ q �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ X Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XI×XM Φ � p′ �❚❚❚❚❚❚ q′ �❥ commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 114 (As we will see in the following section, this homeomorphism is an ana- logue of a collar of the boundary of a manifold, when considering spaces of functions on manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Here we used the pullback diamond inside the commutative diagram XI×XM �❤❤❤❤❤ �❱ ❱ ❱ ❱ ❱ q′ � p′ � XI sX �❥❥❥❥❥❥ tX �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ M p �❣❣❣❣❣❣❣❣❣ q �❚ ❚ ❚ ❚ ❚ ❚ X X Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Let x, x′ ∈ X and z ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a path, x γ−→ x′, in X and m ∈ {x|M|z}, then Φ(γ, m) ∈ {x′|M|z} is in the same path-component as ΓM ⟨γ,constz⟩(m) ∈ {x′|M|z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, recalling the notation in Lemma 94, ΓM ⟨γ,constz⟩ : {x|M|z} → {x′|M|z} is defined from the fibration ⟨p, q⟩: M → X × Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) The following window is fibrant (and further note that all spaces appearing are HF), λ(p,M,q) X := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X XI ×X M p′ � q′ � Z XI sX � tX � M I lX � rZ � Φ−1◦sM � tM � ZI sZ � tZ � X M p � q � Z \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, lX(γ) = p ◦ γ and rZ(γ) = q ◦ γ, if γ ∈ M I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X and Z be HF spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let a HF fibrant span, (p, M, q): X → Z, satisfy the conditions, (1) – (3), just outlined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose xX ⊂ X and zZ ⊂ Z, and that (X, xX) is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x, x′ ∈ xX, γ ∈ {x|XI|x′}, z ∈ zZ, and m ∈ {x′|M|z}, m′ ∈ {x|M|z}, then � PC(γ,m) �� x|XI ×X M|z �� ��� � 2H λ(p,M,q) X (xX,zZ) � (x,z) ��� PCm′� {x|M|z} �� = � 1, if PCΓM ⟨γ,constz⟩(m) � {x|M|z} � = PCm′� {x|M|z} � , 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the natural transformation, 2H λ(p,M,q) X (xX,zZ), of profunctors gives the ap- propriate left-unitor, λ HM (xX ,zZ ) π1(X,xX), for π1(X, xX), in the bicategory vProfGrp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More precisely, the composite of the natural transformations of profunctors, below, from π1(X, xX) to π1(Z, zZ), Idπ1(X,xX) • H M (xX,zZ) ∼ = =⇒ H XI (xX,xX) • H M (xX,zZ) ηXI ,M (xX ,xX ,zZ ) ========⇒ H XI×XM (xX,zZ) 2H λ(p,M,q) X (xX ,zZ ) ========⇒ H M (xX,zZ), A CATEGORIFICATION OF QUINN’S TQFT 115 is Idπ1(X,xX) • H M (xX,zZ) λ HM (xX ,zZ ) π1(X,xX ) =======⇒ H M (xX,zZ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that here the first equivalence is discussed earlier in this section, and the second is in Lemma 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof of the first statement is exactly as in the proof of Lemma 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The second statement follows by passing to the language of profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Comment and Summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this short summary and commentary, we ask again Is H a bifunctor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, and add Is it symmetric monoidal?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There have been a lot of fairly technical results in this section and it is easy to lose track of what they say in toto, so we will step back to look at why they were necessary in the form we gave, using the above questions as a guide.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The answer is in two parts: (i) it is almost, but not quite, and then (ii) from the point of view of the composite ‘constructions’ that we will examining in the next section, the lack of that property does not make any difference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For H to be a (symmetric monoidal) bifunctor, an elementary prerequisite would be that it had a symmetric monoidal bicategory as its domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section, we have constructed a bicategorical type of object, though not quite a bicategory, that we will from now on denote by 2span(HF), following the description starting in page 6 of the Introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The objects of 2span(HF) are homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given homotopy finite spaces, X and Y , the 1-cells, from X to Y , are homotopy finite fibrant spans, (p, M, q): X → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a non-associative composition, •, of 1-cells, obtained via the obvious pullback, see Lemma 55 and Definition 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each homotopy finite space X has a ‘horizontal identity’, given by the path-space fibrant span, (sX, XI, tX): X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given 1-cells (p, M, p′), (q, N, q′): X → Y , the 2-cells, in 2span(HF), connecting them, consist of homotopy finite resolved 2-spans (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3), as below, W = (lX, P, L, Q, rY ): � (p, M, p′): X → Y � =⇒ � (q, N, q′): X → Y � , or, in full, W = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X M p � p′ � Y XI sX � tX � L lX � rY � P � Q � Y I sY � tY � X N q � q′ � Y \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again, homotopy finite resolved 2-spans can be composed horizontally and ver- tically, as described in detail in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' None of these compositions is associative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The vertical composition can be made associative by considering fi- brant resolved 2-spans, up to the equivalence relation in Definition 122, similarly to the construction of the category HFspan in Subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not develop this further, as we do not need it here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') As discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if we apply certain restrictions on the 1-cells (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): X → Y that we allow (which will be automatically satisfied in the cases arising in the construction of the once-extended Quinn TQFT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in Section 6),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we then have ‘vertical identities’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' id(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) : � (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): X → Y � =⇒ � (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' p′): X → Y � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 116 as well as ‘unitor 2-cells’,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' whenever a 1-cell comes equipped with the function space analogue of a collar neighbourbood of the boundary of a manifold,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ρ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q) X : (X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) −−−−−→ Y ) • (Y (sY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='tY ) −−−−−−−→ Y ) =⇒ (X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) −−−−−→ Y ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λ(p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='q) X : (X (sX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='XI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='tX) −−−−−−−→ X) • (X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) −−−−−→ Y ) =⇒ (X (p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='p′) −−−−−→ Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is quite possible that, by considering instead equivalence classes of homotopy finite resolved 2-spans, we could, in this way, obtain a bicategory from 2span(HF), categorifying the category HFspan, but this will not be needed for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similar constructions are in [59, 91] and in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout Section 5, we constructed an ‘assignment’, from now on denoted H = � π1(−, −), H, 2H � : 2span(HF) → vProfGrphf, more precisely a map of 2-truncated globular sets, that gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Each homotopy finite space, X, is sent to its fundamental groupoid, π1(X, X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Given a homotopy finite fibrant span, X (p,M,p′) −−−−−→ Y , we have a Vect- profunctor, as defined in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, H � X (p,M,p′) −−−−−→ Y � : π1(X, X)op × π1(Y, Y ) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) Given a homotopy finite fibrant resolved 2-span, W: (p, M, p′) =⇒ (q, N, q′), as above, we have a natural transformation, of functors π1(X, X)op × π1(Y, Y ) → Vect, as discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, 2HW : H � X (p,M,p′) −−−−−→ Y � =⇒ H � X (q,N,q′) −−−−−→ Y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It follows from the sequence of results in Section 5 that the assignment, H: 2span(HF) → vProfGrphf, preserves all various compositions, and the horizontal identities in 2span(HF) and in vProfGrphf, up to applying appropriate natural isomorphisms, and that vertical identities, and unitors likewise are preserved by H, whenever they exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore H is as close to being a bifunctor as it can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a relative variant of 2span(HF), from now on denoted 2span(HF), where homotopy finite spaces, X, come equipped with subsets, xX ⊂ X, such that (X, xX) is 0-connected (meaning that xX has at least one point in each path- component of X), and the rest of the ‘bicategorical’ structure of 2span(HF) is induced by that of 2span(HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also saw in this section that H can be modified to give a ‘bifunctor’, H: 2span(HF) → vProfGrphf, that gives the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Each pair, (X, xX), is sent to the corresponding fundamental groupoid, π1(X, xX).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We will, in the following section, furthermore suppose that xX is finite, so given that X is homotopy finite, it follows that π1(X, xX) is then a finite groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (2) Given a homotopy finite fibrant span, X (p,M,p′) −−−−−→ Y , xX ⊂ X, and yY ⊂ Y , we thus have a 1-cell, (p, M, p′): (X, xX) → (Y, yY ), in 2span(HF), and a Vect-profunctor, as defined in Notation 113, H(xX,yY ) � X (p,M,p′) −−−−−→ Y � : π1(X, xX)op × π1(Y, yY ) → Vect, A CATEGORIFICATION OF QUINN’S TQFT 117 obtained by restricting H � X (p,M,p′) −−−−−→ Y � : π1(X, X)op × π1(Y, Y ) → Vect to the subgroupoid π1(X, xX)op × π1(Y, yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) Finally, if we are given xX ⊂ X and yY ⊂ Y , and a 2-cell in 2span(HF), W: (p, M, p′) =⇒ (q, N, q′), we have 2-cell in 2span(HF), W: � (p, M, p′): (X, xX) → (Y, yY ) � =⇒ � (q, N, q′): (X, xX) → (Y, yY ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Definition 130, we, then, have a natural transformation of pro- functors, 2H W (xX,yY ) : H(xX,yY ) � X (p,M,p′) −−−−−→ Y � =⇒ H(xX,yY ) � X (q,N,q′) −−−−−→ Y � , obtained by restricting 2HW : H � X (p,M,p′) −−−−−→ Y � =⇒ H � X (q,N,q′) −−−−−→ Y � , to π1(X, xX)op × π1(Y, yY ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section, we also proved that, just as for H: 2span(HF) → vProfGrphf, the relative version H: 2span(HF) → vProfGrphf, preserves all the various compositions, plus the unitors and identities (when they exist) up to natural isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The most challenging calculation concerned the fact that the natural transformations 2H W (−,−) send the horizontal composition of fibrant resolved 2-spans of homotopy finite spaces to that of natural isomorphisms between profunctors, which was dealt with in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also saw in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, buiding from lemmas 114 and 115, that H, and similarly H, takes the product / cartesian monoidal structure in 2span(HF) to the tensor product in vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be discussed further in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, where it will be furthermore written down in the language of symmetric monoidal bifunctors, as in Definition 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to define our once-extended versions of Quinn’s TQFT, we will only need to use 2span(HF) as a ‘half-way house’ between the bicategory, 2Cob(n,n+1,n+2), of 2-cobordisms, that we will introduce in detail in the next section, and vProfGrphf, the second part of this process being given by H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the symmetric monoidal bi- category, vProfGrphf, the images of our possible problem ‘equations’, that do not hold in 2span(HF), are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, in defining H, and similarly H, and checking that it preserves the horizontal and vertical compositions and identi- ties, we have that the composite assignment will be a bifunctor, and, later on, will similarly have that it is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, however, we are getting a bit ahead of ourselves and do need to define and study the once-extended TQFT and its variants in some more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What we can claim to know for certain at this stage of the paper is that the resulting basic form of that extended TQFT will be a bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More will be revealed shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Once-extended versions of Quinn’s TQFT Let X be a space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section, we will frequently abbreviate ιX k : X → X × [0, 1] to ιk, hence ιk(x) = ιX k (x) = (x, k), usually for k = 0 or k = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions for the bicategory 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bicategory, 2Cob(n,n+1,n+2), is that of closed smooth n-manifolds, (n + 1)-cobordisms between manifolds, and equivalence classes of (n + 2)-extended cobordisms connecting (n + 1)-cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details of the construction are in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2] and [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some other important definitions are here in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, A CATEGORIFICATION OF QUINN’S TQFT 118 and we will give an overview in what follows, so as to set out the conventions we will be using57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bicategory, 2Cob(n,n+1,n+2), is thus defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) The class of objects of 2Cob(n,n+1,n+2) is the class of all smooth closed (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' compact and with empty boundary) n-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Given closed smooth n-manifolds, Σ and Σ′, a 1-morphism, (i, S, j): Σ → Σ′, is a cospan, in the category of smooth manifolds and smooth maps, as below, and will be called a (n + 1)-cobordism, Σ i �■ ■ ■ ■ ■ ■ Σ′ j �tttttt S .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This should be such that S is a compact smooth (n + 1)-manifold, possibly with a non-empty boundary, and the universally defined map, ⟨i, j⟩: Σ ⊔ Σ′ → S, gives a diffeomorphism, Σ ⊔ Σ′ ∼= ∂(S), the boundary of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) The composition of the 1-morphisms (i, S, j): Σ → Σ′ and (i′, S′, j′): Σ′ → Σ′′, denoted (i, S • S′, j′) := (i, S ⊔Σ S′, j′), is given by considering the pushout, in CGWH, included as the diamond in the commutative diagram, Σ i � i �❉ ❉ ❉ ❉ ❉ ❉ ❉ Σ′ j �sssssssss i′ �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ Σ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' j′ � j′ �✇✇✇✇✇✇✇ S k �❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ S′ k′ �ssssssss S ⊔Σ′ S′ (As already mentioned, in this paper, we implicitly choose a natural reali- sation for all limits and colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, we took the obvious choice, S ⊔Σ′ S′ = � (S × {0}) ∪ (S′ × {1}) � /j(s) ∼ i′(s), for all s ∈ Σ′, with the quotient topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We note that the pushout is formed in CGWH, so initially we forget the smooth structure on the given manifolds, and consider just their underlying structure as topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The smooth structure on S ⊔Σ′ S′ is then inserted afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As recalled in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, in order to give a smooth structure to S ⊔Σ S′, we could, for instance, consider collars of Σ′ in S and S′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' How- ever the collars are not part of the structure given here to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This issue can be resolved as in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2 and §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2], either by consid- ering “halations”, where collars essentially become part of the cobordism information, or applying the axiom of choice for classes, to endow each cobordism with appropriate collars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not say more on this issue (and essentially will ignore it when we come to compose extended cobor- disms, below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can safely do this as our constructions depend only on the underlying topological manifolds of the smooth manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (4) Given closed smooth n-manifolds, Σ1 and Σ2, and cobordisms, (i1, S, i2): Σ1 → Σ2 and (i′ 1, S′, i′ 2): Σ1 → Σ2, 57As in previous sections, we make no assumption that orientations exist on the manifolds, cobordisms, nor now on the extended cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 119 the 2-morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [K]: � (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i2): Σ1 → Σ2 � =⇒ � (i′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′ 2): Σ1 → Σ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' between them,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' are given by equivalence classes of diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the category of manifolds and smooth maps,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of the form (49) below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' called (n + 2)- extended cobordisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' K: � (i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i2): Σ1 → Σ2 � =⇒ � (i′ 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′ 2): Σ1 → Σ2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (49) K = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i1 � ιΣ1 0 � S iS � Σ2 i2 � ιΣ2 0 � Σ1 × I iE � K Σ2 × I iW � Σ1 ιΣ1 1 � i′ 1 � S′ iS′ � Σ2 i′ 2 � ιΣ2 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here K is a compact smooth (n + 2)-manifold with corners, called the support of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The E on the middle right pointing map is there to indicate that the arrow is ‘pointing’ east in the diagram, and the W, similarly, is pointing west.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Dually to the ideas of windows and fibrant resolved 2-spans, see Defini- tion 117 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, the frame, fr(K), of an extended cobordism, K, as in (49), is defined to be (50) fr(K) := colim \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i1 � ιΣ1 0 � S Σ2 i2 � ιΣ2 0 � Σ1 × I Σ2 × I Σ1 ιΣ1 1 � i′ 1 � S′ Σ2 i′ 2 � ιΣ2 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a canonically defined map, fK : fr(K) → K, as we had for HF resolved 2-spans, hence similarly called the filler of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is required that fK provides a diffeomorphism fr(K) ∼= ∂(K), the boundary of the manifold with corners, K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', page 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (5) Two extended cobordisms, K, K′ : � (i1, S, i2): Σ1 → Σ2 � =⇒ � (i′ 1, S′, i′ 2): Σ1 → Σ2 � , so with the same frame, are called equivalent if there exists a diffeomor- phism, f : K → K′, between the supports of K and K′, that makes the diagram below commute, fr(K) id � fK � K f � fr(K′) fK′ � K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (6) The horizontal and vertical compositions of extended cobordisms are done via the obvious horizontal and vertical pushouts, dually to the case of HF resolved 2-spans, as treated in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see also [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2] and [91, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As already mentioned, we are omitting the details on how to A CATEGORIFICATION OF QUINN’S TQFT 120 construct smooth structures on the resulting topological manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will skip these, and refer to [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our constructions of once-extended Quinn TQFTs do not take into account smooth struc- tures on manifolds, but we note that the construction of the bicategory 2Cob(n,n+1,n+2) does make use of their existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') These two compositions of extended cobordisms descend to their equivalence classes, which defines the horizontal and vertical compositions of 2-morphisms in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (7) Given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, its vertical identity is the equivalence class of the extended cobordism, id2 (i1,S,i2) := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i1 � ιΣ1 0 � S ιS 0 � Σ2 i2 � ιΣ2 0 � Σ1 × I iE � S × I Σ2 × I iW � Σ1 ιΣ1 1 � i1 � S ιS 1 � Σ2 i2 � ιΣ2 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here iE(s, t) = (i1(s), t), for s ∈ Σ1 and t ∈ I, and, similarly, iW (s, t) = (i2(s), t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (8) Given a smooth compact n-manifold, Σ, the horizontal identity of Σ is id1 Σ := (ιΣ 0 , Σ × I, ιΣ 1 ): Σ → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (9) Given an (n + 1)-cobordism, (i1, S, i2): Σ1 → Σ2, we have left and right unitor (n + 2)-extended cobordisms, λ′(i1,S,i2) Σ1 : (ιΣ1 0 , Σ1 × I, ιΣ1 1 ) • (i1, S, i2) =⇒ (i1, S, i2), and ρ′(i1,S,i2) Σ2 : (i1, S, i2) • (ιΣ2 0 , Σ2 × I, ιΣ2 1 ) =⇒ (i1, S, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The support of both is S × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will explain the construction of the left unitor extended cobordism, λ′(i1,S,i2) Σ1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The construction of the right unitor extended cobordism is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the (n + 1)-cobordism, � i′ 1, (Σ1 × I) ⊔Σ1 S, i′ 2 � = (ιΣ1 0 , Σ1 × I, ιΣ1 1 ) • (i1, S, i2), and also an explicit isomorphism of cospans, S Φ � Σ1 i1 �♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ i′ 1 �▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Σ2 i2 �◆◆◆◆◆◆◆◆◆◆◆◆◆ i′ 2 �qqqqqqqqqqq (Σ1 × I) ⊔Σ1 S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that to construct such a homeomorphism, Φ: S → (Σ1 × I) ⊔Σ1 S, we need a collar for the inclusion of Σ1 in S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The left unitor extended cobordism is defined by tweaking the vertical identity of (i1, S, i2): Σ1 → A CATEGORIFICATION OF QUINN’S TQFT 121 Σ2, as shown in the diagram58, λ′(i1,S,i2) Σ1 := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i′ 1 � ιΣ1 0 � (Σ1 × I) ⊔Σ1 S ιS 0 ◦Φ−1 � Σ2 i′ 2 � ιΣ2 0 � Σ1 × I iE � S × I Σ2 × I iW � Σ1 ιΣ1 1 � i1 � S ιS 1 � Σ2 i2 � ιΣ2 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The equivalence classes of the left and right unitor extended cobordisms give the left and right unitors in 2Cob(n,n+1,n+2), as denoted below, λ(i1,S,i2) Σ1 = [λ′(i1,S,i2) Σ1 ]: (ιΣ1 0 , Σ1 × I, ιΣ1 1 ) • (i1, S, i2) =⇒ (i1, S, i2), and ρ(i1,S,i2) Σ2 = [ρ′(i1,S,i2) Σ2 ]: (i1, S, i2) • (ιΣ2 0 , Σ2 × I, ιΣ2 1 ) =⇒ (i1, S, i2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In addition to the above basic structure, we note that, in the classical setting, the category, Cob(n,n+1), has the structure of a symmetric monoidal category with the coproduct / disjoint union, ⊔, as the tensor product, as recalled in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, and that the extended form, 2Cob(n,n+1,n+2), similarly, has a symmetric monoidal bicate- gory structure, again having ⊔ as its tensor product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='15 of Carboni, Kelly, Walters and Wood, [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An explicit proof is in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will revisit this structure in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, particularly §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A once-extended version of Quinn’s TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, n will be a non- negative integer, and B a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These will be the standard as- sumptions throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider an (n, n + 1)-cobordism, between closed smooth n-manifolds, as in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, viewed as a cospan in the category CGWH, (i, S, j) := \uf8eb \uf8ed Σ i �■ ■ ■ ■ ■ ■ Σ′ j �tttttt S \uf8f6 \uf8f8 , so the nodes only encode the data of the underlying topological manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying the contravariant mapping space functor, B(−) : CGWH → CGWH, sends this cospan to a span in CGWH, whose nodes contain the corresponding spaces of maps from the topological manifolds into B, (i∗, BS, j∗) := \uf8eb \uf8ec \uf8ed BS i∗ �ssssss j∗ �▲ ▲ ▲ ▲ ▲ ▲ BΣ BΣ′ \uf8f6 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 146.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This span, (i∗, BS, j∗), of function spaces is a fibrant span in which all the spaces appearing are homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 58We will see, later on in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, several instances of this type of construction, which will be studied for use in defining further structure on 2Cob(n,n+1,n+2), and verifying that that structure satisfies the required equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 122 Now consider an extended (n, n + 1, n + 2)-cobordism with 2-cospan diagram59 as follows, K := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i1 � ι0 � S iN � Σ2 i2 � ι0 � Σ1 × I iE � K Σ2 × I iW � Σ1 ι1 � i′ 1 � S′ iS � Σ2 i′ 2 � ι1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying the same contravariant functor, B(−), to K gives a dual ‘window’, BK := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed BΣ1 � i∗ 1 � ι∗ 0 BS� i∗ N BΣ2 � i∗ 2 � ι∗ 0 BΣ1×I � i∗ E BK BΣ2×I � i∗ W BΣ1 � ι∗ 1 � i′∗ 1 BS′� i∗ S BΣ2 � i′∗ 2 � ι∗ 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recalling the definition of fibrant resolved 2-spans in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we get the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 147.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The window, BK, of mapping spaces is a fibrant resolved 2-span, in which all the spaces appearing are homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, the applica- tion of B(−) preserves the compositions of all cobordisms and extended cobordisms, sending them to the corresponding compositions of fibrant spans, as in Definition 56, and of fibrant resolved 2-spans, as in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The vertical units in 2Cob(n,n+1,n+2), as well as the horizontal identities, and unitors, are also sent to those of §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, when passing to the mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that CGWH is cartesian closed, so BΣ1×I ∼= (BΣ1)I, canonically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will prove Lemmas 146 and 147 together as the proofs are related.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It may be useful to compare with the proof of our earlier Lemma 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will continue to refer to the mapping space picture as being ‘dual’ to the other one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibrancy of the dual span follows from the fact that the inclusion of Σ⊔Σ′ ∼= ∂(S) into S is a cofibration and, similarly, for the dual window, the inclusion of fK : fr(K) ∼= ∂(K) into K is a cofibration, so the dual map, f ∗ K : BK → Bfr(K) ∼= fr(BK), is a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (In that last step, we again used the fact that CGWH is cartesian closed, so the mapping space contravariant functor, B( ) : CGWH → CGWH, send colimits to limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') For the latter reason, all compositions are pre- served (up to isomorphism) when going from cobordisms and extended cobordisms to fibrant spans and fibrant resolved 2-spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to prove that all spaces in BK are homotopy finite, we use the fact that all compact smooth manifolds can be given the structure of a finite CW-complex, and use, once again, Lemma 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Vertical units, horizontal units, and unitors, are preserved by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We also note the following, that once again follows from the fact that CGWH is cartesian closed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 59As before this is a diagram in CGWH, and this will be the same for all subsequent diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 123 Lemma 148.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two manifolds, Σ1 and Σ2, in CGWH, we have a natural isomorphism, BΣ1⊔Σ2 ∼= BΣ1 × BΣ2, and this is true also for 1- and 2-cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ This interprets as saying that ‘taking the mapping space’ converts the ⊔-monoidal structure of CGWH to the − × − one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall, now, the construction of the bicategory, vProfGrphf, defined in Subsec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8, particularly §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 149 (The once-extended Quinn TQFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended Quinn TQFT, denoted 2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, is defined to be the bifunctor given by: if Σ is a closed n-manifold, then 2Q0 B(Σ) := π1(BΣ, BΣ), which defines 2QB on objects;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if (i, S, j): Σ → Σ′ is an (n + 1)-cobordism, then: 2Q1 B � Σ (i,S,j) −−−−→ Σ′ � := H � BΣ (i∗,BS,j∗) −−−−−−−→ BΣ′� : π1(BΣ, BΣ) ↛ π1(BΣ′, BΣ′), where we are using the notation, H, from Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, particularly Definition 111;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and the equivalence class of an extended cobordism, as in Equation (49), K: � (i1, S, i2): Σ1 → Σ2 � =⇒ � (i′ 1, S′, i′ 2): Σ1 → Σ2 � , is sent to the natural transformation of functors, π1(BΣ1, BΣ1)op × π1(BΣ2, BΣ2) → Vect, defined as (using the construction in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, particularly Lemma 129), 2Q2 B([K]) := 2HBK : H � BΣ1 (i∗ 1,BS,i∗ 2) −−−−−−→ BΣ2 � =⇒ H � BΣ1 (i′∗ 1 ,BS′,i′∗ 2 ) −−−−−−−−→ BΣ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It should perhaps be noted that the name we have used here needs justifying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have not as yet shown that the structure outline above does give a once extended TQFT as that will require a proof that the bifunctor is symmetric monoidal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That will be shown later (see Theorem 176 in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the constructions60 in Section 5, combined with the previous lemmas, it follows that we do indeed have a bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fact that 2QB preserves the composition of cobordisms is in Proposition 116, that 2QB preserves the horizontal composition of extended cobordisms follows from Proposition 139, and that 2QB preserves the vertical composition of extended cobordisms is dealt with by Lemma 142.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Preservation of vertical identities follows from Lemma 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, preservation of horizontal identities and unitors follows from the discussion in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, particularly Lemma 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note also that if Σ is a smooth closed manifold, then the groupoid, 2Q0 B(Σ) = π1(BΣ, BΣ), is homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows since the function space, BΣ, is homo- topy finite (Lemma 79) and so, given a pair of objects, f, f ′ : Σ → B, of π1(BΣ, BΣ), the set of arrows from f to f ′, in π1(BΣ, BΣ), is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 60It may be useful to refer to the summary of Section 5 in Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 124 Remark 150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let 2Cob′(n,n+1,n+2) be obtained from 2Cob(n,n+1,n+2), by consid- ering the 2-cells to be extended cobordisms, therefore not considering the latter to be up to equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Because the vertical composition of extended cobordisms is not associative, 2Cob′(n,n+1,n+2) is then not a bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However, we still have com- positions, units and unitors, so 2Cob′(n,n+1,n+2) is, similarly to 2span(HF), a 2-truncated cubical set with compositions, see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, the map- ping space construction B−, in Lemma 147, gives rise to a map of 2-truncated globular sets, B(−) : 2Cob′(n,n+1,n+2) → 2span(HF), which preserves all compositions, units and unitors up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once- extended Quinn TQFT arises from the composite of assignments, below, using the notation of Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8, 2Cob′(n,n+1,n+2) B(−) −−−→ 2span(HF) H −→ vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the remark below, we use, from items (13) and (15) in the discussion on page 16, that, if X is a CGHW space, and x ∈ X, then the path-component in X to which x belongs, will be denoted PCx(X), and that we denote the set of all such path-components by �π0(X) = {PCx(X) : x ∈ X}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note, again, that different x ∈ X may induce the same PCx(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We also note the definition of the homotopy content, χπ(B), of a homotopy finite space B, given in Definition 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 151.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can give a more explicit description of 2QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On objects, 2QB sends a closed n-manifold, Σ, to the fundamental groupoid, π(BΣ, BΣ), of the function space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a cobordism, (i1, S, i2): Σ1 → Σ2, then, if f1 : Σ1 → B and f2 : Σ2 → B are continuous functions, we have 2Q1 B(i1, S, i2)(f1, f2) = Lin � �π0({f1|BS|f2}) � , where Lin: Set → Vect is the free vector space61 functor, and {f1|BS|f2}, in full {f1|B(i∗ 1,S,i∗ 2)|f2}, is the space of maps, H : S → B, such that the diagram, B Σ1 f1 �❧ i1 �❙ ❙ ❙ ❙ ❙ ❙ Σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f2 �❙❙❙❙❙❙ i2 �❦❦❦❦❦❦ S H � , commutes, given with the induced CGWH topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given paths, γ1 : f ′ 1 → f1 in BΣ1 and γ2 : f2 → f ′ 2 in BΣ2, the linear map, 2Q1 B(i1, S, i2)(f ′ 1 [γ1] −−→ f1, f2 [γ2] −−→ f ′ 2) : Lin � �π0({f1|BS|f2}) � → Lin � �π0({f ′ 1|BS|f ′ 2}) � , is defined from the functor, F(BS) : π1(BΣ1 × BΣ2, BΣ1 × BΣ2) → CGWH/ ≃, obtained from the path-space fibration, ⟨i1, i2⟩∗ : BS → BΣ1 × BΣ2 ∼= BΣ1⊔Σ2, in the usual way (see [87, Chapter 7], as reviewed in Lemma 96), and then by applying �π0 : CGWH/ ≃→ Set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' finally linearising by applying Lin: Set → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that the path γ1 : f ′ 1 → f1 must be inverted before applying F(BS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') An explicit description can be obtained from the comments just after Definition 111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 61Recall we are working over an arbitrary subfield of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 125 Given an extended cobordism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' K := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ1 i1 � ι0 � S1 iN � Σ2 i2 � ι0 � Σ1 × I iE � K Σ2 × I iW � Σ1 ι1 � j1 � S2 iS � Σ2 j2 � ι1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then the natural transformation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of profunctors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2Q2 B([K]): 2Q1 B((i1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i2): Σ1 → Σ2) =⇒ 2Q1 B((j1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' j2): Σ1 → Σ2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is such that,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if f1 : Σ1 → B and f2 : Σ2 → B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given H1 ∈ {f1|BS1|f2} and H2 ∈ {f1|BS2|f2},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then we have the following formula for the matrix elements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (51) � PCH1 � {f1|BS1|f2} � | � 2Q2 B([K]) � (f1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f2) | PCH2 � {f1|BS2|f2} �� = χπ \uf8eb \uf8ec \uf8ec \uf8ed \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 T : K → B �������� T ◦ iN = H1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' T ◦ iS = H2 ∀s ∈ Σ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∀t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1] : T (iE(s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' t)) = f1(s) ∀s′ ∈ Σ2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∀t ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1] : T (iW (s′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' t)) = f2(s′) \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe \uf8f6 \uf8f7 \uf8f7 \uf8f8 χπ� PCH2 � {f1|BS2|f2} �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (It follows from the construction in Section 5, in particular Lemma 125, that we are indeed considering homotopy contents only of homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Note that, unless B is a finite set with the discrete topology, then for Σ, a closed smooth n-manifold, the groupoid, 2Q0 B(Σ) = π1(BΣ, BΣ), is uncountable, since it has an uncountable set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However 2Q0 B(Σ) is homotopy finite, so we can extract an equivalent finite subgroupoid from it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Namely, if we choose a finite subset, f Σ ⊂ BΣ, containing at least one element for each path-component, then π1(BΣ, f Σ) will be a finite groupoid, equivalent to 2Q0 B(Σ) = π1(BΣ, BΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter fact will be used in the next subsection to construct a finitary version of the once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A finitary version of the once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For technical and historical reasons in the applications of the above theory, it is often useful to replace groupoids that have possibly infinitely many objects, by more finitary, but equivalent, ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are several useful ways of doing this, for instance, using triangulations of the manifolds, and CW decompositions of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will, however, reduce the size of the groupoids by a different means as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As always, let B be a homotopy finite space, and n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 152 (B-decorated manifold).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A B-decorated n-manifold, Σ = (Σ, fΣ), is given by a closed smooth n-manifold, Σ, called the underlying manifold of Σ, together with a finite subset, fΣ, of BΣ, containing at least one function, f : Σ → B, for each path component of the space, BΣ, of functions from Σ to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Σ be a closed smooth manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall, [94], that Σ has a finite CW- decomposition62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since B is homotopy finite, BΣ is homotopy finite (Lemma 79), 62which can be obtained using various means, A CATEGORIFICATION OF QUINN’S TQFT 126 and hence it has a finite number of path-components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we can see that all closed (smooth) manifolds possess a B-decoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 153 (2Cob (n,n+1,n+2) B ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define a bicategory, 2Cob (n,n+1,n+2) B , as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The objects are B-decorated n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given B-decorated n-manifolds, Σ = (Σ, f Σ) and Σ′ = (Σ′, fΣ′), the 1- morphisms, (i, S, j): Σ → Σ′, are given by (n+ 1)-cobordisms, (i, S, j): Σ → Σ′, with no additional struc- ture on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The (n + 1)-cobordism, (i, S, j): Σ → Σ′, associated to a 1- morphism (i, S, j): Σ → Σ′, will be called the underlying (n+1)-cobordism of that 1-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Given 1-morphisms, (i, S, j), (i′, S′, j′): Σ → Σ′, the 2-morphisms, [K]: � (i, S, j): Σ → Σ′� =⇒ � (i′, S′, j′): Σ → Σ′� , are given by equivalence classes of extended cobordisms, K: � (i, S, j): Σ → Σ′� =⇒ � (i′, S′, j′): Σ → Σ′� , with the equivalence relation as in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As for 1-morphisms, we define the underlying 2-morphism of 2Cob(n,n+1,n+2), associated to such a 2-morphism of 2Cob (n,n+1,n+2) B , by forgetting the B-decoration on the n- manifolds Σ and Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The rest of the bicategory structure in 2Cob (n,n+1,n+2) B is induced, in the obvious way, by that of the undecorated case, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by composing the underlying cobordisms or underlying 2-morphisms of the 1- and 2-morphisms in 2Cob (n,n+1,n+2) B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, the composition below, (Σ, f Σ) (i,S,j) −−−−→ (Σ′, fΣ′) (i′,S′,j′) −−−−−→ (Σ′′, f Σ′′) simply gives (Σ, f Σ) (i,S•S′,j′) −−−−−−−→ (Σ′′, fΣ′′), where (i, S • S′, j′) gives the composite of cobordisms as in item (3) on page 118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For convenience, we recall that the bicategory, vProfGrpfin, defined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, is the sub-bicategory of vProfGrp, whose objects are the finite groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a B-decorated manifold, Σ = (Σ, f Σ), the pair (BΣ, fΣ) is, by definition, 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From Lemmas 116 and 139, it follows that the bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, induces a bifunctor, 2QB : 2Cob (n,n+1,n+2) B → vProfGrphf, by restricting from π1(BΣ, BΣ) to π1(BΣ, f Σ), and leaving the rest of the structure unaltered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will give this structure in more detail shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since we assume that Σ is a closed smooth manifold, as above, it follows that BΣ is homotopy finite, and, thus, that the groupoid π1(BΣ, f Σ) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This leads to the following definition in which we are again using the notation of Definition 130 and from Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, in particular, from notational comment 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 127 Definition 154 (The finitary once-extended Quinn TQFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary once- extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, is the bifunctor defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Σ = (Σ, f Σ) is a B-decorated n-manifold, then 2Q 0 B(Σ) := π1(BΣ, fΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is a finite groupoid as f Σ is finite, B is a HF space, and thus so is BΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') If (i, S, j): Σ = (Σ, f Σ) → Σ′ = (Σ′, f Σ′) is a 1-morphism of 2Cob(n,n+1,n+2), whose underlying cobordism is (i, S, j): Σ → Σ′ = � Σ i �❘ ❘ ❘ ❘ ❘ ❘ Σ′ j �❧❧❧❧❧❧ S � , then 2Q 1 B � (Σ, f Σ) (i,S,j) −−−−→ (Σ′, fΣ′) � := H(f Σ,f Σ′) � BΣ (i∗,BS,j∗) −−−−−−−→ BΣ′� : π1(BΣ, fΣ) ↛ π1(BΣ′, fΣ′) abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = H(f Σ,fΣ′ ) � (i∗, BS, j∗) � : π1(BΣ, f Σ) ↛ π1(BΣ′, fΣ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Concretely the functor, 2Q 1 B � (Σ, f Σ) (i,S,j) −−−−→ (Σ′, f Σ′) � : π1(BΣ, f Σ)op × π1(BΣ′, fΣ′) → Vect, is the restriction to π1(BΣ, fΣ)op × π1(BΣ′, f Σ′), of the functor, 2Q1 B � Σ (i,S,j) −−−−→ Σ′ � : π1(BΣ, BΣ)op × π1(BΣ′, BΣ′) → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A 2-morphism, [K]: � (i1, S, i2): (Σ1, fΣ1) → (Σ2, f Σ2) � =⇒ � (i′ 1, S′, i′ 2): (Σ1, fΣ1) → (Σ2, fΣ2) � , arising as the equivalence class of an extended cobordism, K: � (i1, S, i2): Σ1 → Σ2) � =⇒ � (i′ 1, S′, i′ 2): (S′ : Σ1 → Σ2) � , is sent to the natural transformation, 2HBK (f Σ1,fΣ2 ) : H(fΣ1 ,fΣ2 )(i∗ 1, BS, i∗ 2) =⇒ H(f Σ1 ,fΣ2 )(i′∗ 1 , BS′, i′∗ 2 ), of functors, π1(BΣ1, fΣ1)op × π1(BΣ2, fΣ2) → Vect, as in Definition 130.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, we point out that as an extended TQFT is defined to be a symmetric monoidal bifunctor, this definition will be completed only once we have established the existence of that structure63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That will take a little time and will be completed with Theorem 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof that we have indeed defined a bifunctor 2QB, follows as for the earlier case of 2QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, the crucial fact that 2QB preserves the compo- sition of the 1-morphisms and the horizontal compositions of the 2-morphisms of 63and we recall that being a symmetric monoidal bifunctor requires extra structure as it is not just a property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 128 2Cob (n,n+1,n+2) B , follows from Proposition 116 and Proposition 139.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is for these cases that we need to impose that each pair, (BΣ, fΣ), is 0-connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 155 (The dependence on decorations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let, as before, Σ be a closed smooth n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary once-extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, does not give a value to Σ itself, except when Σ is given a B-decoration, f Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two B-decorations, f Σ and f ′ Σ, of Σ, then there exists a canonically defined, and invertible, profunctor, 2Q 0 B(Σ, f Σ) ↛ 2Q 0 B(Σ, f ′ Σ), defined by 2Q 1 B � (Σ, f Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′ Σ) � : 2Q 0 B(Σ, f Σ) ↛ 2Q 0 B(Σ, f ′ Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These profunctors associated to pairs of B-decorations of Σ are “functorial” in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider decorations fΣ, f ′ Σ and f ′′ Σ of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have 1-morphisms in 2Cob (n,n+1,n+2) B , (Σ, fΣ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′ Σ), (Σ, f ′ Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′′ Σ), and the following, which is homeomorphic to their composite, (Σ, f Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′′ Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying Lemma 116, we have a natural isomorphism of profunctors, 2Q 1 B � (Σ, f Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′ Σ) � 2Q 1 B � (Σ, f ′ Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′′ Σ) � =⇒ 2Q 1 B � (Σ, f Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′′ Σ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It can be proved that these natural isomorphisms satisfy appropriate relations when we consider four different B-decorations of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, this shows that the profunctor associated to a change of B-decoration is always invertible up to 2- isomorphism, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is an adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that we have a bifunctor, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, the profunctors associated to moving from one B-decoration of a manifold to another one are, furthermore, compatible with the profunctors associated to the 1-morphisms of 2Cob (n,n+1,n+2) B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita valued extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We continue working with our chosen field, κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary theory, as given in the previous section, takes values in a bicat- egory of Vect-valued profunctors between finite κ-linear categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To be more easily able to use more usual representation theoretic methods and ideas, it can be convenient to replace this bicategory by one that is better know within the rep- resentation theoretic setting, namely that of finite dimensional algebras (with 1), bimodules and morphisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we will first review the construction of an algebra from a linear category, as given by Mitchell, [90], §7, and then look at it in detail for Lin(Γ), the linear category associated to a (finite) groupoid, Γ, obtained by applying the free vector space functor to the morphism sets of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, the resulting algebra is the well known groupoid algebra, [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We look, in some detail, at the relationship between bimodules over a category algebra and profunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some of this is folklore, and is quite difficult to find explicitly in the literature, yet it seems important for the A CATEGORIFICATION OF QUINN’S TQFT 129 understanding of the relationship between the Prof-valued and the Mor-valued extended TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will define the Morita bicategory, Mor, or, more exactly, Morκ, (also some- times denoted Alg2), of algebras, bimodules and the bimodule morphisms, these latter being often known as intertwiners in a representation theoretic context, and will examine its relation to vProf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will then see how to define a Morita valued extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebra of a small linear category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In section 7, (page 33), of the classic paper, [90], by Mitchell, it was shown how to associate a ring to a small additive category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That construction easily extends to a κ-linear version for any κ-linear category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let C be a (small) κ-linear category, having C0 as its set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set [C] to be the set of C0 × C0 matrices, c, where, for p, q ∈ C0, the (p, q)-entry, denoted cp,q, is an element of the vector space, C(p, q), of arrows from p to q, and each row and column has only a finite number of non-zero entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the addition in each C(p, q), together with the composition from C, we can give [C] the structure of a κ-algebra, which will not usually be commutative, nor, in general, unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have that, as a vector space, [C] = � p,q∈C0 C(p, q), and the multiplication is given by g · f = � g ◦ f if domain(g) = codomain(f) 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Although, in general, [C] will not have a multiplicative identity, each object p ∈ C0 gives an idempotent matrix, 1p, namely the matrix having the identity morphism on p in the (p, p)-position and zeroes elsewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If C0 is finite, then � p∈C0 1p is, however, a multiplicative identity for [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebra, [C], is an example of a generalised matrix algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This algebra is called the category algebra of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any element, c = (cp,q), in [C] can be written as a sum of matrices of form cp,q, where the matrix cp,q is to be zero in all positions except the (p, q) position, where it is, no surprise, cp,q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This sum is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The element, cp,q, clearly has domain equal to p and codomain equal to q, so, in a completely classical way, the product c · 1p = � r,s cr,s · 1p = � s cp,s, whilst 1q · c = � r cr,q, so 1q · c · 1p = cp,q, and, in particular, we have the useful equation: 1q · cp,q · 1p = cp,q, which we will use shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will usually be considering this when C is the κ-linearisation of a (usually finite) groupoid, Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Γ is finite, more generally if Γ has a finite set of objects, then the resulting ‘groupoid algebra’ will be unital.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It has a well known description in terms of the arrows of Γ, which we include in case it is found easier to understand, as it is written in a slightly less abstract way;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see also [15, 16, 95, 123] and [34] for various versions and the development of further theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This groupoid algebra, Lin(2)(Γ), has as its underlying vector space, Lin(Γ1), that is the free vector space over the set of morphisms of Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It thus comes equipped with a natural choice of basis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The product in Lin(2)(Γ) is given on generators by (x g−→ y)(x′ g′ −→ y) := δ(y, x′)(x gg′ −−→ y), A CATEGORIFICATION OF QUINN’S TQFT 130 where δ(y, x′) is 1, if the two objects, y and x′, are equal, and is zero otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The multiplicative unit of Lin(2)(Γ) is given by � x∈Γ0(x idx −−→ x), which makes sense since Γ0 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Examples: The following examples are well known, but worth recalling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 156.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Γ is a group, G, thought of as a single object groupoid in the usual way, and then further thought of as (a basis for) the corresponding κ-linear category, then Lin(2)(Γ) is just the usual group algebra, κ[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is thus also a Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 157.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Returning to the general case for the moment, if P is a (finite) pre- ordered set, and C is the linearisation of the corresponding small category, then [C] is the incidence algebra of the poset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As specific examples, if P = {1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' < n}, then [C] is the algebra of n × n upper triangular matrices over κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we replace the given preorder by the discrete preorder, so p ≤ q here means p = q, then the corresponding [C] is the algebra of diagonal matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, on the other hand, we replace the preorder by the codiscrete preorder (in which p ≤ q for every pair of elements (p, q)), then [C] = Mn(κ), the full algebra of n × n matrices over the field κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The Quantum Double as a groupoid algebra) Let G be a finite group, then we know that its group algebra, κ[G], is a Hopf algebra in a natural way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Drinfel’d double of that Hopf algebra is a groupoid algebra for a naturally defined groupoid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see, for example, [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is an action of G on itself by conjugation, g a−→ aga−1, and we can form the action groupoid of this action64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This has the elements of G as its objects and the arrows have form (g, a) : g → aga−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is variously written AUT (G) or G � G in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use the former of these for the moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Composition is by group multiplication (in reverse order).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We write [AUT (G)] for the groupoid algebra of the linearisation of AUT (G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is given in detail in, for instance, [34, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The product on the basis elements, as above, is given by (g, a)(g′, a′) = δ(aga−1, g′)(g, aa′), where g, g′, a, a′ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we define a comultiplication, ∆(x, g) = � yz=x (y, g) ⊗ (z, g), and a counit, ǫ(x, g) = δ(x, 1G), then, for suitable and quite evident definitions of an antipode and an R-matrix, the resulting object is a quasi-triangular Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is clear, from standard descriptions of the ‘double construction’, that this is D(G) the Drinfel’d double or quantum double of the Hopf algebra, κ[G].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is worth noting that this is the untwisted version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The twisted version is ex- amined in [123].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Another useful reference giving further insight to this construction is [34], that we mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 159.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The link between groupoid algebras, for groupoids having finitely many objects, and Hopf algebra-type structure is stronger than just the previous example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To discuss this briefly, we will need to give the idea of a weak bialgebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a κ-module, B, having both an algebra and a coalgebra structure, subject to some compatibility conditions (which can be found in, for example, [15, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If Γ is a groupoid having finitely many objects, and we take B = Lin(Γ1), the ‘free vector space’ on the set of arrows of Γ, in the terminology of [16], then the 64We will give a more general form of action groupoid later;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see page 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 131 groupoid algebra of Γ has a coproduct, ∆ : B → B × B, given by the diagonal, ∆(f) = f ⊗ f, whilst the counit sends all basic generators to the identity element, 1, of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A weak Hopf algebra, as defined in, for example, [16], is a weak bialgebra equipped with an antipode map, S : H → H, subject to some axioms generalising the proper- ties required of an antipode in the definition of Hopf algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid algebra of a groupoid having finitely many objects has an antipode, given on the basic generators by S(f) := f −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Because of this, weak Hopf algebras are sometimes called quantum groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Weak Hopf algebras have a nice and well understood representation theory, some references for which can be found in the paper, [15], already cited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (For more on this see, for instance, [15, 16, 95].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') As this last example applies to the type of groupoid algebras that we will be producing from the once-extended Quinn TQFTs, it is particularly promising for future developments of the theory using the tools from the theory of weak Hopf algebras / quantum groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Non-functoriality of [−] and Morita equivalences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This subsection examines more properties of this situation, but they will not be immediately needed for the main theme of this paper, so can be left aside on first reading, moving on to section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, where the bicategory, Mor is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We would recommend that they be at least skimmed in a later reading as they provide further insights on the algebraic mechanisms involved later on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A functor, F : C → D, does induce a linear map, [F] : [C] → [D], but, in general, this map will not preserve multiplication, so [−] is not a functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The more-or-less minimal example for this is to take C to be the (κ-linearisation of the) discrete category on the set having just two elements, say a and b, and D to be the corresponding construction on a singleton set, {c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Take F : C → D to be the functor induced by the unique map from {a, b} to {c}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we write 1a for the identity on a, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', then [C] is the direct sum of two copies of κ, but with 1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1b = 0, since the composite of a and b in C is not defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Over in [D], which is, itself, isomorphic to κ, [F](1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [F](1b) = 1c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1c = 1c, so certainly [F](1a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1b) ̸= [F](1a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [F](1b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will examine this slightly odd situation in a bit more detail shortly, as it is perfectly manageable given the approach that we are using.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In any case, the following result of Mitchell, [90], Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, makes one realise that there is a lot of power in the category algebra construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the statement, we think of the category algebra as a linear category having just a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 160.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose C is a linear category having only finitely many objects, then C and [C] are Morita equivalent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can expand this as follows, if we define C−Mod to be Funcκ(C, Vect), the set of κ-linear functors from C to Vect, then C−Mod and [C]−Mod are equivalent categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This, thus, says that, as far as linear representations are concerned, C and [C] are encoding the same information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will sketch out a proof of this as it contains some ideas that help one to understand what is happening here, and hence why the ‘linearisation’ versus ‘cat- egorification’ process is so useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (A full proof is given in [90] on page 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A discussion of the ideas can be found online in the n-Category Caf´e, (May 14, 2014), in a post, Categories vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Algebras, by Tom Leinster;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') First we construct a functor, [−], from C −Mod to [C]−Mod, so suppose that M : C → Vect is a κ-linear functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set [M] = ⊕q∈C0M(q), A CATEGORIFICATION OF QUINN’S TQFT 132 the direct sum of all the image vector spaces of the functor, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is finite dimensional if each M(q) is, as C0 is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It has a [C]-module structure in a natural and fairly obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If c = (cp,q) is an element of [C], then cp,q : p → q is in C, so M(cp,q) : M(p) → M(q) is a linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now, if m = (mp) is an element of [M], we define c · m = � p,q M(cp,q)(mp).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The linearity and functoriality of M ensures that this does give a [C]-module structure to [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is easily seen to define a functor, [−], as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This forms part of the claimed equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The other direction, starting from a [C]-module and ending up with a κ-linear functor from C to Vect, is not so obvious, although it is, in fact, a generalisation of a well known process from elementary linear algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose N is a [C]-module and that p is an object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The element, 1p ∈ [C], is idempotent, so multiplication by it gives an idempotent linear map from N to itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can thus split N as 1pN ⊕ (1 − 1p)N, and we can repeat this with each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We get N ∼= ⊕p∈C01pN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set � N(p) to be the summand, 1pN, and show that this is the object part of the required functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If cp,q : p → q in C, then, as before, let cp,q ∈ [C] be the matrix having cp,q in position (p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If n ∈ � N(p), then 1p · n = n, and cp,q · n = 1q · cp,q · 1p · n, which is in 1qN = � N(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define � N(cp,q) : � N(p) → � N(q) by � N(cp,q)(n) = cp,q · n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Linearity of this map is automatic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof that � N : C → Vect is a functor is fairly routine, as is that of the functoriality of the construction, � (−) : [C]−Mod → C−Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally it should be fairly clear65 that this is the required quasi-inverse for [−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark: This proof is a mild categorification of that of the elementary result that the internal and external descriptions of direct sum amount to the same thing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have left ‘to the reader’ the detailed verification that the above constructions do yield an equivalence between C−Mod and [C]−Mod, as it is fairly routine to give a direct proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will, in fact, investigate that equivalence by a separate route.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this, we recall that [C], as it is a κ-algebra, can be considered as a κ-linear category in its own right, namely one having a single object, ∗, and with [C](∗, ∗) being the set of elements of [C] itself, with composition being the multiplication in [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not make any notational distinction between the κ-algebra, [C], and the linear category, [C], at least where no confusion is likely to arise by so doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is well known that two κ-algebras, R and S, are Morita equivalent if there are bimodules, RAS and SBR, such that the functors, A ⊗S − and B ⊗R −, form an adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What is also clear it that this should generalise to κ-linear categories and it does.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It does, however, seem a bit difficult to find a simple pub- lished proof of this, as it is a special case of some very wide ranging generalisations, whose generality we do not need, or, in fact, want here as our aim is to justify and interpret some calculations in a specific case of that general theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It does, however, suggest that we try to find ‘bimodules’, CA[C] and [C]BC, with similar properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What are such ‘bimodules’ to be?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They are just another name for Vect-valued profunctors, which, in the case of interest, would give A : C ↛ [C] and B : [C] ↛ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This observation, and quite a bit of what follows, is adapted from the n-Category Caf´e discussion, (May 14, 2014), [79], as mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The ideas, there and here, were largely given by Karol Szumi�lo, but with a few additional features and verifications added here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We should add that any errors should be attributed to us, and not to him.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Before that, however, we will give the pair of profunctors as suggested above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 65i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' this is ‘left to the reader’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 133 Earlier, on and around page 36, we saw that, in Prof or vProf, the identity profunctor on a category, C, was the double Yoneda embedding, C(−, −) : Cop × C → Set, or, of course, with codomain Vect if C is a κ-linear category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need, here, a functor, A : Cop × [C] → Vect, and an obvious candidate can be derived from that Yoneda based functor, by applying the [−]-construction to one side of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We, therefore, define a functor, A, as required, by A(p, ∗) = ⊕q∈C0C(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall that, here, ∗ is the unique object of the κ-linear category, [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formula is clearly (contravariantly) functorial in p, so it remains to see how some ∗ c−→ ∗, acts on A(p, ∗), where c is a matrix, (cr,s), and each cr,s ∈ C(r, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p be an object of C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let xp be an element of A(p, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we wrote before, given another object q, its q component is some xp,q ∈ C(p, q), and then (xp · c)p,s = � q∈C xp,q · cq,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We want to calculate the composite, C A ↛ [C] N −→ Vect, for a [C]-module, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will write N for both the module, and the functor, N : [C] → Vect, although we should remember that N is also N(∗), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the functor evaluated on the single object of the algebra (considered as a linear category).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' What should this mean?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composite of a profunctor and a functor?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can interpret this as being A • ϕN : C ↛ Vect, so giving us a profunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The notation ϕN is explained in Example 33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That would be a first step, thus we want to examine the corresponding functor, A • ϕN : Cop × Vect → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We evaluate it on a pair of objects, (p, V ), with p ∈ C0 and V being a vector space over κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formula for the composition in vProf gives (A • ϕN)(p, V ) = ˆ ∗ A(p, ∗) ⊗ Vect(N, V ), but we note that the coend is ‘integrating’ over the one object category correspond- ing to [C], so is just the term being ‘integrated’ divided by the diagonal action of the algebra, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', it is (⊕q∈C0C(p, q) ⊗ Vect(N, V ))/ ≃, where the action of any cr,s, which is homogeneous with value cr,s, on the right hand side, Vect(N, V ), is by the action on N, so if v : N → V , then (cr,s · v)(n) = v(cr,s · n), whilst on the left hand side, it is by post-composition by cr,s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any element, (cp,q, v) in this direct sum is ≃-equivalent to one of the form (1p, w), by factoring the cp,q as 1p · cp,q, and then shifting the cp,q across to the other side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we do this to 1p itself, we find that (1p, w) ≃ (1p, 1p · w), so is determined by the restriction of the linear map, w, to the direct summand 1pN, which we have denoted above by � N(p), as 1p · w is the composition of w with the projection onto that direct summand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, (A • ϕN)(p, V ) ∼= Vect( � N(p), V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to see that this isomorphism is natural in both N and V , so (A • ϕN) is a representable profunctor, represented by � N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To summarise, the composite profunctor, C A ↛ [C] N −→ Vect, is ‘really’ the functor � N, as we hoped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 134 We now turn to the profunctor, B : [C] ↛ C, so B : [C]op × C → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given the success of the formula for A above, the ‘obvious’ formula for B is B(∗, q) = ⊕p∈C0C(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This certainly gives a functor, [C]op × C → Vect, (and thus a profunctor) as hoped for, and, to get the analogue of our earlier calculation, we will think of a functor, M : [C] → Vect, as a profunctor, ϕM : [C] ↛ Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have (B • ϕM)(∗, V ) ∼= ˆ q ⊕p∈C0C(p, q) ⊗ Vect(M(q), V ), for q ∈ C0 and a vector space, V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is isomorphic to ⊕p∈C0Vect(M(p), V ) by one of the forms of the co-Yoneda lemma, and this, in turn, is Vect([M], V ), up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These isomorphisms are natural, so (B•ϕM) ∼= ϕ[M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composite profunctor, (B • ϕM), is thus representable, and is ‘really’ [M].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This sets up the two functors, [−] and � (−), on the categories of ‘modules’, as being given by the profunctors A and B, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The final steps to explore, in this investigation of Theorem 160, are to calculate the composites, A • B : C ↛ C and B • A : [C] ↛ [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are (slightly careful) manipulations involving the coend formulation of profunctor composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) A • B ∼= C(−, −), the unit profunctor on C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) B • A ∼= [C](∗, ∗) ∼= [C], the unit profunctor / bimodule on [C].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) We take p, q ∈ C0, then A • B(p, q) = ˆ ∗ A(p, ∗) ⊗ B(∗, q) = ˆ ∗ ⊕r∈C0C(p, r) ⊗ ⊕s∈C0C(s, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As this coend is over (the single object category) [C], it can be calculated as (⊕r∈C0C(p, r)) ⊗[C] (⊕s∈C0C(s, q)), so as a tensor product over (the algebra), [C], then, given the form of the mul- tiplication in [C], it is clear that this tensor product is isomorphic to the vector space, C(p, q), and as all the isomorphisms are natural in p and q, we thus have that A • B ∼= C(−, −), the unit profunctor on C as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) This part is easier: B • A(∗, ∗) = ˆ q B(∗, q) ⊗ A(q, ∗) = ˆ q ⊕p∈C0C(p, q) ⊗ ⊕r∈C0C(q, r) ∼= ⊕p,rC(p, r) ∼= [C], which is, of course, the same as [C](∗, ∗), as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This resolves, at least in part, the problem that we noted earlier, namely that [−] is not a functor as such, at least in the most obvious sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose, however, that F : C → D is a κ-linear functor, then there is an ‘induced’ way to get from [C] to [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It can be given by the composite profunctor, BC • ϕF • AD : [C] ↛ C ↛ D ↛ [D], where we have indicated the ‘versions’ of the profunctors, A and B, by adding suitable suffices, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', AD being the A profunctor for D, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As this is a A CATEGORIFICATION OF QUINN’S TQFT 135 profunctor between two single object linear categories, it is ‘just’ a left [C]-, right [D]-bimodule (determined up to isomorphism).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This, in fact, shows clearly that the bicategory of algebras, bimodules, and bimod- ule morphisms has some better properties than the category of algebras and algebra homomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The original functor induces a bimodule, but, in general, not a homomorphism, between the two category algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We could have got to this in- duced bimodule without going via the Morita context, but the route we have taken has some advantages for what we will be needing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the above construction for a functor extends easily to handling a profunctor, H : C ↛ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any such profunctor can be whiskered by suitable units and counits, A and B, to give BC • H • AD : [C] ↛ C ↛ D ↛ [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The discussion that we just gave can be used to prove that we have a bifunctor from the bicategory of linear categories, Vect-enriched profunctors between them, and enriched natural transformations, to the bicategory of algebras, bimodules and bimodule maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It sends C to [C], and H : C ↛ D to the composite profunctor above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This construction will be clarified in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will deal with a particular case of this latter construction in the following section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next turn to the bicategory of algebras, and describe it in a bit more detail, as it and related structures are the target for our next version of the once-extended TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bicategory, Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next give the detailed definition of the bicategory of algebras, bimodules, and bimodule morphisms / intertwiners, that we have been using in a fairly sketchy form for some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In so doing, we will shift our notation to put the actions of the algebras on the bimodules into a more central role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bicategory is sometimes denoted Alg or Alg2 in the literature, but we will denote it by Mor, and refer to it as the Morita bicategory, as it is the natural and classical setting for Morita equivalence, an adjoint equivalence in Mor, in the sense of bicategory theory, being precisely a classical Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We follow [10, 63, 106], as well as more classical sources on bicategories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 164.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita bicategory, Mor = Morκ, is the bicategory such that: the objects of Mor are unital κ-algebras;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given algebras A and B, 1-morphisms A ↛ B are (A, B)-bimodules, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To give some more detail, and for a notational recall, M will be a κ-vector space equipped with a left A-representation / action, ⊲, and a right B-representation, ⊳, that are compatible, meaning that given a ∈ A, b ∈ B and m ∈ M, we have that (a⊲m) ⊳ b = a⊲(m ⊳ b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the 2-morphisms, F : (M : A ↛ B) =⇒ (N : A ↛ B), are given by (A, B)- bimodule maps, F : M → N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and, finally, the horizontal composite of a compatible pair of 2-morphisms, A M1 � N1 � ⇓ F1 B M2 � N2 � ⇓ F2 C , A CATEGORIFICATION OF QUINN’S TQFT 136 is A M1⊗BM2 � N1⊗BN2 � ⇓ F1 ⊗B F2 C , where ⊗B is the usual tensor product over the algebra, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are also ’well known’ horizontal units and unitors, completing the con- struction of the bicategory Mor, whose explicit description is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that vProfGrpfin is the sub-bicategory of vProfGrphf, whose objects are finite groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The constructions of the previous section (§6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3) give a bi- functor, Lin(2) : vProfGrpfin → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bifunctor sends: each finite groupoid, Γ = (s, t: Γ1 → Γ0, id), to its groupoid algebra, Lin(2)(Γ), see page 129;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and if given groupoids, Γ = (s, t: Γ1 → Γ0, id) and Γ′ = (s, t: Γ′ 1 → Γ′ 0, id), and a profunctor, H: Γ ↛ Γ′, hence a functor H: Γop ×Γ′ → Vect, we consider the bimodule, Lin(2)(H), to have underlying vector space, Lin(2)(H) := � x∈Γ0, y∈Γ′ 0 H(x, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the rest of the bimodule structure on Lin(2)(H), we let a ∈ Γ0 and b ∈ Γ′ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Below, we will not distinguish between an element, v(a,b) ∈ H(a, b), and its image under the obvious inclusion of H(a, b) into Lin(2)(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The left and right actions of the algebras, Lin(2)(Γ) and Lin(2)(Γ′), on Lin(2)(H) are such that, given v(a,b) ∈ H(a, b), we have, given (x g−→ y) ∈ Γ1 and (x′ g′ −→ y′) ∈ Γ′ 1, (x g−→ y)⊲v(a,b) = � H � x g−→ y, b idb −−→ b � (v(a,b)), if y = a, 0, if y ̸= a, hence, if v(y,b) ∈ H(y, b), then (x g−→ y)⊲v(y,b) ∈ H(x, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the right action, v(a,b) ⊳ (x′ g′ −→ y′) = � H � a ida −−→ a, x′ g′ −→ y′� (v(a,b)), if x′ = b, 0, if x′ ̸= b, so, if v(a,x′) ∈ H(a, x′), then v(a,x′) ⊳ (x′ g′ −→ y′) ∈ H(a, y′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 165.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bimodule, Lin(2)(H): Lin(2)(Γ) ↛ Lin(2)(Γ′), is an instance of the general construction mentioned at the end of §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, namely Lin(2)(H) is isomorphic to the composite BC • H • AD : [C] ↛ C ↛ D ↛ [D], where, here, C is Lin(2)(Γ) and D is Lin(2)(Γ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This can help when checking, for instance, preservation, up to invertible 2-morphisms, of horizontal composition for the candidate bifunctor, Lin(2) : vProfGrpfin → Mor, see below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that if a ∈ Γ0 and b ∈ Γ′ 0, then, for v(a,b) ∈ H(a, b), we have that (a ida −−→ a)⊲v(a,b) = H � a ida −−→ a, b idb −−→ b � (v(a,b)) = v(a,b) and v(a,b) ⊳ (b idb −−→ b) = H � a ida −−→ a, b idb −−→ n � (v(a,b)) = v(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 137 The remaining details of the verification that the above construction does give a bifunctor, Lin(2) : vProfGrpfin → Morκ, will mostly be left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The key property that Lin(2) preserves horizontal compositions of 1-morphisms, up to a canonical natural equivalence, is given by the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider finite groupoids, Γ, Γ′, and Γ′′, and profunctors, H: Γ ↛ Γ′ and H′ : Γ′ ↛ Γ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a canonical isomorphism of � Lin(2)(Γ), Lin(2)(Γ′′) � bimodules, I : Lin(2)(H • H′) =⇒ Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As mentioned above, in Remark 165, this follows from the calculations in the previous section, and in particular on the properties of the composite profunctors, A • B and B • A, as given in Lemma 161, but we will give a direct proof, so as to accustom the reader to the links between profunctor and bimodule composition arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first see what happens at the level of underlying vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let x ∈ Γ0 and z ∈ Γ′′ 0, then (H • H′)(x, z) = ˆ y∈Γ′ 0 H(x, y) ⊗ H′(y, z) = � � y∈Γ′ 0 H(x, y) ⊗ H′(y, z) � / ≃ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, fixing x ∈ Γ0 and z ∈ Γ′′ 0, the linear equivalence relation66, ≃, is generated by, for y, y′ ∈ Γ′ 0, v(x,y) ∈ H(x, y) and v′ (y′,z) ∈ H(y′, z), and an arrow, y g−→ y′, in Γ′ 1, v(x,y) ⊗ H′(y g−→ y′, z 1z −→ z)(v′ (y′,z)) ≃ H(x 1x −→ x, y g−→ y′)(v(x,y)) ⊗ v′ (y′,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter relation means exactly that, given y, y′ ∈ Γ′ 0, v(x,y) ∈ H(x, y) and v′ (y′,z) ∈ H(y′, z), and an arrow, y g−→ y′ in Γ′ 1, we have v(x,y) ⊗ ((y g−→ y′)⊲v′ (y′,z)) ≃ (v(x,y) ⊳ (y g−→ y′)) ⊗ v′ (y′,z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that Lin(2)(H • H′) = � x∈Γ0,z∈Γ′′ 0 (H • H′)(x, z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, we have Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) = � � x∈Γ0,z∈Γ′′ 0 � y,y′∈Γ′ 0 H(x, y) ⊗ H′(y′, z) � / ∼ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here the linear equivalence relation, ∼, is such that, given x ∈ Γ0, z ∈ Γ′′ 0, y, y′ ∈ Γ′ 0, v(x,y) ∈ H(x, y) and v′ (y′,z) ∈ H(y′, z), we have v(x,y) ⊗ ((w g−→ w′)⊲v′ (y′,z)) ∼ (v(x,y) ⊳ (w g−→ w′)) ⊗ v′ (y′,z), for arbitrary (w g−→ w′) ∈ Γ′ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly we have a bimodule map, I : Lin(2)(H • H′) =⇒ Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H), sending the equivalence class of v(x,y) ⊗ v′ (y,z) ∈ � x∈Γ0,z∈Γ′′ 0 � y∈Γ′ 0 H(x, y) ⊗ H′(y, z), 66i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', an equivalence relation whose quotient is a vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 138 under ≃, to the equivalence class of v(x,y) ⊗ v′ (y,z) ∈ � x∈Γ0,z∈Γ′′ 0 � y,y′∈Γ′ 0 H(x, y) ⊗ H′(y′, z), under ∼, and we claim that I is a bijection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If y, y′ ∈ Γ′ 0 are not equal, and v(x,y) ∈ H(x, y) and v′ (y′,z) ∈ H(y′, z), then v(x,y) ⊗ v′ (y′,z) ∼ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is because, (on using Remark 166), v(x,y) ⊗ v′ (y′,z) = � v(x,y) ⊳ (y 1y −→ y) � ⊗ v′ (y′,z) ∼ v(x,y) ⊗ � (y 1y −→ y) � ⊲v(y′,z) � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, I is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now define a bimodule map, I′ : Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) =⇒ Lin(2)(H • H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a bilinear map, I′′ : Lin(2)(H) × Lin(2)(H′) → Lin(2)(H • H′), given by, if x ∈ Γ0, y, y′ ∈ Γ0 and z ∈ Γ′′ 0, and also v(x,y) ∈ H(x, y) and v′ (y′,z) ∈ H(y′, z), then, I′′(v(x,y), v′ (y′,z)) = � [v(x,y) ⊗ v′ (y′,z)]≃, if y = y′, 0, if y ̸= y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is clearly balanced, considering the right and left actions of Lin(2)(Γ′), so I′′ descends to a linear map, I′ : Lin(2)(H) ⊗Lin(2)(Γ′) Lin(2)(H′) → Lin(2)(H • H′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, I′ ◦ I = id, and so, in particular, I is injective as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rest of the details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita-valued once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the results of the pre- vious sections, we can take our finitary version of the once-extended Quinn TQFT that we gave in subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, and reflect it into the bicategory Mor, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As always, we let n be a non-negative integer and B be a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 168 (The Morita-valued once-extended Quinn TQFT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Morita valued once-extended Quinn TQFT, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor, is defined as the following composite of bifunctors, 2Cob (n,n+1,n+2) B 2QB −−−→ vProfGrpfin Lin(2) −−−−→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to simplify the notation, we will use the same notation, 2Q Mor B , for all components of the bifunctor 2Q Mor B .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 169.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We follow here an approach found in [30, Subsection 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3]) Let Σ be a closed smooth n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two B-decorations, f Σ and f ′ Σ, of Σ, the argument in Remark 155 gives a canonically defined invertible bimodule connecting the algebras, 2Q Mor B (Σ, fΣ) and 2Q Mor B (Σ, f ′ Σ), namely 2Q Mor B � (Σ, f Σ) (ιΣ 0 ,Σ×I,ιΣ 1 ) −−−−−−−−→ (Σ, f ′ Σ) � : 2Q Mor B (Σ, f Σ) ↛ 2Q Mor B (Σ, f ′ Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All comments in Remark 155 pass over to the Morita setting with the obvious mod- ifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the previous remark, one can prove that, if Σ is a closed smooth manifold, then all the algebras, 2Q Mor B (Σ, fΣ), where fΣ is a B-decoration of Σ, are Morita equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, such Morita equivalences can be canonically chosen, given a pair of decorations of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 139 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structure in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We fix a non- negative integer n throughout this subsection and the following as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The central result of this paper is that one can categorify the finite total homotopy TQFT of Quinn, [101], in a sensible way to get a once-extended TQFT, 2QB : 2Cob(n,n+1,n+2) −→ vProfGrphf, where B is a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From there we have shown that the resulting theory can be cut down in size to be more finitary by various means such as the introduction of decorations, and can be linked up with better known ‘algebraic’ bicategories such as Mor, which are frequently met in representation theoretic contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Following Schommer-Pries, [106], Lurie, [82], and others, we have taken a once- extended TQFT to be a symmetric monoidal bifunctor, as above, but we remark that the existing definitions do not agree on the target / codomain bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have defined 2QB, and have shown it to be a bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is, however, one further step to complete the proof that these constructions give once-extended TQFTs, and that is to prove 2QB, and its cousins, are symmetric monoidal bi- functors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this, we have to specify the symmetric monoidal structures on the cobordism bicategory, 2Cob(n,n+1,n+2), and will also recall that of vProfGrphf, which was formally proved to exist in [63].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that being a symmetric monoidal bifunctor is a structure, not a property, and refer the reader to the sketch in Definition 30 and to [106, Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5] for a more detailed description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Sometimes the extra structure, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', that beyond being a bifunctor, is ‘evident’, but in our case that extra categorical structure encodes some of the ‘geometric’ structure, for instance cobordisms, and 2-cobordisms, and we do need to have the transition between the various contexts made explicit to allow the naturality of the constructions to be made clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A preliminary result towards the construction of the symmetric monoidal structure in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details of the construction of the symmetric monoidal structure in the bicategory 2Cob(n,n+1,n+2), using the language of sym- metric monoidal pseudo-double categories [63], can be found in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In [82, Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ], it is stated that, in the case of 2Cob(n,n+1,n+2) (or, more ex- actly, Lurie’s analogue of this), the monoidal structure is straightforward, as “the tensor product operation is simply given by disjoint union of manifolds”, just as in the more classical case of Cob(n,n+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Although correct, this statement hides some important details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The disjoint union of manifolds, cobordisms and extended cobordism indeed gives rise to a bifunctor, by abuse of language denoted67 ⊔: 2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) → 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A monoidal bicategory is however not just the tensor product and unit, but also the associator, unitors and with additional pentagonators, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', as sketched in Definition 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, we need the tensor product to be symmetric, so need to specify a braiding, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This may seem excessive detail to give, but is needed as there is a slight trap that has to be avoided, as we will now see.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In general, as we saw in Definition 26, in a monoidal bicategory, (A, ⊗, I, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), the structure is given by morphisms in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, the monoidal associator is an adjoint equivalence, so for each triple, A, B, C, of objects in A, we have that αCBA : (C ⊗ B) ⊗ A → C ⊗ (B ⊗ A), 67It is important to note that this bifunctor is not strict, in the sense that the natural isomor- phism, ϕ, in item (3) of Definition 20 is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 140 whilst α∗ CBA : C ⊗ (B ⊗ A) → (C ⊗ B) ⊗ A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have, in the situation that A = 2Cob(n,n+1,n+2), that we need to specify cobordisms, (C ⊔ B) ⊔ A i �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ C ⊔ (B ⊔ A), j �❧❧❧❧❧❧❧❧ M for all triples of n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is not difficult, but does involve some techni- calities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two ends of the required cobordisms are not equal, although they are naturally homeomorphic (and diffeomorphic if we include consideration of the smooth structure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The required (n + 1)-manifold, M, will be, topologically, a cylinder, but some care is needed with the labelling of its ends, and throughout we need to remember that a cobordism is not just an (n + 1)-manifold, but has extra structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Definition 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that an analogous situation was encountered when discussing the bicat- egory Span(C), in the list in Examples 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We cannot directly use the diffeomorphism between the two sides, (C ⊔ B) ⊔ A and C ⊔ (B ⊔ A), as the associator, as that would be a morphism in C = CGWH or Diff n, but not in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have to convert that isomorphism to a cobordism before checking that it works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this, and for similar later situations, we need a result which is, in some sense, a dual of Lemma 63, the context for which we will set up next.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 4, we saw, page 62, that denoting by Diff n, the category of closed n-manifolds and diffeomorphisms between them, we have a functor, I′ : Diffn → Cob(n,n+1), which is the identity on objects, and such that I′(f : Σ → Σ′) is the equivalence class of the cospan, I′(f : Σ → Σ′) = \uf8eb \uf8ec \uf8ed Σ ιΣ 0 �❑ ❑ ❑ ❑ ❑ ❑ ❑ Σ′ ιΣ′ 1 ◦f −1 �rrrrrrr Σ × I \uf8f6 \uf8f7 \uf8f8 , or equally well of the cospan with f used in the left (co)leg, as in Remark 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a cospan is always cofibrant in the following sense, which is dual to the earlier definition of fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A cospan, A i �❘ ❘ ❘ ❘ ❘ ❘ B, j �❦❦❦❦❦❦ M in CGWH is said to be cofibrant if the induced map, � i j � : A ⊔ B → M, is a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A thorough discussion on cofibrant cospans can be found in [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) In this case, both i and j will be cofibrations as well, and inter- preting cofibrations more or less as inclusions of locally flat submanifolds, it also says that the two submanifolds, i(A) and j(B), of M, have empty intersection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) We note that these cofibrant cospans are the cofibrant objects in the category CGWHΛop, where, as in Remark 49, CGWH is given the Hurewicz / Strøm model structure but, now, CGWHΛop is given the projective model category structure, so that weak equivalences and fibrations are objectwise;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 49 for the dual case of fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 141 (iii) It is also of note that we could have replaced the role of CGWH with that Hurewicz / Strøm model category structure, by many other suitably structured model categories in the above remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From a terminological point of view, it is perhaps also worth noting that the projection, σ : A × I → A, gives a morphism, (ι0, A × I, ι1) → (idA, A, idA), which is a weak equivalence in Cospan(CGWH) := CGWHΛop, but the idenitity cospan is not a cofibrant, so in passing to cofibrant cospans, we are ‘resolving’ the cospans by cofibrant ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We may sometimes work with cofibrant cospans in CGWH, although the objects that are central to our study will be the objects coming from cobordisms and ex- tended cobordisms between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A stand-alone treatment of Cospan(CGWH), containings some of the discussion above is in the already mentioned references, [117, 118].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need an adapted categorified version, I : Diff n → 2Cob(n,n+1,n+2), of the construction, I′ : Diff n → Cob(n,n+1), given in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, page 62, that we mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the latter construction works in more generality, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', at the cospan level with homeomorphisms rather than diffeomorphisms, but then the smoothness of the corresponding cobordism would be in doubt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We suppose that f is a diffeomorphism from X to Y and write I(f) for the cobordism represented by the cofibrant cospan below (52) I(f) := \uf8eb \uf8ec \uf8ec \uf8ed X ιY 0 f �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιY 1 �①①①①①①①① Y × I \uf8f6 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that this is the cobordism and not just the equivalence class determined by it, so, for instance, the corresponding right (co)leg version, using f −1 is distinct from this, although there is an invertible 2-cobordism connecting them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This I does not give a ‘functor’ from Diffn to 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The reason is, essentially, that the horizontal composition in 2Cob(n,n+1,n+2) is that of a bicate- gory, not a category, since we are now not taking cobordisms up to diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore I does not preserve composition in a direct way, as is easy to show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' How- ever, we instead have a pseudo-functor I : Diff n → 2Cob(n,n+1,n+2), in the sense that we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that we have diffeomorphisms, X f−→ Y g−→ Z, and thus two cobordisms / cospans I(f): X → Y and I(g): Y → Z, as well as I(gf): X → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can form I(f)• I(g) by the usual pushout and can put all this A CATEGORIFICATION OF QUINN’S TQFT 142 into a diagram as follows: (53) X ιY 0 f �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ιZ 0 gf � Y ιZ 0 g �▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ▲ ιY 1 �rrrrrrrrrr Z ιZ 1 �②②②②②②②② ιZ 1 � Y × I ΨY g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ℓ �❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ ❑ Z × I ΨZ g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � r �ssssssssss PO(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f) � Ψg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f �✤ ✤ ✤ Z × I The pushout,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' PO(g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' can be given,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' up to isomorphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by (Y × I) ⊔ (Z × I)/ ∼,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for all y ∈ Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1) ∼ (g(y),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will often abuse notation a bit and write this as (Y × I) ⊔Y (Z × I)/ ∼, leaving explicit mention of the equivalence relation ∼ from the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similarly we will usually omit mention of ℓ and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Other models for the pushout could be used, but this seems the easiest to handle so we will use this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The important thing to remember is that the pushout is defined by a universal property so, as we said, it is only determined up to isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However, as before in this paper, we implicitly chose a natural realisation for each pushout, here by considering the obvious quotient of the disjoint union.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to see that there is a map, Ψg,f : PO(g, f) → Z × I, given by ΨY g,f(y, t) = (g(y), t/2), and ΨZ g,f(z, t) = (z, (t + 1)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the pushout in (53) is a pushout in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order for this construction to be usable in the context of 2Cob(n,n+1,n+2), we must put a smooth structure on PO(g, f), and also possibly modify Ψg,f : PO(g, f) → Z × I, slightly, in order that it is smooth at the junction, where the cylinders Y × I and Z × I join.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These however can be easily handled using the usual mechanisms of collars, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', so we will not concern ourselves more with this aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The map, Ψg,f, is a homeomorphism, with inverse, Ψ′ g,f, given by Ψ′ g,f(z, t) = � (g−1(z), 2t), for 0 ≤ t ≤ 1/2, (z, 2t − 1), for 1/2 ≤ t ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Up to now, of course, this construction is very similar to what we used in our earlier section, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, to show that the uncategorified version of the construction gave a functor from Diffn to Cob(n,n+1), except that, as we already mentioned, we are now not taking the quotient of cobordisms by diffeomorphism (relative to the boundary).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However it is not yet quite in the right form to be used for extended cobordisms, so as to give an extended cobordism / 2-cobordism between I(f)•I(g) and I(gf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For that we use an analogue of the I-construction one dimension up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 143 In general, suppose we have an isomorphism of cobordisms, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a diffeomor- phism, f, making the diagram below commute, (54) M f ∼ = � X i �① ① ① ① ① ① ① i′ �❋ ❋ ❋ ❋ ❋ ❋ ❋ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' j �❋❋❋❋❋❋❋ j′ �①①①①①①① N We can expand this out as a map of cospans, (55) X i � idX � M f � Y j � idY � X i′ � N Y, j′ � to which we apply the same idea as in the I-construction to each vertical diffeo- morphism to get (56) J (f) := \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed X i � ιX 0 � M ιN 0 ◦f � Y j � ιY 0 � X × I i′×I � N × I Y × I j′×I � X i′ � ιX 1 � N ιN 1 � Y ιY 1 � j′ � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This corresponds to an (n + 2)-extended cobordism / 2-cospan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to equiva- lence classes, we get a 2-morphism, [J (f)] : (i, M, j) =⇒ (i′, N, j′), in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is a vertically invertible 2-morphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To prove this one uses a categorified version of the argument used to show functoriality of I on page 62, and, in particular, that, if f : M → N, in (54), is a diffeomorphism, then [I(f)], in (56) is an invertible 2-cobordism class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now suppose that we have diffeomorphisms of cospans, f : (i, M, j) → (i′, N, j′) and g : (i′, N, j′) → (i′′, P, j′′), as below, M f ∼ = � X i �① ① ① ① ① ① ① i′ �❋ ❋ ❋ ❋ ❋ ❋ ❋ Y j �❋❋❋❋❋❋❋ j′ �①①①①①①① N and N g ∼ = � X i′ �① ① ① ① ① ① ① i′′ �❋ ❋ ❋ ❋ ❋ ❋ ❋ Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' j′ �❋❋❋❋❋❋❋ j′′ �①①①①①①① P We can then compose them to get gf : (i, M, j) → (i′′, P, j′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 2-cospans, J (f): (i, M, j) =⇒ (i′, N, j′) and J (g): (i′, N, j′) =⇒ (i′′, P, j′′), equally well compose, using the vertical composition given by the obvious pushout diagram, which fits into a diagram analogous to the diagram, (53), above, but, of course, replacing X, Y , and Z, with M, N and P, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We leave the enterprising reader to extend this diagram to include what happens to the vertical cospans, A CATEGORIFICATION OF QUINN’S TQFT 144 X → X × I ← X, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The composite 2-cospan will be of form, X � � M � Y � � (X × I) ⊔X (X × I) � L (Y × I) ⊔Y (Y × I) � X � � P � Y, � � in which L is given by the pushout, N ιP 0 ◦g � ιN 1 � P × I � N × I � L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are diffeomorphisms, (X × I) ⊔X (X × I) ∼ = −→ X × I, extending the obvious one from I ⊔{∗} I → I, and the discussion given after (53) carries over to the setting here, giving an equivalence between J (f)#1J (g) and J (gf), so [J (f)]#1[J (g)] = [J (gf)], in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, when M = N and f is the identity diffeomorphism on M, [J (f)] is the vertical identity on (i, M, j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With the functoriality property, we get that if we have an arbitrary diffeomor- phism, f, the 2-morphism, [J (f)], will be invertible, as claimed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It should, now, be more-or-less clear that we have a pseudo-functor, I : Diffn → 2Cob(n,n+1,n+2), so we refer back to page 27 for a checklist of structure and properties needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that I is contravariant due to our notational convention for composition of cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In this setting, for each manifold, X, considered as an object of Diff n, we have that I(X) is that same object considered as an object of 2Cob(n,n+1,n+2), but note that we will write X instead of I(X) most of the time in this context;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each diffeomorphism, f : X → Y , we have a 1-morphism / cobordism, I(f): X → Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each composable pair, X f−→ Y g−→ Z, an invertible 2-morphism, [J (Ψg,f)]: I(f) • I(g) =⇒ I(gf);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and for each object, X, I(idX) is the chosen identity cobordism on X, so there is no problem with the identities (and similarly none with the left and right unitors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This leaves us just to check compatibility of I with the associator in 2Cob(n,n+1,n+2), namely that, given a triple of composable diffeomorphisms, X f−→ Y g−→ Z h−→ W, A CATEGORIFICATION OF QUINN’S TQFT 145 the diagram (57) (I(f)#0I(g))#0I(h) [J (g,f)]#0I(h) � a � I(gf)#0I(h) [J (h,gf)] �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ I(hgf), I(f)#0(I(g)#0I(h)) I(f)#0[J (h,g)] � I(f)#0I(hg) [J (hg,f)] �❥ commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we have written hgf for the value of (hg)f and h(gf), which, of course, are equal, and have abbreviated J (Ψg,f) to J (g, f) for ease of labelling the diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, to emphasise that, here, it is the horizontal composi- tion that is being used, we have replaced the convenient, but ‘generic’, symbol for composition, •, by the more specific one, #0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will formalise this in a proposition for ease of reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 174.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a pseudo-functor, I : Diffn → 2Cob(n,n+1,n+2), given as the identity on objects, and, if f : X → Y is a diffeomorphism between manifolds, then I(f) is given by the cospan, X ιY 0 f �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ Y, ιY 1 �②②②②②②②② Y × I or, equivalently, as the corresponding cobordism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rest of the structure of I is explained in some detail below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The proof of the above will take the form of a reasonably informal exploration of I and J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As it is quite long and a bit technical, we suggest that, if the reader is willing to accept the truth of the above proposition, then such an impatient reader who wants to see the result in action should skip that discussion and go to page 154.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The associativity 2-morphism, a, in the bicategory 2Cob(n,n+1,n+2), ap- pearing in the diagram in (57), is the (equivalence class of the) image under J of the natural homeomorphism68 that comes from the two iterative ways of forming the colimit of the diagram (58) Y ιY 1 �②②②②②②②② ιY 0 g �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ Z ιZ 1 �②②②②②②②② ιZ 0 h �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ Y × I Z × I W × I, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I) and (Y × I) ⊔Y ((Z × I) ⊔Z (W × I)), so, in the left hand biassed one, first constructing the pushout on the left span, followed by that on the right, and, similarly, for the right hand biassed one, doing the right one first in the evident way69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 68which needs smoothing 69We note that the notation used, ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', does not specify how the equivalence relations used are defined, and is just a shorthand referring back to the diagram and the maps used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 146 Given our triple, X f−→ Y g−→ Z h−→ W, however, we can form a colimit of the resulting zig-zag, (58), in a non-biassed way which leads to a direct triple construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That non-biassed model of the colimit is formed from the disjoint union, (Y × I) ⊔ (Z × I) ⊔ (W × I), (which can be given without reference to any pairwise, and thus iterative, formation of such a disjoint union) by quotienting by the evident equivalence relation, generated by (y, 1) ∼ (g(y), 0) and (z, 1) ∼ (h(z), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will write this model for the colimit of the zigzag diagram as (Y × I) ⊔Y (Z × I) ⊔Z (W × I), that is, without (extra) parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This and the other two that we have used are all related by unique diffeomorphisms, given by the fact that they all satisfy the universal property of colimits with respect to the diagram, (58).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can use this unbiassed form of the colimit as a means to construct a ‘multiple’ composite, which it seems reasonable to write as I(f)#0I(g)#0I(h), from X to W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is the cospan obtained from (59) X ιY 0 f �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ Y ιZ 0 g �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ιY 1 �②②②②②②②② Z ιZ 1 �②②②②②②②②② ιW 0 h �❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ ❊ W, ιW 1 �✇✇✇✇✇✇✇✇ Y × I � Z × I � W × I � Colim where Colim = (Y × I) ⊔Y (Z × I) ⊔Z (W × I), by composing to get X → Colim ← W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In diagrams, we will sometimes abbreviate Colim to just C, or, perhaps, to that letter with suffices / superfices to indicate which setting we are using it in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is merely as this takes less space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Returning to the associativity constraint,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' appearing in (57),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' it is fairly evident that as the diffeomorphisms given by the universal properties of a colimit are unique with that property,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have that the associativity 2-morphism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � I(f)#0I(g) � #0I(h) a=⇒ I(f)#0 � I(g)#0I(h) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in (57),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' can be factored as the composite � I(f)#0I(g) � #0I(h) a ⇐⇒ I(f)#0I(g)#0I(h) a ⇐⇒ I(f)#0 � I(g)#0I(h) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where we have abused notation by writing a for the equivalence class of the images under J of each of the relevant unique diffeomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These 2-morphisms do very little as they simply remove or add parentheses to objects and morphisms, but are needed to enable certain diagrams to be constructed without evident clashes with regard to the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will see shortly that this also helps subdivide diagram (57) into two smaller more manageable pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As, for various choices of g and f, [J (g, f)] will occur repeatedly in the coming pages, it will be useful to give it in a lot more detail and to consider generalisations as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is really just a question of plugging the specification of Ψg,f, in (53), A CATEGORIFICATION OF QUINN’S TQFT 147 into the J -construction, but rather than having to work through that each time we need it and for the various combinations of composites that we will be using, we will give the basic form here to act as a template.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have diffeomorphisms, X f−→ Y g−→ Z, as before, and then [J (g, f)] is the equivalence class of the extended cobordism / 2-span, (60) X ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ιX 0 � (Y × I) ⊔Y (Z × I) d � Z ιZ 1 � ιZ 0 � X × I (ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf))×I � (Z × I) × I Z × I ιZ 1 ×I � X ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf) � ιX 1 � (Z × I) ι(Z×I) 1 � Z, ιZ 1 � ιZ 1 � where d(y, s) = (g(y), s/2, 0) and d(z, s) = (z, (s + 1)/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that we have put � (ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf))×I � (x, t) = (gf(x), 0, t), whilst the other maps are hopefully evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we also have to consider composable triples of diffeomorphisms, for the com- patibility diagram, (57), it is natural also to consider the direct way of getting from the unbiassed I(f)#0I(g)#0I(h) to I(hgf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Analogously to the case with just two inputs, we have a diffeomorphism, with domain the unbiassed colimit, Φh,g,f : (Y × I) ⊔Y (Z × I) ⊔Z (W × I) → W × I, given by morphisms: (i) � α : Y × I → W × I, α(y, s) = (hg(y), s/3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) � β : Z × I → W × I, β(z, s) = (h(z), (s + 1)/3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and (iii) � γ : W × I → W × I γ(w, s) = (w, (s + 2)/3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Adapting diagram (59), we have (61) X ιY 0 f �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf) � Y ιZ 0 g �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ιY 1 �②②②②②②②② Z ιZ 1 �①①①①①①①①① ιW 0 h �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ ιY 1 �①①①①①①①①① W ιW 1 �✇✇✇✇✇✇✇✇✇ ιW 1 � Y × I α �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ Z × I β � W × I γ �❧❧❧❧❧❧❧❧❧❧❧❧❧❧ W × I and the induced map, Φh,g,f : Colim → W × I, gives a diffeomorphism of cospans, from I(f)#0I(g)#0I(h) to I(hgf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can, thus, form [J (Φh,g,f)], which we will relabel [J (h, g, f)], and which will be a 2-morphism, again from I(f)#0I(g)#0I(h) to I(hgf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This fits into a subdivided version of the compatibility diagram / cocycle A CATEGORIFICATION OF QUINN’S TQFT 148 equation,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (57): (62) (I(f)#0I(g))#0I(h) [J (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f)]#0I(h) � � a � I(gf)#0I(h) [J (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='gf)] �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ I(f)#0I(g)#0I(h) [J (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f)] � I(hgf),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' I(f)#0(I(g)#0I(h)) I(f)#0[J (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g)] � � a � I(f)#0I(hg) [J (hg,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f)] �❥ and clearly commutativity of this subdivided form will imply commutativity of (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have that [J (h, g, f)] is the equivalence class of the 2-span, in the following diagram, where we recall that C is the Colim in (59), (63) X ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ιX 0 � C ι(W ×I) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Φh,g,f � W ιW 1 � ιW 0 � X × I (ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf))×I� (W × I) × I W × I ιW 1 ×I � X ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf) � ιX 1 � (W × I) ι(W ×I) 1 � W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιW 1 � ιW 1 � This is, of course, easy to calculate by analogy with previous diagrams, but it will be very useful to have it explicitly given to help in the comparison with the 2-morphism that results from the composite around the top of the compatibility diagram, or, for that matter, that around the bottom half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In a diagram as here, we will often omit a specific notation for the composite of an inclusion into a coproduct followed by the quotient to a pushout, or more general colimit, thus, in the above, we have written ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f : X → C for the composite X ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f −−−→ Y × I ֒→ (Y × I) ⊔ (Z × I) ⊔ (W × I) quot −−−→ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Now in order to show the compatibility diagram, (57), or (62), commutes, it will pay to take at least some of the other main 2-morphisms apart to see what they are actually doing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is really included as an exercise in ‘book-keeping’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The idea behind the fact that the top face of the diagram in (62) commutes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is that both compositions yield manifolds homeomorphic to W × I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and each composition is obtained by gluing rectangles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Y × I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Z × I2 and W × I2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' along the way indicated in the schematic pictures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (64) and (65),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' just below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and making use of the homeomorphisms f : X → Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' g : Y → Z and h: Z → Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' to perform the A CATEGORIFICATION OF QUINN’S TQFT 149 identifications: (64) Y ×I Z×I W×I X×I W×I Z × I2 W × I2 Z×I W×I X×I W×I W × I2 W×I and (65) Y ×I Z×I W×I X×I W×I Y × I2 Z × I2 W × I2 Y ×I Z×I W×I X×I W×I W × I2 W×I Looking first at [J (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f)]#0I(h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in (62),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' corresponding to the top bit of the schematic picture in (64),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we note that its (vertical) domain is � I(f)#0I(g) � #0I(h),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' which is a cospan from X to W,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' whose underlying (n + 1)-cobordism is L := ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note again, for later comparison purposes, that this means (y, 1) ∼ (g(y), 0), (z, 1) ∼ (h(z), 0), and the maps making it a cospan are x �→ (f(x), 0) and w �→ (w, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This L is a parenthesised version of C, which we recall is C = (Y × I) ⊔Y (Z × I) ⊔Z (W × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The (vertical) codomain of [J (g, f)]#0I(h), is I(gf)#0I(h), which has as its middle object, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', underlying (n+1)-cobordism, (Z ×I)⊔Z (W ×I), where (z, 1) ∼ (h(z), 0), and the end maps are x �→ (gf(x), 0) and w �→ (w, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Putting this together, and adding a bit more detail, we get [J (g, f)]#0I(h) is of the form below, which completes the description of the top bit of the schematic A CATEGORIFICATION OF QUINN’S TQFT 150 picture in (64), (66) X ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ιX 0 � L down � W ιW 1 � ιW 0 � X × I (ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf))×I � M W × I ιW 1 ×I � X ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf) � ιX 1 � N ιN 1 � W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιW 1 � ιW 1 � Here L = ((Y × I) ⊔Y (Z × I)) ⊔Z (W × I), M = ((Z × I) × I) ⊔Z×I ((W × I) × I), and N = (Z × I) ⊔Z (W × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The morphism labelled ‘down’ satisfies down(y, s) = (g(y), s/2, 0), down(z, s) = (z, (s + 1)/2, 0), and down(w, s) = (w, s, 0), with this last being in the cofactor ((W ×I)×I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As was said before, we will often, in this sort of context, omit a specific notation for the composite of an inclusion into a coproduct followed by the quotient to a colimit such as a pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, for instance, we have written ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf) for the composite map, X ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf) −−−−−→ (Z × I) inc −−→ (Z × I) ⊔ (W × I) quot −−−→ (Z × I) ⊔Z (W × I), where the second map, quot, is the quotient map from the coproduct to the pushout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The final part of the composite along the top face of (62), which corresponds to the bottom bit of the sketch in (64), is obtained by vertical composition with [J (h, gf)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again, to include some ‘book-keeping’, we note that [J (h, gf)] is the equivalence class of the 2-cospan, (67) X ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf) � ιX 0 � (Z × I) ⊔Z (W × I) δ � W ιW 1 � ιW 0 � X × I (ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf))×I � (W × I) × I W × I ιW 1 ×I � X ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf) � ιX 1 � W × I ι(W ×I) 1 � W, ιW 1 � ιW 1 � where δ(z, s) = (h(z), s/2, 0) and δ(w, s) = (w, (s + 1)/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Of course, the middle of the vertical domain / top row is exactly N from earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A CATEGORIFICATION OF QUINN’S TQFT 151 If we now calculate the entire � [J (g, f)]#0I(h) � #1[J (h, gf)], sketched in dia- gram (64), we first construct, where the notation is defined in (53), (68) X ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ιX 0 � L down � W ιW 1 � ιW 0 � X × I (ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf))×I � M W × I ιW 1 ×I � X ιX 1 � ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf)� ιX 0 � (Z × I) ⊔Z (W × I) ιN 1 � δ � W ιW 1 � ιW 1 � ιW 0 � X × I (ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf))×I � (W × I) × I W × I ιW 1 ×I � X ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf) � ιX 1 � (W × I) ι(W ×I) 1 � W, ιW 1 � ιW 1 � and then, taking pushouts ‘vertically in the central belt’, we get (69) X ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='ιY 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='f � ιX 0 � L λ � W ιW 1 � ιW 0 � X × I µ � P W × I ν � X ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf) � ιX 1 � W × I ι(W ×I) 1 � W, ιW 1 � ιW 1 � where we have simplified the expressions using X × I ∼= (X × I) ⊔X (X × I) in the left hand side, and similarly for W × I on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These diffeomorphism will be important later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to explore more thoroughly the description of P here, as the iden- tifications used in its construction are not all of the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have M = ((Z × I) × I) ⊔Z×I ((W × I) × I), so a point in M is either of form (z, s, t) or (w, s, t), where (z, 1, t) ∼ (h(z), 0, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The straightforward way to form P is to first take M ⊔ ((W × I) × I) and then form a quotient, so in addition to the two forms of element we had from M, we have elements in the second copy of ((W ×I)×I), and we will write a typical element there as (w, s, t)2, to indicate this is in the second copy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To get the quotienting relations, we take typical elements of N = (Z × I) ⊔Z (W × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These will be either of form (z, s) or (w, s), and we have (z, 1) ∼ (h(z), 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The image of (z, s) in M is (z, s, 1) and similarly for (w, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We then recall that δ(z, s) = (h(z), s/2, 0)2, and δ(w, s) = (w, (s + 1)/2, 0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The equivalence relation on M ⊔ ((W × I) × I) is generated by (z, s, 1) ∼ (h(z), s/2, 0) and (w, s, 1) ∼ (w, (s + 1)/2, 0)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will return to this very shortly to provide a diffeomorphism from P to another space which is easier to handle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Changing P by that diffeomorphism will not change the class of the 2-cospan and thus just gives a different representing 2-cospan for that 2-morphism in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 152 We have, however, still to describe70 the three arrows, λ, µ, and ν, in diagram (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) λ is defined on L as being the morphism labelled ‘down’ from L to M, composed with the inclusion of M = ((Z × I) × I) ⊔Z×I ((W × I) × I) into M ⊔N ((W × I) × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recalling that L = � (Y × I) ⊔Y (Z × I) � ⊔Z (W × I), this gives λ(y, s) = (g(y), s/2, 0), λ(z, s) = (z, (s + 1)/2, 0), and λ(w, s) = (w, s, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) µ is defined via the diffeomorphism from X × I to (X × I) ⊔X (X × I), so has different defining rules on X × [0, 1/2] and on X × [1/2, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If 0 ≤ t ≤ 1/2, µ(x, t) is given by diagram (66), so is ((ιZ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (gf))×I)(x, 2t) which is (gf(x), 0, 2t) in the ((Z × I) × I) cofactor of M, then ‘included’ into P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If 1/2 ≤ t ≤ 1, then the relevant diagram is (67) and so µ(x, t) = ((ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf)) × I)(x, 2t − 1) = (hgf(x), 0, 2t − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, (iii) for ν, again we have different descriptions for ν(x, t) in the two evident cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If 0 ≤ t ≤ 1/2, ν(w, t) = (w, 1, 2t), whilst if 1/2 ≤ t ≤ 1, ν(w, t) = (w, 1, 2t − 1)2, so within the other copy of (W × I) × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now want to change P by a diffeomorphism, ϕ : P → Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will result in an equivalent 2-cospan, as we can compose each of the four maps in (69) which have codomain P, with this ϕ to get the result we want.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The idea is to deform the 2-cospan in diagram (69) within its equivalence class so as to look more like J (h, g, f), making comparison with that of diagram (63) easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A rough guide to what we want to obtain is given by the schematic picture in (65), representing the final result of the middle path from � I(f)#0I(g) � #0I(h) to I(hgf), in (62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We return to look again at M = ((Z × I) × I) ⊔Z×I ((W × I) × I), whose schematic picture is that of the top bit of the sketch in (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is formed in very much the same way as the middle term (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' underlying (n + 1)-cobordism) of I(f)#0I(g), where we linked that to Z × I;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see the comments just after (60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we try the same idea here, we have a diffeomorphism, ϕ′, between M and (W × I) × I given by ϕ′(z, s, t) = (h(z), s/2, t) ϕ′(w, s, t) = (w, (s + 1)/2, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that ϕ′(z, 1, t) = (h(z), 1/2, t) = ϕ′(h(z), 0, t), so this assignment is a continuous map from M to (W × I)× I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We leave the reader to check that this is a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That it can then be smoothed to a diffeomorphism is then clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can induce a diffeomorphism, ϕ : P → (W × I) × I, which we can take to be the ϕ : P → Q of our previous discussion71, by first forming ϕ′ ⊔N ((W × I) × I), 70It may help to look again at the schematic figure in (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 71thus setting Q = (W × I) × I, A CATEGORIFICATION OF QUINN’S TQFT 153 followed by again using ((W × I) × I) ⊔(W×I) (W × I) × I) ∼= (W × I) × I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That this is compatible with the identifications is routine and is left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We do need now to calculate the deformed model of the 2-cospan of diagram (69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As ι(W×I) 1 ends up in a part of ((W × I) × I) left untouched by the diffeo- morphism, this does not change its specification (although, of course, its codomain does change).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is also easy to check that ϕµ is ((ιW 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (hgf)) × I), whilst ϕν is ιW 1 ×I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, the deformed version of diagram (69) is almost the same as diagram (63), differing from it only in the downward pointing arrow, which is ϕλ in the first of these, and ι(W×I) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Φh,g,f in (63).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Calculation of ϕλ gives ϕλ(y, s) = (hg(y), s/4, 0) ϕλ(z, s) = (h(z), (s + 1)/4, 0) and ϕλ(w, s) = (w, (s + 1)/2, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Comparison with the formula for ι(W×I) 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Φh,g,f, referring to page 147 for that for Φh,g,f, we have that they differ only by a reparametrisation of (W ×I)×{0} within (W × I) × I, and that can be performed by a diffeomorphism of (W × I) × I, which is fixed on the other three faces, (W × {0}) × I, (W × {1}) × I, and (W × I) × {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That gives us a diffeomorphism between the supports of the extended cobordism corresponding to [J (g, f)]#0I(h))#1[J (h, gf)] and that given by [J (h, g, f)], after inserting of parentheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can therefore conclude that [J (h, g, f)] = ([J (g, f)]#0I(h))#1[J (h, gf)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is clear how to adapt the above argument to show that [J (h, g, f)] = (I(f)#0[J (h, g)])#1[J (hg, f)], which concludes the proof of Proposition 174, as well as providing confirmation that I and J together act as a categorification of the I-construction from the unextended case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We note that there is a contravariant version of this Proposition that can also be useful and in which f is sent to the cospan, Y ιY 0 �❋ ❋ ❋ ❋ ❋ ❋ ❋ ❋ X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ιY 1 f �①①①①①①①①① Y × I The proof is more or less the same as that of the above, with some obvious changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 175.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, if g : Y → X is the inverse diffeomorphism of f : X → Y , then I(f) • I(g) ∼= idX, and I(g) • I(f) ∼= idY , in 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These can be used to prove that, in the bicategory 2Cob(n,n+1,n+2), I(f) forms part of an adjoint equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 154 Finally, if f : A → B and g : C → D are diffeomorphisms, then we can form f ⊔ g : A ⊔ C → B ⊔ D, and it is easy to see that I(f ⊔ g) ∼= I(f) ⊔ I(g), again by the diffeomorphism coming from (B ⊔ D) × I ∼= (B × I) ⊔ (D × I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This implies that the pseudo-functor, I : Diffn → 2Cob(n,n+1,n+2), is compatible with the coproduct monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we will use this in the case when one of the two diffeomorphisms is the identity on the corresponding object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A sketch of the construction of the symmetric monoidal structure in the bi- category, 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' After this technical diversion, we can return to the problem of the monoidal associators in the monoidal bicategory, 2Cob(n,n+1,n+2), that we started discussing on page 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to formalise things a little more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this, it may be helpful to give a reference for a fairly standard form of the axioms for a monoidal bicategory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use Johnson and Yau, [68], §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, as a basic reference and will, in general, use their terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For each triple of objects, A, B, C in 2Cob(n,n+1,n+2), we seek a cobordism, αCBA, of form (C ⊔ B) ⊔ A i �◗ ◗ ◗ ◗ ◗ ◗ ◗ ◗ C ⊔ (B ⊔ A) j �♠♠♠♠♠♠♠♠ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have, for n-manifolds, A, B, C, a diffeomorphism, aCBA : (C ⊔ B) ⊔ A → C ⊔ (B ⊔ A), and note that, as (CGWH, ⊔, ∅), forms a monoidal category, these satisfy the pentagon axiom, so for A, B, C and D, the diagram, (70) (D ⊔ (C ⊔ B)) ⊔ A � D ⊔ ((C ⊔ B) ⊔ A) �❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ((D ⊔ C) ⊔ B) ⊔ A �♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ D ⊔ (C ⊔ (B ⊔ A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (D ⊔ C) ⊔ (B ⊔ A) �❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ commutes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now write αCBA := I(aCBA), using the notation defined in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that we have a pseudo-functor, I : Diffn → 2Cob(n,n+1,n+2), as shown in Proposition 174, whenever we have two composable diffeomorphisms, f and g, we have a 2-morphism, [J (Ψg,f)]: I(f) • I(g) =⇒ I(gf), A CATEGORIFICATION OF QUINN’S TQFT 155 which satisfies the cocycle identity in (57).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying this to the arrows in (70), we can then derive an expression for the required pentagonator: (D ⊔ (C ⊔ B)) ⊔ A � D ⊔ ((C ⊔ B) ⊔ A) �❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ((D ⊔ C) ⊔ B) ⊔ A �♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ ♣ �❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ❲ ⇓πDCBA D ⊔ (C ⊔ (B ⊔ A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (D ⊔ C) ⊔ (B ⊔ A) �❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ ❣ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, as I : Diff n → 2Cob(n,n+1,n+2) is a pseudo-functor, this pentag- onator then satisfies a higher order cocycle identity, as in [61, Page 61] and [58, Page 10], when we have five (closed and smooth) n-manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, given (n + 1)-cobordisms, (iA, K, jA′): A → A′, (iB, M, jB′): B → B′, and (iC, N, jC′): C → C′, we have a natural 2-morphism in 2Cob(n,n+1,n+2), fitting inside the diagram below, (71) (C ⊔ B) ⊔ A αCBA � ✙✙✙✙� α2 NMK � (iC,N,jC′)⊔(iB,M,jB′) � ⊔(iA,K,jA′) � (C′ ⊔ B′) ⊔ A′ αC′B′A′ � C ⊔ (B ⊔ A) (iC,N,jC′)⊔ � (iB,M,jB′)⊔(iA,K,jA′) � � C′ ⊔ (B′ ⊔ A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, in the diagram above, we have abbreviated the notation, putting α2 NMK = α� (iC,N,jC′),(iB,M,jB′ ),(iA,K,jA′ ) �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This 2-morphism, α2 N,M,K, arises from the obvious diffeomorphism between the (n+1)-cobordisms obtained from the two paths, from (C ⊔B)⊔A to C′ ⊔(B′ ⊔A′), in the diagram above, together with the construction in Equation (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That diffeo- morphism underpins the naturality of the associativity constraints in Cob(n,n+1), where the diagram consisting of the 1-dimensional arrows in (71) would commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the monoidal bicategory 2Cob(n,n+1,n+2), this diffeomorphism is unsurprisingly promoted to being a part of the symmetric monoidal bicategory structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Together with the associator 1-morphisms, αCBA, the class of all 2-morphisms, α2 NMK, defines a pseudo-natural transformation of bifunctors, α: � 2Cob(n,n+1,n+2)�3 → 2Cob(n,n+1,n+2), called the associator pseudo-natural transformation, as shown below, � 2Cob(n,n+1,n+2)�3 ⊔× � 2Cob(n,n+1,n+2)� � � 2Cob(n,n+1,n+2)� ×⊔ � ✗✗✗✗� α � 2Cob(n,n+1,n+2)�2 ⊔ � � 2Cob(n,n+1,n+2)�2 ⊔ � 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have two different bifunctors, from � 2Cob(n,n+1,n+2)�4 to 2Cob(n,n+1,n+2), defined as ⊔◦(⊔×id)◦ � ⊔×id×id � and as ⊔◦(id×⊔)◦ � id×id×⊔ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Two different pseudo-natural transformations between these bifunctors can be constructed using the associator pseudo-natural transformation, α, above, by considering the two different paths in diagram (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The class of all pentagonators, πDCBA, then defines A CATEGORIFICATION OF QUINN’S TQFT 156 a modification between the corresponding pseudo-natural transformations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This “pentagonator modification” satisfies its own cocycle identity, where we have five copies of 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The equation satisfied is in [61, Page 61] and [58, Page 10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can similarly use that the unit object in (CGWH, ⊔, ∅), comes with natural isomorphisms, ∅ ⊔ A ℓA −→ A and A ⊔ ∅ rA −−→ A, to obtain cospans, λA := I(ℓA) and ρA := I(rA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are just the obvious ones, but linking them with the construction of the pseudo-functor explicitly means that certain diagrams will immediately do what we need, without further checking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Middle Unity Axiom gives that (A ⊔ ∅) ⊔ B aA,∅,B� rA⊔B � A ⊔ (∅ ⊔ B) A⊔ℓB � A ⊔ B = � A ⊔ B, commutes, so, on applying I, we get a specific modification72, µA,B : (idA ⊔ λB) ◦ αA,∅,B → ρA ⊔ idB, in which ◦ stands for the composition of cospans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The evident commutative diagram, (∅ ⊔ A) ⊔ B ℓA⊔B � a∅,A,B �❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ A ⊔ B ∅ ⊔ (A ⊔ B) ℓA⊔B �r r r r r r r r r r r after application of I gives a left 2-unitor, and the reverse / adjoint of r, r∗ A : A → A ⊔ ∅, likewise gives the right 2-unitor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fact that the pasting diagrams for these modifications work as required follows from the (trivially commutative) diagrams in (CGWH, ∅, ⊔) itself, on ap- plication of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All this works in CGWH, but we note that if the objects are smooth manifolds, the structure gives corresponding cobordisms as required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Turning to the braiding, R, on 2Cob(n,n+1,n+2), the structural 1-morphisms are obtained as the image under I of the braiding, τA,B : A ⊔ B ∼= B ⊔ A, in CGWH, given by the universal property of the coproduct, so given closed smooth n-manifolds, A and B, we put RA,B = I(τA,B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As for the case of the associator pseudo-natural transformation, given cobordisms, (iA, M, jA′): A → A′ and (iB, K, jB′): B → B′, we have an extended cobordism, R2 M,N = R� (iA,M,jA′),(iB,K,jB′ ) �, 72An explicit formula can be written down using the explicit construction of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, however, that formula is not that useful in itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 157 fitting into the commutative diagram, A ⊔ B RA,B � ✘✘✘✘� R2 M,N (iA,M,jA′ )⊔(iB,K,jB′ ) � A′ ⊔ B′ RA′,B′ � B ⊔ A (iB,K,jB′ )⊔(iA,M,jA′ ) � B′ ⊔ A′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again, this extended cobordism arises from the obvious diffeomorphism between the two composite cobordisms from A⊔B to B′ ⊔A′, obtained from the two paths from A⊔B to B′⊔A′ in the diagram above, on applying the construction in diagram (56).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Similarly to the associator natural transformation, this diffeomorphism underpins the naturality of the braiding in Cob(n,n+1), but is now promoted to a crucial bit of structure in the symmetric monoidal bicategory 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Together with the RA,B, the class of all R2 M,N defines a pseudo-natural transformation of bifunctors, R, fitting into the diagram below, 2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) ⊔ �❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ τ � ✞✞✞✞� R 2Cob(n,n+1,n+2) × 2Cob(n,n+1,n+2) ⊔ � 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here the bifunctor τ is obtained simply by swapping coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Moreover, this is part of an adjoint equivalence, as in [60, page 4234].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As in the case of the associator natural transformation, to finish constructing a braiding in the monoidal bicategory 2Cob(n,n+1,n+2), we still need to specify modifications as in [60, page 4235], which we will not need explicitly here, and also check the remaining axioms for a braided monoidal bicategory, see loc cit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, this braiding satisfies the axioms for a braided monoidal bicategory to be a symmetric monoidal bicategory, which can be found in [62, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definitions].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This finishes the sketch of the construction of the symmetric monoidal structure on 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structure of the bifunctor 2QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let B be a homotopy finite space, and recall that we fix a non-negative integer, n, throughout this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The basic case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now sketch the proof of the fact that the bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, can be given the structure of a symmetric monoidal bifunctor, with respect to the symmetric monoidal structure, ⊔, in 2Cob(n,n+1,n+2), whose construction we just sketched73, and the symmetric monoidal structure in vProfGrphf, outlined in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter symmetric monoidal structure is a particular case of that of the bicategory of Vect-enriched productors, which is discussed in [63, Corollary 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], in a more general context, and using the language of symmetric monoidal pseudo double categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We know the structure of Prof (and of vProf, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), as bicategories (see Subsec- tion 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8), and this induces the bicategory structure in vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The monoidal structure in vProfGrphf is essentially given as follows: 73see §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2 for a quick outline, A CATEGORIFICATION OF QUINN’S TQFT 158 on objects (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', homotopy finite groupoids) it is given by the usual cartesian product of groupoids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if F : A0 ↛ B0 and G: A1 ↛ B1 are 1-morphisms in vProfGrphf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Vect-profunctors, then F ⊗G: A0×A1 ↛ B0×B1 is given by the composite functor74, (A0×A1)op×(B0×B1) ∼ = −→ (Aop 0 ×B0)×(Aop 1 ×B1) F ×G −−−→ Vect×Vect −⊗Vect− −−−−−−→ Vect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' on 2-morphisms, the rule is (α ⊗ β)(A0,A1),(B0,B1) = α(A0,B0) ⊗ β(A1,B1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With this information, we can construct a bifunctor, ⊗: vProfGrp × vProfGrp → vProfGrp, which is the starting point for the construction of the symmetric monoidal struc- ture on the bicategory vProfGrp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The remaining bits of structure look after themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial component for our discussion of the symmetric monoidal structure of the bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, is the discussion in Lem- mas 114 and 115, and in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, which we need to transfer from 2span(HF) to 2Cob(n,n+1,n+2), by using the mapping space construction B(−);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Remark 150 and Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8 for notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notation for the additional bits of structure that we will give to 2QB follows the pattern of the notation of Definition 30, though we will add a prime to all structure morphisms, to distinguish the notation here from that already used in the context of 2span(HF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first construct a pseudo-natural transformation of bifunctors, fitting into the diagram, � 2Cob(n,n+1,n+2)�2 2QB×2QB � ⊔ � ✗✗✗✗� χ′ � vProfGrphf)2 ⊗ � 2Cob(n,n+1,n+2) 2QB � vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given closed smooth n-manifolds, X and X′, the cartesian closed structure of CGWH gives a natural isomorphism of groupoids, m′ (X,X′) : π1(BX, BX) × π1(BX′, BX′) → π1(BX⊔X′, BX⊔X′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We hence have a profunctor, using the construction in Example 33, χ′ (X,X′) : π1(BX, BX) × π1(BX′, BX′) ↛ π1(BX⊔X′, BX⊔X′), defined as χ′ (X,X′) := ϕm′ (X,X′ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, given cobordisms, (i, Σ, j): X → Y and (i′, Σ′, j′): X′ → Y ′, and hence a cobordism, (i ⊔ i′, Σ ⊔ Σ′, j ⊔ j′) : X ⊔ X′ → Y ⊔ Y ′, we have a 2-morphism in vProfGrphf, π1(BX, BX) × π1(BX′, BX′) H � (i∗,BΣ,j∗) � ⊗H � (i′∗,BΣ′,j′∗) � � χ′ (X,X′) � ✚✚✚✚� χ′ ((i,Σ,j),(i,Σ′ ,j′)) π1(BY , BY ) × π1(BY ′, BY ′) χ′ (Y,Y ′) � π1(BX⊔X′, BX⊔X′) H �� (i⊔i′)∗,BΣ⊔Σ′ ,(j⊔j′)∗�� � π1(BY ⊔Y ′, BY ⊔Y ′), 74see page 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 159 or changing notation, 2QB(X) ⊗ 2QB(X′) 2QB � (i,Σ,j) � ⊗2QB � (i′,Σ′,j′) � � χ′ (X,X′) � ✙✙✙✙� χ′ ((i,Σ,j),(i,Σ′ ,j′)) 2QB(Y ) ⊗ 2QB(Y ′) χ′ (Y,Y ′) � 2QB(X ⊔ X′) 2QB �� (i⊔i′,Σ⊔Σ′,j⊔j′) �� � 2QB(Y ⊔ Y ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This natural isomorphism of profunctors is obtained from Lemma 115, together with the fact that we have a homeomorphism of HF spans, � (i ⊔ i′)∗, BΣ⊔Σ′, (j ⊔ j′)∗� ∼= � i∗ × i′∗, BΣ × BΣ′, j∗ × j′∗� , arising from the cartesian closed structure of CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By applying Lemma 133, it follows that χ′ is a pseudo-natural transformation, χ′ : ⊗ ◦(2QB × 2QB) → 2QB ◦ ⊔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will sketch the construction of the rest of the symmetric monoidal structure on 2QB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have two bifunctors L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' R: � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2)�3 → vProfGrphf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' defined by composition along the boundary left and right paths,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in the two diagrams below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as in [61,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' page 67],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2)�3 id×⊔ � ⊔×id �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ✚✚✚✚� χ′×2QB 2QB×2QB×2QB � � vProfGrphf �3 ⊗×id � � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2))2 ⊔ � ⇐= α � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2))2 ⊔ �✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ ❴❴❴❴� χ′ 2QB×2QB � � vProfGrphf �2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ⊗ �✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐✐ 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2) 2QB � vProfGrphf ω′ ❴� � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2)�3 id×⊔ � ✚✚✚✚� 2QB×χ′ 2QB×2QB×2QB � � vProfGrphf �3 ⊗×id � id×⊗ �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ � 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2))2 ⊔ � ✕✕✕✕� χ′ 2QB×2QB � � vProfGrphf)2 ⇐= α ⊗ � � vProfGrphf �2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ⊗ �❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥❥ 2Cob(n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='n+2) 2QB � vProfGrphf,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as well as two natural transformations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' connecting L and R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as defined in the two diagrams above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that we have used the same notation for the associator modification of 2Cob(n,n+1,n+2) and of vProfGrphf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The constructions, as discussed in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, especially in Notation 134 and Lemma 135, give a modification, ω′, as shown above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given manifolds X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X′ and X′′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and abbreviating,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for a topological space X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' π(X) = π1(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' X),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have that A CATEGORIFICATION OF QUINN’S TQFT 160 ω′ (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′′) is the obvious natural isomorphism of profunctors in the diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � π(BX) × π(BX′) � × π(BX′′) D �❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧❧ A �P P P P P P P P P P P P P P P P P π(BX) × � π(BX′) × π(BX′′) � E � ❴❴❴❴� ω′ (X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′′) π(BX⊔X′) × π(BX′′) B � π(BX) × π(BX′⊔X′′) F �❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ π � B(X⊔X′)⊔X′′� C �♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦ π � BX⊔(X′⊔X′′)� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we have considered the associator, αCGWH,⊔ X,X′,X′′ : (X ⊔ X′) ⊔ X′′ → X ⊔ (X′ ⊔ X′′), in the category CGWH, with the disjoint union monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The arrows in the above diagram are labelled as follows: A = ϕm′ (X,X′) × idπ(BX′′ );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' B = ϕm′ (X⊔X′ ,X′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' C = ϕπ � B � αCGWH,⊔ (X,X′,X′′) �−1� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' D = ϕ αGrp (π(BX ),π(BX′ ),π(BX′′ ));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' E = idπ(BX) × ϕm′ (X′,X′′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and F = ϕm′ (X,X′⊔X′′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This diagram is the result of applying the ϕ(−)-construction, given in Example 33 in which a functor is converted into a profunctor, to an evident commutative diagram at the groupoid level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this reason, the modification, ω′, satisfies the cocycle equation in [61, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3], or [58, page 17], when we are given manifolds X′, X′, X′′ and X′′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, is also compatible with the unitor natural transformations in 2Cob(n,n+1,n+2) and in vProfGrphf, as well as with the braiding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from considerations analogous to those we have just given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We state for the sake of reference: Theorem 176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bifunctor, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' what we have called the once-extended Quinn TQFT75 of Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, can be upgraded to being a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 75The name used is now justified by this theorem A CATEGORIFICATION OF QUINN’S TQFT 161 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A symmetric monoidal structure for the B-decorated case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bicategory 2Cob(n,n+1,n+2) induces an obvious symmetric monoidal structure on the bicat- egory 2Cob (n,n+1,n+2) B , defined in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The tensor product of (Σ, f Σ) with (Σ′, gΣ′) is given by (Σ, fΣ) ⊗ (Σ′, gΣ′) = (Σ ⊔ Σ′, f Σ ⊗ gΣ′), where f Σ ⊗ gΣ′ := � ⟨φ, φ′⟩ | φ ∈ fΣ and φ′ ∈ gΣ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The discussion in that subsection can easily be adapted for the case where we have decorated manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows again from the calculations in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We hence have: Theorem 177.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bifunctor, 2QB, defined in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, (Definition 154), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', what we have called the finitary once-extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, can be upgraded to being a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A symmetric monoidal structure for the Morita valued once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' So far in this section, we have sketched proofs of the facts that both bi- functors, 2QB : 2Cob(n,n+1,n+2) → vProfGrphf, and 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, can be given the structure of a symmetric monoidal bifunctor, hence they are fully fledged once-extended TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now sketch the construction of the symmetric monoidal structure of the Morita valued once-extended Quinn TQFT, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor, defined in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, we show that 2Q Mor B is a fully fledged once- extended TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that the Morita valued once-extended TQFT is obtained as a composite of bifunctors, 2Cob (n,n+1,n+2) B 2QB −−−→ vProfGrpfin Lin(2) −−−−→ Mor, it will be sufficient to prove that the latter arrow, from vProfGrpfin to Mor, can be upgraded to being a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That Lin(2) can be given a symmetric monoidal structure is a purely categorical / algebraic exercise, so we will just give a sketch of that claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our main tool is Mitchell’s theorem, here Theorem 160 on page 131, for a linear category C, with finitely many objects, using the approach that we took of constructing a bifunctor [−] : C−Mod → [C]−Mod as well as the two profunctors, AC and BC, giving a Morita equivalence;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see the discussion in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, starting on page 131, and especially Remark 163.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, we will show how that theory allows one to define a bifunctor from the bicategory vProf, of Vect-enriched categories and Vect-enriched profunctors between them, to Mor, which is the ‘reflector’ onto the sub-bicategory corresponding to Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that no finiteness or other restrictions are needed for this, which is why this construction works on the whole of vProf, and not just in vProfGrpfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define a bifunctor, [−]: vProf → Mor, as follows: A CATEGORIFICATION OF QUINN’S TQFT 162 remembering that vProf 0 consists of linear categories, we have that [−]0 : vProf 0 → Mor0 sends C to the algebra [C];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if H : C ↛ D is a Vect-valued profunctor, then [H]: [C] ↛ [D] is the com- posite bimodule, obtained by the following composition of profunctors76, as discussed in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, particularly in Remark 163, BC • H • AD : [C] ↛ C ↛ D ↛ [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall that this bimodule is determined only up to natural isomor- phism as it is given by multiple use of coends.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The question of the lack of associativity of the composition is absorbed into that as well, but a standard form of representing bimodule can be given, namely, [H] = � p∈C,q∈D H(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The left and right algebra actions of [C] and [D] are as discussed in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, following our conventions for profunctors77, H : C ↛ D corresponds to a functor H : Cop × D → Vect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next assume given H : C ↛ D and K : D ↛ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We do not expect that composition will be preserved by [−], so look at the two ways of producing things, namely [H • K] and [H] • [K].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Firstly [H • K] = BC • H • K • AE, and then [H] • [K] = BC • H • AD • BD • K • AE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We will ignore any problems arising from composition being non-associative in vProf, as these can be handled using associators, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', in a standard way, completely analogous to handling non-associativity of tensor products in Mor, which is a special case of this one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We recall, from Lemma 161, that the profunctors AD •BD and D(−, −), from D to itself, are naturally isomorphic, so we have an invertible 2-cell [H] • [K] ⇒ [H • K], in Mor, as required, and moreover, this satisfies the appropriate cocycle identity, given a triple of composable bifunctors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that explicit formulae can be given for this 2-cell in terms of the direct sum over the objects of C and E, with tensoring78 over the algebra [D].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also need an invertible 2-cell, id[H] ⇒ [idH].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The domain of this is the identity morphism on [H], as a bimodule, and the righthand side is the whiskered composite, BC •idH •AD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is easily checked to be isomorphic to the identity on [H], for instance using that [H] ∼= ⊕p∈C,q∈DH(p, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We omit the verification that this structure determines a bifunctor, [−]: vProf → Mor, as that verification is quite long, whilst not being that full of insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We next ask if [−] is monoidal, or more precisely whether it can be given a monoidal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We start with F : A ↛ B and G: C ↛ D, and also have F ⊗ G: A × C ↛ B × D, given by (F ⊗ G)((a, c), (b, c)) = F(a, b) ⊗κ G(c, d), 76Note once again our composition convention for profunctors, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 77Again see Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 78In those terms, the invertibility of the 2-cell is related to the distributivity of tensors and direct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 163 so [F ⊗ G] = ⊕(a,b) ⊕(c,d) F(a, b) ⊗κ G(c, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, [F] ⊗ [G] = � ⊕(a,b) F(a, b) � ⊗κ � ⊕(c,d) G(c, d) � , and these two are naturally isomorphic by the standard argument relating tensors and direct sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Interaction of [−] with the monoidal units is easy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In vProf, the monoidal unit is the single object linear category having a 1-dimensional vector space as its endomorphism ring, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', it is actually an algebra in its own right, being essentially a copy of κ itself, and applying [−] does nothing to it!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have that [−] is a normalised monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That [−] respects the symmetric monoidal bicategory structure (up to specified isomorphisms) is then, once again, a result of the natural isomorphisms linking tensor products of direct sums with direct sums of tensor products, when that is suitably interpreted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We leave it to the reader explicitly to write down the full symmetric monoidal bifunctor structure of [−]: vProf → Mor, as just outlined, in the language of Defi- nition 30, similarly to that which we did for the bifunctor 2QB : 2Cob(n,n+1,n+2) → vProfGrphf in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that [−] is essentially the identity bifunctor when it is restricted to Mor (which can be considered as a sub-bicategory of vProf).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not explore further this relationship between vProf and Mor as this might distract from the use we will be making of the basic reflection bifunctor, [−].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The implication of the above is that the ‘Mor-valued’ once-extended TQFT, developed in Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, is, as we claimed, actually a fully-fledged once-extended TQFT, as it can be given the structure of a symmetric monoidal bifunctor from 2Cob (n,n+1,n+2) B to Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This proves: Theorem 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bifunctor, 2QB, defined in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', what we called the Morita valued once-extended Quinn TQFT, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' can be upgraded to being a symmetric monoidal bifunctor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Part 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Calculations for classifying spaces of ω-groupoids We now have a TQFT and a once-extended TQFT that depend on the choice of a homotopy finite space, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this Part, in Section 7, we will show that we have the means for efficiently calculating the values of such TQFTs and once- extended TQFTs, if we restrict to homotopy finite spaces B that are classifying spaces of homotopy finite crossed complexes (the latter being equivalent to strict ω-groupoids, [27, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], that are themselves homotopy finite).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Section 8, we will further use these ideas to calculate the values for specific examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, let n be a non-negative integer, and B be a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have for s, a complex parameter, Quinn’s finite total homotopy TQFT, Q(s) B : Cob(n,n+1) → VectC, A CATEGORIFICATION OF QUINN’S TQFT 164 (Definition 83), as well as the finitary once-extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin (Definition 153), and the Morita valued once-extended Quinn TQFT, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor, (Definition 154).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These can all be explicitly computed, combinatorially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This can be achieved in various ways, for example, by passing to the category of simplicial sets, or similar combinatorial models for homotopy theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The calculations of Q(s) B , 2QB and 2Q Mor B , using the category of simplicial sets, require only finite calculations, since, by Ellis’ theorem, [46], path-connected spaces with a finite number of non-trivial homotopy groups, all of which are finite, can be represented (up to homotopy) by finite simplicial groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will examine this in a separate paper, as it requires some development of other techniques, especially around triangulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the remainder of this paper, we will outline the interesting special cases of such explicit combinatorial computations for the case in which B is the classifying space, BA, of a homotopy finite crossed complex, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We apply the tools of the homotopy theory of crossed complexes, developed for instance in [27, 116], and show that their use yields explicit formulae for Q(s) B and 2QB, and, as a consequence, for 2Q Mor B , as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A particularly simple case is when A is a finite crossed complex, with a unique 0-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The formulae we will obtain for Q(s) B : Cob(n,n+1) → VectC, in this case, extend those of our previous paper, [53], (which only dealt with the case of closed manifolds), in the particular case of a trivial homology class.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will summarise the necessary parts of the theory of crossed complexes and will also give some new results that will be needed to enable the computations to proceed without difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category of crossed complexes, which strictly includes the category of strict 2-groups, [7], and [27, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5], is equivalent to the category of (strict) omega-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a precise statement and proof of this see [27, §13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter fact justifies the title of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed complexes also model strict ∞-groupoids via the nerve construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It should be noted that homotopy finite crossed complexes do not model all homotopy finite spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The specification of the homotopy types thus classified is slightly complicated, and will not be needed here, so will not be recalled in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such crossed complexes do, however, model all 2-types, X, that is, all spaces, X, such that πi(X, x) = 0 if i ≥ 3, and for all possible choices of base- points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The explicit formulae we will construct, thus apply to the TQFTs Q(s) B , and the extended TQFTs 2QB and 2Q Mor B , where B is the classifying space of a finite 2-group (equivalent to a crossed module), as mentioned above, and as in [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the last part of this paper leads to a construction of TQFTs and extended TQFTs derived from discrete higher gauge theory based on a 2-group;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [124], and also [34, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the extended case, the formulae are similar to those derived from the ‘tube algebras’ considered in [32] and [33, Section 3], in the context of excitations of strict 2-group topological phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 165 In more detail, in the coming section, we will review some of the basics of the homotopy theory of crossed complexes, and their classifying spaces, and then prove some refinements of well know results in the literature, which will lead to the explicit formulae for TQFTs and once-extended TQFTs derived from crossed complexes mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter work will be done in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that subsections 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 – 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 contain no new results, and essentially fol- low [25, 26, 27, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, on fibrations of crossed complexes, revisits definitions and results from [22, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial new result, refining the main theo- rem in [26], concerns a crossed complex model for the fibre of the restriction map (BA)|S| → (BA)|T |, where A is a crossed complex, and T is a subcomplex of a sim- plicial set, S, ( | − | here denoting geometric realisation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This result will be used, later, to write down, in all detail, TQFTs and extended TQFTs derived from ho- motopy finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is done in Section 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Before that, the results in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6 are essentially in [49, 53], and they allow for a simple calculation of the homotopy content of finite crossed complexes, akin to the well known for- mula for the Euler characteristic of a finite CW-complex, as the alternating sum of cardinalities of the sets of i-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed complexes: their homotopy theory and their classifying spaces The main sources for this section are [27], and / or some of the sources already listed, which are summarised therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A new source, that we hope will be available shortly, is our (pre) preprint, [54], which handles some of those aspects of the theory of crossed complexes that were not included in the main source, cited above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This section will, naturally, consist of lots of definitions, with some commentary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition of crossed complexes, and related notions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first need some useful terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By a set over X, we will mean a set, Y , together with a surjective map, β : Y → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid right-action of a groupoid, Γ = (s, t: Γ1 → Γ0), on a set over Γ0, (Y, β), is an operation which, given y ∈ Y and an arrow, (β(y) γ−→ x) ∈ Γ1, associates y ⊳ γ, also denoted y ⊳ (β(y) γ−→ x), in Y , with β(y ⊳ g) = x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is such that if β(y) = a, we always have: � y ⊳ (a γ−→ b) � ⊳ (b γ′ −→ c) = y ⊳ (a γγ′ −−→ c), y ⊳ (a 1a −→ a) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid action of Γ gives rise to a functor Γ → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Various equivalent formulations of the definition of crossed complex can be found in [12, 13, 20, 27, 116], and many other places in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that Baues, in [12], and [13], prefered to call them crossed chain complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For convenience, we recall one form of the definition here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 179 (Crossed complex, [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='iii]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed complex, A = (An)n∈Z+ 0 , is given by: a set, A0, called the set of objects of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a groupoid, A1 = (s, t: A1 1 → A0), with object set A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for each integer n ≥ 2, a totally disconnected groupoid, An = (β : A1 n → A0), A CATEGORIFICATION OF QUINN’S TQFT 166 with object set A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' whenever n ≥ 2, a groupoid map ∂n = ∂ : An → An−1, which is required to restrict to the identity on the set of objects A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and a groupoid right-action, ⊳, of A1 on all the underlying sets, β : An → A0, over A0, for all n ≥ 2, which is required to preserve the composition and the identities in each An, where n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x, y ∈ A0, a = (x a−→ x) ∈ A1(x, x), and g = (x g−→ y) ∈ A1(x, y), we also write a ⊳ g := g−1ag ∈ A1(y, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This data is to satisfy the following additional conditions: (1) for all n ≥ 3, given (x a−→ x) ∈ A1 n with x ∈ A0, then ∂(∂(x a−→ x)) = 1x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) for n ≥ 2, and given any x, y ∈ A0 and g ∈ A1(x, y), if a ∈ An(x, x), we have ∂(a ⊳ g) = ∂(a) ⊳ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is sometimes called the first Peiffer condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (3) for any x ∈ A0 and a, b ∈ A2(x, x), then a ⊳ ∂(b) = b−1ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is sometimes called the second Peiffer condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (4) If n ≥ 3, then given any x ∈ A0, and a ∈ A2(x, x), b ∈ An(x, x), we have b ⊳ ∂(a) = b, and so ∂(A2) ≤ A1 acts trivially on all An for n ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (5) If x ∈ A0 and n ≥ 3, then each group, An(x, x), is abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our notation, A = (An)n∈Z+ 0 , for a crossed complex leaves the boundary maps, ∂, and the actions of the groupoid, A1, implicit in the notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Another useful way of picturing this structure is: A = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 n ∂−→ A1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 2 ∂−→ A1 1 t ⇒ s A0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will usually refer to the arrows in An, as morphisms, making clear in which dimension they are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We may also call them n-morphisms or n-arrows if, in some context, that will be clearer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A0 is a singleton, and hence the groupoid, A1, has just a single object, {∗}, as will often be the case, we may also write this as: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ An ∂−→ An−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A2 ∂−→ A1 β→{∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, here, we have identified each groupoid, A1 i , with its group of mor- phisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter type of crossed complex is sometimes referred to as being reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two sources, [12, 13], restrict attention to such reduced crossed complexes, calling them ‘crossed chain complexes’, but we will need the non-reduced variety as we will be considering the crossed complexes corresponding to function spaces, where the restriction to a reduced case would be very unnatural.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a reduced crossed complex is thus a chain complex of groups, such that Ai is abelian if i ≥ 3, together with actions of the group, A1, on all the groups in higher dimensions and, of course, satisfying some other axioms, as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For later use, we will also need the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 180 (Crossed modules of groupoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed module of groupoids is given by a 2-truncated crossed complex, that is, one in which each groupoid, An, for n > 2, consists just of identity loops on each object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is easy to give a direct definition of this notion, independently of that of crossed complex, but, for the moment, we leave that to the reader to find in the sources or to concoct by throwing away axioms and structure in dimensions greater than 2 A CATEGORIFICATION OF QUINN’S TQFT 167 in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For ease of reference, we will give an explicit definition of the group (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' reduced) version later on in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will define truncation more formally shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Maps between crossed complexes, see, for instance, [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Let A = (An)n∈Z+ 0 and B = (Bn)n∈Z+ 0 be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed complex map, f = (fn)n∈Z+ 0 : A → B, is given by a set map, f0 : A0 → B0, and groupoid maps fi : Ai → Bi, where i ≥ 1, that restrict to f0 on objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These are required to preserve the actions of A1 and B1, and the boundary maps in A and B, in the obvious way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed complexes and the maps between them form a category which will be denoted Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As is shown in [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2], Crs is closed under all small limits and colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following is essentially in [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='vi].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a positive integer n, a crossed complex, A = (Ai)i∈Z+ 0 , is said to be n-truncated if, for i > n, the groupoids, Ai, have only identity morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We thus have that, for i > n, each of the groupoids, Ai, is a discrete / trivial groupoid on the set of objects, A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have inclusion functors, ι1 : Grp → Crs, sending a groupoid to the obvious 1-truncated complex, and ι0 : Set → Crs, sending a set to the obvious 0-truncated crossed complex, which is thus discrete in all dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, we can consider the inclusion, ιn, of the subcategory of n-truncated crossed complexes into Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All of these will be treated as if they are inclusions of subcategories, and then the ιn may, sometimes, be omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, we may think of groupoids as 1- truncated crossed complexes, and so once we have defined the classifying space BA of a crossed complex, A, then we may treat the construction of a classifying space of a group or groupoid as being a particular case of that more general construction, without making additional comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These inclusion functors have right adjoints, denoted Tn, so that T1 and T0 (respectively), send A = (An)n∈Z+ 0 to the groupoid, A1, and to the set, A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The functors, ι1 : Grp → Crs and ι0 : Set → Crs, also have very useful left adjoints79, π1 : Crs → Grp, the fundamental groupoid functor, and π0 : Crs → Set, the set of components functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In simple terms, these are given as follows: Definition 183 (The functors, π1(A, A0) and π0(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fundamental groupoid of A is defined as: π1(A) := A1/∂(A2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We will sometimes denote π1(A) by the more suggestive π1(A, A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We also put: π0(A) := π0(A1), the set of connected components of the groupoid at the base of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As the notation indicates, π1(A, A0) is a groupoid with one object for each element of A0, and we note that π0(A) = π0(π1(A, A0)), the set of components of the fundamental groupoid of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As the functor, π0, is left adjoint to ι0 : Set → Crs, it comes with a unit map, ηA : A → ι0 � π0(A) � , 79The other functors ιn also have left adjoints, but we will not be using them here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 168 which is a quotient map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This quotient map is, thus, really part of the definition of π0(A), although it is often not mentioned explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This map, ηA, is, in fact, also the coequaliser morphism of the evident pair of maps from AI to A, being evaluation at 0 and 1 respectively, analogously to the situation in the topological case that we used earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here by AI, we mean the crossed complex, CRS(I, A), of maps from the unit interval groupoid, which we denote here by I, thought of as a 1-truncated crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This groupoid, I, which we will meet again very shortly, is the fundamental groupoid of the unit inter- val, [0, 1], based at the two end points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will see the construction of CRS(A, B) in general a bit later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we noted above, we will often, even usually, omit the ι0 from the notation, thus tacitly thinking of a set as the set of objects of the corre- sponding 0-truncated crossed complex, which will be denoted by the same letter, and we will do the analogous thing for ι1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will return to this topic below, but before that we need: Definition 184 (Fibres of a crossed complex map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 and B = (Bn)n∈Z+ 0 be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f = (fn)n∈Z+ 0 : A → B be a crossed complex map, and let b ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre of f : A → B, at b, is the sub-crossed complex, f −1(b) = (Cn)n∈Z+ 0 , of A, with object set C0 = f −1 0 (b), and such that Cn consists of those elements, a ∈ An, such that fn(a) = 1Bn b , the identity of the groupoid, Bn, at b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As in the above definition, let b ∈ B0, then we let ˆb denote the sub-crossed complex of B with object set, {b}, and only identity arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We clearly have a pullback diagram f −1(b) � inc � A f � ˆb inc � B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Inclusions of crossed complexes, with the obvious meaning, will sometimes be denoted inc, provided no confusion arises.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Just as in the topological case, the ‘set’, π0(A), is not just a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As given above, it is a quotient of A0, so its ‘elements’ would naturally be thought of equivalence classes of elements of that set of objects of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' That, however, forgets that π0(A) is also a quotient of A, and the quotienting map, ηA, is really part of the structure of this ‘set of connected components’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now give some formal definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 185 (PCx(A) and � π0(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ A0, we define the crossed com- plex, PCx(A) := (Bn)n∈Z+ 0 , in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set, B0, of objects of PCx(A) consists of all elements in A0 connected to x in the groupoid, A1, so if a ∈ B0, there is an arrow a → x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For each positive integer n, the set of morphisms in Bn consists of the morphisms in An connecting elements in B0, so either, if n = 1, connecting two elements of B0 in the groupoid A1, or, if n > 1, in the set of loops, An(b, b), for some b ∈ B0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We call the crossed complex PCx(A), the path-component of x in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also may write � π0(A) = {PCx(A) | x ∈ A0}, the collection of these path- components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that different elements x ∈ A0 may induce the same PCx(A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed complex is said to be path-connected if it only has one path-component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 169 We leave it to the reader to check that indeed PCx(A) is a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, for c ∈ A0, PCc(A) is the fibre of ηA over ηA(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 186.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a crossed complex A, there is an obvious epimorphism of crossed complexes, � B∈� π0(A) B → π0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This epimorphism maps the crossed complex, PCc(A), to the element ηA(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is almost a bit of ‘double think’, but, later on, it will be useful to distinguish between π0(A) and � π0(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Remark 187.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have, in fact, a canonical isomorphism of crossed complexes, � B∈� π0(A) B −→ A, where on the cofactor, B = (PCc(A) → π0(A)), the map is the inclusion of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This says that any crossed complex, A, is a coproduct of its connected components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Just as with spaces, crossed complexes come with a notion of homotopy groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They are defined to be the obvious homology groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 188 (Homotopy groups of crossed complexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let c ∈ A0, and let n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define πn(A, c) to be the group, ker � ∂ : An(c, c) → An−1(c, c) � /im � ∂ : An+1(c, c) → An(c, c)) � , thus, in particular, π1(A, A0)(c, c) = π1(A, c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fundamental crossed complexes of filtered spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Many of the prime examples of crossed complexes come from filtered spaces, and, in particular, from CW-complexes considered with their natural skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a more com- plete view of this, see [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='i] and the development of related ideas there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first need to say what we mean by a filtered space, and the appropriate notion of maps between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 189.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A filtered space, X∗, is a CGWH space, X, together with an increasing sequence, X0 ⊆ X1 ⊆ X2 ⊆ · · · ⊆ Xn ⊆ · · · ⊆ X, of subspaces of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A filtered map, f : X∗ → Y∗, between filtered spaces, is a continuous map, f : X → Y , of the ambient spaces, such that f(Xi) ⊆ Yi for all i ∈ Z+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let Fil denote the category of filtered spaces and filtered maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a crossed complex functor, Π: Fil → Crs, sending a filtered space, X∗, to its fundamental crossed complex, Π(X∗), defined as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 190.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fundamental crossed complex, Π(X∗), of a filtered space X∗, is specified by the following: the set of objects of Π(X∗) is Π(X∗)0 = X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the groupoid, Π(X∗)1, is given by the fundamental groupoid, π1(X1, X0), of X1, with set of base-points X0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if n ≥ 2 and x ∈ X0, then Π(X∗)n(x) = Π(X∗)n(x, x) := πn(Xn, Xn−1, x), the usual relative homotopy group based at x80;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 80Recall from page 14, that if G is a groupoid, then G(x) is the same as G(x, x), and is the vertex group of a groupoid at an object x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 170 for each n ≥ 2, and each x ∈ X0, the boundary map, ∂ : Π(X∗)n(x) → Π(X∗)n−1(x), of Π(X∗) is the map appearing at the relevant position in the long homotopy exact sequence of the triple (Xn, Xn−1, Xn−2), and the action of π1(X1, X0) is the standard one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We direct the reader to [27, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='v] for more explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Most filtrations appearing in this paper will be skeletal filtrations of CW-complexes, X, and, to start with, these may be denoted Xsk := (X0 ⊆ X1 ⊆ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), where Xi is the i-skeleton of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, X, and all the spaces, Xi, are CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, a filtered map, f : X → Y , between CW-complexes with the skeletal filtrations, is called cellular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will give quite a few examples, so as to fix some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First we consider some basic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let I = [0, 1], with the standard CW-decomposition with two 0- cells, at 0 and 1, and one 1-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If n ≥ 1, we let Sn have the CW-decomposition with one 0-cell, denoted ∗, for concreteness at the south pole, and one n-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Dn+1 have the CW-decomposition for which Sn is subcomplex, and we have an additional (n + 1)-cell attaching along the identity map, Sn → Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have: Π(Isk) ∼= ι1(π1(I, {0, 1})).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, in π1(I, {0, 1}), we have objects 0 and 1, and only two non-identity morphisms, 0 (0,1) −−−→ 1 and 1 (1,0) −−−→ 0, so this is exactly the ‘unit interval groupoid,’ that we denoted by I above, but now considered as a crossed complex in the way that we mentioned earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All the higher dimensional groupoids are discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Π(S1 sk) ∼= · · · → 0 → 0 → 0 → Z → ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Π(D2 sk) ∼= · · · → 0 → 0 → Z id −→ Z → ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here the action of Z ∼= π1(S1, ∗) on π2(D2, S1, ∗) ∼= Z is the trivial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If n ≥ 2, then Π(Sn sk)n = Z, Π(Dn+1 sk )n = Z and Π(Dn+1 sk )n+1 ∼= Z, and all other groupoids are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hence: Π(Sn sk) ∼= · · · → 0 → 0 −→ Z → 0 → · · · → 0 → ∗, Π(Dn+1 sk ) ∼= · · · → 0 → Z id −→ Z → 0 → · · · → 0 → ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need the fundamental crossed complex of the n-simplex, with its skeletal filtration, and, more generally, of the geometric realisation of simplicial sets, where the crossed complex can also be specified algebraically in terms of the combinatorial description of the simplicial set, by using the crossed complex version of the homotopy addition lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A good source for the details on these topics is §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='10 of [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Before we look at those families of examples, we will need to recall a few results on the more general case of CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that referring to a CW-complex means that the space comes with a specified CW-decomposition, and the cells used, attaching and characteristic maps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', are regarded as part of the structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, it will be useful, for the sake of the exposition, to be a bit more restrictive, and to consider what we will call special CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 171 Definition 192 (Special CW-complex).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A special CW-complex is a CW-complex, X, for which the attaching maps of all n-cells, for n ≥ 2, are such that the unique 0-cell of Sn−1 is sent to a 0-cell of Xn−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that it follows that the attaching maps and characteristic maps of each cell of a special CW-complex are cellular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let CW be the category of CW-complexes, (each given a specified CW-decomposition), and cellular maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let sCW be the full subcategory, of CW, whose ob- jects are the special CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fundamental crossed complex functor, Π: Fil → Crs, restricts to functors, Π: CW → Crs, and Π: sCW → Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fundamental crossed complex of a CW-complex is ‘free’ on its cells [20, Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us explain what this means, for the sake of exposition, in the particular case of special CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following is adapted from [27, Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='19 and Corollary 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First the intuitive idea, before we give a few more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that X is a CW-complex, then, for each 0-cell / vertex, x, we have Π(X∗)n(x, x) = πn(Xn, Xn−1, x), which, if n ≥ 3, is well known to be a free module, over the fundamental group π1(X, x), free on the set of n-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, π1(X1, X0), it- self, is a free groupoid, over the set of 1-cells, of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For n = 2, π2(X2, X1, x) is a free crossed π1(X1, X0)-module, [27, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The idea of ‘freeness’ in the crossed complex case is that we can completely specify a crossed complex map, f : Π(X∗) → A, by saying where that generating set of n-cells is to be sent in each dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Of course, that assignment should be compatible with all the actions, boundary maps, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', so requires quite a bit more work, and a bit more notation, to make this precise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a special CW-complex, X, and n ∈ Z+ 0 , we let C(X, n) be the set of n-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given an n-cell, c ∈ C(X, n), we let Dn c = Dn and Sn−1 c = Sn−1, (the latter being empty if n = 0), and let ic: Sn−1 c → Dn c be the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Supposing that n ≥ 1, let ψc : Sn−1 → Xn−1 be the attaching map of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let φc : Dn c → Xn be the characteristic map of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The inclusion of the (n − 1)-skeleton Xn−1 into Xn, the n-skeleton of X, will be denoted ιn : Xn−1 → Xn, in the two diagrams below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a pushout diagram in the category CGWH, � c∈C(X,n) Sn−1 c � c∈C(X,c) ψc � � c∈C(X,n) ic � � c∈C(X,n) Dn c � c∈C(X,c) φc � Xn−1 ιn � Xn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The vertical arrows arise from universal properties of disjoint unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Note that all maps appearing in the diagram above are cellular if X is a special CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 172 The diagram below is a pushout in the category of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � c∈C(X,n) Π(Sn−1 c,sk ) � c∈C(X,c) Π(ψc) � � c∈C(X,n) Π(ic) � � c∈C(X,n) Π(Dn c,sk) � c∈C(X,c) Π(φc) � Π(Xn−1 sk ) Π(ιn) � Π(Xn sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here Sn−1 c,sk and Dn c,sk denote the skeletal filtrations of Sn−1 c and Dn c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Furthermore, we have a natural isomorphism (of functors from sCW to Crs), Π(Xsk) ∼= colimn(Π(Xn sk), Π(ιn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This implies the freeness criteria we mentioned before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More exactly, we note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (For more details, see [27, Definition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='13] and [13, Chapter III], also reviewed in [49, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1], and in [50] for fundamental crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Let X be a special CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A way to state the freeness of Π(Xsk) = (Π(Xsk)n)n∈Z+ 0 on the cells of X is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid, Π(Xsk)1 = π1(X1, X0), is the free groupoid on the graph corresponding to the 1-skeleton of X1 of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, Π(Xsk)1 is the free groupoid on the set of 1-cells of X, and their attaching maps in X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The totally disconnected groupoid Π(Xsk)2, with X0 as its set of objects, is the top groupoid of the free crossed π1(X1, X0)-module, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [27, 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='ii] on the attaching maps for the 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For an explicit description of what this means, see [31, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If n ≥ 3, then the totally disconnected abelian groupoid, Π(Xsk)n, is a free π1(X1, X0)-module over the set of n-cells of X, and the boundary map, ∂ : Π(Xsk)n → Π(Xsk)n−1, is derived from the attaching maps of the n- cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the sake of interpretation, we note that if X is a reduced CW-complex, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if it has a unique 0-cell, then Π(Xsk) is a reduced crossed complex, which means it has a unique object, and the group, Π(Xsk)1, is free on the set of loops making up the 1-skeleton of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Baues, [13], refers to such a crossed complex as being totally free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a special CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given an (n+1)-cell, c ∈ C(X, n+1), we have an induced map of pointed spaces, ψc : (Sn, ∗) → (Xn, ψc(∗)), and we let ι′(c) ∈ πn(X, ψc(∗)) be the element given by the image of the generating element of πn(Sn, ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives an element, ι(c) ∈ πn(X, Xn−1, ψc(∗)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let, now, A = (.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ An ∂−→ An−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A2 ∂−→ A1 t ⇒ s A0) be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It follows from our previous discussion and, for instance from [27, page 238], that crossed complex maps, f = (fn)n∈Z+ 0 : Π(Xsk) → A, are A CATEGORIFICATION OF QUINN’S TQFT 173 in one to one correspondence with sequences of maps, � f ′ n : C(X, n) → An)n∈Z+ 0 , such that, for each n and c ∈ C(X, n), we have fn−1(ι(c)) = ∂f ′ n(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This allows for the inductive construction of maps, Π(Xsk) → A, by giving their values on the cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopy of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A source for much of this review of the homotopy of maps of crossed complexes is [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The particular case of homotopy of crossed modules (of groupoids) is in [28] and, also in [53, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A version for reduced crossed complexes, as already mentioned, there called crossed chain complexes, is given by Baues, [13, page 98].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will give a short description of the notion of homotopy of crossed complex maps, focusing on showing the particular explicit formulae that we will need to write down the TQFTs and extended TQFTs derived from finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopy of crossed complex maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Throughout this subsection, we fix two crossed complexes, A = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 n ∂−→ A1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 2 ∂−→ A1 1 t ⇒ s A0, and B = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ B1 n ∂−→ B1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ B1 2 ∂−→ B1 1 t ⇒ s B0, so we have groupoids, A1 = (s, t: A1 1 → A0), B1 = (s, t: B1 1 → B0), and totally disconnected groupoids, An = (β : A1 n → A0) and Bn = (β : B1 n → B0), if n ≥ 2, where β is the map that says where an arrow, in A1 n or B1 n, is based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are several equivalent ways of defining the notion of homotopy between morphisms of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a crossed complex, A, we can form the tensor product, I ⊗ A, in a way we will see shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here, as before, we have used I to refer to the unit interval crossed complex, Π(Isk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') This gives a model for a ‘cylinder’ on A, so then a homotopy between two maps, f and f ′ : A → B, will be a morphism, h: I ⊗ A → B, satisfying some fairly obvious conditions as in [27];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Theorem 200, below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For future use, we note that the ‘cylinder’ structure, (e0, e1, σ), on I ⊗ A, is induced from the morphisms, 0 and 1, from the terminal groupoid, {∗}, to I and the terminal morphism I termI −−−−→ {∗}, so for instance, σ = (termI)A ∗ = term ⊗ A81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Alternatively, we can use the internal ‘hom’, which we will meet in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, and form AI = CRS(I, A), which we have seen before, and which acts as a cocylinder on A, so is the analogue of the path space construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This leads to a homotopy being seen as a morphism from A to BI, as it can with spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is another definition of homotopy, which is the crossed complex analogue of the notion of homotopy of morphisms of chain complexes often given in books on Homological Algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This does not need additional constructions to make it work and, in fact, is needed to make sense of the construction, CRS, of the internal hom, and then of the tensor, so we start with this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The idea is that we start with both a morphism f and an ‘f-homotopy’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a homotopy that ends at f, and then obtain the ‘other end’ of the homotopy from that input;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [20, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The restricted version on ‘crossed chain complexes’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', reduced crossed complexes, is given on page 98 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 81For more precision on cylinder functors, see [69] and, in this particular case, the hopefully forthcoming [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 174 Definition 195.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a crossed complex map, f = (fn)n∈Z+ 0 : A → B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An f-homotopy, H = (hn)n∈Z+ 0 , or 1-fold f-homotopy, is given, in low dimensions, by a set map, h0 : A0 → B1 1, such that t ◦ h0 = f0, so h0 looks like x ∈ A0 h0 �−→ � s(h0(x)) h0(x) −−−→ f0(x) � ∈ B1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a set map, h1 : A1 1 → B1 2, such that β(h1(g)) = f0(t(g)), and h1 looks like (x g−→ y) ∈ A1 1 h1 �−→ � f0(y) h1(g) −−−→ f0(y) � ∈ B1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This map, h1, is to be such that, if the morphisms / 1-arrows, g and g′, in A1 can be composed, then h1(gg′) = � h1(g) ⊳ f1(g′) � h1(g′), so, algebraically, it is a form of derivation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In higher dimensions, we have, if n ≥ 2, a groupoid map, hn : An → Bn+1, which, on objects, restricts to f0, such that, given x, y ∈ A0, if a ∈ An(x) and g ∈ A1(x, y), then hn(a ⊳ g) = hn(a) ⊳ f1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We denote the set of 1-fold f-homotopies by CRS1(A, B)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the setting of this definition, given f = (fn)n∈Z+ 0 : A → B, and H = (hn)n∈Z+ 0 ∈ CRS1(A, B)f, it then follows that we have a crossed complex map, f ′ = (f ′ n)n∈Z+ 0 : A → B, defined by: f ′ 0(x) = s(h0(x)), if x ∈ A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f ′ 1 � x g−→ y) = h0(x) f(x g−→ y) ∂ � f0(y) h1(g) −−−→ f0(y) � h0(y)−1, if (x g−→ y) ∈ A1 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f ′ n(a) = � fn(a) hn−1(∂(a)) ∂(hn(a)) � ⊳ h1(β(a))−1, if n ≥ 2 and a ∈ A1 n;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [27, Exercise 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We put s(H, f) = f ′ and t(H, f) = f, and also write f ′ (H,f) −−−→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We may say that (H, f) is a crossed complex homotopy from f ′ to f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We put CRS0(A, B) = Crs(A, B), the set of crossed complex maps from A to B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As the notation indicates in its use of s and t, we have a groupoid CRS1(A, B) = � s, t: CRS1(A, B)1 → CRS0(A, B) � , whose objects are the maps, f : A → B, and the morphisms from f ′ to f are the f-homotopies such that f ′ (H,f) −−−→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of f ′′ (H′,f ′) −−−−→ f ′ and f ′ (H,f) −−−→ f, denoted f ′′ (J,f) −−−→ f, with J = (jn)n∈Z+ 0 , is such that, if H = (hn)n∈Z+ 0 and H′ = (h′ n)n∈Z+ 0 , then j0(x) = h′ 0(x) h0(x), if x ∈ A0, and jn(x g−→ y) = hn(x g−→ y) � h′ n(x g−→ y) ⊳ h0(y) � , if n ≥ 1, and (x g−→ y) ∈ A1 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 196.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If considering possible higher order analogues of this theory, we note that the proof that J is indeed an f-homotopy uses the second Peiffer condition, given in Definition 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the more general case of 2-crossed complexes, which is in many ways similar but where the second Peiffer condition need not hold, cofibrancy conditions are needed, on A, to concatenate homotopies, see [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 175 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The internal hom functor, CRS(−, −), in the category of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let, as before, A and B be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid, CRS1(A, B) = � s, t: CRS1(A, B)1 → CRS0(A, B) � , can be ‘ex- tended’ to a crossed complex, denoted, CRS(A, B) = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' δ−→ CRSn(A, B)1 δ−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' δ−→ CRS2(A, B)1 δ−→ CRS1(A, B)1 t ⇒ s CRS0(A, B) � , by considering k-fold homotopies between crossed complex maps for each k ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This construction is explicitly given in both [25] and [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' With regard to this, we will give some explicit formulae and results that we will need later in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a crossed complex map, f = (fn)n∈Z+ 0 : A → B, and let k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A k-fold f-homotopy, Hk = (hk n)n∈Z+ 0 = (hk 0, hk 1, hk 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), is given by: the choice of an element hk 0(x) ∈ Bk(f0(x)) for each x ∈ A0, so we have a mapping hk 0(x): A0 → B1 k, given in the form x ∈ A0 hk 0 �−→ (f0(x) hk 0(x) −−−→ f0(x)) ∈ Bk(f0(x));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given (x g−→ y) ∈ A1 1, the choice of an element, hk 1(x g−→ y) ∈ Bk+1(f0(y)), to be such that, if g and g′ can be composed in A1, then hk 1(gg′) = � hk 1(g) ⊳ f1(g′) � hk 1(g′);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' given n ≥ 2, and x ∈ A0, a function, hk n : A1 n → B1 n+k, satisfying β(hk n(a)) = f0(β(a)), for all a ∈ An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This mapping, hk n : A1 n → A1 n+k, is to be such that, given any x ∈ A0, the re- striction of hk n to An(x) is a group homomorphism, An(x) → Bn+k(f0(x)), and further, if x, y ∈ A0, a ∈ An(x) and (x g−→ y) ∈ A1 1, then hk n � a ⊳ (x g−→ y) � = hk n � a) ⊳ (f0(x) f1(g) −−−→ f0(y)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let CRSk(A, B)f denote the set of all k-fold f-homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : A → B be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using the obvious point-wise product of k-fold f-homotopies, as in [27, Definition 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5], we have that the CRSk(A, B)f has a group structure, and that is abelian if k ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given k ≥ 2, we have a totally disconnected groupoid, CRSk(A, B) := \uf8eb \uf8edβ : � f : A→B CRSk(A, B)f → CRS0(A, B) \uf8f6 \uf8f8 , with object set, CRS0(A, B) = Crs(A, B), the set of crossed complex maps, f : A → B, and with the obvious map, β : � f : A→B CRSk(A, B)f → CRS0(A, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 198.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : A → B be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let H2 = (h2 0, h2 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ) be a 2-fold f-homotopy, then δ(H2) = (δ(h2 0), δ(h2 1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), defined by: δ(h2 0)(x) := ∂ � h1 0(x) � , for each x ∈ A0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' δ(h2 1)(x g−→ y) := � h2 0(x) �−1 ⊳ � f0(x) f1(g) −−−→ f0(y) � h2 0(y) ∂(h2 1(x g−→ y)), where (x g−→ y) ∈ A1 1, A CATEGORIFICATION OF QUINN’S TQFT 176 and given n ≥ 2 and a ∈ A1 n, δ(h2 n)(a) = ∂ � h2 n+1(a) � h2 n(∂(a)) � (−1)n� , is an f-homotopy, f (δ(H),f) −−−−−→ f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is proved by explicit calculations, using the second Peiffer condition from Definition 179.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Similarly in higher dimensions, for n ≥ 3, we have groupoid maps, δ: CRSn(A, B) → CRSn−1(A, B), which, again, restrict to the identity on the set of objects, together with ac- tions of the groupoid, CRS1(A, B), on all of the totally disconnected groupoids, CRSn(A, B), for n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives rise to a crossed complex, CRS(A, B), the in- ternal hom in the category of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again for the details, see [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='i].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following groupoid will play a key role later on when we give explicit de- scriptions of the once-extended Quinn TQFT derived from a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A and B be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have the following groupoid, where we are using the notation of Definition 183, π1 � CRS(A, B), CRS0(A, B) � = π1 � CRS(A, B) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set of objects of π1 � CRS(A, B), CRS0(A, B) � is, thus, the set, CRS0(A, B) = Crs(A, B), of crossed complex maps, f from A to B, and given f, g : A → B, the set of arrows from f to g is given by all equivalence classes of homotopies, f [(H,g]) −−−−→ g, connecting f and g, where homotopies are considered up to the equivalence given by 2-fold homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Tensor product and homotopies of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial property of the crossed complexes, CRS(A, B), where A and B are crossed complexes, is that they vary functorially in both positions, so we have a functor, CRS(−, −) : Crsop × Crs → Crs, sending (A, B) to CRS(A, B);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This functor, CRS(−, −), acts as an ‘internal hom’, that is to say that CRS(A, B) behaves like the “object of morphisms” from A to B, so CRS(−, −) is analogous to the mapping space functor that we mentioned on page 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note some explicit formulae below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 199.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B is a crossed complex, we have a functor, CRS(−, B): Crsop → Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It sends a crossed complex, A, to CRS(A, B), and a crossed complex map, f : A′ → A, to the crossed complex map, φ∗ : CRS(A, B) → CRS(A′, B), such that: (1) each crossed complex map, φ: A → B, is sent to the composite, φ ◦ f : A′ → B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and (2) given a positive integer k, a crossed complex map, φ: A → B, and a k-fold φ-homotopy, hk = (hk 0, hk 1, hk 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), then f ∗(hk) := (hk 0 ◦f, hk 1◦f, hk 2 ◦f, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), which is a k-fold (φ ◦ f)-homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This corresponds to ‘pre-composition with f’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is also a functor, CRS(A, −), with the ‘opposite’ properties, and which on morphisms gives ‘post-composition’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 177 Given crossed complexes, A and A′, we can also form their tensor product, A⊗A′;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' again, for details, see [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='iii] and [116, Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a functor, ⊗CRS : Crs × Crs → Crs, sending (A, A′) to A ⊗ A′, and an exponential law, Crs(A ⊗ A′, B) ∼= Crs(A, CRS(A′, B)), that holds naturally in A and B, showing that the functor ‘tensor product with A′’, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', −⊗A′, is left adjoint to the functor, CRS(A′, −), derived from the internal hom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives Crs the structure of a monoidal closed category, [27, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, the tensor product is symmetric, so Crs with the above tensor is a symmetric monoidal closed category, see [27, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='16], but also take note of the discussion after that result in that source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a crossed complex, A, we have morphisms, which look like the inclusions of the ends of a cylinder, and will here be denoted i0, i1 : A → Π(Isk) ⊗ A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [116, page 203].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We note that, as this notation, i0, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', is overcharged, occurring in several contexts, often with different meanings, we will sometimes replace i0 by e0(A), or ιA 0 , etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', depending on the other use of symbols in the setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') If k ≥ 2, we also have a canonical inclusion, i: A → Π(Dk sk) ⊗ A, as Π(Dk sk) is a reduced crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, there is a morphism from Π(Isk) ⊗ A to A, which is a partial inverse to the ‘end inclusion’ morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that Π(Isk)⊗ A behaves exactly like a cylinder on A, and can be used to define a notion of homotopy between morphisms in Crs, which, thankfully, coincides with the one that we introduced earlier, where we used the abbreviation I for Π(Isk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All this is very thoroughly discussed in [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='i], and we note: Theorem 200.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A and B be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : A → B be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a canonical correspondence between homotopies, f ′ (f,H) −−−→ f, and commutative diagrams in Crs of form: A f ′ �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ i0 � Π(Isk) ⊗ A H′ � A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i1 � f �ttttttttttt B If k ≥ 2, we have a canonical correspondence between k-fold f-homotopies, Hk, and commutative diagrams in Crs of form A f ′ �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ i � Π(Dk sk) ⊗ A H′ � B □ We could also use H′ in the first bullet point to get a morphism, h: A → CRS(I, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This involves first using the symmetry of the tensor product to get a morphism from A ⊗ I to B and then the adjunction to give a homotopy in that form, h: A → CRS(I, B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we write BI for the codomain of that map, then, as we mentioned earlier, BI has the structure of a ‘cocylinder’, or ‘path space’ object in Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will not pursue this here, but it is developed further in both [27] and [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the geometric interpretation of the tensor product, the following is important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 178 Theorem 201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ([27, Theorem 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1]) Let X and Y be CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Give X ×Y the usual structure of a CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a natural isomorphism of crossed complexes, Π � (X × Y )sk � ∼ = −→ Π(Xsk) ⊗ Π(Ysk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A version of this result, for the slight variant of the tensor product used by Baues, is also discussed on page 92 of [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homotopies and totally free crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To be able to work fairly simply with the above notions of homotopy between crossed complex maps, we will need to be able to construct homotopies in ways analogous to the ‘induction up the skeleton’ methods used in many topological contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is intuitively clear that this should be the case, but, as ever, writing out the detailed statement and proof that matches that intuition needs a bit of care.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The detailed result in Lemma 202, below, is crucial for what follows and is one such statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is given a direct proof in [49, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], for the case when the CW-complex, X, has a single 0-cell, so is ‘reduced’, and in [53, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1] for general CW-complexes, but, there, for the particular case in which A is a crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Related results are also to be found in [27, Corollary 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], [29, Proposition 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 I] and [13, Chapter III, §4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to simplify the notation and the exposition, we state the result only in the case when A has a single object, so A is itself a reduced crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Otherwise further compatibility relations are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will assume that X is a special CW-complex, as in Definition 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given an n ≥ 0, we let C(X, n) be the set of n-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 202 (Free construction of crossed complex homotopies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that A = � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 n ∂−→ A1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 2 ∂−→ A1 1 β−→ A0 = {∗} � is a reduced crossed complex, thus with a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(Xsk) → A be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let k be a non-negative integer, then k-fold f-homotopies are uniquely specified by their value on the elements of Π(Xsk), defined from the cells of X, the latter in the sense explained in Remark 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly this means that: (1) there exists a one-to-one correspondence between 1-fold f-homotopies and sequences of maps, (m1 i : C(X, i) → A1 i+1)i∈Z+ 0 , and note that there are no further compatibility conditions between the maps, m1 i : C(X, i) → A1 i+1, and the boundary maps of Π(Xsk) and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will denote this bijection by (72) (m1 i )i∈Z+ 0 �→ Extend1 X � (m1 i )i∈Z+ 0 , f � ∈ CRS1(Π(Xsk), A)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) If k ≥ 2, there exists a one-to-one correspondence, between k-fold f- homotopies and sequences of maps, (mk i : C(X, i) → A1 i+k)i∈Z+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We denote this bijection by: (mk i )i∈Z+ 0 �→ Extendk X � (mk i )i∈Z+ 0 , f � ∈ CRSk(Π(Xsk), A)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 179 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows essentially from Remark 193, whose nomenclature we use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For k = 1, given (m1 i : C(X, i) → A1 i+1)i∈Z+ 0 , then Extend1 X � (m1 i )i∈Z+ 0 , f � is the unique 1-fold f-homotopy that takes the value, m1 i , on the set of i-cells of X (or more precisely on the elements of Π(X)i given by them).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The existence and uniqueness of Extend1 X � (m1 i )i∈Z+ 0 , f � follows from elementary techniques, since Π(Xsk) is free on the set of cells of X, in the sense explained in Remark 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As mentioned before, more details can be found in [49, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], in the case when X has a unique 0-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A similar argument is valid when k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a sequence of maps, (mk i : C(X, i) → A1 i+k)i∈Z+ 0 , then Extendk X � (mk i )i∈Z+ 0 , f � is the unique k-fold f-homotopy that takes the value, mk i , on the set of i-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 203.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A perhaps more conceptual proof of Theorem 202 follows by combin- ing Theorems 200 and 194, using the fact that the crossed complex, Π � (X ×I)sk � ∼= Π(Xsk) ⊗ Π(Isk), is free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The same argument works for k ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The more combina- torial approach we have given is, however, often better for use with our presentation of extended TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 202 naturally leads to the following definition, in which k is a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 204 (k-fold X-homotopy sequence).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a special CW-complex and let C(X, i) be the set of i-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 n ∂−→ A1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 2 ∂−→ A1 1 β→ A0 = {∗} be a reduced crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A k-fold X-homotopy sequence is a sequence of maps, (mk i )i∈Z+ 0 , in the category of sets, where mk i : C(X, i) → A1 i+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Theorem 202, given a crossed complex map, f : Π(X) → A, there is a bijection (72) between k-fold f-homotopies and k-fold X-homotopy sequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will need a generalisation of Theorem 202 for when (X, Y ) is a CW-pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let X be a special CW-complex, Y be a subcomplex of X, with ι: Y → X denoting the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also consider a reduced crossed complex, A, as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 205.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(Xsk) → A be a crossed complex map and let k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a k-fold f-homotopy, hk = (hk j )j∈Z, we define the k-fold Y -homotopy sequence, denoted Restrictk Y (hk) = (mk j : C(Y, j) → A1 j+k)j∈Z+ 0 , to be the restriction of hk to the elements of Π(X)j defined by the j-cells of Y ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Remark 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a k-fold Y -homotopy sequence, nk = (nk j : C(Y, j) → A1 j+k)j∈Z+ 0 , we define the k-fold X-homotopy sequence, denoted Expandk(nk, Y, X) = (mk j : C(X, j) → A1 j+k)j∈Z+ 0 , to be such that, given j ∈ Z+ 0 , then mk j coincides with nk j over C(Y, j) ⊆ C(X, j), and otherwise mk j takes as values the identity element of A1 j+k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use the above notation in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 180 Lemma 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a special CW-complex and A = (An)n∈Z+ 0 be a reduced crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(Xsk) → A be a crossed complex map and k be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1) Given a k-fold f-homotopy, hk, we have Extendk X � Restrictk X(hk), f � = hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (2) Let Y be a subcomplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Π(ι): Π(Ysk) → Π(Xsk) be the induced map, which induces a crossed complex map, going in the other direction, via “restric- tion”, Π(ι)∗ : CRS � Π(Xsk), A � → CRS � Π(Ysk), A � , (see Notation 199).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a k-fold f ◦ Π(ι)-homotopy, hk, where we note that f ◦ Π(ι): Π(Ysk) → A, we have that: Π(ι)∗� Extendk� Expandk� Restrictk Y (hk), Y, X � , f �� = hk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the freeness of Π(Xsk), that of Π(Ysk), and Lemma 202.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is well known that inclusions of CW-complexes are cofibrations, both in the standard (Quillen/Serre) model category structure, and in the Strøm Hurewicz model category structure on the category of (compactly generated and weak Hausdorff) topological spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally one has the notion of a relative CW complex, in which one has a pair, (X, Y ), of spaces where X is obtained from Y by attaching cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is an analogous notion for crossed complexes, called a morphism of relatively free type in [27, section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='iii].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The idea is that a morphism, f : A → B, of crossed complexes, is of relatively free type if it can be constructed by forming a succession of pushouts, which attach copies of various dimensional Π(Dn+1 sk ), by attaching maps, Π(Sn sk) → A(k), to the kth step of the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that if A j−→ B is a crossed complex morphism of relatively free type, then it is a cofibration in the model category structure on Crs given by Brown and Golasi´nski, [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The corresponding cofibrant objects are the ‘crossed complexes of free type’ of [27, section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='iii], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the free crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More on the Brown- Golasi´nski model category structure can be found in [54], which is our (hopefully forthcoming) summary of the homotopy theory of crossed complexes from a point of view near to that of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notion, discussed above, is treated on page 109 of [13], and is related to the cofibration category structure on Crs that is developed both there, and in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From either viewpoint, Lemma 206 can be used to prove that the fundamental crossed complex functor preserves cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The classifying space of a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To set the scene for this section a little, recall that lattice models of quantum field theory are finite ap- proximations to quantum field theories, which are set up on a ‘lattice’ on a closed (2-)manifold, X, say;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [71, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Typically in simple cases, one ‘colours’ the edges of the lattice with elements of a finite group, G, subject to some ‘flatness’ con- ditions relating to the ‘plaquettes’ of the manifold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the squares cut out by the embedded lattice, where we impose that the product of the elements around a square is the identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we replace the lattice by a triangulation, which seems sensible if given dimensions higher than 2, then again we can consider G-colourings of the edges of the triangulation, TM, with the flatness conditions coming from the 2-dimensional faces of the triangulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This interprets, after a little bit of work, as thinking of a G-colouring as being a simplicial map from TM to the classifying space, BG, of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here we are skating over quite a few important details, but will return to them later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A CATEGORIFICATION OF QUINN’S TQFT 181 We recall that, classically, a ‘classifying space’, BG, of a group, G, classifies principal G-bundles / G-torsors, so given a space, X, one has a bijection between isomorphism classes of G-torsors on X and homotopy classes of maps from X to the space, BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that for a discrete group, G, one method of construction of such a classifying space is as the geometric realisation of the ‘nerve’ of the corresponding single object groupoid, sometimes denoted G[1] or BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The correspondence pulls back a universal G-bundle over BG along a (representative of a) class of maps in [X, BG] and thus of maps from X to BG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, as in the case of interest here, G is discrete, these are also flat connections on X, and this gives geometric meaning to a G-colouring, which is a more algebraic / combinatorial analogue of this same notion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will see how to construct an analogue of the nerve for crossed complexes and will link that to A-colourings for a crossed complex A, in similar way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In [53], we considered colourings by elements of a finite crossed module / 2-group, or, more generally, of a (homotopy finite) crossed complex, A, and such colourings corresponded to simplicial maps from TM to the classifying space, BA, of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As this section extends the approach in [53], we will recall some of the theory of such colourings and of the corresponding classifying spaces, but, in addition, will explore and refine slightly different aspects of the homotopy theoretic results linking that construction of classifying spaces with the homotopy theory of simplicial sets, and of function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In fact, a few of the results that we will prove in detail do not seem to be published in the literature on crossed complexes, although they are related to others that are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fundamental crossed complex of a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We freely use the notion of a simplicial set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g, [41, 55, 86] and numerous other places in the literature and on-line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The category of simplicial sets will be denoted by Simp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let S be a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a non-negative integer n, we let Sn denote the set of n-simplices of S, the face maps by di := dn i : Sn → Sn−1, for 0 ≤ i ≤ n, similarly the degeneracy maps by sj := sn j : Sn → Sn+1, for 0 ≤ i ≤ n, and then let Snd n denote the set of non-degenerate n-simplices of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that the geometric realisation, |S|, of S is canonically a special CW- complex, with one n-cell for each non-degenerate n-simplex of S;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [55, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let S and S′, now, be simplicial sets, then the geometric realisation of a simplicial map, f : S → S′, is a cellular82 map, |f|: |S| → |S′|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [55, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To simplify notation, we will, from now on, tend to insist that given a simplicial set, S, the notation |S| will stand both for the topological space |S|, the geometric realisation of S, but also for |S|, with its CW-decomposition, hence considered as a filtered space with the skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will therefore sometimes omit the suffix ‘sk’ and replace |S|sk by |S|, when no confusion should result from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Combining geometric realisation with the fundamental crossed complex functor, we, therefore, have a functor, Π: Simp → Crs, which we will refer to as the fundamental crossed complex functor, sending a simplicial set |S| to Π(S) := Π(|S|), where |S| is given its skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use both notations, Π(S) and Π(|S|), for the fundamental crossed complex of a simplicial set, S, depending on the requirements of the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is worth noting that there are constructions of 82Actually, |f|: |S| → |S′| is furthermore regular, so sending open cells of |S| to open cells of |S′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 182 Π(S), up to isomorphism, that do not use the geometric realisation, using instead purely algebraic techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [54], once it is available, Sauvageot’s thesis, [104, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2], and [26, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A-colourings of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Re- mark 194, crossed complex maps, f, from Π(|S|) to A can be specified, uniquely, by giving the value of f on those elements of Π(|S|) given by the non-degenerate simplices of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to develop this a bit further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We start by fixing some terminology and notation, and by recalling some results on simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let ∆ be the simplex category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that the objects of ∆ are non-negative integers, n, or more exactly the finite ordinals, [n] = {0 < 1 < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' < n}, and the morphisms from [m] to [n], are the non-decreasing maps, {0 < · · · < m} → {0 < · · · < n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A simplicial set, S, is then a functor, S : ∆op → Set, and, if n is a non-negative integer, the set, Sn, of n simplices is the image, S(n), of [n] under S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let, again, n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let ∆(n) be the n-simplex, here considered as a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is defined to be the representable functor, ∆(n)(m) := ∆([m], [n]), that is, ∆(n) : ∆op → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set of m-simplices of ∆(n) is, thus, the set of non-decreasing maps, σ, from [m] to [n].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such an m simplex can be represented by a string, (a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , am), where ak = σ(k), and thus we have, 0 ≤ a0 ≤ a1 ≤ · · · ≤ am ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An m-simplex, σ, is non-degenerate if the map, σ, is injective, so we have 0 ≤ a0 < a1 < · · · < am ≤ n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can express any simplicial set as a coend, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', as a colimit, of copies of standard simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives, in its simplest form, S ∼= ˆ n Sn × ∆(n), where, here, Sn×∆(n) is, in fact, just shorthand for the coproduct of copies of ∆(n) labelled by the n-simplices of S, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the Sn-fold copower of ∆(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This interprets as taking lots of labelled copies of the various standard simplices, and then glueing them along common faces, also taking into account the degeneracies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Returning to the standard n-simplex, ∆(n), this has a unique non-degenerate n-simplex, namely that given by the identity map, id[n] : [n] → [n], which is, of course, non-decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All other simplices in ∆(n), of any dimension, are images of this n-simplex under the face and degeneracy maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is often useful to note that, given a simplicial set, S, the set, Sn, of n-simplices of S is in bijective correspondence with the set, Simp(∆(n), S), where, if σ ∈ Sn, we have a unique map, ˜σ : ∆(n) → S, which, in dimension n, sends id[n] to σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The behaviour of that map on any other simplex of ∆(n) is completely determined by that specified assignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The geometric realisation, |∆(n)|, of ∆(n) is the geometric n-simplex, ∆n, with the obvious CW-decomposition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see, for instance, [55, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will often write ∆n for |∆(n)|, and will thus think of it as both a CW-complex and as the corresponding A CATEGORIFICATION OF QUINN’S TQFT 183 filtered space, using the skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need the fundamental crossed complex, Π([n]) := Π(|∆(n)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Following on from the coend description of a simplicial set, S, we note that we have a similar formula for its geometric realisation, |S| ∼= ˆ n Sn × ∆n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the fact that geometric realisation is a left adjoint, so preserves colimits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will use this shortly to get a similar coend formula for Π(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a crossed complex, A = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 n ∂−→ A1 n−1 ∂−→ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∂−→ A1 2 ∂−→ A1 1 t ⇒ s A0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 208.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An A-colouring of a simplicial set, S, is defined to be a map of crossed complexes, f : Π(S) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This definition needs taking apart.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will recall how this corresponds to the intuition of labelling the simplices of S with elements of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We start with the basic example, namely the case S = ∆(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This should reinforce the description of A-colourings given in [53, section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and, in particular, subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='32], whilst giving it in a bit more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Remark 194, crossed complex maps, f : Π([n]) → A, are determined by what they do on the non-degenerate simplices of ∆(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we have said, the idea is that this is a labelling of the edges, and higher dimensional faces of ∆(n), by elements of appropriate dimension within A, and these are to satisfy some com- patibility rules which are analogues of higher dimensional cocycle rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such a crossed complex map consists of the following information (and note that we will tacitly be using the homotopy addition lemma from [26, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 99] here): a map, f0 : {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , n} = ∆(n)nd 0 → A0, so picking out n + 1 objects of A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a map, f1 : ∆(n)nd 1 → A1 1, such that if (a, b) ∈ ∆(n)nd 1 , so 0 ≤ a < b ≤ n, d0(a, b) = b, and d1(a, b) = a, then t(f1(a, b)) = f0(d0(a, b)), and s(f1(a, b)) = f0(d1(a, b)), and the element f1(a, b) of A goes between the images of the vertices, a and b, f0(a) f1(a,b) −−−−→ f1(b) ∈ A1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' a map, f2 : ∆(n)nd 2 → A1 2, such that for each (a, b, c) ∈ ∆(n)2, so with 0 ≤ a < b < c ≤ n, we have: β(f2(a, b, c)) = f0(a) = f0(d1d2(a, b, c)), so the image is in the vertex group, A1 2(a), and, as the edges fit together as follows83: f0(b) f2(b,c) �● f2(a,b,c) f0(a) f1(a,b) �① ① ① ① ① ① ① ① f1(b,c) � f0(c), the boundary of this element being: ∂(f2(a, b, c)) = f1(a, b) f1(b, c) f1(a, c)−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 83remember that d0(a, b, c) = (b, c), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 184 a map f3 : ∆(3)nd → A1 3, which is such that, given (a, b, c, d) ∈ ∆(n)3, β(f3(a, b, c, d)) = f0(a) = f0(d1d2d3(a, b, c, d)), so once again it is in the appropriate dimension of the vertex crossed com- plex over f0(a), whilst the boundary84 is ∂(f3(a, b, c, d)) = � f2(b, c, d) ⊳ f1(a, b)−1� f2(a, b, d) f2(a, c, d)−1 f2(a, b, d)−1, and for 3 < i ≤ n, a map, fn : ∆(n)nd i → A0 i , such that if (a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai) ∈ ∆(n)i, then β(fi(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai)) = f0(a0) = f0(d1d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' di(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai)), whilst the boundary, ∂(fi(a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai)) = � fi−1(di(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai)) ⊳ f1(a0, a1)−1� n � j=1 � fi−1(dj(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai)) �� (−1)j� , where we note that (a0, a1) = d2d3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' di(a0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , ai).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following makes the link with [53, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition], which we can take as an alternative, more algebraic definition of an A-colouring of a simplicial set S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 209 (Algebraic definition of A-colouring).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An A-colouring of S is given by a sequence of maps, fn : Snd n → An, for n = 0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , satisfying the same relations as in the ∆(n) case, above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that, on an n-simplex, σ, the composite, f ◦ ˜σ is an A-colouring of ∆(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We may refer to the data for this form of the definition as giving an algebraic A-colouring of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Before continuing, we note the following useful Theorem, which is Proposition 2·2 of [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 210.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let S be a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let |S| be its geometric realisation, with the skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a natural isomorphism Π(|S|) ∼= ˆ n Sn × Π([n]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In other words, Π(S) is obtained by glueing copies of the basic Π([n]) together by the same rules as used for expressing S as a coend as on page 182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note the use of Sn × Π([n]) as shorthand for the Sn-fold copower of Π([n]), analogously to its use in the previous coend formulae.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In [26], this Theorem is proved as a consequence of the higher homotopy van Kampen theorem, [27, Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This also follows from Remark 194, using the explicit cell decomposition of |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is also a consequence of the fact that Π : Simp → Crs is a left adjoint, a fact that can be proved directly, without using the van Kampen theorem, but using that Π has an algebraic description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will introduce the right adjoint of Π very shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 84The diagram below shows that we can divide the boundary of ∆(3) into two parts, b � d0 d2 �❃❃❃❃❃❃❃❃ c � b � c � a � � d a � � d3 d1 ��������� d This explains where this formula comes from, as the boundary is the difference between the two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 185 We have the following, which again follows from Remark 194, and that explicit cell decomposition of |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some more discussion is to be found in [53, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two definitions of A-colouring of a simplicial set are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, crossed complex maps, f : Π(|S|) → A, are in natural one-to-one correspondence with algebraic A-colourings of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ More generally, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let K and L be disjoint subcomplexes of S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let i(K,S) : K → S and i(L,S): L → S be the inclusion maps (in the category of simplicial sets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(|K|) → A and f ′ : Π(|L|) → A be crossed complex maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a one-to- one correspondence between crossed complex maps, h: Π(|S|) → A, that make the diagram below commute, A Π(|K|) Π(i(K,S)) �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ f �s s s s s s s s s s Π(|L|) f ′ �❏❏❏❏❏❏❏❏❏❏ Π(i(L,S)) �✉✉✉✉✉✉✉✉✉ Π(|S|), h � and (algebraic) A-colourings of S extending those colourings of K and L given by f and f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Review of nerves and classifying spaces of a crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Source ma- terial for classifying spaces of crossed complexes can be found in [26], and [18], also in [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='10] and [53], as well as in various other of the sources cited earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The classifying space functor, B : Crs → CGWH, is defined as the composite of a nerve functor, N : Crs → Simp, and the geometric realisation functor from Simp to CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This nerve functor, in fact, goes back to Blakers, [14], so precedes the publication of much of the foundational sources on crossed complexes by some time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is an example of the general procedure for producing analogues of the singular complex functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly we have a functor, Π ◦ ∆: ∆ → Crs, sending [n] ∈ ∆ to Π([n]) = Π(|∆(n)|), which, on varying n, gives a cosimplicial crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 213.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The nerve of a crossed complex, A, is the simplicial set, N(A), whose set of n-simplices is N(A)n := Crs(Π([n]), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The maps between the different dimensions are induced from ∆: ∆ → Crs, so, if µ : [m] → [n] in ∆, the corresponding map, N(A)µ, from N(A)n to N(A)m is Crs(Π(µ), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This defines the nerve functor N : Crs → Simp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For us, one of the most useful facts about the nerve functor is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 214 (Brown-Higgins).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The nerve functor, N : Crs → Simp, is right adjoint to the fundamental crossed complex functor, Π: Simp → Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' See [26, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Given a simplicial set, S, and crossed complex, A, we, thus, have a bijection, φA S : Crs(Π(|S|), A) → Simp(S, N(A)), A CATEGORIFICATION OF QUINN’S TQFT 186 natural in both S and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Another fundamental fact that is underpins the use of the nerve is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 215.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a crossed complex, then N(A) is a Kan simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' See [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 216.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Kan complexes are often called ∞-groupoids, or, more exactly, weak ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Kan complexes of form N(A) for a crossed complex, A, are special, however, as noted on page 100 of [26], and correspond to strict ∞-groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The relationship generalises that between bicategories, which are sometimes called weak 2-categories and (strict) 2-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A good introduction to strict ∞-groupoids is given in [2], §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The equivalence with crossed complexes is briefly discussed in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2 of that paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The original dis- cussion ot this equivalence is in [26], but from a slightly different point of view, namely a globular rather than a simplicial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The term used in that source is ω-groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Another approach, which explores a more simplicial view, is given by Verity, [120], in his proof of the Street-Roberts conjecture, which originated in some notes from 1978 by John Roberts, developed from his approach to algebraic quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Verity uses the theory of complicial sets, which closely resembles that of the T -complexes mentioned in [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Complicial sets model strict ∞-categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An introduction to that theory can be found in [103].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As stated above, the classifying space construction that we will be using is ob- tained from the nerve by taking geometric realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 217 (Classifying space of a crossed complex [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The classifying space, BA, of a crossed complex, A, is defined as the geometric realisation, |N(A)|, of N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, A0 ∼= N(A)0, and so each object a ∈ A0 of A gives rise to a 0-simplex of N(A), therefore to a vertex of the CW-complex, which will be denoted ˜a ∈ BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given simplicial sets, K and L, their simplicial mapping space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the function complex of [86, §6], will be denoted SIMP(K, L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that Simp becomes a cartesian closed category with this function space construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we have SIMP(X, Y )0 = Simp(X, Y ), the set of simplicial set maps from X to Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As is well known, if K and L are simplicial sets, with L a Kan complex, then we have a natural weak homotopy equivalence, |SIMP(K, L)| → TOP(|K|, |L|) = |L||K|85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly, this weak homotopy equivalence sends the equivalence class of (f : K × ∆(n) → L, s), seen as an element of |SIMP(K, L)|, so s ∈ |∆(n)|, to the function |K| → |L|, such that k �→ |f|(k, s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An explicit proof that this is a weak homotopy equivalence is in [53, page 131].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This clearly is natural in K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a simplicial map, f : K → L, the weak homotopy equivalence sends the corresponding vertex of SIMP(K, L) to the geometric realisation, |f|: |K| → |L|, of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The technical results collected up, for convenience, in the next theorem are due to Brown–Higgins, [26], and Tonks, [115, 116].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They are discussed in the cubical, as opposed to the simplicial, setting by Brown–Higgins–Sivera in [27], and, to some extent, in a simplicial setting in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 219 (Brown–Higgins;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Brown–Higgins–Sivera;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Tonks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let A = (An)n∈Z+ 0 be a crossed complex, and take S to be a simplicial set, then: 85as usual | | : Simp → CGWH denotes geometric realisation, and TOP(|K|, |L|) = |L||K| denotes the corresponding function space in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 187 (1) there is a natural isomorphism of groupoids, π1(A) := π1(A, A0) ∼= π1(BA, � A0), where � A0 = {˜a | a ∈ A0}86, and hence, (2) there is a natural bijection, π0(A) ∼= π0(BA);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (3) let a ∈ A0, and let n be a positive integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a natural isomorphism, πn(A, a) ∼= πn(BA, ˜a), preserving the actions of π1(A, A0) and π1(BA, ˜ A0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (4) There is a weak homotopy equivalence of simplicial sets, ηA S : N � CRS(Π(|S|), A) � → SIMP(S, N(A)), which, at the level of 0-simplices, coincides with the bijection, φA S : Crs(Π(|S|), A) → Simp � S, N(A) � , given by the adjunction Π ⊣ N of Proposition 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This weak homotopy equiva- lence is natural in S87, and also in A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (5) There is a weak homotopy equivalence, ηA S : ��N � CRS(Π(|S|), A) ��� → TOP(|S|, BA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This weak homotopy equivalence is natural in both S and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We will use the results in the previous theorem without giving a proof here, rather we note 1) 2) and 3) form parts of [26, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for 4), see [26, Theorem A] and [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For 5), we refer again to [26, Theorem A] and [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ], and then proceed by composing with the canonical weak homotopy equivalence, |SIMP(S, N(A))| → TOP(|S|, |N(A)|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(|S|) → A be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The adjunction Π ⊣ N gives a simplicial set map, φA S (f): S → N(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its geometric realisation gives a continuous map, |φA S (f)|: |S| → BA = |N(A)|, and then ηA S ( ˜f) = |φA S (f)|, where ˜f is the vertex of the classifying space, |N(CRS(Π(|S|), A))|, corresponding to f, following Notation 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that item (5), above, links the classifying space of the crossed complex mapping space (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' internal hom) with the topological mapping space, from the realisation of S to the classifying space of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is the starting point, in the setting with B = BA, for computing the Quinn finite total homotopy TQFT, and its extended versions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 220.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a natural isomorphism of groupoids, π1 � CRS(Π(|S|), A), CRS0(Π(|S|), A) � ∼= π1 � TOP(|S|, BA), � |φA S (f)| : f ∈ CRS0(Π(|S|), A) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the first point of Theorem 219, together with the weak homotopy equivalence, ηA S : |NCRS(Π(|S|), A)| → TOP(|S|, BA)), since ηA S is in- jective on the set { ˜f : f ∈ CRS0(Π(|S|), A)}88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 86using Notation 218 87We note however that ηA S is not simplicially natural in S, for which fact see [116] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 88Note that if a weak homotopy equivalence, g : X → Y , between spaces, is injective on X0 ⊂ X, then g induces an isomorphism of groupoids π1(X, X0) ∼ = π1(Y, g(X0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 188 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrations of crossed complexes and profunctors from fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrations of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall the notion of fibrations of groupoids, which was originally given in [19], and is discussed in [27, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We then turn to fibrations of crossed complexes, which is, also, given in that second source, [27, Definition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 221.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G′ = (s, t: G′ 1 → G′ 0) and G = (s, t: G1 → G0) be groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A map, (f1, f0): G′ → G, of groupoids, is said to be a fibration of groupoids if, given any x ∈ G0, and x′ in G′ 0 with f0(x′) = x, and any arrow in G of form (y g−→ x) ∈ G1, so ending at x, there exists at least one arrow, (y′ g′ −→ x′) ∈ G′ 1, with f1 � y′ g′ −→ x′� = (y g−→ x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This, then, is a ‘path lifting’ or, more precisely, an ‘arrow lifting’ condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 and B = (Bn)n∈Z+ 0 be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A map, f = (fn)n∈Z+ 0 : A → B, of crossed complexes is called a fibration of crossed complexes if (1) f1 : A1 → B1 is a fibration of groupoids, and (2) given any integer n ≥ 2, and any x ∈ A0, the group homomorphism, An(x, x) → Bn(f0(x), f0(x)), induced by fn : An → Bn, is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An important link with Kan fibrations of simplicial sets is given in the following result;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [26, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a proof in the cubical, as opposed to simplicial set, setting, see [27, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 223.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: A → B be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following are equiva- lent: p: A → B is a fibration, the induced map on nerves, N(p): N(A) → N(B), is a Kan fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The next result will play a major role later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It appears to be new, however not unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall Notation 199 for the induced map in this setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a CW-complex, and Y be a subcomplex of X with ι: Y → X denoting the inclusion, which induces a crossed complex map, Π(ι): Π(Y ) → Π(X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The induced crossed complex map between internal homs, Π(ι)∗ : CRS(Π(X), A) → CRS(Π(Y ), A), is a fibration of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We point out that this map goes in the opposite direction as we are applying CRS(Π(−), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') When A has a unique object, which is our main case of interest, a proof follows directly from the second point of Lemma 206.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This argument can be easily adapted for the case when A has more than one object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ A CATEGORIFICATION OF QUINN’S TQFT 189 Remark 225.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A model category theoretical proof of Lemma 224 follows from the fact that the category, Crs, of crossed complexes is a monoidal model category, which was observed by Sauvageot, in [104], and the fact that the crossed complex map, Π(Y ) → Π(X), induced by the inclusion, is a cofibration89 in that structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [27, Proposition 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This point of view is explored in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By [26, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, ii], given p: A → B, a fibration of crossed complexes, p has the right-lifting property with respect to the map, Π({0}) → Π(I), induced by inclusion, where, as usual, I = [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let I × I be given the usual product CW-decomposition, and let U be made of the left, right and bottom sides of I × I, with the obvious skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The map, Π(U) → Π � I × I � , induced by the inclusion, is a trivial cofibration of crossed complexes, and hence has the left-lifting property with respect to all fibrations of crossed complexes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From this, we can easily prove the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 226 (The functor derived from a fibration of crossed complexes).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 and B = (Bn)n∈Z+ 0 be crossed complexes, and let p: A → B be a fibration between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a functor, Fp : π1(B, B0) → Set, in full Fp = F(p: A→B), such that Fp sends b ∈ B0 to π0(p−1(b)), where the crossed complex p−1(b) is the fibre of p: A → B, at b ∈ B0, as in Definition 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a morphism, [γ]: b → b′, in π1(B, B0), the map, Fp([γ]): π0(p−1(b)) → π0(p−1(b′)), is defined from the right-lifting property of, p: A → B, with respect to the map Π({0}) → Π(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ This lemma is a version of Lemma 96 for crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that a result as strong as Lemma 96 is not likely to hold, since fibrations of crossed complexes, as defined above, do not necessarily satisfy the full homotopy lifting condition, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' they are not necessarily Hurewicz fibrations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [22, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrant spans of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notion of fibrant span of spaces, of course, adapts to other contexts, and can be formulated in terms of fibrant objects in a category of form, CΛ, where C is any reasonably structured model category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we can define the notion of fibrant spans of crossed complexes, which we will discuss below, for completeness, even though most of what is written below will not be used directly in the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The theory of fibrant spans of crossed complexes very closely parallels that of fibrant spans of spaces, so we will not give proofs of results if the proofs of the spatial case are easily adapted to this other context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 227.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A span, B p �❥❥❥❥❥❥ q �❚ ❚ ❚ ❚ ❚ ❚ A0 A1, of crossed complexes, is said to be fibrant if the induced map, ⟨p, q⟩ : B → A0 × A1, is a fibration of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 89See [27, section 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1] and [22] for a more detailed discussion of cofibrations and trivial cofibrations of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We just need that inclusions, so cofibrations, of CW-complexes are sent by Π to cofibrations of crossed complexes and similarly for trivial cofibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 190 We note that, given any crossed complex, A, any projection from a product, A×A′ → A will be a fibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It follows that in a fibrant span of crossed complexes, both p and q will be fibrations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 228.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have two potential meanings for ‘fibration’, and thus for ‘fibrant span’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As was mentioned earlier, the category, Crs, has a model category structure, given by Brown and Golasi´nski, [22], where fibrations are as in Definition 222.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have a cylinder-cocylinder based homotopy structure in which ‘fibration’ means an analogue of a Hurewicz fibration, and in which the relevant analogue of ‘weak equivalence’ is that of homotopy equivalence with respect to the adjoint cylinder- cocylinder pair, A ⊗ I and AI := CRS(I, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, at the time of writing, it does not seem to be known if this latter theory does actually give a model category structure on Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is clear, and explicitly proved in both [22] and [54], that any Hurewicz fibration is a fibration in the sense used here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two notions agree on all cofibrant crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can use the definitions and constructions of the injective model category structure on CrsΛ, induced from either structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Recall also Remark 49 for con- text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In our applications of these fibrant spans, we will almost always be handling homotopy equivalences rather than weak equivalences, so many of these technical ‘difficulties’ will evaporate!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 229.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given any morphism, f : A → B, of crossed complexes, we can use the analogy of the mapping cocylinder construction to replace f by a fibration, Nf := A ×B BI ρ(f) −−−→ B, just as in the spatial case, (see Example 50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying this to an ordinary span, (f, M, g) := � M f �✐✐✐✐✐✐ g �❯ ❯ ❯ ❯ ❯ ❯ A B � , and taking ϕ = ⟨f, g⟩, we can replace this span by a fibrant one, Nϕ �❥❥❥❥❥❥ �❚ ❚ ❚ ❚ ❚ ❚ A B, which is homotopy equivalent to the original one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We call this fibrant span a fibrantly resolved span corresponding to (f, M, g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 230.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a crossed complex, then we can apply the above process to the identity span on A, and note that AI sA �❥❥❥❥❥❥ tA �❚ ❚ ❚ ❚ ❚ ❚ A A is a fibrant span, since the morphism, AI → A × A, is a (Hurewicz) fibration of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 231.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, in a span, (p, M, p′) : B → B′, of crossed complexes, all three crossed complexes are homotopy finite90, then we say (p, M, p′) is a homotopy finite span of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following is evident.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 232.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If (p, M, p′) is a homotopy finite span, then there is a natural ho- motopy equivalence between it and a homotopy finite fibrant span.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 90The definition of a homotopy finite crossed complex is the obvious one, and will be given formally later in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 191 It should be clear that the usual composition of spans (here ‘of crossed com- plexes’), extends to fibrant spans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B, B′ and B′′ be crossed complexes, and (p, M, p′) : B → B′ and (p′′, M′, p′′′) : B′ → B′′ be fibrant spans, then in the obvious pullback diagram (analogous to (11)), the composite span is fibrant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If, moreover, we start with homotopy finite fibrant spans, then their composite is also a homotopy finite fibrant span of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ This, thus, gives a composite span B (p,M,p′)•(p′′,M′,p′′′) −−−−−−−−−−−−−−→ B′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The notion of equivalence of fibrant spans (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition (57)), is easily adapted to the context of crossed complexes, and the results on resolved identity, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Def- inition (59), and the following discussion) adapt to Crs without difficulty, and so will not be given again here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We will explore some of this theory of fibrations and fibrant spans in the hopefully forthcoming [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor H(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y,Z:A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a special CW-complex, with Y and Z being two disjoint subcomplexes of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let A be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a natural isomorphism of crossed complexes, Π(Ysk ⊔ Zsk) ∼= Π(Ysk) ⊔ Π(Zsk), so it follows from the explicit definition of CRS( , A) in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, and also from the closed monoidal structure in Crs, that we have a natural isomorphism of crossed complexes, CRS � Π(Ysk) ⊔ Π(Zsk), A � ∼= CRS � Π(Ysk), A � × CRS � Π(Zsk), A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note also that given crossed complexes, C = (Cn)n∈Z+ 0 and C′ = (C′ n)n∈Z+ 0 , there is a natural isomorphism of groupoids, π1(C × C′, C0 × C′ 0) ∼= π1(C, C0) × π1(C′, C′ 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying Lemma 224, the map, Π(ι): Π(Ysk ⊔Zsk) ∼= Π(Ysk)⊔Π(Zsk) → Π(Xsk), induced by inclusion, gives a fibration of crossed complexes, p: CRS(Π(Xsk), A) → CRS(Π(Ysk ⊔ Zsk), A) ∼= CRS(Π(Ysk), A) × CRS(Π(Zsk), A), where p := Π(ι)∗ = CRS(Π(ι), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 226 then gives a functor91, F(p) : π1 � CRS(Π(Ysk), A) � × π1 � CRS(Π(Zsk), A) � ∼= π1 � CRS(Π(Ysk), A) × CRS(Π(Zsk), A) � → Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This leads to the following profunctor, whose construction mimics that of the profunctor, H(X′ (p,M,p′) −−−−−→ Y ′): π1(X′, X′) ↛ π1(Y ′, Y ′), associated to a fibrant span, X′ (p,M,p′) −−−−−→ Y ′, of CGWH spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Subsection 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X, Y , Z and A be as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 91If B is a crossed complex, we use two notations for the fundamental groupoid of B, namely π1(B) and π1(B, B0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If W is CW-complex, and A is a crossed complex, we hence abbreviate π1 � CRS(Π(Wsk), A), Crs(Π(Wsk), A) � to π1 � CRS(Π(Wsk), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 192 Definition 234 (The profunctor, H(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y,Z:A)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The profunctor, H(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y,Z:A) : π1 � CRS(Π(Ysk), A) �op × π1 � CRS(Π(Zsk), A) � → Set, is defined as the composite, π1 � CRS(Π(Ysk), A) �op × π1 � CRS(Π(Zsk), A) � (( )−1×id) −−−−−−−→ π1 � CRS(Π(Ysk), A) � × π1 � CRS(Π(Zsk), A) � F (p) −−−→ Set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Item (2) of Lemma 206 gives a way to understand the fibration, p = Π(ι)∗, combinatorially, and hence can be used to write down, explicitly, the functor F(p), and, hence, the profunctor H(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='Y,Z:A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the previous definition, put X = [0, 1], Y = {0} and Z = {1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let also G = (s, t: G1 → G0) be a groupoid, and take A = ι1(G) to be that groupoid, considered as a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let ∗ be the crossed complex with a unique object, and only identity morphisms, so Π({0}) ∼= ∗ and Π({1}) ∼= ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly, CRS0 � ∗, ι1(G) � = G0, and moreover the groupoid, CRS1 � ∗, ι1(G) � , of crossed complex maps from ∗ to ι1(G), and homotopies between them, is canonically isomorphic to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The crossed complex CRS � ∗, ι1(G) � is easily seen to only have identity morphisms at levels n ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore, as it would be expected, we have CRS � ∗, ι1(G) � ∼= ι1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let I be the unit interval groupoid, hence, as observed in Example 191, Π([0, 1]) ∼= ι1(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By Theorem 200, morphisms ι1(I) → ι1(G) are in one-to-one correspon- dence with arrows g : x → y, in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Unsurprisingly, the arrows in the groupoid CRS1 � ι1(I), ι1(G) � , i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the homotopies between crossed complex maps, from ι1(I) to ι1(G), are in bijection with diagrams, in G, as below, (73) x hL � g � y hR � x′ h−1 L g hR � y′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, the two vertical arrows, hL : x → x′ and hR : y → y′, are taken from the groupoid G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid composition in CRS1 � ι1(I), ι1(G) � is given by the obvious vertical composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, for n ≥ 2, all morphisms in CRSn � ι1(I), ι1(G) � are identity morphisms, for all 2-fold homotopies are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The crossed complex fibration, of crossed complex mapping spaces, obtained from the inclusion of {0, 1} in the interval [0, 1] (with the obvious CW-decomposition), denoted p: CRS � Π([0, 1]), ι1(G) � → CRS � Π({0, 1}), ι1(G) � ∼= CRS � Π({0}), ι1(G) � × CRS � Π({1}), ι1(G) � , is, thus, at the level of groupoids, given by the groupoid map from CRS1 � ι1(I), ι1(G) � → G × G that chooses the two vertical arrows in (73).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, we can see that, π1 � CRS(Π({0}), ι1(G)) � ∼= G and π1 � CRS(Π({1}), ι1(G)) � ∼= G, and moreover, that the profunctor H � [0,1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='{0},{1}:ι1(G) � is given by the profunctor in Example 32, that is, the identity profunctor on G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 193 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fibrations of crossed complexes and fibrations of mapping spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The follow- ing theorem, which generalises the last point of Theorem 219, will be fundamental in giving explicit calculations of TQFTs from crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This result appears to be new, however is not unexpected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let S be a simplicial set and T a subcomplex of S, with i(T,S) : T → S being the inclusion map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Its geometric realisation, |i(T,S)|: |T | → |S|, is an inclusion of CW-complexes (by, for instance, [55, Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='38]), hence it induces a crossed complex map, Π(i(T,S)): Π(|T |) → Π(|S|), in fact a cofibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 224, then, gives a fibration of crossed complexes between the appropriate internal homs, Π(i(T,S))∗ : CRS(Π(|S|), A) → CRS(Π(|T |), A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(|T |) → A be a crossed complex map and, using Definition 184, con- sider the crossed complex obtained as the fibre over f ∈ CRS0(Π(|T |), A) = Crs(Π(|T |, A) in this fibration, Π(i(T,S))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will denote this crossed complex by CRS(f)(Π(|S|), A) := (Π(i(T,S))∗)−1(f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This latter crossed complex, of course, fits inside the pull-back diagram below, where ˆf is the crossed complex with object set {f}, and only identity arrows in all dimensions, (74) CRS(f)(Π(|S|), A) � inc � CRS(Π(|S|), A) Π(i(T,S))∗ � ˆf inc � CRS(Π(|T |), A) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since |i(T,S)|: |T | → |S| is an inclusion of CW-complexes, hence a cofibration, we have a mapping space fibration of CGWH topological spaces, |i(T,S)|∗ : TOP(|S|, BA) → TOP(|T |, BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have the crossed complex map, f : Π(|T |) → A, and this gives rise, via the usual adjunction, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proposition 214, followed by geometric realisation, to a continuous map, |φA T (f)|: |T | → |BA|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre of |φA T (f)|: |T | → |BA| under the mapping space fibration will be denoted TOP(|φA T (f)|)(|S|, BA) := (|i(T,S)|∗)−1(|φA T (f)|), and we have a pullback diagram in CGWH, TOP(|φA T (f)|)(|S|, BA) � inc � TOP(|S|, BA) |i(T,S)|∗ � {|φA T (f)|} inc � TOP(|T |, BA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Brown–Higgins/Brown–Higgins–Sivera/Tonks weak homotopy equivalence, ηA S : ��N � CRS(Π(|S|), A) ��� → TOP(|S|, BA), in Theorem 219, restricts to a weak homotopy equivalence, |N � CRS(f)(Π(|S|), A) � | → TOP(|φA T (f)|)(|S|, BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 194 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since the nerve functor N : Crs → Simp is a right adjoint, it preserves limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Applying N to diagram in (74), we have a pull-back diagram in Simp92, N � CRS(f)(Π(|S|), A) � � inc � N � CRS(Π(|S|), A) � N(Π(i(T,S))∗ � N( ˆf) inc � N � CRS(Π(|T |), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The geometric realisation functor, Simp → CGWH, preserves finite limits, [55, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='16], so applying geometric realisations to the previous diagram, yields a pull-back diagram in CGWH, (75) |N � CRS(f)(Π(|S|), A) � | � inc � |N � CRS(Π(|S|), A) � | |N(Π(i(T,S))|)∗| � |N( ˆf)| inc � |N � CRS(Π(|T |), A) � | .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have another commutative diagram in CGWH, arising from the natural- ity93, on varying the simplicial set, S, see [18, Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='] and [26, Theorem A], of the weak homotopy equivalence, ηA S : |N � CRS(Π(|S|), A) � | → TOP(|S|, BA)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives a commutative diagram, (76) |N � CRS(Π(|S|), A) � | ηA S � ���N � Π(i(T,S))∗���� � TOP(|S|, BA) |i(T,S)|∗ � |N � CRS(Π(|T |), A) � | ηA T � TOP(|T |, BA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that diagrams (75) and (76) share one of their vertical arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Next, we note the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given that Π(i(T,S))∗ : CRS(Π(|S|), A) → CRS(Π(|T |), A) is a fibration of crossed complexes (by Lemma 224), then its nerve N � Π(i(T,S))∗� : N � CRS(Π(|S|), A) � → N � CRS(Π(|T |), A) � is a fibration of simplicial sets, and so its geometric realisation is a fibra- tion of CGWH topological spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [55, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='25]94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The left downwards arrow of diagram (76) is, therefore, a fibration in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since the inclusion, |iT,S|: |T | → |S|, is a cofibration, the right downwards arrow of diagram (76) is a fibration in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The two horizontal maps in diagram (76) are weak homotopy equivalences in CGWH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since (76) commutes, the map ηA S : |N(CRS(Π(|S|), A))| → TOP(|S|, BA) sends fibres to fibres.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As f : Π(|T |) → A is a crossed complex map, we have that ηA T ( ˜f) = |φA T (f)|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the notation in Remark 218.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the fourth point of 92In general, let A = (An)n∈Z+ 0 and B = (Bn)n∈Z+ 0 be crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If B is a sub-crossed complex of A, meaning that each groupoid, Bn, is a subgroupoid of An, then we have a natural inclusion of N (B) in N (A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 93Recall [116] and [18, Page 177] that the weak homotopy equivalence is only natural with respect to simplicial maps, but not natural in the enriched sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 94Note that geometric realisations of Kan fibrations are only sure to have the homotopy lifting property with respect to homotopies whose domain is a CGWH space, see [55, Page 185].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 195 Theorem 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The map, ηA S , restricts to a map on the corresponding fibres, which we denote g′ : � |N(Π(i(T,S))|∗�−1 ( ˜f) → TOP(|φA T (f)|)(|S|, BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Making use of the pull-back diagram (75), and the uniqueness95 of pull-backs, this map, g′, gives rise to another map, in CGWH, g : |N(CRS(f)(Π(|S|), A))| → TOP(|φA T (f)|)(|S|, BA), arising from the canonical homeomorphism, given by the uniqueness of pull-backs, |N(CRS(f)(Π(|S|), A))| ∼= � |N(Π(i(T,S))|∗�−1 ( ˜f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This g is exactly the map we want to prove is a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') We will show that g′, and hence g, is a weak homotopy equivalence, which yields the statement of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this we use the homotopy long exact sequences of the two vertical fibrations of diagram (76).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The pair of weak equivalences ηA S , of the total spaces, and ηA T , of the base spaces, together with g′, the map on the fibre, maps the first long exact sequence to the latter one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore the five-lemma proves that g induces an isomorphism for all homotopy groups, and hence g is a weak homotopy equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Example 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider again the inclusion, ι, of {0, 1} into [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G be a group, viewed as a crossed complex, via ι1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibration of mapping spaces, arising from the inclusion ι, will be denoted P : TOP � [0, 1], BG � → TOP � {0, 1}, BG � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The crossed complex map, p: CRS � Π([0, 1]), G � → CRS � Π({0, 1}), G � , induced by Π(ι): Π({0, 1}) → Π([0, 1]) is made explicit in Example 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is only one crossed complex map f : Π({0, 1}) → G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre of the projection, p, at f, is the crossed complex CRS(f)(Π([0, 1]), G), whose set of objects is the underlying set of G, and with only identity morphisms at all orders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From this fact, we can see that the fibre of the fibration, P, over the map |φG {0,1}(f)| → BG, the map that sends both 0 and 1 to the unique vertex of BG, is homotopic to the classifying space of CRS(f)(Π([0, 1]), G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre, P −1(|φG {0,1}(f)|), is, thus, the disjoint union of contractible spaces, one for each element of G, as it one should expect, since the classifying space BG is an aspherical space, and π1(BG) ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Computing the homotopy content of a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The results in this subsection are essentially all in [49], or [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A = (An)n∈Z+ 0 is a finite connected crossed complex, then, in order to de- termine the homotopy content of the classifying space, BA, one does not need to compute the homotopy groups of BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The computation can be reduced to an alter- nating product of cardinalities of sets of certain morphisms in An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This fact (and its proof) is similar to the fact that the Euler characteristic of a finite CW-complex, X, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the alternating sum of the ranks of its homology groups, Hi(X), can be computed as �∞ i=1(−1)ini, where ni is the number of i-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we will see later, for this reason, the formula for the Quinn TQFT, and its once-extended ver- sions, greatly simplify when B is the classifying space of a finite crossed complex, with a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some notation will need be defined to explore this in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 95up to isomorphism A CATEGORIFICATION OF QUINN’S TQFT 196 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finite and homotopy finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will start by defining what it means for a crossed complex to be finite, and more generally homotopy finite, which follows naturally from the framework already introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 238.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We say that A = (An)n∈Z+ 0 is finite if all the groupoids, An, are finite, and there exists m ∈ N such that the groupoids, An, contain only identity arrows for n ≥ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We say that A is homotopy finite if A has only a finite number of path compo- nents, each of which with a finite number of non-trivial homotopy groups, all of which are finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Clearly, if a crossed complex is finite then it is homotopy finite, but not con- versely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the relationship between crossed complexes and simplicially enriched groupoids, which we will not give here, one can use the results of [46] to show that, if A is a homotopy finite, then it is weakly homotopy equivalent to a finite one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following result will be implicitly used several times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a special finite CW-complex and let A be a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set, Crs � Π(Xsk), A � , of crossed complex maps from Π(Xsk) to A, is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows directly from Remark 194.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Definition 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a path-connected homotopy finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define the homotopy content of A to be χπ(A) := ∞ � i=1 |πi(A, c)|(−1)i, where c is any object of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, if B is any homotopy finite crossed complex, we define χπ(B) := � A∈� π0(B) χπ(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here � π0(B) is the set of path-components of B;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Definition 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 219 immediately implies the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a homotopy finite crossed complex, then its classifying space, BA, is a homotopy finite space, and χπ(A) = χπ(BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Notation 242.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x ∈ A0 and n ∈ Z+, define Θx n(A) ∈ Z+ to be the cardinality of the set of morphisms, in the groupoid An, with source x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 243.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that, fixing n ∈ Z+ 0 , then Θx n(A) ∈ Z+ depends only on the path component in A, or equivalently in π1(A), to which x ∈ A0 belongs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following result appears in [49, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a finite crossed complex, then χπ(A) = � x∈A0 � ∞ � i=1 � Θx i (A) �(−1)i � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 197 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from a telescopic calculation, similar to the proof of Theorem 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial point in the proof, allowing us to pass from a sum over path components of A to a sum over objects of A, is that, if (s, t: G1 → G0) is a finite groupoid, then given x ∈ G0, the cardinality of the set of morphisms in G, with source x, is equal to |G(x, x)| |[x]|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here [x] is the set of objects of G connected to x by a morphism of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Full details are in [49, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Definition 245.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A finite crossed complex, A = (An)n∈Z+ 0 , will be called homoge- neous if, given a non-negative integer n, and an object x of A, the value of Θx n(A) depends only on n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This means that there exists, for each non-negative integer n, a positive integer, Θn(A), such that, for each x ∈ A0, Θx n(A) = Θn(A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A is homogeneous, define Θ(A) = ∞ � i=1 (Θn(A)) � (−1)i� ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that path-connected finite crossed complexes are automatically homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Corollary 246.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose that A = (An)n∈Z+ 0 is homogeneous, and so, in particular, finite, then χπ(A) = Θ(A) |A0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homotopy content of CRS � Π(Xsk), A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a finite CW-complex, and Y a subcomplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 247 (K(n, X) and K(n, X, Y )).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We set K(n, X) to be the number of n-cells of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' More generally, let Y be a subcomplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An n-cell, c, of X is said to be internal to (X, Y ), if it is not in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let K(n, X, Y ) be the number of n-cells of X that are internal to the pair (X, Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By applying the results in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, and, in particular, Lemma 202, together with the notions and notation introduced earlier in this Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, we have: Lemma 248.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a finite crossed-complex with a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a finite special CW-complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The crossed complex CRS(Π(Xsk), A) is homogeneous, in the sense of Definition 245, and, in particular, finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, we have that, for a positive integer j, Θj � CRS(Π(Xsk), A) � = ∞ � i=0 |Ai+j|K(i,X).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, χπ� CRS(Π(Xsk), A) � = ��Crs(Π(Xsk), A) �� ∞ � j=1 � ∞ � i=0 |Ai+j|K(i,X) �(−1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ This is the special case in which Y is empty, of the following more general result, in which we let (X, Y ) be a finite CW-pair, with both X and Y being special CW-complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 198 Let f : Π(Ysk) → A be a crossed complex map and, as before, let Π(i): Π(Ysk) → Π(Xsk) be induced by the inclusion i: Y → X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider also the induced fibration96 (Lemma 224) between the ‘internal homs’, Π(i)∗ : CRS � Π(Xsk), A � → CRS � Π(Ysk), A � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 249.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The fibre of Π(i)∗ at f 97, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', the crossed complex, CRS(f)� Π(Xsk), A � := (Π(i)∗)−1(f), is homogeneous, and if j is a positive integer, then Θj � CRS(f)(Π(Xsk), A) � = ∞ � i=0 |Ai+j|K(i,X,Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular χπ� CRS(f)(Π(Xsk), A) � = ��� g : Π(Xsk) → A | g ◦ Π(i) = f ��� ∞ � j=1 � ∞ � i=0 |Ai+j|K(i,X,Y ) �(−1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that the above crossed complex was already considered in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, in the case in which (X, Y ) was the geometric realisation of a simplicial pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from Lemmas 202, 206 and 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The particular case when we have a CW-triad98, (X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Y, Z), of finite special CW- complexes will be very useful when we come to write down explicit formulae for TQFTs derived from crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall, from page 54, that by this, Quinn means that Y and Z are disjoint subcomplexes of a special CW-complex, X, which then implies that Y ⊔ Z is a subcomplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The example to have in mind is a triangulated cobordism, M : S → S′, where X = M, Y = S and Z = S′, although we will shortly consider more general CW-decompositions than triangulations in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Let i(Y,X) : Y → X and i(Z,X) : Z → X be the inclusion maps, so we have a cellular map, � i(Y,X) i(Z,X) � : Y ⊔ Z ∼= Y ∪ Z → X, which gives the inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given crossed complex maps, f : Π(Ysk) → A and f ′ : Π(Zsk) → A, we can combine them into a map, � f f ′ � : Π � (Y ∪ Z)sk � ∼= Π(Ysk) ⊔ Π(Zsk) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The set of objects of the crossed complex, CRS �� f f ′ �� (Π(Xsk), A) = (Π( � i(Y,X) i(Z,X) � )∗)−1⟨f, f ′⟩ 96(of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') 97(See Definition 184.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') 98in the sense of Quinn, see page 54, A CATEGORIFICATION OF QUINN’S TQFT 199 is the set of crossed complex maps, h: Π(Xsk) → A, that make the diagram, A Π(Ysk) Π(i(Y,X)) �❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ ❏ f �s s s s s s s s s s Π(Zsk) f ′ �❑❑❑❑❑❑❑❑❑❑ Π(i(Z,X)) �ttttttttt Π(Xsk), h � commute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 250.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let X be a finite special CW-complex with Y and Z, two disjoint subcomplexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A = (An)n∈Z+ 0 be a finite reduced crossed complex, so with a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(Ysk) → A and f ′ : Π(Zsk) → A be crossed complex maps, then χπ � CRS �� f f ′ �� (Π(Xsk), A) � = ��{h: Π(Xsk) → A : h ◦ Π(i(Y,X)) = f and h ◦ Π(i(Z,X)) = f ′} �� ∞ � j=1 � ∞ � i=0 |Ai+j|K(i,X,Y ∪Z) �(−1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the discussion just before the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Finally, let X be a special CW-complex, A = (An)n∈Z+ 0 a finite reduced crossed complex, and let f : Π(Xsk) → A be a crossed complex map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By passing to the path- component, PCf(CRS(Π(X)sk, A)), of f in the crossed complex CRS(Π(X), A), we have, within the same context as before: Lemma 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(Xsk) → A be a crossed complex map, then χπ� PCf(CRS(Π(Xsk), A) � = |[f]CRS(Π(Xsk),A)| ∞ � j=1 � ∞ � i=0 |Ai+j|K(i,X) �(−1)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here [f]CRS(Π(Xsk),A) denotes the homotopy class of f (the set of all crossed complex maps, Π(Xsk) → A, that are homotopic to f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' TQFTs and once-extended TQFTs derived from homotopy finite crossed complexes As before, let n be a non-negative integer and let A be a homotopy finite (often finite and reduced) crossed complex, so its classifying space, BA, is a homotopy finite space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that, as usual, we are working over a subfield, κ, of C, as we have to be able to invert non-zero integers when working with the homotopy content of spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Usually we think of using κ as being Q, but that is just the minimal case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section, we use the techniques of the homotopy theory of crossed com- plexes, that were recalled and slightly refined in the previous section, to give explicit formulae for: A CATEGORIFICATION OF QUINN’S TQFT 200 Quinn’s finite total homotopy TQFT, see Definition 83, where s ∈ C, Q(s) BA : Cob(n,n+1) → VectC, for which we give explicit formulae in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and the finitary once-extended Quinn TQFT, in Definition 154, 2QBA : 2Cob (n,n+1,n+2) BA : 2Cob (n,n+1,n+2) BA → vProfGrpfin, which we will treat in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will lead to explicit formulae for the Morita valued once-extended Quinn TQFT from Definition 168, 2Q Mor BA : 2Cob (n,n+1,n+2) BA → Mor, which is discussed in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will consider our manifolds, Σ, to be provided with what we call simplicial stratifications, fΣ : |XΣ| → Σ, where XΣ is a finite simplicial set (see Definition 255), and, analogously, we can define simplicial stratifications of cobordisms and of extended cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All formulae for the Quinn and the once-extended Quinn TQFTs will be given in terms of such simplicial stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Picking n-manifolds equipped with simplicial stratifications, leads naturally to another variant, 2Cob (n,n+1,n+2) st , of the bicategory 2Cob(n,n+1,n+2), whose ob- jects are pairs, (Σ, iXΣ), where iXΣ : |XΣ| → Σ is a simplicial stratification, of the (closed and smooth) n-manifold Σ, and with the rest of the bicategory structure induced, in the obvious way, from that of 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (In particular, cobor- disms and extended cobordisms do not come with chosen simplicial stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 and §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, we will also address the construction of two closely related symmetric monoidal bifunctors, where A is a finite crossed complex, 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, and 2Q Mor A : 2Cob (n,n+1,n+2) st → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal bifunctors, just mentioned, do not attach a value to an n-manifold, Σ, unless it is equipped with either a BA-decoration or a simplicial stratification, even though the associated groupoids and algebras are unique, up to a canonical invertible profunctor / Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to approach the literature on the subject, such as [106, 10, 11], in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, we will address how to get rid of this latter dependence on simplicial stratification of the n-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This step, however, is non-canonical, and requires the use of the axiom of choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives rise to once-extended TQFTs, � 2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, and � 2QMor A : 2Cob(n,n+1,n+2) −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The well known (1, 2, 3)-extended TQFT sending S1 to the quantum double of the group algebra of a finite group, [10, 92, 100], is an example of this latter construc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As recalled in the beginning of Part 4, homotopy finite crossed complexes do not model the homotopy types of all homotopy finite spaces B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The constructions in this section will not, therefore, give formulae for all possible Quinn TQFTs, A CATEGORIFICATION OF QUINN’S TQFT 201 and once-extended Quinn TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They do, however, provide formulae for those derived from, for instance, finite 2-types B, most relevant for higher gauge theory;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' for instance, Baez and Schreiber, [8, 3], Baez and Huerta, [6] or Faria Martins and Picken, [52], where the links between 2-groups / crossed modules and higher gauge theory are summarised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Conventions and nomenclature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section, given CGWH spaces, M and N, the space of functions from M to N (with the CGWH topology) will be denoted both by N M and TOP(M, N), whichever is more convenient for the formula in question;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the overall conventions are otherwise as in Subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If M is a CGWH space, and x ∈ M, then PCx(M) denotes the path-component of x in M, with the induced CGWH topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Also recall, see item (15) on page 16, that � π0(M) denotes the set of k-ified path-components of M, which we recall is, as a set, in one-to-one correspondence99 with π0(M).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If f : M → N is a map, then PCf(TOP(M, N)) will be the set of functions from M to N that are homotopic to f, and PCf(TOP(M, N)) is given the CGWH topology induced from TOP(M, N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 252 (Finite Simplicial set).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A simplicial set, X, is called finite if it only has a finite number of non-degenerate simplices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If a simplicial set X is finite, then, as we recalled earlier, its geometric realisation, |X|, is naturally a finite CW-complex, with one i-cell for each non-degenenate i- simplex of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, |X| will be a special CW-complex, in the sense of Definition 192.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also have a relative notion in which (X, Y ) is a pair of finite simplicial sets, meaning that Y is a sub-simplicial set of X, and then |Y | is naturally a subcomplex of the CW-complex, |X|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Notation 253.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let (X, Y ) be a pair of finite simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Extending Notation 247, we note that K(i, |X|) will also be the number of non-degenerate i-simplices of X, and K(i, |X|, |Y |), the number of non-degenerate i-simplices of X that are not in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will extend the use of the notation, removing the geometric realisations signs for convenience, so that: K(i, X) will denote be the number of non-degenerate i-simplices of X, and K(i, X, Y ) will denote the number of non-degenerate i-simplices of X, that are not in Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This notation will appear in the formulae for the TQFTs, and the once-extended TQFTs, derived from finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Classically, see, for instance, [64, pp 107], an (abstract) simplicial complex, K, is defined to be given by a sequence, K = (K0, K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), consisting of a set, K0, the set of vertices of K, together with, for each i ∈ Z+, a subset, Ki, of the set of subsets of K0 that have cardinality i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The elements of Ki are called the i-faces (or vertices if i = 0) of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By definition, these are to have the property that if F is a subset of cardinality j of some i-face of K, then F will be itself a j-face of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 254.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A triangulation of a (smooth) manifold, M, is a homeomorphism, f : |K| → M, where K is a simplicial complex and |K| is its geometric realisation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that our definition of a triangulation of M makes no reference to the smooth structure in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is not required as our construction of TQFTs only makes use of the underlying topological manifold of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Otherwise we would need to consider smooth and regular triangulations of M as, for example, in [94, Chapter II].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 99The framework of this paper makes it natural to distinguish between π0 and � π0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 202 If a simplicial complex, K = (K0, K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ), is additionally provided with a total order on the set, K0, of its vertices, then K gives rise to a simplicial set, K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, given a non-negative integer i, we have a one-to-one correspondence between the i-faces of K and the non-degenerate i-simplices of K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This well known construction appears, for example, in [41, Examples 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3] and [53, §1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that we have a canonical homeomorphism, |K| ∼= |K′|, between the geometric realisations of the simplicial complex and of the simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The geometric realisation of K′ does not depend on the choice of an order on K0, up to canonical homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this paper, it will be convenient to consider a more general variant of “tri- angulations”, defined in the broader context of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will call these simplicial stratifications of the manifold, as in the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 255 (Simplicial stratifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Σ be a compact (smooth) n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let XΣ be a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A simplicial stratification of Σ is a homeomorphism, fΣ : |XΣ| → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We denote such a simplicial stratification of Σ by (XΣ, fΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that since a simplicial set has a compact geometric realisation if, and only if, it is finite, XΣ must always be a finite simplicial set in the above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') More generally, consider an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, between the closed (smooth) n-manifolds, Σ and Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A simplicial stratification (of the cobor- dism) is given by a (Quinn) triad, (YM;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XΣ, X′ Σ′), of simplicial sets, so that the sim- plicial sets, XΣ, and X′ Σ′, are subcomplexes of the simplicial set, YM, and XΣ ∩X′ Σ′ is empty, together with a homeomorphism, (fΣ, gM, f ′ Σ′), of cospans in CGWH, (77) |XΣ| |j| �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ fΣ � |X′ Σ′| |j′| �❥❥❥❥❥❥❥❥❥❥❥ f ′ Σ′ � |YM| gM � Σ i �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′ �✐✐✐✐✐✐✐✐✐✐✐✐✐ M Here j : XΣ → YM and j′ : X′ Σ′ → YM denote the obvious simplicial inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Simplicial stratifications of a cobordism, (i, M, i′): Σ → Σ′, arise, for instance, from triangulations of M that extend given triangulations of Σ and Σ′, and which, furthermore, are equipped with a total order on the set of vertices of the triangula- tion of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Since simplicial stratifications can, in general, be chosen to be smaller than triangulations, they have advantages over the triangulated form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Any simpli- cial stratification can be processed to give a finer triangulation, but typically with more parts to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' TQFTs from homotopy finite and finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this subsection, we will work over the field C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For X a simplicial set, and A a crossed complex, we recall, from (Brown–Higgins) Lemma 214, that there is a natural bijection, φA X : Crs � Π(|X|sk), A � → Simp � X, N(A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here |X|sk is the realisation of X, with its skeletal filtration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also need to use the Brown–Higgins–Sivera–Tonks Theorem, [18, 26, 27, 115, 116], that is given here as Theorem 219, and which gives that we have a natural map100 in CGWH, which is a weak homotopy equivalence, (78) ηA X : ��N � CRS(Π(|X|sk), A) ��� → TOP(|X|, BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 100A map, ηA Y : |CRS(Π(Ysk), A)| → TOP(Y, BA), can also be defined, [26], when Y is a CW- complex, with its cellular decomposition, however the map is, in that case, only specified up to homotopy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 203 Moreover, using the notation in Notation 218, and the discussion following Theorem 219, if f : Π(|X|sk) → A is a crossed complex map, ηA X( ˜f) = |φA X(f)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If f, f ′ : Π(|X|sk) → A are homotopic crossed complex maps, then the objects f, f ′ ∈ CRS0(Π(|X|sk, A) are connected by an arrow in CRS(Π(|X|sk), A), and, thus, ˜f and ˜f ′ belong to the same path-component of ��N � CRS(Π(|X|sk), A) ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The weak homotopy equivalence, (78), then gives that the two maps, |φA X(f)|, |φA X(f ′)|: |X| → BA, are homotopic maps between CGWH spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that π0 � CRS(Π(|X|sk), A) � is just the set of homotopy classes of crossed complex maps from Π(|X|sk) to A, for which see Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we saw in Lemma 241, if A is homotopy finite, then its classifying space, BA, is homotopy finite, and we can consider Quinn’s finite total homotopy TQFT, of [101, Lecture 4], or more generally as explored here in Subsection 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, Q(s) BA : Cob(n,n+1) → Vect with base-space B = BA, and in which s is a complex parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our main case of study is when A is finite and reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, the formulae for Q(s) BA become particularly simple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using Proposition 88 and Theorem 90, the knowledge of that restricted case is enough to give the calculation in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Recall that it is known that any homotopy finite crossed complex is weak equivalent to a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This can be derived from Ellis’ theorem, [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In the theorem below, if X is a simplicial set, we put Π(X) = Π(|X|sk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also note that | − | is used to denote both the cardinality of a finite set and the geometric realisation of a simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Which one it is, should be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will assume, now, that A is a homotopy finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This A will be fixed throughout the discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, let n be a non-negative integer, and Σ be a closed smooth n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a simplicial stratification, (XΣ, iXΣ), of Σ, so with XΣ a simplicial set, and iXΣ : |XΣ| → Σ, a homeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a crossed complex map, f : Π(XΣ) → A, we define f := |φA X(f)| ◦ i−1 XΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here, as φA X(f) is a simplicial map, φA X(f) : XΣ → N(A), we have that f : Σ → BA is a continuous map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have s ∈ C, a fixed parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have an isomorphism of vector spaces, C � π0 � CRS(Π(XΣ), A) �� T (iXΣ ,A) −−−−−−→ Q(s) B (Σ) = C � �π0(BΣ) � , such that, given f : Π(XΣ) → A and on denoting the homotopy class of the crossed complex map, f, by [f]CRS(Π(XΣ),A), we have T (iXΣ, A) � [f]CRS(Π(XΣ),A) � = PCf(BΣ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Suppose given an (n + 1)-cobordism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′): Σ → Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' between the closed n- manifolds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Σ and Σ′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and a simplicial stratification of the cobordism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 204 obtained from the cospan of simplicial sets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' XΣ j �❙ ❙ ❙ ❙ ❙ ❙ X′ Σ′ j′ �❦❦❦❦❦❦ YM together with a homeomorphism of cospans in CGWH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (79) |XΣ| |j| �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ iXΣ � |X′ Σ′| |j′| �❥❥❥❥❥❥❥❥❥❥❥ iX′ Σ′ � |YM| iYM � Σ i �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Σ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′ �✐✐✐✐✐✐✐✐✐✐✐✐✐ M Given crossed complex maps f : Π(XΣ) → A, f ′ : Π(X′ Σ′) → A, let f = |φA XΣ(f)| ◦ i−1 XΣ : Σ → BA, and f ′ = |φA X′ Σ′ (f ′)| ◦ i−1 X′ Σ′ : Σ′ → BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 256 (Quinn’s finite total homotopy TQFT, Q(s) B for B = BA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the formula for the matrix elements of Quinn’s finite total homotopy TQFT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Q(s) B ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' which is as follows,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � PCf(BΣ) | Q(s) B � [(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′)] � | PCf ′(BΣ′) � = χπ�� f|BM|f ′�� � χπ� PCf(BΣ) ��s � χπ� PCf ′(BΣ′) ��1−s ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as before,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � f|BM|f ′� = \uf8f1 \uf8f4 \uf8f2 \uf8f4 \uf8f3 H : M → B ������� B Σ f �❦ ❦ ❦ ❦ ❦ ❦ i �❙ ❙ ❙ ❙ ❙ ❙ Σ′ f ′ �❙❙❙❙❙❙ i′ �❦❦❦❦❦❦ M H � commutes \uf8fc \uf8f4 \uf8fd \uf8f4 \uf8fe ⊂ BM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' with the induced CGWH topology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' each factor can be calculated as follows101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First of all, (80) χπ�� f|BM|f ′�� = χπ � CRS �� f f ′ �� (Π(YM), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A is finite, and reduced (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' with a single object), then 102 (81) χπ�� f|BM|f ′�� = �������� \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 h: Π(YM) → A �������� A Π(XΣ) f �❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ Π(j) �❯ ❯ ❯ ❯ ❯ ❯ Π(X′ Σ′) f ′ �❱❱❱❱❱❱❱❱ Π(j′) �❤❤❤❤❤ Π(YM) h � commutes \uf8fc \uf8f4 \uf8f4 \uf8fd \uf8f4 \uf8f4 \uf8fe �������� ∞ � k=1 � ∞ � i=0 |Ai+k|K(i,YM,XΣ∪X′ Σ′ ) �(−1)k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 101For notation see §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 102Here recall that K(i, YM, XΣ ∪ X′ Σ′) is the number of non-degenerate i-simplices of the simplicial set YM that are neither in XΣ nor in X′ Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 205 Continuing with the evaluation of the other terms in the formula for � PCf(BΣ) | Q(s) B � [(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′)] � | PCf ′(BΣ′) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' we have χπ� PCf(BΣ) � = χπ� PCf(CRS(Π(XΣ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (82) and χπ� PCf ′(BΣ′) � = χπ� PCf ′(CRS(Π(X′ Σ′),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A)) � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (83) so (by Lemma 251),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' if A is finite and reduced,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' then χπ� PCf(BΣ) � = ��[f]CRS(Π(XΣ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A) �� ∞ � k=1 � ∞ � i=0 |Ai+k|K(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='XΣ) �(−1)k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' χπ� PCf ′(BΣ′) � = ��[f]CRS(Π(X′ Σ′ ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A) �� ∞ � k=1 � ∞ � i=0 |Ai+k|K(i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='X′ Σ′ ) �(−1)k ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (84) where,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as above,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [f]CRS(Π(XΣ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A)| denotes the homotopy class of the crossed complex map,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' f,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' analogously,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [f ′]CRS(Π(X′ Σ′ ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='A)| is that of f ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This result follows from the general discussion in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 and §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the crucial ingredient is the Brown–Higgins–Sivera–Tonks weak ho- motopy equivalence, ηA S : |N(CRS(Π(S), A))| → TOP(|S|, BA), in item (5) of Theorem 219, where S is a simplicial set and A is a crossed complex, and its refinement in Theorem 236, together with the result that the homotopy groups of a crossed complex coincide with those of its geometric realisation, for which see again Theorem 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, Equation (80) follows from Theorem 236, applied to the simplicial inclusion � j j′ � : XΣ ⊔ X′ Σ′ → YM, and the crossed complex map, � f f ′ � : Π(XΣ) ⊔ Π(X′ Σ′) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We then apply Lemma 241.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally (81) follows from Lemma 250, and Equation (84) follows from Lemma 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 257 (Independence from simplicial stratifications).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a homotopy finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that, by construction, all formulae for the Quinn fi- nite total homotopy TQFT, Q(s) BA, in the previous theorem are independent of the chosen simplicial stratifications of the n-manifolds, Σ and Σ′, and of the (n + 1)- cobordism, (i, M, i′): Σ → Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is no need to make use of Alexander moves, or equivalently of Pachner moves, to prove triangulation-independence103, as done for instance in [84, 9, 119].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is because the formulae were directly derived to give quantities that are, by construction, topologically invariant, and related to the homotopy content of function spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 258.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using the previous theorem together with Lemma 212, we can see that the calculations of Quinn’s finite total homotopy TQFT, Q(s) BA, for A a finite crossed complex, and for given simplicial stratifications of the manifolds and cobordisms concerned, could in theory be computed in finite time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 103nor, for this paper, independence from simplicial stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 206 We expect that the techniques just shown will also be applicable for computing, explicitly, TQFTs derived from finite crossed complexes, which are, furthermore, equipped with a cohomology class valued in U(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The existence of these TQFTs, generalising Dijkgraaf-Witten TQFTs [43], was suggested in Remark 92, and they were treated in [53], in the particular case of closed manifolds and crossed modules, using similar techniques to those of this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Our approach here can likely also be adapted to give concrete formulae for homotopy quantum field theories derived from (classifying spaces of) crossed complexes, possibly equipped with appropriate cohomology classes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These homotopy quantum field theories are treated in [108], and also in [98, 99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also expect that similar techniques as used in this subsection can be used to give formula for Quinn’s finite total homotopy TQFT, Q(s) B , in the case when B is the classifying space of a finite simplicial group, in which case we would obtain concrete formulae for all types of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Since finite simplicial groups model all pointed homotopy finite spaces [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') This would likely yield formulae similar to those in [97].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended TQFTs derived from finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this section and the next, we will work over the field of rational numbers Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also fix a finite crossed complex, A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', actually finite, not just homotopy finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here we will freely use the notation and results from Section 6, particularly Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, as well as of our review of the homotopy theory of crossed complexes, and their classifying spaces, BA, in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let B = BA, which we recall is a homotopy finite space, by Theorem 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let n be a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will give explicit formulae for some instances of the finitary once-extended Quinn TQFT, 2QB : 2Cob (n,n+1,n+2) B → vProfGrpfin, and consequently of its Morita version, 2Q Mor B : 2Cob (n,n+1,n+2) B → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will also treat a few variants of these once-extended TQFTs, as mentioned in the beginning of this current section, Section 8, namely: 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, 2Q Mor A : 2Cob (n,n+1,n+2) st → Mor, � 2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, and � 2QMor A : 2Cob(n,n+1,n+2) −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This will be done at the same time as we examine the dependence of the formulae that we have give on the choice of the simplicial stratification of an n-dimensional manifold that is being used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The BA-decoration of a manifold arising from a simplicial stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Σ be a closed (and as usual smooth) n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall, Definition 152, that, given a HF space, B, a B-decoration, fΣ, of Σ, is given by a finite subset, f Σ, of the function space, BΣ, of functions from Σ to B, containing at least one function, f : Σ → B, from each path component of BΣ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If A CATEGORIFICATION OF QUINN’S TQFT 207 B = BA, the classifying space of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Simplicial stratifications of Σ (see Definition 255) naturally give rise to B-decorations of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can see this as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let XΣ be a finite simplicial set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall that the Brown–Higgins–Siviera– Tonks Theorem, here Theorem 219, provides a weak homotopy equivalence, ηA XΣ : |CRS(Π(XΣ), A)| → TOP(|XΣ|, BA), so suppose that we have a simplicial stratification of Σ, given by a homeomorphism iXΣ : |XΣ| → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed complex map, f : Π(XΣ) → A, gives rise to a continuous map |φA XΣ(f)|: |XΣ| → BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Noting the comments after Theorem 219, we have |φA XΣ(f)| = ηA XΣ( ˜f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The weak homotopy equivalence, ηA XΣ, hence gives a BA- decoration of Σ, defined by (85) f Σ � iXΣ : |XΣ| → Σ, A � := � |φA XΣ(f)| ◦ i−1 XΣ | f ∈ CRS0(Π(XΣ), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that CRS0(Π(XΣ), A), the set of crossed complex maps from Π(XΣ) to A, is finite, by Lemma 239.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will frequently abbreviate the notation and put fΣ � iXΣ : |XΣ| → Σ, A � abbr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' = f Σ(iXΣ, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicit formulae for the finitary once-extended Quinn TQFT for B = BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this subsection we give, explicitly, the various formulae obtained by taking the identification, up to natural isomorphism, of the various parts of the once-extended Quinn TQFT, in the case in which B = BA, in terms of crossed complexes and related groupoids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We give that description first, but, for ease of reference in the later examples, we will state this more formally in a theorem at the end of the description.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that to get such a description one has to suppose given simplicial stratifications of the manifolds, cobordisms and 2-cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will discuss the dependence on these choices in a later subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is analogous, of course, to taking triangulations, so as to get ‘lattice models’ and ‘state sum’ models, as we mentioned in Remark 257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The finitary once-extended Quinn TQFT, in Definition 154, 2QBA : 2Cob (n,n+1,n+2) BA → vProfGrpfin, can be specified / described, up to isomorphism, as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (i) If Σ is a closed n-manifold, and iXΣ : |XΣ| → Σ is a simplicial stratification of Σ, then we have a canonical isomorphism of groupoids, (86) 2Q 0 BA � Σ, fΣ � iXΣ, A) � ∼= π1 � CRS(Π(XΣ), A), CRS0(Π(XΣ), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (ii) Given an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, between the closed n- manifolds, Σ and Σ′, consider a simplicial stratification of the cobordism, (i, M, i′), derived from a co-span of simplicial sets, XΣ j �❘ ❘ ❘ ❘ ❘ ❘ X′ Σ′ , j′ �❦❦❦❦❦❦ YM together with a homeomorphism of cospans in CGWH, |XΣ| |j| �❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ ❚ iXΣ � |X′ Σ′| |j′| �❥❥❥❥❥❥❥❥❥❥❥ iX′ Σ′ � |YM| iYM � Σ i �❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Σ′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' i′ �✐✐✐✐✐✐✐✐✐✐✐✐✐ M A CATEGORIFICATION OF QUINN’S TQFT 208 The simplicial stratifications, iXΣ : |XΣ| → Σ, and iX′ Σ′ : |X′ Σ′| → Σ, of Σ and Σ′, respectively, yield BA-decorations, fΣ(iXΣ, A), of Σ, and fΣ′(iX′ Σ′ , A), of Σ′, giving the associated 1-morphism in 2Cob (n,n+1,n+2) BA , � Σ, fΣ(iXΣ, A) � (i,M,i′) −−−−−→ � Σ′, fΣ′(iX′ Σ′ , A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using Definitions 154 and 234, we have a natural isomorphism of profunc- tors, (87) 2Q 1 BA �� Σ, fΣ(iXΣ, A) � (i,M,i′) −−−−−→ � Σ′, fΣ′(iX′ Σ′ , A) �� ∼= Lin ◦ H(|YM|sk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='|XΣ|sk,|X′ Σ′|sk:A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Recall also that Lin: Set → Vect is the free vector space functor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') (iii) Finally,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' at the level of 2-morphisms,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' consider an (n,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' n + 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' n + 2)-extended cobordism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (88) K = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed Σ i � ι0 � M iN � Σ′ i′ � ι′ 0 � Σ × I iE � K Σ′ × I iW � Σ ι1 � j � M ′ iS � Σ′ j′ � ι′ 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and also a diagram,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (a ‘co-window’),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of finite simplicial sets,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' WK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' as below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (89) WK = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed XΣ i � ι0 � YM iN � X′ Σ′ i′ � ι′ 0 � XΣ × ∆(1) iE � ZK X′ Σ′ × ∆(1) iW � XΣ ι1 � j � Y ′ M′ iS � X′ Σ′ j′ � ι′ 1 � \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' together with a homeomorphism of diagrams,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (90) g : |WK| → K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here |WK| is obtained by applying geometric realisation to all components of WK, and, in order to simplify notation, all ‘components’ of g will be denoted g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider, also, the frame, fr(WK), of WK, a simplicial set, (defined exactly as was done in (50) for extended cobordisms), together with the filler, gWK : fr(WK) → ZK, which is a map of simplicial sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that g : |WK| → K gives simplicial stratifications for Σ and for Σ′, which extend to simplicial stratifications of Σ× I and Σ′ × I, and moreover to simplicial stratifications of the (n + 1)-cobordisms, (i, M, i′): Σ → Σ′ and (j, M ′, j′): Σ → Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The homeomorphism, |fr(WK)| → fr(K) ∼= ∂K, induced by g : |WK| → K, then also provides a simplicial stratification of ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The simplicial stratification of K that g gives then restricts to the latter simplicial stratification over ∂K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given two crossed complex maps, H : Π(YM) → A, and H′ : Π(Y ′ M′) → A, A CATEGORIFICATION OF QUINN’S TQFT 209 suppose that H and H′ agree on Π(XΣ) and on Π(X′ Σ′), that is to say that H and H′ coincide when composed with Π(i): Π(XΣ) → Π(YM) and with Π(j): Π(XΣ) → Π(Y ′ M′), and also coincide when composed with Π(i′): Π(X′ Σ′) → Π(YM) and with Π(j′): Π(X′ Σ′) → Π(Y ′ M′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let f : Π(XΣ) → A be equal to H◦Π(i) = H′◦Π(j), and f ′ : Π(X′ Σ′) → A be H ◦ Π(i′) = H′ ◦ Π(j′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define the following continuous maps, (where we recall that all com- ponents of g : |WK| → K are denoted g), H = ��φA YM (H) �� ◦ g−1 : M → BA, H′ = ���φA Y ′ M′ (H′) ��� ◦ g−1 : M ′ → BA, f = ��φA XΣ(f) �� ◦ g−1 : Σ → BA, f ′ = ���φA X′ Σ′ (f ′) ��� ◦ g−1 : Σ′ → BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let p: XΣ × ∆(1) → XΣ and p′ : X′ Σ′ × ∆(1) → X′ Σ′ be the simplicial projections, inducing crossed complex maps, Π(p): Π(XΣ×∆(1)) → Π(XΣ) and Π(p′): Π(X′ Σ′×∆(1)) → Π(X′ Σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, the crossed complex maps, H : Π(YM) → A, H′ : Π(Y ′ M′) → A, f ◦ Π(p): Π(XΣ × ∆(1)) → A, f ′ ◦ Π(p′): Π(X′ Σ′ × ∆(1)) → A, combine104 into one, that will be denoted [H, H′]: Π(fr(WK)) → A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We then have the following description of the matrix entries105, (91) � PCH � {f|BM|f ′} � | � 2Q2 B([K]) � (f,f ′) | PCH′� {f|BM′|f ′} �� = χπ � CRS � [H,H′] �� Π(WK), A �� χπ � PCH′ � CRS �� f f ′ ��� Π(Y ′ M′), A ��� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that to simplify expressions slightly we have, in the final formulae, written B for BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a finite crossed complex, and n a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The structures specified in (i), (ii), and (iii), above, give the finitary once-extended Quinn TQFT, in Definition 153, 2QBA : 2Cob (n,n+1,n+2) BA → vProfGrpfin, if we restrict to the objects of 2Cob (n,n+1,n+2) BA of the form � Σ, f Σ(iXΣ, A) � , where iXΣ : |XΣ| → Σ is a simplicial stratification of a closed smooth n-manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 104Here we are implicitly using the fact that the fundamental crossed complex functor preserves certain colimits, including the colimit defining fr(WK), for which see [27, 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='i Coproducts with amalgamation].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 105It may help to look back at the notation developed at the end of Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6, so, for in- stance, CRS � [H,H′] �� Π(ZK), A � denotes the fibre, as in Definition 184, over [H, H′]: Π(fr(WK)) → A, of the crossed complex map, CRS�Π(ZK), A� → CRS�Π(fr(WK)), A�, induced by the inclusion of fr(WK) into ZK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 210 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For the most part, the proof is essentially as in Theorem 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance Equation (86) follows from Lemma 220, and Equation (91) follows from Lemma 236.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Equation (87) follows from the fact that106 we have a weak homotopy equivalence, ηA YM : ��N(CRS(Π(YM), A)) �� → TOP(|YM|, BA), by (Brown–Higgins–Sivera–Tonks) Theorem 219.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 260.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we assume, furthermore, that A is reduced, then the crossed complexes appearing in (91) are homogeneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the discussion at the end of Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular (as in Theorem 256), we can obtain explicit formulae for their homotopy content, similar to (81) and (84), using Lemmas 250 and 251.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Dependence of the formulae on simplicial stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We need to address the dependence of the formulae in Theorem 259 on the choice of simplicial strat- ifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We freely use Remark 155, where the dependence on decorations was discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let Σ be a closed (and, as usual, smooth) n-manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we choose different simplicial stratifications, iXΣ : |XΣ| → Σ and jYΣ : |YΣ| → Σ, of Σ, then the corre- sponding BA-decorations of Σ, f Σ(iXΣ, A) and f Σ(jYΣ, A), as defined in Equation (85), will, in general, be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Nevertheless, we have a canonically defined invertible profunctor, 2Q 0 BA � Σ, f Σ(iXΣ, A) � ↛ 2Q 0 BA � Σ, fΣ � jYΣ, A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This profunctor is obtained from 2QBA : 2Cob (n,n+1,n+2) BA → vProfGrpfin, as 2Q 1 BA �� Σ, f Σ(iXΣ, A) � (ι0,Σ×I,ι1) −−−−−−−→ � Σ, fΣ(jYΣ, A) �� , where ι0(a) = (a, 0) and ι1(a) = (a, 1), for all a ∈ Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, these profunctors, which connect the groupoids associated to different simplicial stratifications of Σ, are functorial, so compose well with respect to further changes in the simplicial stratification, and are compatible with the profunctors associated to cobordisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A precise statement follows from Theorem 261, below, and is as in Remark 155.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, the formula, in Equation (87), for the profunctors associated to an (n + 1)-cobordism, (i, M, i′): Σ → Σ′, does not depend on the simplicial stratification, iYM : |YM| → M, of M, extending that of Σ and Σ′, as shown in Equation (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (The simplicial stratifications of Σ and Σ′ were part of the given data and so are themselves fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Likewise, in Equation (91), the formula for some of the matrix elements associ- ated to the natural transformation of profunctors provided by an (n + 2)-extended cobordism, K in (88), required the choice of a co-window of simplicial sets, as in (89), and a homeomorphism of diagrams, g : |WK| → K, as shown in (90).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The final result depended, however, only on the simplicial stratifications of Σ and of Σ′ that g : |WK| → K induces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is to say that the value in (91) depends neither on the simplicial stratifications of M and M ′, extending those of Σ and Σ′, nor on the simplicial stratification of the (n + 2)-manifold with border K, extending those of M, M ′, Σ × I and Σ′ × I, that g : |WK| → K gives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 106We expect to give more discussion for why the profunctors in (87) are equivalent in [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 211 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The bifunctor, 2QA : 2Cob (n,n+1,n+2) st → vProfGrpfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some of discussion concerning the (in)dependence of the formulae for the finitary once-extended Quinn TQFT, 2QBA : 2Cob (n,n+1,n+2) BA → vProfGrpfin, given in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, with respect to the simplicial stratifications, can be repackaged in a new version of the finitary once-extended Quinn TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, n is a non-negative integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We first define a variant, 2Cob (n,n+1,n+2) st , of the bicategory 2Cob(n,n+1,n+2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The objects of 2Cob (n,n+1,n+2) st are pairs, (Σ, iXΣ), where iXΣ : |XΣ| → Σ is a simplicial stratification of the (closed and smooth) n-manifold Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 1-morphisms, (Σ, iXΣ) → (Σ′, iX′ Σ′), are given by (n + 1)-cobordisms, (i, M, j): Σ → Σ′, without any chosen simplicial stratification on M (ex- tending the one that already exists on Σ and Σ′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The 2-morphisms, in 2Cob (n,n+1,n+2) st , � (i, M, j): (Σ, iXΣ) → (Σ′, iX′ Σ′) � =⇒ � (i′, M ′, j′): (Σ, iXΣ) → (Σ′, iX′ Σ′ ) � , are given by equivalence classes of extended cobordisms107, K: � (i, M, j): Σ → Σ′� =⇒ � (i′, M ′, j′): Σ → Σ′� , without any chosen simplicial stratification on K, in (88), extending the one that already exists on Σ and Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The rest of the bicategory structure for 2Cob (n,n+1,n+2) st is induced from that of the bicategory 2Cob(n,n+1,n+2), in the obvious way, as in the construction of the bicategory 2Cob (n,n+1,n+2) B , in Definition 153.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a finite crossed complex, A, we therefore have a bifunctor, VA : 2Cob (n,n+1,n+2) st −→ 2Cob (n,n+1,n+2) BA , which, on objects, is such that108 � Σ, iXΣ � VA �−→ � Σ, f(iXΣ, A) � , and on 1-morphisms, VA� (i, M, j): (Σ, iXΣ) → (Σ′, iX′ Σ′ ) � = (i, M, j): � Σ, f(iXΣ, A) � → � Σ′, f(iX′ Σ′ ,A) � , and analogously for 2-morphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The symmetric monoidal structure of 2Cob (n,n+1,n+2) BA , which is naturally de- rived from that of 2Cob(n,n+1,n+2), was briefly explained at the end of Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the tensor product of two BA-decorated n-manifolds is (Σ, fΣ) ⊗ (Σ′, gΣ′) = (Σ ⊔ Σ′, f Σ ⊗ gΣ′), where f Σ ⊗ gΣ′ := � ⟨φ, φ′⟩ | φ ∈ fΣ and φ′ ∈ gΣ′ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Here, given φ: Σ → BA and φ′ : Σ′ → BA, ⟨φ, φ′⟩: Σ ⊔ Σ′ → BA is defined from the universal property of disjoint unions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Analogously, we can define a symmetric monoidal structure in the bicategory 2Cob (n,n+1,n+2) st , where the tensor product of two closed, smooth, n-manifolds, Σ 107as for 2Cob(n,n+1,n+2) 108Given a simplicial stratification, iXΣ : |XΣ| → Σ, of Σ, the corresponding BA-decoration, f(iXΣ, A) ⊂ TOP(Σ, BA) is as defined by Equation (85).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 212 and Σ′, provided with simplicial stratifications, iX : |XΣ| → Σ and i′ X′ : |X′ Σ′| → Σ′, is given by (Σ, iX) ⊗ (Σ′, i′ X′) := � Σ ⊔ Σ′, (iX ⊔′ i′ X′): |XΣ ⊔ X′ Σ′| → Σ ⊔ Σ′� , where, explicitly, the homeomorphism iX ⊔′ i′ X′ is defined as the composite |XΣ ⊔ X′ Σ′| ∼ = −→ |XΣ| ⊔ |X′ Σ′| iX⊔i′ X′ −−−−−→ Σ ⊔ Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the fact that Π(XΣ ⊔ X′ Σ′) ∼= Π(XΣ) ⊔ Π(X′ Σ′), it can moreover be proved that VA is compatible with the symmetric monoidal structures of 2Cob (n,n+1,n+2) st and 2Cob (n,n+1,n+2) BA .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This discussion implies the following: Theorem 261 (The bifunctor 2QA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a (symmetric monoidal) bifunctor, denoted 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, which is defined as the following composite of bifunctors, 2Cob (n,n+1,n+2) st VA −−−→ 2Cob (n,n+1,n+2) BA 2QBA −−−−−→ vProfGrpfin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Note that 2QA is now decorated with a crossed complex A, rather than with its classifying space BA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is because the step VA depends on the crossed complex A, and not only on its classifying space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Morita valued once-extended TQFTs derived from finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicit formulae for the Morita valued version of the once-extended Quinn TQFT, 2Q Mor BA : 2Cob (n,n+1,n+2) BA → Mor, in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, can be derived from Theorem 259, by applying the general constructions from Subsection 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As above, A denotes a fixed finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing from groupoids, Γ, to their groupoid algebras, Lin(2)(Γ), as in §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, and with Σ a closed smooth n-manifold, iXΣ : |XΣ| → Σ being a simplicial stratification of Σ109, we have a canonical isomorphism of finite dimensional algebras, (92) 2Q Mor BA � Σ, fΣ(iXΣ, A) � ∼= Lin(2)� π1(CRS(Π(XΣ), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These finite dimensional algebras associated to a closed n-manifold, Σ, with a simplicial stratification, depend, explicitly, on the chosen simplicial stratification of Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This dependence is, however, in a quite ‘mild’ way, exactly as for the case of 2QBA outlined in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We follow quite closely the discussion of this point, both the ideas and approach, given in [30, 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 Morita equivalence].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If we choose two simplicial stratifications, iXΣ : |XΣ| → Σ and jYΣ : |YΣ| → Σ, of Σ, then there exists a canonically defined invertible bimodule, 2Q Mor BA � Σ, f Σ(iXΣ, A) � ↛ 2Q Mor BA � Σ, f Σ(jYΣ, A) � , connecting the algebras thus obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This bimodule is, itself, obtained by apply- ing 2Q Mor BA to the identity cobordism, Σ ι0 �❚ ❚ ❚ ❚ ❚ ❚ Σ ι1 �❥❥❥❥❥❥ Σ × I , 109so, in particular, XΣ is a finite simplicial set A CATEGORIFICATION OF QUINN’S TQFT 213 where the source and target are provided with the BA-decorations, fΣ(iXΣ, A) and f Σ(jYΣ, A), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By construction, these bimodules compose well if we make further changes to the simplicial stratification, or, for that matter, if we reverse the ‘direction’ of the cylinder exchanging ι0 and ι1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is then easily seen (the exposition is exactly as in Remark 155 and §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5) that these bimodules are invertible in the sense of the theory of Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, all algebras obtained by considering different simplicial stratifications of Σ will be Morita equivalent, and a canonically defined Morita equivalence ex- ists connecting each pair of these algebras.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover the bimodules associated to changes of simplicial stratifications are natural with respect to the bimodules as- sociated to cobordisms, (i, M, j): Σ → Σ′, where both Σ and Σ′ have a simplicial stratification, and hence a given BA-decoration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, we have, Theorem 262 (The bifunctor 2Q Mor A ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a (symmetric, monoidal) bifunc- tor, 2Q Mor A : 2Cob (n,n+1,n+2) st → Mor, obtained by the following composition of bifunctors, 2Cob (n,n+1,n+2) st VA −−−→ 2Cob (n,n+1,n+2) BA 2QBA −−−−−→ vProfGrpfin Lin(2) −−−−→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' “Absolute” once-extended TQFTs derived from finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before we take A to be a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As was noted earlier in an analogous context, the once-extended TQFTs, 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, and 2Q Mor A : 2Cob (n,n+1,n+2) st −→ Mor, do not give a value to a closed smooth n-manifold Σ, unless Σ is given a sim- plicial stratification, iΣ : |XΣ| → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In order to construct bifunctors whose do- main is 2Cob(n,n+1,n+2), and whose target, unlike that of the once-extended Quinn TQFT110, 2QBA : 2Cob(n,n+1,n+2) −→ vProfGrphf, only outputs finite groupoids and finite dimensional algebras, we must specify a symmetric monoidal bifunctor, from 2Cob(n,n+1,n+2) to 2Cob (n,n+1,n+2) st .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If n = 0, the latter is quite easy to do, as 0-dimensional manifolds Σ essentially have only one simplicial stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For instance, we can observe that a 0- dimensional manifold Σ has a simplicial stratification given by the obvious bijection |XΣ| → Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here XΣ is a simplicial set whose set of 0-simplices is Σ, and all i- simplices with i > 0 are degenerate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, this gives a symmetric monoidal bifunctor 2Cob(0,1,2) → 2Cob(0,1,2) st .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For n ≥ 1, we need to use the axiom of choice, for classes, to construct a sym- metric monoidal bifunctor 2Cob(n,n+1,n+2) → 2Cob (n,n+1,n+2) st .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (As we mentioned at the beginning of Section 8, this step is non-canonical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') For instance, this can be done by picking a simplicial stratification, iΣ : |XΣ| → Σ, of each connected com- pact smooth manifold Σ, and then, if Σ′ is a not-necessarily connected manifold, the decomposition of Σ′ into path-components provides a simplicial stratification of Σ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 110of Definition 149 A CATEGORIFICATION OF QUINN’S TQFT 214 Theorem 263.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let A be a finite crossed complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have once-extended TQFTs, � 2QA : 2Cob(n,n+1,n+2) −→ vProfGrpfin, and � 2QMor A : 2Cob(n,n+1,n+2) −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These can be ‘normalised’ so that if Σ is any chosen path-connected (and, as usual, closed and smooth) n-manifold, and iΣ : |XΣ| → Σ is a simplicial stratification of Σ, then � 2QA(Σ) ∼= π1 � CRS(Π(XΣ), A), CRS0(Π(XΣ), A) � , and � 2QMor A (Σ) ∼= Lin(2)� π1 � CRS(Π(XΣ), A), CRS0(Π(XΣ), A) �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We compose the chosen bifunctor 2Cob(n,n+1,n+2) → 2Cob (n,n+1,n+2) st with either 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, or 2Q Mor A : 2Cob (n,n+1,n+2) st −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Under the conditions of the previous theorem, we note that we will always have that the state space of Quinn’s finite total homotopy TQFT, Q(s) BA : 2Cob(n,n+1,n+2) → VectC, on Σ, is canonically isomorphic to the free vector space on the set of components of the groupoid π1 � CRS(Π(XΣ), A), CRS0(Π(XΣ), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In other words, Q(s) BA(Σ) = C � π0 � CRS(Π(XΣ), A), CRS0(Π(XΣ), A) �� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see Theorem 256.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This makes it again clear in what sense the once-extended Quinn TQFT is a categorification of Quinn’s finite total homotopy TQFT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some explicit calculations for the once-extended TQFTs derived from finite groupoids and 2-groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the remainder of this paper, we will give some examples of the algebras that the once-extended TQFTs111, 2QA : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, and 2Q Mor A : 2Cob (n,n+1,n+2) st −→ Mor, assign to some n-dimensional manifolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These will be for low dimensions, n = 0, 1, 2, and when A is the crossed complex given by a finite group, a finite groupoid, or a crossed module of finite groups (equivalently a finite 2-group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebras we assign to loops and surfaces are particular cases of ‘tube algebras’ considered in [32], [30, Chapters 10 and 13] and [33, Section 3], in the context of models for excitations of topological phases, derived from discrete higher gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Understanding these algebras was one of the initial motivations for this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Note that this will also determine the state spaces of the associated Quinn finite total homotopy TQFTs, as observed in Remark 264.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 111in theorems 261, 262 and 263, above A CATEGORIFICATION OF QUINN’S TQFT 215 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The simplest example: the (0, 1, 2)-extended TQFT derived from a finite groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall from Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 that we can think of a groupoid as a 1- truncated crossed complex, leading to a functor, ι1 : Grp → Crs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We recall that this sends a groupoid, G = (s, t: G1 → G0), to the crossed complex, ι1(G) = · · · → � a∈G0 {ida} → · · · → � a∈G0 {ida} → G1 t ⇒ s G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, if G is a group, we then have ι1(G) = · · · → {1G} → · · · → {1G} → G → {∗}, and so the classifying space, Bι1(G), of the crossed complex, ι1(G), is the usual simplicially defined classifying space, BG, of the group G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a non-negative integer n, a complex number s ∈ C, and a finite group, G, we thus have the Quinn finite total homotopy TQFT, Q(s) BG : Cob(n,n+1) → VectC, that is obtained from the classifying space of G, and which coincides with the Dijkgraaf-Witten TQFT with trivial cocycle112;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us now turn to the extended case, and look at what happens when n = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Each 0-dimensional manifold is trivially diffeomorphic to the disjoint union of copies of the singleton manifold, S = {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On unpacking the construction in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we obtain that the groupoid, π1 � CRS � Π{∗}, ι1(G) � , CRS0 � Π{∗}, ι1(G) �� , is isomorphic to the groupoid G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This leads to the following result, essentially in [75, 76].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are, thus, considering the identification 2Cob(0,1,2) ∼= 2Cob (0,1,2) st , mentioned in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Theorem 265 ((0,1,2)-extended TQFTs derived from finite groupoids).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For G a finite groupoid, there are once-extended TQFTs, � 2Qι1(G) : 2Cob(0,1,2) → vProfGrpfin, and � 2QMor ι1(G) : 2Cob(0,1,2) → Mor, such that � 2Qι1(G) � {∗} � ∼= G, and � 2QMor ι1(G) � {∗} � ∼= Lin(2)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here Lin(2)(G) is the groupoid algebra of G, over Q, so is the usual rational group algebra of G, if G is a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The remaining parts of the specification of these (0, 1, 2)-extended TQFTs can be obtained from Theorem 259 and Example 235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 112Note that, if considered just in the oriented case, so all the manifolds are orientable, Quinn’s TQFT can naturally be twisted by cohomology classes, as we noted in Remark 92, using the same method as in [43, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In that case, one would recover Dijkgraaf-Witten TQFT in full generality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 216 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' One composes the already mentioned equivalence, 2Cob(0,1,2) ∼ = −→ 2Cob(0,1,2) st , with either 2Qι1(G) : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, or 2Q Mor ι1(G) : 2Cob (n,n+1,n+2) st −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Remark 266.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The existence of the (0,1,2)-extended TQFT, � 2QMor ι1(G) : 2Cob(0,1,2) → Mor, in the previous theorem, follows from [106, Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='52 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The key reason is that oriented (0, 1, 2)-extended TQFTs are given by separable symmet- ric Frobenius algebras, and groupoid algebras of finite groupoids can be given such a structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see [75, Examples 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1 and 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We give some details, following the notation and conventions of [106, Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='61 and Definition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='62 in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A symmetric Frobenius algebra, (A, λ, e), by convention here over Q, is given by an associative Q-algebra, A, with 1, together with: a Q-linear map λ: A → Q, satisfying that λ(ab) = λ(ba), for all a, b ∈ A, an element e ∈ A ⊗ A, so e = � i xi ⊗ yi, satisfying that given any w ∈ A, we have � i (wxi) ⊗ yi = � i xi ⊗ (yiw).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, λ and e should satisfy the following compatibility condition, � i λ(xi) ⊗ yi = 1A = � i xi ⊗ λ(yi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We also recall that an algebra A is called separable if there exists an element e ∈ A ⊗ A, written e = � i x′ i ⊗ y′ i, similarly satisfying that � i (wx′ i) ⊗ y′ i = � i x′ i ⊗ (y′ iw), for any w ∈ A, and moreover such that � i x′ iy′ i = 1A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A groupoid algebra, Lin(2)(G), of a finite groupoid, is a separable algebra, for in- stance via113 e = � ( x g−→ y ) 1 Nx (x g−→ y) ⊗ (y g−1 −−→ x) ∈ Lin(2)(G) ⊗ Lin(2)(G), where the sum is extended to all morphisms (x g−→ y) in G, and given an object x, in G, Nx is the number of morphisms in G with source x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a finite groupoid, G, the data that makes Lin(2)(G) a separable symmetric Frobenius algebra, [75], is as shown below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' λ: Lin(2)(G) → Q is defined by λ(x g−→ y) = � 1, if (x g−→ y) = (x idx −−→ x) 0, otherwise, whilst, as above, 113Note that an algebra being separable is a properly, and not a structure, unlike that of an algebra be given the structure of a Frobenius algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 217 e = � ( x g−→ y ) (x g−→ y) ⊗ (y g−1 −−→ x) ∈ Lin(2)(G) ⊗ Lin(2)(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By [106, Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='54], since groupoid algebras of finite groupoids are ∗-algebras, via the inversion of morphisms, and that ∗-structure is compatible with the Frobenius structure, Lin(2)(G) gives rise to a unoriented (0,1,2)-extended TQFT, therefore to a (0,1,2)-extended TQFT as we have defined in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Using the explicit formulae in Theorem 259, we can then see that given a fi- nite groupoid, G, the (0, 1, 2)-extended TQFT constructed, as in [106, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6], from the separable symmetric Frobenius algebra � Lin(2)(G), λ, e � coincides with � 2QMor ι1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The results in this paper, in particular, provide a homotopy-theoretical definition for these oriented once-extended TQFT derived from finite groupoids, in terms of homotopy orders of function spaces, but our construction, here, applies to consid- erably more general settings than that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Example 267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For a positive integer k, we let I(k) be the groupoid with set of objects {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' , k}, and a single morphism k → k′, given objects k and k′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that all these groupoids are homotopy equivalent to I(1) ∼= {∗}, the final groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A quick calculation shows that � 2QMor I(k) ({∗}) ∼= MQ(k), the algebra of k × k matrices with entries in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We note that this is Morita equiv- alence to MQ(1), that is to Q itself;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see also Example 157 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1, 2, 3)-extended TQFT derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now consider (1, 2, 3)- TQFTs derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The same analysis can be modified to handle finite groupoids, though the formulae become a bit more complicated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let us choose a particular simplicial stratification of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We consider S1 with a simplicial stratification, iS1 : |XS1| → S1, where the simplicial set, XS1, has a single 0-simplex and a single non-degenerate one-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Therefore, iS1 : |XS1| → S1 gives a CW-decomposition of S1, with a unique 0-cell and a unique 1-cell114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='. This naturally gives simplicial stratifications for arbitrary disjoint unions of S1, by using the obvious disjoint unions of this simplicial stratification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Looking at Equation 86, we can now determine the groupoid, π1 � CRS(Π(XS1), A), CRS0(Π(XS1), A) � ∼= π1 � CRS(Π(S1 sk), A), CRS0(Π(S1 sk), A) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (We are using the notation of Example 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') The following definition is well known and ‘classical’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 268 (Action groupoid).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let the group G have a left-action, •, on a non-empty set X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The action groupoid, X � G, or.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' in full, X �• G, has X as its set of objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given x, y ∈ X, the set of morphisms, from x to y, is given by the set of all pairs, (x, g), where g ∈ G is such that g • x = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of the morphisms in X � G is then such that � x (x,g) −−−→ g • x � composed with � g • x (g•x,h) −−−−−→ (hg) • x � 114We could consider CW-decompositions of S1 with more than one 0-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The algebras thereby obtained would then, in general, be different, however a natural Morita equivalence con- nects them all;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The case of S1 decomposed by using multiple 0-cells is reminiscent of the calculations in [74, II-d] and in [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Credit is due here to discussions with Alex Bullivant, including [30, Theorem 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2] A CATEGORIFICATION OF QUINN’S TQFT 218 is � x (x,hg) −−−−→ (hg) • x � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Remark 194, or directly by definition, we can see that crossed complex maps from Π(S1 sk) ∼= · · · → {0} → {0} → {0} → Z → {∗} to ι1(G) = · · · → {1G} → · · · → {1G} → G → {∗}, are in one-to-one correspondence with elements g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, by unpacking the construction in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we have an isomorphism of groupoids, π1 � CRS(Π(S1 sk), A), CRS0(Π(S1 sk), A) � ∼ = −→ G � G, where G � G is the action groupoid of the left-action of G on itself by conjugation, that we met back in Example 158.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given the above, for G a finite group and considering S1 with the simplicial stratification, iS1 : |XS1| → S1, with a single 0-simplex, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 269.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended TQFTs, 2Qι1(G) : 2Cob (1,2,3) st → vProfGrpfin, and 2Q Mor ι1(G) : 2Cob (1,2,3) st → Mor, are such that 2Qι1(G)(S1, iS1) ∼= G � G, and 2Q Mor ι1(G)(S1, iS1) ∼= Lin(2)(G � G), where the groupoid algebra is taken over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ By the previous theorem, it follows that 2Qι1(G) essentially gives the ω = 1 case of the (1, 2, 3)-extended TQFT constructed by Morton in [92], categorifying Dijkgraaf-Witten theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This (1,2,3)-extended TQFT 2Qι1(G) is also found in [100].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally note that we can apply Theorem 263 to the simplicial stratification iS1 : |XS1| → S1 of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives that we have (albeit non-canonical) once- extended TQFTs, � 2Qι1(G) : 2Cob(1,2,3) → vProfGrpfin, and � 2QMor ι1(G) : 2Cob(1,2,3) → Mor, such that � 2Qι1(G)(S1) ∼= G � G, and � 2QMor ι1(G)(S1) ∼= Lin(2)(G � G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we recalled in Example 158, the algebra Lin(2)(G � G) coincides with the quantum double of the group algebra of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' some extra discussion on this is found in [123] and also in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the argument leading to Theorem 269 gives another proof (and provides a homotopy theoretical underpinning for) the A CATEGORIFICATION OF QUINN’S TQFT 219 fact that, if G is a finite group, then there exists a Morita valued (1,2,3)-extended TQFT sending S1 to the quantum double of the group algebra of G, see [10, 92].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Towards (2, 3, 4)-extended TQFT derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will briefly discuss (2, 3, 4)-extended TQFTs derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is an infinite number of diffeomorphism classes of surfaces, therefore, the once-extended TQFTs, 2Qι1(G) : 2Cob (2,3,4) st → vProfGrpfin, and 2Q Mor ι1(G) : 2Cob (2,3,4) st → Mor, will a priori require115 an infinite set of data to be specified, even if the algebras and profunctors associated to surfaces are specified only up to a natural Morita equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For this paper, we will focus only on some examples of the algebras assigned to S2 and T 2 = S1 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We could consider all other surfaces (orientable and non-orientable), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' by choosing the usual CW-decompositions with unique 0 and 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We choose a simplicial stratification, iS2 : |XS2| → S2, where XS2 has a single 0-simplex and a single non-degenerate 2-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let S2 sk be the induced CW- decomposition of S2, already considered in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In this case, all crossed complex morphisms from Π(S2 sk) ∼= · · · → {0} → · · · → {0} → Z → {1} → {∗} to ι1(G) = · · · → {1G} → · · · → {1G} → {1G} → G → {∗}, are trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' However, the possible crossed complex homotopies are in bijection with G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is then easy to see that π1 � CRS � Π(S2 sk), ι1(G) � , CRS0 � Π(S2 sk), ι1(G) �� ∼= G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We now determine the groupoid associated to the 2-torus T 2 = S1 × S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There is a simplicial stratification of the 2-disk D2, with two non-degenerate 2-simplices, meeting along a diagonal edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Identifying boundary edges in the classical way, this gives a simplicial stratification of the 2-torus, T 2, here denoted iT 2 : |XT 2| → T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We let T 2 sk be the induced CW-decomposition of T 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A quick calculation reveals that the groupoid, CRS1 � Π(T 2 sk), ι1(G)), of crossed complex maps from Π(T 2 sk) to ι1(G), together with the homotopies between them, is isomorphic to the action groupoid, X � G, where X = � (a, b) ∈ G × G | ab = ba � , with the left-action of G given by g •(a, b) := (gag−1, gbg−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' All 2-fold homotopies of crossed complex maps from Π(T 2 sk) to ι1(G), are trivial, so we also have that π1 � CRS � Π(T 2 sk), ι1(G) � , CRS0 � Π(T 2 sk), ι1(G) �� ∼= X � G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From the construction in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, particularly §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4, we then obtain: 115This issue will likely disappear if one further categorification level is introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 220 Theorem 270 ((2,3,4)-TQFTs derived from finite groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended TQFTs,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2Qι1(G) : 2Cob (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4) st → vProfGrpfin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and 2Q Mor ι1(G) : 2Cob (2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4) st → Mor,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' are such that (where the simplicial stratifications,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iS2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of the 2-sphere and,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iT 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of the 2-torus are as described above),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 2Qι1(G)(S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iS2) ∼= G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and 2Q Mor ι1(G)(S2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iS2) ∼= Q(G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' where Q(G) = Lin(2)(G) is the group algebra of the group G over Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' whilst 2Qι1(G)(T 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iT 2) ∼= X � G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' and 2Q Mor ι1(G)(T 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' iT 2) ∼= Lin(2)(X � G),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the groupoid algebra over Q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' of the action groupoid X � G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ By applying Theorem 263, it therefore follows that: Theorem 271.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G be a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There are once-extended TQFTs, � 2Qι1(G) : 2Cob(2,3,4) → vProfGrpfin, and � 2QMor ι1(G) : 2Cob(2,3,4) → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These can be normalised so that � 2Qι1(G)(S2) ∼= G, and � 2QMor ι1(G)(S2) ∼= Q(G), whilst � 2Qι1(G)(T 2) ∼= X � G, and � 2QMor ι1(G)(T 2) ∼= Lin(2)(X � G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed modules of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Crossed modules of groupoids (Definition 180) are well known to model all homotopy 2-types;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [85] or [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' They can be considered as 2-truncated crossed complexes, and so can be used as the algebraic ‘base’ of some once-extended TQFTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' For convenience, we give the definition explicitly, but will restrict to the reduced case, hence for crossed modules of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Such algebraic structure will, thus, model connected homotopy 2-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 272 (Crossed module of groups).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed module, G = (∂ : E → G, ⊳), of groups is given by: A CATEGORIFICATION OF QUINN’S TQFT 221 a group homomorphism ∂ : E → G, together with a right-action, ⊳, of G on E by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This action is such that: (1) ∂(a ⊳ g) = g−1 ∂(a) g, for all a ∈ E, g ∈ G (which is called the 1st Peiffer relation), (2) a⊳ ∂(e) = e−1 a e, for all a, e ∈ E (which is called the 2nd Peiffer relation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The definition of crossed modules of groups is classical, going back at least to Whitehead’s original papers on CW-complexes and crossed complexes, [121, 122].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recent treatments are in [20, 27, 7, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As is well know, the category of crossed modules is equivalent to the category of 2-groups;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' [27, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5] and [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crossed module, G = (∂ : E → G, ⊳), of groups gives rise to a reduced crossed complex, ι2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Explicitly ι2(G) has the form, (93) ι2(G) = · · · → {1} → · · · → {1} → E ∂−→ G → {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The classifying space, BG, of a crossed module, G, is, by definition, the same as the classifying space of the crossed complex ι2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As usual, we may sometimes use the same symbol for a crossed module and the corresponding crossed complex, thus suppressing the notation for the inclusion functor, ι2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If the crossed module, G, is finite, meaning that both E and G are finite, then BG will be homotopy finite, and for n, a non-negative integer, we have an (n, n+1)- TQFT, as in Definition 83, where s ∈ C is an arbitrary parameter, Q(s) BG : Cob(n,n+1) → VectC, an (n, n + 1, n + 2)-extended TQFT, as in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4 2Qι2(G) : 2Cob (n,n+1,n+2) st −→ vProfGrpfin, and the corresponding Morita valued form of the latter, as in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5, 2Q Mor ι2(G) : 2Cob (n,n+1,n+2) st −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Finally, we have, non-canonical, once-extended TQFTs, as in Theorem 263, � 2Qι2(G) : 2Cob(n,n+1,n+2) −→ vProfGrpfin, and � 2QMor ι2(G) : 2Cob(n,n+1,n+2) −→ Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The closed-manifold case of Quinn’s finite total homotopy TQFT, Q(s) BG, where G is a finite crossed module, was addressed in [53], in the context of homologically twisted Yetter TQFT, and it coincides with the Yetter homotopy 2-type TQFT of [124].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Formulae for the entire TQFT, Q(s) BG, can be obtained as a particular case of the formulae in Subsection 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, which work more generally for homotopy finite crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A more recent paper, [108], addresses closely related homotopy quantum field theories derived from finite crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In the remainder of this paper, we will give some explicit formulae for the once- extended TQFTs derived from finite crossed modules, in the cases n = 0, n = 1, and n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our description of those once-extended TQFTs will be parallel to the discussion in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2 and §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, of (0, 1, 2), (1, 2, 3)- and (2, 3, 4)-extended TQFTs derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A crucial difference, however, is that, when moving to the crossed module case, 2-fold homotopies may become non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 222 Remark 273.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G be a crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In discrete higher gauge theory, as treated in [31, 93], given an (in this case closed and smooth) n-manifold Σ, which with a CW-decomposition will give the skeletally filtered manifold Σsk, we have, a 2-groupoid, 2Gauge(Σsk, G), of 2-gauge G-configurations, supported on Σsk, gauge transformations, connecting 2-gauge G-configurations, and 2-gauge transformations between gauge transformations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' again see [31, 93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This 2-groupoid was addressed from the point of view of differential-geometric higher gauge theory in [3, 6, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Passing to the more general language of crossed complexes, 2Gauge(Σsk, G) is the 2-groupoid associated to the underlying crossed module of the crossed complex, CRS � Π(Σsk), ι2(G) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular the groupoid π1 � CRS � Π(Σsk), ι2(G) � , CRS0 � Π(Σsk), ι2(G) �� , which the once-extended Quinn TQFT � 2Qι2(G) associates to Σ, with that particu- lar CW-decomposition, can be interpreted as the groupoid of 2-gauge G-connections supported in Σsk, and gauge transformations, considered up to 2-gauge transforma- tions, between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Remark 274.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is very likely that there exists an (n, n + 1, n + 2, n + 3)-extended TQFT hence sending an n-manifold Σ, with a CW-decomposition, to the 2-groupoid 2Gauge(Σsk, G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We expect to address this in a future publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some of the explicit calculations of the groupoids that the once-extended TQFTs derived from crossed modules assign to the circle, the 2-sphere and the 2-torus can also be found in [93], in the language of double groupoids, of 2-connections on a manifold, and in [32, 33], in the language of tube algebras, whose irreducible representations model excitations of higher gauge theory models for topological phases of matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (0,1,2)-extended TQFTs derived from finite crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If G = (∂ : E → G, ⊳) is a crossed module of groups, then ∂(E) is a normal subgroup of G, as is clear from the first Peiffer relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, from the facts that Π({∗}) ∼= · · · → {0} → {0} → {0} → {1} → {∗}, and ι2(G) = · · · → {1} → · · · → {1} → E ∂−→ G → {∗}, it is easy to see the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If G = (∂ : E → G, ⊳) is a crossed module of groups, then the groupoid, CRS1 � Π({∗}), ι2(G) � , of crossed complex maps from Π({∗}) to ι2(G), and homotopies between them, is isomorphic to G, so, in particular, that groupoid has only one object, and the groupoid, π1 � CRS � Π({∗}), ι2(G) � , CRS0 � Π({∗}), ι2(G) �� , of crossed com- plex maps from Π({∗}) to ι2(G), and 2-fold homotopy classes of homotopies between them, is isomorphic to the quotient group G/∂(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ As a consequence, in this context, we have the following result116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 276 ((0,1,2)-TQFTs derived from finite crossed modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G = (∂ : E → G, ⊳) be a finite crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once-extended TQFTs, � 2Qι2(G) : 2Cob(0,1,2) → vProfGrpfin, 116As in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1, we are considering the identification 2Cob(0,1,2) ∼ = 2Cob (0,1,2) st , of §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') A CATEGORIFICATION OF QUINN’S TQFT 223 and � 2QMor ι2(G) : 2Cob(0,1,2) → Mor, are such that � 2Qι2(G)({∗}) ∼= G/∂(E), and � 2QMor ι2(G)({∗}) ∼= Q � G/∂(E) � , the group algebra of the group, G/∂(E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The remaining parts of the specification of � 2Qι2(G) and � 2QMor ι2(G) can be obtained from Theorem 259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, following on from Remark 266, restricting to the oriented case, � 2QMor ι2(G) is obtained from the following symmetric Frobenius algebra structure on the group algebra Q � G/∂(E) � of G/∂(E), a separable algebra, λ: Lin(2)(G) → Q is defined by λ([g]) = | ker(∂)| δ � [g], 1G/∂(E) � , e = 1 | ker(∂)| � [g]∈G/∂(E) [g] ⊗ [g]−1 ∈ Q � G/∂(E) � ⊗ Q � G/∂(E) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The details are left to the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the (0, 1, 2)-extended TQFTs � 2QMor ι2(G) and � 2QMor ι1(G/∂(E)) are equiv- alent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As we have just seen, the (0, 1, 2)-extended TQFTs derived from finite crossed modules are not more general than the (0, 1, 2)-extended TQFTs derived from finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It is an open problem whether (0,1,2)-extended TQFTs derived from more general homotopy finite spaces, B, can similarly always be reduced to the finite group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (1, 2, 3)-extended TQFTs derived from finite crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We give some explicit formulae for the (1,2,3)-extended TQFTs that can be derived from a finite crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These give rise to (1,2,3)-extended TQFTs therefore associated to finite 2-group higher gauge theory [6, 31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Some preliminaries are needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similar constructions are in [31, 34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The calcu- lations shown below are a particular case of those of [51] and [57], which address the more general case of 2-crossed modules, which are models for pointed homotopy 3-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' First recall that if a group, G, has a right-action on the group E, by automor- phisms, then we can form the semidirect product, G ⋉⊳ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Our convention for the semidirect product will be (h′, e′)(h, e) := � h′ h, e (e′ ⊳ h) � , where h, h′ ∈ G, and e, e′ ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This slightly non-standard convention for semidirect products arises from the con- struction, in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, of the groupoid, CRS1(A, B), of crossed complex maps from the crossed complex, A, to the crossed complex, B, and homotopies between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Fix a crossed module, G = (∂ : E → G, ⊳).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider S1, with the same CW- decomposition as in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, so with a single 0-cell and a single 1-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By applying Remark 194, we can see that crossed complex maps from Π(S1 sk) to ι2(G) are in one-to-one correspondence with elements of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This can also be derived from (93) A CATEGORIFICATION OF QUINN’S TQFT 224 and the fact that Π(S1 sk) ∼= · · · → {0} → {0} → {0} → Z → ∗ ι2(G) = · · · → {1} → · · · → {1} → E ∂−→ G → {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (94) To describe homotopies between these crossed complex maps, we can use Lemma 202, or a direct calculation, to see that, given a map f : Π(S1 sk) → ι2(G), homotopies with source f (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 1-fold f-homotopies) are in one-to-one correspondence with elements of G × E, seen as a set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' To describe how each such crossed complex f- homotopy modifies f : Π(S1 sk) → ι2(G), as explained just after Definition 195, we use the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is motivated by the construction in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, and the following diagram, representing a homotopy of maps from Π(S1 sk) to ι2(G);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' also [98, 99, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here g, h ∈ G and e ∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' h e g h g ∂(e) h−1 Lemma 277.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a left-action, •, of G⋉⊳ E on the underlying set of G, such that, given g ∈ G and (h, e) ∈ G ⋉⊳ E, we have (h, e) • g := h g ∂(e) h−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This is proved by direct calculation, using the first Peiffer identity, as follows: (h′, e′) • � (h, e) • g � = (h′, e′) • � h g ∂(e) h−1� = h′ h g ∂(e) h−1 ∂(e′) h′−1 = h′ h g ∂ � e (e′ ⊳ h) � h−1 h′−1 = � h′ h, e (e′ ⊳ h) � g = � (h′, e′)(h, e) � g, where (h′, e′), (h, e) ∈ G ⋉⊳ E and g ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ Unpacking the construction in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we can see that, for G = (∂ : E → G, ⊳) a crossed module of groups, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 278.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid, CRS1 � Π(S1 sk), ι2(G) � , of crossed complex maps from Π(S1 sk) to ι2(G), and homotopies between them, is isomorphic to the action groupoid, G � (G ⋉⊳ E), where we consider the action, •, of G ⋉⊳ E on G, just defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ In order to describe the groupoid, π1 � CRS � Π(S1 sk), ι2(G) � , CRS0 � Π(S1 sk), ι2(G) �� , of crossed complex maps from Π(S1 sk) to G, and 2-fold homotopy classes of homo- topies between them, we must quotient out the morphisms G � (G ⋉⊳ E) by 2-fold homotopies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using Lemma 202, or a direct calculation, again based on (94), given a crossed complex map, f : Π(S1 sk) → ι2(G), 2-fold f-homotopies are in one- to-one correspondence with elements of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The resulting 2-groupoid is described in detail in [93].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 225 To see what the quotient groupoid looks like, first note that if a ∈ E, (h, e) ∈ G ⋉⊳ E, and g ∈ G, we have (h, e) • g = � h ∂(a), (a−1 ⊳ g) e a � g, where we use the first Peiffer identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 279.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have a right-action ⊳′ of E on the underlying set of the group G × � G ⋊⊳ E � , such that: (g, h, e) ⊳′ a = � g, h ∂(a), (a−1 ⊳ g) e a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (Note that ⊳′ is not necessarily an action by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows because the action, ⊳, of G on E is by automorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ We therefore have an equivalence relation on the set of morphisms of the groupoid G � (G ⋉⊳ E), which preserves source and target maps, given by, for all a ∈ E, � g (g,h,e) −−−−→ (h, e) • g � ∼ � g (g,h,e)⊳′a −−−−−−→ (h, e) • g � = � g � g,h∂(a),(a−1⊳g) e a � −−−−−−−−−−−−−−→ (h, e) • h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 280.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composition of morphisms in the action groupoid G � (G ⋉⊳ E) descends to the quotient under the equivalence relation above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This follows from the general construction of the internal hom, CRS(−, −), in the category of crossed complexes, as in [27, §9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3], and [28] for the case of crossed modules, as was explained here in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A direct proof can be given as follows, showing the importance of the second Peiffer condition for the construction to work as stated117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a chain of composable morphisms in G � (G ⋉⊳ E), g (g,h,e) −−−−→ (h, e) • g � (h,e)•g,h′,e′� −−−−−−−−−→ � (h′, e′)(h, e) � g, and hence their composite is g � g,h′h,e (e′⊳h) � −−−−−−−−−−→ � (h′, e′)(h, e) � g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Given a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' b ∈ E,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the composite of the chain of morphisms in G � (G ⋉⊳ E),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' below,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' g (g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e)⊳′a −−−−−−→ (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e) • g � (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e)•g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='h′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='e′� ⊳′b −−−−−−−−−−−→ � (h′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e′)(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e) � g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' or more explicitly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' g � g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='h∂(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='(a−1⊳g) e a � −−−−−−−−−−−−−−→ h g ∂(e) h−1 � h g ∂(e) h−1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='h′ ∂(b),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' � b−1⊳(hg∂(e)h−1) � e′ b � −−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ � (h′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e′)(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e) � g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' is g � g,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='h′∂(b) h∂(a),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='(a−1⊳g) e a �� b−1⊳(hg∂(e)h−1) � e′ b � ⊳(h∂(a)) � −−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−−→ � (h′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e′)(h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' e) � g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using the first and second Peiffer identities, and the fact that the action, ⊳, of G on E is by automorphisms, this latter morphism in G � (G ⋉⊳ E) simplifies to � g, h′ h ∂ � (b ⊳ h) a � ,(a−1 ⊳ g) e �� b−1 ⊳ (hg∂(e)h−1) � e′ b � ⊳ h a) � = � g, h′ h ∂ � (b ⊳ h) a � , (a−1 ⊳ g) e � b−1 ⊳ (hg∂(e)) � (e′ ⊳ h) (b ⊳ h) a � = � g, h′ h ∂ � (b ⊳ h) a � , (a−1 ⊳ g) � b−1 ⊳ (hg) � e (e′ ⊳ h) (b ⊳ h) a � = � g, h′ h, e e′ ⊳ h � ⊳′ � (b ⊳ h) a � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 117See [57] for how to deal with the case of 2-crossed modules, which are models for homo- topy 3-types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' There the second Peiffer condition is categorified and only holds up to a coherent isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 226 □ This leads to the following definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Definition 281 (The groupoid G �G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G = (∂ : E → G, ⊳) be a crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We define the groupoid, by abuse of notation denoted by G � G, such that the objects of G � G are elements of G;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the morphisms of G �G are equivalence classes of arrows of the form below, where g, h ∈ G and e ∈ E, g [(g,h,e)] −−−−−→ (h, e) • g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here (g, h, e) ∼ (g, h′, e′) if there exists a ∈ E, such that h′ = h ∂(a), and e′ = (a−1 ⊳ g) e a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The composite of the chain of morphisms, g [(g,h,e)] −−−−−→ (h, e) • g �� (h,e)•g,h′,e′�� −−−−−−−−−−−→ � (h′, e′)(h, e) � g, is g �� g,h′ h,e (e′⊳h) �� −−−−−−−−−−−−→ � (h′, e′)(h, e) � g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The following lemma follows by an explicit calculation, using the construction in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, of the internal hom, CRS(−, −), in the category of crossed complexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 282.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider a crossed module G = (∂ : E → G, ⊳), of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The groupoid, π1 � CRS � Π(S1 sk), ι2(G) � , CRS0 � Π(S1 sk), ι2(G) �� , whose objects are crossed complex maps, from Π(S1 sk) to ι2(G), and whose mor- phisms are 2-fold homotopy classes of homotopies between them, is isomorphic to G � G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ The proof of the following theorem follows from exactly the same form of dis- cussion as that in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, where we treated the group case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular the chosen simplicial stratification, of S1, has the form iS1 : |XS1| → S1, where XS1 has a single 0-simplex and a single non-degenerate one-simplex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This yields a simplicial stratification on all finite disjoint unions of S1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theorem 283.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G = (∂ : E → G, ⊳) be a finite crossed module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once- extended TQFTs, 2Qι2(G) : 2Cob (1,2,3) tr → vProfGrpfin and 2Q Mor ι2(G) : 2Cob (1,2,3) tr → Mor, are such that 2Qι2(G)(S1, iS1) ∼= G � G, and 2Q Mor ι2(G)(S1, iS1) ∼= Lin(2)(G � G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Here Lin(2) gives the groupoid algebra over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 227 As a consequence, applying Theorem 263, we have (albeit non-canonical) ex- tended TQFTs, � 2Qι2(G) : 2Cob(1,2,3) → vProfGrpfin, and � 2QMor ι2(G) : 2Cob(1,2,3) → Mor, that can be normalised such that their values on S1 are � 2Qι2(G)(S1) ∼= G � G and � 2QMor ι2(G)(S1) ∼= Lin(2)(G � G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' As before, recall that the rest of the structures of the once-extended TQFTs can be obtained from the discussion in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Towards (2, 3, 4)-extended TQFTs derived from finite crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We resume the notation from §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3 and that simpler case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We will briefly look at (2, 3, 4)-extended TQFTs derived from finite crossed mod- ules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Very similar constructions are in [32] and [33, Section 3], framed in the context of excitations of strict 2-group topological phases, and also in [34], framed in the context of invariants of loop braids derived from discrete higher gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We use the simplicial stratifications of the 2-sphere S2 and of the 2-torus T 2 appearing in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Recall that these yield CW-decompositions of S2, with unique 0- and 2-cells, and of T 2 with a 0-cell, three 1-cells and two 2-cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' If G = (∂ : E → G, ⊳) is a crossed module of groups, then the action, ⊳, of G on E restricts to an action of G on ker(∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By using the second Peiffer relation, the latter action descends to an action of G/∂(E) on ker(∂), where a ⊳ [g] := a ⊳ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Note that Π(S2 sk) ∼= · · · → {0} → {0} → {0} → Z → {1} → {∗}, ι2(G) = · · · → {1} → · · · → {1} → E ∂−→ G → {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (95) Therefore, an explicit calculation118 gives the following in which G = (∂ : E → G, ⊳) is, as before, a crossed module of groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lemma 284.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have an isomorphism of groupoids, CRS1 � Π(S2 sk), ι2(G) � ∼= ker(∂) � G, using the left-action of G on ker(∂), given by g • e := e ⊳ g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We have an isomorphism of groupoids, π1 � CRS � Π(S2 sk), ι2(G) � , CRS0 � Π(S2 sk), ι2(G) �� ∼= ker(∂) � (G/∂(E)), considering the left-action, of G/∂(E) on ker(∂), such that [g] • e := e ⊳ g−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' □ To examine the groupoid, π1 � CRS � Π(T 2 sk), ι2(G) � , CRS0 � Π(T 2 sk), ι2(G) �� , associated to the 2-torus, T 2, we start by unpacking the construction in Subsection 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' We can see that this groupoid is isomorphic to the groupoid, T 2(G), defined as follows: 118for conventions on action groupoids see Definition 268.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' A CATEGORIFICATION OF QUINN’S TQFT 228 the objects of T 2(G) are diagrams of the form below119 ∗ g � � h ∗� h ∗ g � v⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ �⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ e′ e ∗ g, h, v ∈ G e, e′ ∈ E, ∂(e) = g−1h−1v, ∂(e′) = v−1gh ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' the 1-morphisms of T 2(G) are equivalence classes of arrows of the form given below120, where g, h, v ∈ G, and e, e′ ∈ E, and, in addition, x ∈ G and a, b, c ∈ E, ∗ g � � h ∗� h ∗ g � v⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ �⑧ ⑧ ⑧ ⑧ ⑧ ⑧ ⑧ e′ e ∗ (x,a,b,c) −−−−−→ ∗ xg∂(a)x−1 � � xh∂(b)x−1 ∗� x h ∂(b) x−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' ∗ x g ∂(a) x−1 � x v ∂(c) x−1 t t t t t t t t t t t t t t �t t t t t t t t t t t t t t � c−1 e′ (a⊳h) b � ⊳x−1 � a−1 (b−1⊳g) e c � ⊳x−1 ∗ two arrows, given by (x, a, b, c) and (x′, a′, b′, c′) in T 2(G), where x, x′ ∈ G and a, a′, b, b′, c, c′ ∈ E, with the same source and target, as in the example above, are said to be equivalent if there exists a p ∈ E such that (x′, a′, b′, c′) = � x ∂(p), (p−1 ⊳ g) a p, (p−1 ⊳ h) b p, (p−1 ⊳ v) c p � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (In terms of the gauge interpretation mentioned in the footnotes, we are here identifying two gauge transformations between discrete 2-connections if they differ by a 2-gauge transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') Finally, the composition in the groupoid, T 2(G), is induced by the semi-direct prod- uct, G ⋉⊳ (E × E × E), with the product action of G, namely (a, b, c) ⊳ g = (a ⊳ g, b ⊳ g, c ⊳ g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' It follows from the general construction in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='2, that indeed T 2(G) is a groupoid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The direct calculations are as in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Similar calculations appear in [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Analogously to the discussion in §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, but now for a finite crossed module, G = (∂ : E → G, ⊳), of groups, we can get: Theorem 285 ((2,3,4)-TQFTs derived from finite crossed modules).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The once- extended TQFTs, 2Qι2(G) : 2Cob (2,3,4) st → vProfGrpfin, and 2Q Mor ι2(G) : 2Cob (2,3,4) st → Mor, 119For how to interpret these diagrams in terms of fake-flat 2-gauge configurations, see [31, §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5], 120For how to interpret these arrows in terms of gauge transformations between fake-flat 2- gauge configurations, see [31, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='1]), A CATEGORIFICATION OF QUINN’S TQFT 229 are such that 2Qι2(G)(S2, iS2) ∼= ker(∂) � � G/∂(E) � , and 2Qι2(G)(T 2, iT 2) ∼= T 2(G), and, therefore, it follows that 2Q Mor ι2(G)(S2, iS2) ∼= Lin(2) � ker(∂) � � G/∂(E) �� , whilst 2Q Mor ι2(G)(T 2, iT 2) ∼= Lin(2) � T 2(G) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' By applying Theorem 263, we, therefore, have non-canonical once-extended TQFTs, � 2Qι2(G) : 2Cob(2,3,4) → vProfGrpfin, and � 2QMor ι2(G) : 2Cob(2,3,4) → Mor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These can be normalised such that, for the 2-sphere S2, � 2Qι2(G)(S2) ∼= ker(∂) � (G/∂(E)), and therefore � 2QMor ι2(G)(S2) ∼= Lin(2) � ker(∂) � � G/∂(E) �� , and, on the 2-torus, T 2, � 2Qι2(G)(T 2) ∼= T 2(G), from which we get � 2QMor ι2(G)(T 2) ∼= Lin(2) � T 2(G) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Final note: why should we bother with crossed modules?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The TQFTs obtained from finite crossed complexes are strictly more general than the ones that can be obtained from groupoids, equivalently from disjoint unions of finite groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An easy example showing that this is so arises when n = 4, and Q(s) BG : Cob(n,n+1) → Vect, for G a crossed module, with classifying space BG = Bι2(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the CW-decomposition of S4 with a single 0-cell and a single 4-cell, and no other cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Consider the CW-decomposition of S2 with a single 0-cell and a single 2-cell, and the product CW-decomposition on S2 × S2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The latter CW- decomposition hence has a unique 0-cell, no 1-cells, two 2-cells, no 3-cell, and one 4-cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' These CW-decompositions, of S4 and of S2 × S2, are easily seen to arise from simplicial stratifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Moreover, we have Π(S4 sk) ∼= {0} → {0} → · · · → {0} → Z → {0} → {0} → {1} → {∗} and Π � (S2 × S2)sk � ∼= {0} → {0} → · · · → {0} → Z → {0} → Z ⊕ Z → {1} → {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Let G be a finite groupoid, and recall ι1(G) = · · · → � a∈G0 {ida} → · · · → � a∈G0 {ida} → G1 t ⇒ s G0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An explicit calculation gives that the crossed complexes, CRS � Π(S4), ι1(G) � and CRS � Π(S2 × S2), ι1(G) � , A CATEGORIFICATION OF QUINN’S TQFT 230 each are isomorphic to ι1(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the state spaces, Q(s) BG(S4) and Q(s) BG(S2 × S2), where BG = Bι1(G), each have dimension given by the cardinality of the set of components of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' On the other hand, if G = (∂ : E → G, ⊳) is a finite crossed module of groups, then ι2(G) = · · · → {1} → · · · → {1} → E ∂−→ G → {∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An explicit calculation gives that π1 � CRS � Π(S4 sk), ι2(G) � , CRS0 � Π(S4), ι2(G) �� ∼= G/∂(E), a groupoid with a single object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In particular, the dimension of Q(s) BG(S4) is always 1, independently of the crossed module G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Again, from an explicit calculation very similar to that of §8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='3, we can see that π1 � CRS � Π � (S2 × S2)sk � , ι2(G) � , CRS0 � Π � (S2 × S2)sk � , ι2(G) � ∼= � ker(∂) ⊕ ker(∂) � � � G/∂(E) � , where we consider the action of G/∂(E) on ker(∂) ⊕ ker(∂) such that [g] ⊲ (a, b) = (a⊳ g−1, b⊳ g−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' (This is well defined given the second Peiffer condition on crossed modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=') In particular, the dimension of Q(s) BG(S2 × S2) is given by the number of orbits of the right-action of G on ker(∂) × ker(∂).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' This gives, in general, a value different from dim � Q(s) BG(S4) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' References [1] Mathieu Anel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The elementary infinity-topos of truncated coherent spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' arXiv preprint, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' URL: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='org/abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='02082.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 43 [2] Dimitri Ara and Fran¸cois M´etayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' The Brown-Golasi´nski model structure on strict ∞- groupoids revisited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Homology Homotopy Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', 13(1):121–142, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 5, 186 [3] John Baez and Urs Schreiber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Higher gauge theory: 2-connections on 2-bundles, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='48550/arXiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='hep-th/0412325.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 11, 73, 201, 222 [4] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Baez and James Dolan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' From finite sets to Feynman diagrams.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' In Mathematics unlimited—2001 and beyond, pages 29–50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Berlin: Springer, 2001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3, 44, 45 [5] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Baez, Alexander E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Hoffnung, and Christopher D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Walker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Higher dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' VII: Groupoidification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Theory Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Categ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=', 24:489–553, 2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3, 4, 54, 55, 56, 57, 64 [6] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Baez and John Huerta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' An invitation to higher gauge theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Gen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Relativ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Gravi- tation, 43(9):2335–2392, 2011.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 3, 11, 73, 201, 222, 223 [7] John C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Baez and Aaron D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Lauda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Higher-dimensional algebra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' V: 2-Groups.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='org/abs/2107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='10526.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' 24, 29, 30, 31, 154 [69] Klaus Heiner Kamps and Timothy Porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Abstract Homotopy and Simple Homotopy The- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' World Scientific Publishing Co.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content=' Inc.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='uk Email address, Timothy Porter: t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='porter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='maths@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} +page_content='com (Faria Martins) School of Mathematics, University of Leeds, Leeds, LS2 9JT, United Kingdom (Porter) Ynys Mˆon / Anglesey, Cymru / Wales, ex-University of Bangor' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE0T4oBgHgl3EQflwHV/content/2301.02491v1.pdf'} diff --git a/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/2301.04738v1.pdf.txt b/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/2301.04738v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a5f983a1fb4d56b1b78627941ae08da47c45229f --- /dev/null +++ b/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/2301.04738v1.pdf.txt @@ -0,0 +1,381 @@ +arXiv:2301.04738v1 [math.DS] 11 Jan 2023 +Upper bounds for the Hausdorff dimension of Weierstrass curves. +Ted Alexander and Tommy Murphy +Abstract - We produce an upper bound for the Hausdorff dimension of the graph of a Weierstrass- +type function. Whilst strictly weaker than existing results, it has the advantage of being directly +computable from the theory of hyperbolic iterated function systems (IFS). +1 +Introduction +The concept of Hausdorff dimension is intimately bound up with the study of fractals; for +instance Mandelbrot’s well-known assertion that a fractal is characterized by the Hausdorff +dimension strictly exceeding the topological dimension. +In some special cases it is easy to +compute. The Koch curve can be defined using four translated contractions of itself. Since +the scaling factor is 1/3 and four non-warping contractions are used, the Hausdorff dimension +of the Koch Curve is exactly log3 4. In contrast, it is a remarkable fact that for arguably the +earliest known example of a fractal, namely Weierstrass’ monster, the precise computation of the +Hausdorff dimension was an open problem until quite recently. The essential difficulty is that +the contraction mappings defining the fractal have uneven warping, which complicates matters +significantly. +The Hausdorff dimension of the graph of the function +Wa,b(x) = +∞ +� +n=0 +an cos(2πbnx) +where x ∈ R, b ∈ N, and 1 +b < a < 1, was long conjectured to be D := 2 + logb a. This was +settled by Shen [12] in 2018. The classical examples of Weierstrass were of the form b ∈ N and +ab + 1 > 3π +2 . These became famous in the mathematical world as they were the first published +examples of functions which are everywhere continuous yet nowhere differentiable. +Throughout this paper b ∈ N and 1 +b < a < 1. Let φ be a C1 function defined on [0, 1], and +also denote by φ its Z-periodic extension to R. Set +wφ +a,b(x) = +∞ +� +n=0 +anφ(bnx). +If F ⊂ R2, let dimH[F] denote the Hausdorff dimension. +Given a function w : R → R, +graph(w) ⊂ R2 will denote the graph of the function. We can now state Shen’s Theorem: +Theorem 1.1 (Shen) There exists a K0 = K0(φ, b) > 1 such that if 1 < ab < K0, then +dimH[graph(wφ +a,b)] = D. +1 + +−2 +−1 +1 +2 +−2 +−1 +1 +2 +Figure 1: A classical Weierstrass curve +Clearly the classical examples of Weierstrass follow on setting φ(x) = cos(2πx) and choosing a +and b appropriately. Since D > 1, this in particular produces many examples of fractals. +Shen’s work is the culmination of many years of research beginning with the work of Bescovitch- +Ursell [5]. Of particular interest to us is the well-known estimate +dimH[graph(wφ +a,b)] ≤ D. +(1) +The argument to establish this is standard, but indirect; see Section 2 of [1] for details. One uses +the fact dimH[F] ≤ dimB[F], where dimB denotes the box-counting dimension and F ⊂ R2. +The box dimension of graph(w) is then estimated via studying local oscillations in terms of +H¨older continuity. Consequently, the main question in the field has been to understand lower +bounds for the Hausdorff dimension, and Shen’s theorem answers this question for a wide family +of examples. We also refer the interested reader to the related works [2], [8], [9], and [11]. +Our result is the following. +Theorem 1.2 If |φ′(x)| ≤ 1 and a2 + a +b < 1, then +dimH[graph(wφ +a,b)] ≤ logh(1/b), +where +h = +� +� +� +�1 +2 +� +2 +b2 + a2 + +� +4 +b4 + a4 +� +. +It is not hard to see that logh(1/b) > D; write D = logb(b2a) and change the base of the +logarithm on the right hand side. This means our bound is always worse than Equation (1). +We already know that this must be the case since Shen’s Theorem states that D actually is +the Hausdorff dimension for a wide class of examples. The merit of our result is that it avoids +estimating the Hausdorff dimension via the approach of estimating the box-counting dimension, +but rather uses the theory of iterated function systems (IFS). Our main technical achievement +is the observation that there is a global upper contraction bound on the IFS determined by φ +under our assumptions. +2 + +Many standard examples of IFS are given by linear transformations, written as 2×2 matrices +with constant coefficients (see [4] for many such examples). The techniques of our proof will +also apply in these instances. The case of Weierstrass curves was more interesting to us as the +coefficients of the matrix vary, so estimating the contraction factors is harder. We would expect +further examples of fractals could also be analyzed in this framework. +2 +Setup +Here we set up some basic notation and definitions. Throughout we work in the standard metric +space W = [0, 1] × R ⊂ R2. Standard texts explaining the basics of IFS are [3] and [6], following +the foundational work of Hutchinson [10]. Let {Si}b +i=1 be contraction mappings on W with +contraction factors {ui}b +i=1. The class of non-empty compact subsets of W equipped with the +associated Hausdorff metric then has associated contraction mappings, also denoted Si, with +the same contraction factors. Let F be the invariant set for {Si}, i.e. +F = +b� +i=1 +Si(F). +The basic idea underlying the theory of IFS [3] is that the existence and uniqueness of F is +granted by the Banach fixed-point Theorem. +Definition 2.1 Given a set F ⊂ R2 with δ-covers Ui, we define the Hausdorff s-content Hs(F) +to be Hs(F) = inf � +i |Ui|s, where the infimum is taken over all such possible δ-covers. The +Hausdorff dimension dimH(F) is defined to be the infimal positive s such that Hs(F) is finite. +Lemma 2.2 dimH[F] ≤ s, where �b +i=1 us +i = 1. +Proof. See Theorem 8.8/Exercise 8.5 of [6]. The open set condition required is satisfied taking +V to be a small open tubular neighbourhood of F\{x = 0, 1}. +□ +3 +Contraction mappings associated to Weierstrass curves IFS +Endow W ⊂ R2 with its usual metric space structure. It is standard [1] to rewrite the graph of +a Weierstrass curve as an IFS using the mappings +Si(x, y) = +�x + i − 1 +b +, ay + φ +�x + i − 1 +b +�� +1 ≤ i ≤ b. +(2) +There are various related definitions of an IFS in the literature. Our definition, following [3], +is sometimes referred to as a hyperbolic IFS: each Si is a contraction mapping. In [1] and [2], +Equation (2) defines a smooth nonlinear system with two negative Lyapunov exponents which +they also call an IFS. This is a little different to our definition because the mappings (2) are +not assumed to be contraction mappings. However, under our additional assumptions each Si is +a contraction mapping and so we can apply some standard techniques to bound the Hausdorff +dimension. +Lemma 3.1 Under the assumptions of Theorem 1.2, each Si is a contraction mapping. +3 + +Proof. Choose distinct points x1 = (x1, y1) and x2 = (x2, y2) in W. We need to show that +d(Si(x1), Si(x2)) < d(x1, x2). +(3) +The left-hand side is +d(Si(x1), Si(x2)) = +��∆x +b +�2 ++ +� +a∆y + φ +�x1 + i +b +� +− φ +�x2 + i +b +��2 +where ∆x = x1 − x2 and ∆y = y1 − y2. Since φ ∈ C1, applying the mean value theorem there +is a positive number c < 1 so that +����φ +�x1 + i − 1 +b +� +− φ +�x2 + i − 1 +b +� ���� = c +b|∆x|. +Plugging this in and expanding, Equation (3) beomes +� +1 + c2 +b2 +(∆x)2 + a2(∆y)2 + 2ac +b ∆x∆y < +� +(∆x)2 + (∆y)2. +(4) +Applying the AM-GM inequality, +���� +2ac +b ∆x∆y +���� ≤ ac +b +� +(∆x)2 + (∆y)2 +� +. +(5) +Squaring both sides of Equation (4), applying the triangle inequality and Equation (5), and +splitting the (∆x)2 and (∆y)2 terms, we see Equation (3) will follow if we show that +1 + c2 +b2 ++ ac +b < 1 +and +a2 + ac +b < 1. +(6) +Noting that the left-hand side of both of these inequalities is an increasing function of c, which +is the value of the derivative of φ at some point, and |φ′| ≤ 1, we see c ≤ 1 which leads to +2 +b2 + a +b < 1 and +a2 + a +b < 1. +(7) +The first equation always holds, since |a| < 1 and b ∈ N > 1. The second equation holds as that +is precisely the assumption on the coefficients in the statement of the main theorem. +□ +4 +Proof of Theorem 1.2 +Armed now with the knowledge that our the mappings Si are contraction mappings, the strategy +of our proof is to apply Lemma 2.2 to estimate the Hausdorff dimension. +Proof. +From Lemma 2.2, it is clear that we need to estimate the contraction factors ui. +Following the lines of the proof of Lemma 3.1, choose distinct points x1 = (x1, y1) and x2 = +(x2, y2) ∈ W. Then d2(Si(x1), Si(x2)) can be written in matrix form as +� +∆x +∆y +� � 1+c2 +b2 +ac +b +ac +b +a2 +� � ∆x +∆y +� +. +(8) +4 + +Now view v = (∆x, ∆y) as an element of R2: the question is how to extremize +√ +vT Av, +where T denotes the transpose and A is the positive definite symmetric matrix +� 1+c2 +b2 +ac +b +ac +b +a2 +� +. +(9) +The alert reader will note that this is not a matrix with constant coefficients, since c is deter- +mined, via the Mean-Value Theorem, by x1 and x2 and so ultimately depends upon x1 and x2. +Our proof proceeds by fixing c, so that Equation (9) is regarded as a fixed symmetric matrix +A. It is a standard fact that a positive definite symmetric matrix has positive real eigenvalues +and that +√ +vT Av ≤ +√ +λ∥v∥, where λ denotes the largest eigenvalue of A. We then vary obtain +an upper bound that is independent of c. For two distinct points x1, x2 ∈ W there will be a +corresponding c in the formula for d(Si(x1), Si(x2)) and thus a corresponding matrix of the form +(9). As our upper bound is independent of c we can thus estimate the contraction factor of Si. +A straightforward computation shows the eigenvalues of this matrix are is +λ± = 1 +2 +�1 + c2 +b2 ++ a2 +� +± 1 +2 +��1 + c2 +b2 ++ a2 +�2 +− 4a2 +b2 . +If there is only one eigenvalue, +� +1+c2 +b2 ++ a2�2 +− 4a2 +b2 += 0 which implies that ab = 1 ± +√ +−c2, +an immediate contradiction because ab is real. So, there cannot be one repeated eigenvalue +and hence there must be two distinct eigenvalues. For our purposes, we need only the larger +eigenvalue to establish the upper bound. Hence we focus on +λ = 1 +2 +�1 + c2 +b2 ++ a2 +� ++ 1 +2 +��1 + c2 +b2 ++ a2 +�2 +− 4a2 +b2 . +Note this is an increasing function of c. As |φ′| ≤ 1, we set c = 1 to obtain +λmax = 1 +2 +� +2 +b2 + a2 + +� +4 +b4 + a4 +� +. +This directly implies an upper bound for the contraction factor for each Si is +ui = +� +λmax := h +1 ≤ i ≤ b. +Hence, by Lemma 3.1 an upper bound on the Hausdorff dimension of the graph of w is given by +solving bhs = 1. Equivalently, +s = logh(1/b). +The result now follows. +□ +Acknowledgments +We thank the Department of Mathematics at Cal State Fullerton for encouraging undergraduate +research and for supporting T.A. with a summer research scholarship. T. M. thanks the math- +ematics department at UC Irvine for their hospitality whilst this work was written up. Both +authors thank K. Bara´nski and the anonymous referee for helpful comments. +5 + +References +[1] K. Bara´nski, Dimension of the graphs of the Weierstrass- type functions, Fractal geometry and stochastics +V, 77–91, Progr. Probab., 70, Birkh¨auser/Springer, Cham, 2015. +[2] K. Bara´nski, B. B´ar´any, and J. Romanowska, On the dimension of the graph of the classical Weierstrass +function, Adv. Math, 265 (2014), 32–59. +[3] M.F.Barnsley, Fractals Everywhere, Academic Press, 1993. +[4] M.F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, Proc. Roy. +Soc. London A 399, 243-275 (1985). +[5] A.S. Besicovitch and H. D. Ursell, Sets of fractional dimensions (V): On dimensional numbers of some +continuous curves, J. London Math. Soc. 1 (1937), No. 1, 18–25. +[6] K. Falconer, The geometry of fractal sets, Cambridge University Press, 2010. +[7] K. Falconer, Fractal geometry. Mathematical foundations and applications, Wiley & Sons, 2014. +[8] T.Y. Hu and K.S. Lau, Fractal Dimensions and Singularities of the Weierstrass type functions, Trans. Amer. +Math. Soc., 335 (1993), No. 2, 649–665. +[9] B. R. Hunt, The Hausdorff dimension of graphs of Weierstrass functions, Proc. Amer. Math. Soc 126 (1998), +No. 3, 791–800. +[10] J. Hutchinson, Fractals and self-similarity, Indiana Univ. J. Math. 30, 713–747 (1981). +[11] J. Thim, Continuous Nowhere Differentiable Functions, Masters Thesis, Lulea University of Technology, +(2003). +[12] W. Shen, Hausdorff dimension of the graphs of the classical Weierstrass functions, Math. Zeit, 289 (2018), +No. 1–2, 223–266 +Tommy Murphy +Department of Mathematics, +CSU Fullerton, +800 N. State College Blvd., +Fullerton CA 92831. +E-mail: tmurphy@fullerton.edu +http://www.fullerton.edu/math/faculty/tmurphy/ +Ted Alexander +Department of Mathematics, +CSU Fullerton, +800 N. State College Blvd., +Fullerton CA 92831. +E-mail: tedforpresident@gmail.com +6 + diff --git a/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/load_file.txt b/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8cc8a6a6010775778cc43392f2dde9ff5649e102 --- /dev/null +++ b/VtE3T4oBgHgl3EQf0gtr/content/tmp_files/load_file.txt @@ -0,0 +1,194 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf,len=193 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='04738v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='DS] 11 Jan 2023 Upper bounds for the Hausdorff dimension of Weierstrass curves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Ted Alexander and Tommy Murphy Abstract - We produce an upper bound for the Hausdorff dimension of the graph of a Weierstrass- type function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Whilst strictly weaker than existing results, it has the advantage of being directly computable from the theory of hyperbolic iterated function systems (IFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 1 Introduction The concept of Hausdorff dimension is intimately bound up with the study of fractals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' for instance Mandelbrot’s well-known assertion that a fractal is characterized by the Hausdorff dimension strictly exceeding the topological dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' In some special cases it is easy to compute.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The Koch curve can be defined using four translated contractions of itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Since the scaling factor is 1/3 and four non-warping contractions are used, the Hausdorff dimension of the Koch Curve is exactly log3 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' In contrast, it is a remarkable fact that for arguably the earliest known example of a fractal, namely Weierstrass’ monster, the precise computation of the Hausdorff dimension was an open problem until quite recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The essential difficulty is that the contraction mappings defining the fractal have uneven warping, which complicates matters significantly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The Hausdorff dimension of the graph of the function Wa,b(x) = ∞ � n=0 an cos(2πbnx) where x ∈ R, b ∈ N, and 1 b < a < 1, was long conjectured to be D := 2 + logb a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' This was settled by Shen [12] in 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The classical examples of Weierstrass were of the form b ∈ N and ab + 1 > 3π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' These became famous in the mathematical world as they were the first published examples of functions which are everywhere continuous yet nowhere differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Throughout this paper b ∈ N and 1 b < a < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Let φ be a C1 function defined on [0, 1], and also denote by φ its Z-periodic extension to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Set wφ a,b(x) = ∞ � n=0 anφ(bnx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' If F ⊂ R2, let dimH[F] denote the Hausdorff dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Given a function w : R → R, graph(w) ⊂ R2 will denote the graph of the function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We can now state Shen’s Theorem: Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='1 (Shen) There exists a K0 = K0(φ, b) > 1 such that if 1 < ab < K0, then dimH[graph(wφ a,b)] = D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 1 −2 −1 1 2 −2 −1 1 2 Figure 1: A classical Weierstrass curve Clearly the classical examples of Weierstrass follow on setting φ(x) = cos(2πx) and choosing a and b appropriately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Since D > 1, this in particular produces many examples of fractals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Shen’s work is the culmination of many years of research beginning with the work of Bescovitch- Ursell [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Of particular interest to us is the well-known estimate dimH[graph(wφ a,b)] ≤ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (1) The argument to establish this is standard, but indirect;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' see Section 2 of [1] for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' One uses the fact dimH[F] ≤ dimB[F], where dimB denotes the box-counting dimension and F ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The box dimension of graph(w) is then estimated via studying local oscillations in terms of H¨older continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Consequently, the main question in the field has been to understand lower bounds for the Hausdorff dimension, and Shen’s theorem answers this question for a wide family of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We also refer the interested reader to the related works [2], [8], [9], and [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Our result is the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2 If |φ′(x)| ≤ 1 and a2 + a b < 1, then dimH[graph(wφ a,b)] ≤ logh(1/b), where h = � � � �1 2 � 2 b2 + a2 + � 4 b4 + a4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' It is not hard to see that logh(1/b) > D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' write D = logb(b2a) and change the base of the logarithm on the right hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' This means our bound is always worse than Equation (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We already know that this must be the case since Shen’s Theorem states that D actually is the Hausdorff dimension for a wide class of examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The merit of our result is that it avoids estimating the Hausdorff dimension via the approach of estimating the box-counting dimension, but rather uses the theory of iterated function systems (IFS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Our main technical achievement is the observation that there is a global upper contraction bound on the IFS determined by φ under our assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 2 Many standard examples of IFS are given by linear transformations, written as 2×2 matrices with constant coefficients (see [4] for many such examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The techniques of our proof will also apply in these instances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The case of Weierstrass curves was more interesting to us as the coefficients of the matrix vary, so estimating the contraction factors is harder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We would expect further examples of fractals could also be analyzed in this framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 2 Setup Here we set up some basic notation and definitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Throughout we work in the standard metric space W = [0, 1] × R ⊂ R2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Standard texts explaining the basics of IFS are [3] and [6], following the foundational work of Hutchinson [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Let {Si}b i=1 be contraction mappings on W with contraction factors {ui}b i=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The class of non-empty compact subsets of W equipped with the associated Hausdorff metric then has associated contraction mappings, also denoted Si, with the same contraction factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Let F be the invariant set for {Si}, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' F = b� i=1 Si(F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The basic idea underlying the theory of IFS [3] is that the existence and uniqueness of F is granted by the Banach fixed-point Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='1 Given a set F ⊂ R2 with δ-covers Ui, we define the Hausdorff s-content Hs(F) to be Hs(F) = inf � i |Ui|s, where the infimum is taken over all such possible δ-covers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The Hausdorff dimension dimH(F) is defined to be the infimal positive s such that Hs(F) is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2 dimH[F] ≤ s, where �b i=1 us i = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' See Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='8/Exercise 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='5 of [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The open set condition required is satisfied taking V to be a small open tubular neighbourhood of F\\{x = 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' □ 3 Contraction mappings associated to Weierstrass curves IFS Endow W ⊂ R2 with its usual metric space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' It is standard [1] to rewrite the graph of a Weierstrass curve as an IFS using the mappings Si(x, y) = �x + i − 1 b , ay + φ �x + i − 1 b �� 1 ≤ i ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (2) There are various related definitions of an IFS in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Our definition, following [3], is sometimes referred to as a hyperbolic IFS: each Si is a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' In [1] and [2], Equation (2) defines a smooth nonlinear system with two negative Lyapunov exponents which they also call an IFS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' This is a little different to our definition because the mappings (2) are not assumed to be contraction mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' However, under our additional assumptions each Si is a contraction mapping and so we can apply some standard techniques to bound the Hausdorff dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='1 Under the assumptions of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2, each Si is a contraction mapping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 3 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Choose distinct points x1 = (x1, y1) and x2 = (x2, y2) in W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We need to show that d(Si(x1), Si(x2)) < d(x1, x2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (3) The left-hand side is d(Si(x1), Si(x2)) = ��∆x b �2 + � a∆y + φ �x1 + i b � − φ �x2 + i b ��2 where ∆x = x1 − x2 and ∆y = y1 − y2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Since φ ∈ C1, applying the mean value theorem there is a positive number c < 1 so that ����φ �x1 + i − 1 b � − φ �x2 + i − 1 b � ���� = c b|∆x|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Plugging this in and expanding, Equation (3) beomes � 1 + c2 b2 (∆x)2 + a2(∆y)2 + 2ac b ∆x∆y < � (∆x)2 + (∆y)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (4) Applying the AM-GM inequality, ���� 2ac b ∆x∆y ���� ≤ ac b � (∆x)2 + (∆y)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (5) Squaring both sides of Equation (4), applying the triangle inequality and Equation (5), and splitting the (∆x)2 and (∆y)2 terms, we see Equation (3) will follow if we show that 1 + c2 b2 + ac b < 1 and a2 + ac b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (6) Noting that the left-hand side of both of these inequalities is an increasing function of c, which is the value of the derivative of φ at some point, and |φ′| ≤ 1, we see c ≤ 1 which leads to 2 b2 + a b < 1 and a2 + a b < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (7) The first equation always holds, since |a| < 1 and b ∈ N > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The second equation holds as that is precisely the assumption on the coefficients in the statement of the main theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' □ 4 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2 Armed now with the knowledge that our the mappings Si are contraction mappings, the strategy of our proof is to apply Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2 to estimate the Hausdorff dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' From Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='2, it is clear that we need to estimate the contraction factors ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Following the lines of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='1, choose distinct points x1 = (x1, y1) and x2 = (x2, y2) ∈ W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Then d2(Si(x1), Si(x2)) can be written in matrix form as � ∆x ∆y � � 1+c2 b2 ac b ac b a2 � � ∆x ∆y � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (8) 4 Now view v = (∆x, ∆y) as an element of R2: the question is how to extremize √ vT Av, where T denotes the transpose and A is the positive definite symmetric matrix � 1+c2 b2 ac b ac b a2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' (9) The alert reader will note that this is not a matrix with constant coefficients, since c is deter- mined, via the Mean-Value Theorem, by x1 and x2 and so ultimately depends upon x1 and x2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Our proof proceeds by fixing c, so that Equation (9) is regarded as a fixed symmetric matrix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' It is a standard fact that a positive definite symmetric matrix has positive real eigenvalues and that √ vT Av ≤ √ λ∥v∥, where λ denotes the largest eigenvalue of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' We then vary obtain an upper bound that is independent of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' For two distinct points x1, x2 ∈ W there will be a corresponding c in the formula for d(Si(x1), Si(x2)) and thus a corresponding matrix of the form (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' As our upper bound is independent of c we can thus estimate the contraction factor of Si.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' A straightforward computation shows the eigenvalues of this matrix are is λ± = 1 2 �1 + c2 b2 + a2 � ± 1 2 ��1 + c2 b2 + a2 �2 − 4a2 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' If there is only one eigenvalue, � 1+c2 b2 + a2�2 − 4a2 b2 = 0 which implies that ab = 1 ± √ −c2, an immediate contradiction because ab is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' So, there cannot be one repeated eigenvalue and hence there must be two distinct eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' For our purposes, we need only the larger eigenvalue to establish the upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Hence we focus on λ = 1 2 �1 + c2 b2 + a2 � + 1 2 ��1 + c2 b2 + a2 �2 − 4a2 b2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Note this is an increasing function of c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' As |φ′| ≤ 1, we set c = 1 to obtain λmax = 1 2 � 2 b2 + a2 + � 4 b4 + a4 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' This directly implies an upper bound for the contraction factor for each Si is ui = � λmax := h 1 ≤ i ≤ b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Hence, by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='1 an upper bound on the Hausdorff dimension of the graph of w is given by solving bhs = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Equivalently, s = logh(1/b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' The result now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' □ Acknowledgments We thank the Department of Mathematics at Cal State Fullerton for encouraging undergraduate research and for supporting T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' with a summer research scholarship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' thanks the math- ematics department at UC Irvine for their hospitality whilst this work was written up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Both authors thank K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' Bara´nski and the anonymous referee for helpful comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} 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+page_content=' Zeit, 289 (2018), No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' 1–2, 223–266 Tommy Murphy Department of Mathematics, CSU Fullerton, 800 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' State College Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=', Fullerton CA 92831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' E-mail: tmurphy@fullerton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='edu http://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='fullerton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='edu/math/faculty/tmurphy/ Ted Alexander Department of Mathematics, CSU Fullerton, 800 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' State College Blvd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=', Fullerton CA 92831.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content=' E-mail: tedforpresident@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} +page_content='com 6' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/VtE3T4oBgHgl3EQf0gtr/content/2301.04738v1.pdf'} diff --git 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+Abstract. This paper concerns the physical behaviors of any solutions to the one +dimensional compressible Navier-Stokes equations for viscous and heat conductive +gases with constant viscosities and heat conductivity for fast decaying density at +far fields only. First, it is shown that the specific entropy becomes not uniformly +bounded immediately after the initial time, as long as the initial density decays to +vacuum at the far field at the rate not slower than O +� +1 +|x|ℓρ +� +with ℓρ > 2. Further- +more, for faster decaying initial density, i.e., ℓρ ≥ 4, a sharper result is discovered +that the absolute temperature becomes uniformly positive at each positive time, no +matter whether it is uniformly positive or not initially, and consequently the cor- +responding entropy behaves as O(− log(̺0(x))) at each positive time, independent +of the boundedness of the initial entropy. Such phenomena are in sharp contrast +to the case with slowly decaying initial density of the rate no faster than O( 1 +x2 ), +for which our previous works [34–36] show that the uniform boundedness of the +entropy can be propagated for all positive time and thus the temperature decays to +zero at the far field. These give a complete answer to the problem concerning the +propagation of uniform boundedness of the entropy for the heat conductive ideal +gases and, in particular, show that the algebraic decay rate 2 of the initial density +at the far field is sharp for the uniform boundedness of the entropy. The tools to +prove our main results are based on some scaling transforms, including the Kelvin +transform, and a Hopf type lemma for a class of degenerate equations with possible +unbounded coefficients. +1. Introduction +The compressible Navier–Stokes equations for the ideal viscous and heat conductive +gases read as +∂tρ + div (ρu) = 0, +(1.1) +ρ(∂tu + (u · ∇)u) − µ∆u − (µ + λ)∇div u + ∇p = 0, +(1.2) +Date: January 2, 2023. +2010 Mathematics Subject Classification. 35A01, 35B45, 35Q86, 76D03, 76D09. +Key words and phrases. Uniform positivity of temperature; immediately unboundedness of en- +tropy; global classic solution; compressible Navier–Stokes equations; fast decay density; Kelvin +transform; Hopf type lemma. +1 + +2 +JINKAI LI AND ZHOUPING XIN +cvρ(∂tθ + u · ∇θ) + pdiv u − κ∆θ = Q(∇u), +(1.3) +where the unknowns ρ ≥ 0, u ∈ RN, with N the spatial dimension, θ ≥ 0, and +p = Rρθ, respectively, represent the density, velocity, temperature, and pressure. +Here, R and cv are positive constants, µ and λ are the viscous coefficients, both +assumed to be constants and satisfy the physical constraints µ > 0 and 2µ+Nλ > 0, +κ is the heat conductive coefficient, assumed to be a positive constant, and Q(∇u) +is a quadratic term of ∇u given as +Q(∇u) = µ +2|∇u + (∇u)T|2 + λ(div u)2. +By the Gibbs equation θDs = De + pD( 1 +ρ), where s is the specific entropy and +e = cvθ is the specific internal energy, it holds that p = Ae +s +cv ργ for some positive +constant A, where γ − 1 = R +cv . It is clear that γ > 1. In terms of ρ and θ, the specific +entropy s can be expressed as +s = cv +� +log R +A + log θ − (γ − 1) log ρ +� +, +(1.4) +satisfying +ρ(∂ts + u · ∇s) − κ +cv +∆s = κ(γ − 1)div +�∇ρ +ρ +� ++ 1 +θ +� +Q(∇u) + κ|∇θ|2 +θ +� +, +(1.5) +in the region where both ρ and θ are positive. +As the governing system in the gas dynamics, the compressible Navier–Stokes +equations have been studied extensively. One of the central concepts in the mathe- +matical theory for the compressible Navier–Stokes equations is the vacuum, which, +if occurs, means that the density vanishes at either some interior points or on the +boundary or at the far fields. Indeed, the possible presence of vacuum is one of the +main difficulties in the theory of global well-posedness of general solutions to the +compressible Navier–Stokes equations. Note that the equation (1.5) for the entropy +is highly degenerate and singular near the vacuum, it is even more difficult to analyze +the dynamic behavior of the entropy in the presence of vacuum. Due to this, most +of the mathematical theories developed in the existing literatures on the compress- +ible Navier–Stokes equations in the presence of vacuum are for system (1.1)–(1.3) +regardless of the entropy. +There are extensive literatures on the mathematical studies concerning the com- +pressible Navier–Stokes equations (1.1)–(1.3). In the one-dimensional case, the corre- +sponding theory is satisfactory and in particular the global well-posedness has been +known for long time. In the absence of vacuum, for which the information of the +entropy follows from that of the density and the temperature directly by (1.4), the +global well-posedness of strong solutions was established by Kazhikov–Shelukin [24] +and Kazhikov [25], which were later extended in the setting of weak solutions, see, +e.g., [2, 23, 58, 59]; large time behavior of solutions with general initial data was + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +3 +proved by Li–Liang [32]. In the presence of vacuum, but without considering the en- +tropy, the corresponding global well-posedness were established by the first author of +this paper in [29, 30], for both heat conductive and non-heat conductive ideal gases. +As shown by Hoff–Smoller [17], for the one-dimensional compressible Navier–Stokes +equations, no vacuum can be formed later in finite time from non-vacuum initial +data, while such a result remains open in the multidimensional case. +In the multi-dimensional case, the mathematical theory for the compressible Navier– +Stokes equations is less complete than that in the one-dimensional case. The break- +through for the global existence of finite energy weak solutions with general initial +data and possible vacuum, to the isentropic compressible Navier–Stokes equations, +was achieved by Lions [37, 38]. The results of Lions [37, 38] were later improved +by Feireisl–Novotn´y–Petzeltov´a [12], Jiang–Zhang [22], and more recently Bresch– +Jabin [1]. For the full compressible Navier–Stokes equations, the global existence of +variational weak solutions was proved by Feireisl [14], under some assumptions on +the equations of states. The uniqueness of weak solutions is still a challenging open +problem. If the initial datum is suitably regular, then the compressible Navier–Stokes +equations admit a unique local strong or classic solution, see [21, 39, 46, 48, 50, 51, 53] +for the case in the absence of vacuum, and [5–7, 15, 18, 31, 49] for the case in the pres- +ence of vacuum. However, the corresponding global existence with general initial data +may not be expected, due to the recent finite time blow up results by Merle–Rapha’el– +Rodnianski–Szeftel [44, 45], where for the three-dimensional isentropic compressible +Navier–Stokes equations with spherical symmetry, regular solutions with finite time +singularities are constructed for a class of initial data with far field vacuum. Indeed, +up to now, global strong or classical solutions are established only under some ad- +ditional conditions on the initial data: the case with small perturbed initial data +around non-vacuum equilibriums was achieved by Matsumura–Nishida [40–43], and +later developed in many works, see, e.g., [3, 4, 8–11, 16, 26, 47, 52]; while the case with +initial data of small energy but allowing large oscillations and vacuum was proved by +Huang–Li–Xin [20] and Li–Xin [33] for the isentropic system, and later generalized +to the full system in [19, 28, 54]. +It is worth pointing out that there are some significant differences in the mathe- +matical theories for the compressible Navier–Stokes equations between the vacuum +and non-vacuum cases and new phenomena may occur depending on the locations +and states of vacuum. In the non-vacuum case, the solutions can be establish in both +the homogeneous and inhomogeneous spaces depending on the properties of the ini- +tial data, and the solution spaces guarantee the uniform boundedness of the entropy. +However, these may fail in general in the presence of vacuum. Indeed, in the case +that the density has compact support, the solution can be established in the homo- +geneous spaces, see, e.g., [5–7, 15, 18, 20, 31], but not in the inhomogeneous spaces, +see Li–Wang–Xin [27]. Further more, the blowup results of Xin [56] and Xin–Yan +[57] imply that the global solutions established in [19, 28, 54] must have unbounded +entropy, if initially there is an isolated mass group surrounded by the vacuum region. + +4 +JINKAI LI AND ZHOUPING XIN +However, it is somewhat surprising that if the initial density vanishes only at far fields +with a rate no more than O( 1 +|x|2), then, as for the non-vacuum case, the solutions can +be established in both the homogeneous and inhomogeneous spaces, and the entropy +can be uniformly bounded, see the recent works by the authors [34–36]. +It should be noted that since system (1.1)–(1.3) is already closed, one can indeed +establish self-contained mathematical theories for it, as already developed in the pre- +vious works mentioned above. However, since the second law of the thermodynamics +is not taken in to account, these theories are insufficient from the physical point of +view. Therefore, some new theories are needed to provide information for the entropy +in the presence of vacuum to meet the physical requirements. However, due to the +lack of the expression and high singularity and degeneracy of the governing equation +for the entropy near the vacuum region, in spite of its importance, the mathematical +analysis of the entropy for the viscous compressible fluids in the presence of vac- +uum was rarely carried out before. Mathematical studies towards this direction has +been initiated in our previous works [34, 35] and further developed in [36], where +the propagation of the uniform boundedness of the entropy and the inhomogeneous +Sobolev regularities was achieved for the compressible Navier–Stokes equations, with +or without heat conductivities, in the presence of vacuum at the far fields, under the +crucial condition that the initial density decays to vacuum at the rate no faster than +O( 1 +|x|2). +In this paper, we continue our studies on the dynamic behavior of the entropy in +the presence of vacuum. Different from the cases considered in [34–36], where the +density decays slowly to the vacuum at far fields, in the current paper, we investigate +the case with fast decaying density at the far fields. For simplicity, we study the +one-dimensional case in the current paper while leave the multi-dimensional case as +future works. It will be shown in this paper that, in sharp contrast to the cases +with slowly decaying density in [34–36], the uniform boundedness of the entropy can +not be propagated by the compressible Navier–Stokes equations for viscous and heat +conductive ideal gases with constant viscosities and heat conductivities, if the initial +density decays faster than the order O( +1 +|x|ℓρ ) at the far fields with ℓρ > 2. Since the +uniform boundedness of the entropy has already been established in [34–36] if the +decay rate is less than O( 1 +|x|2), our results in this paper reveal that the decay rate 2 of +the initial density at the far field is sharp for the uniform boundedness of the entropy. +Surprisingly, in case that the initial density decays faster than the order O( 1 +x4), some +sharper results can be achieved: the temperature is uniformly positive immediately +after the initial time, for any general nonnegative (not identically zero) initial tem- +perature, and, as a result, the entropy tends to infinity at the order O(− log(̺0(x))) +at any positive time. +Consider the Cauchy problem to the one-dimensional compressible Navier–Stokes +equations for viscous and heat conductive ideal gases +ρt + (ρu)x += +0, +(1.6) + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +5 +ρ(ut + uux) − µuxx + px += +0, +(1.7) +cvρ(θt + uθx) + pux − κθxx += +µ(ux)2, +(1.8) +where p = Rρθ, subject to the initial condition +(ρ, u, θ)|t=0 = (ρ0, u0, θ0). +(1.9) +The main results of this paper will be stated and proved in the Lagrangian co- +ordinates; however, since the velocity of the solutions obtained in this paper have +Lipschitz regularities in the spatial variable, the results can be transformed back to +those in the Eulerian coordinates. +Define the coordinate transform between the Lagrangian coordinate y and the +Eulerian coordinate x as x = η(y, t) satisfying +� +∂tη(y, t) = u(η(y, t), t), +η(y, 0) = y. +Set +̺(y, t) := ρ(η(y, t), t), +v(y, t) := u(η(y, t), t), +ϑ(y, t) := θ(η(y, t), t), +and +J := J(y, t) = ηy(y, t). +Then, it holds that +Jt = vy, +J|t=0 ≡ 1, +J̺ = ̺0, +with ̺0 := ρ0. +We still use s to denote the specific entropy in the Lagrangian +coordinates. Then, it follows from (1.4) that +s(y, t) = cv +� +log R +A + log ϑ(y, t) − (γ − 1) log ̺0(y) + (γ − 1) log J(y, t) +� +, +(1.10) +for any y ∈ R and t ∈ [0, ∞). +Then, in the Lagrangian coordinates, the system (1.6)–(1.8) becomes +Jt += +vy, +(1.11) +̺0vt − µ +�vy +J +� +y + πy += +0, +(1.12) +cv̺0ϑt + vyπ − κ +�ϑy +J +� +y += +µ|vy|2 +J , +(1.13) +where π = R ̺0 +J ϑ. The initial data can be taken as +(J, v, ϑ)|t=0 = (1, v0, ϑ0), +(1.14) +where v0 = u0 and ϑ0 = θ0. +The following conventions will be used throughout this paper. For 1 ≤ q ≤ ∞ and +positive integer m, Lq = Lq(R) and W 1,q = W m,q(R) denote the standard Lebesgue +and Sobolev spaces, respectively, and Hm = W m,2. For simplicity, Lq and Hm denote + +6 +JINKAI LI AND ZHOUPING XIN +also their N product spaces (Lq)N and (Hm)N, respectively. ∥u∥q is the Lq norm of u, +and ∥(f1, f2, · · · , fn)∥X is the sum �N +i=1 ∥fi∥X or the equivalent norm +��N +i=1 ∥fi∥2 +X +� 1 +2. +The main results of this paper are the following three theorems. The first one +yields the global existence of a solution to the Cauchy problem (1.11)–(1.13), subject +to (1.14). +Theorem 1.1. Let the initial density ̺0 be given such that 0 < ̺0 ∈ L1(R)∩W 2,∞(R) +and +|̺′ +0| + |̺′′ +0| ≤ K1̺0 +on R, +(H1) +for a positive constant K1. Assume that (v0, ϑ0) satisfies ϑ0 ≥ 0 on R and +(√̺0v0, √̺0v2 +0, v′ +0, v′′ +0, √̺0ϑ0, √̺0ϑ′ +0, √̺0ϑ′′ +0) ∈ L2(R), +G′ +0 +√̺0 +∈ L2(R), (1.15) +lim +y→−∞ +|v′ +0(y)| +� +̺0(y) ++ lim +y→+∞ +|v′ +0(y)| +� +̺0(y) +< +∞, +(1.16) +where G0 := µv′ +0 − R̺0ϑ0. +Then, there is a global solution (J, v, ϑ) to (1.11)–(1.13), subject to (1.14), satis- +fying inf(y,t)∈R×(0,T) J > 0, θ ≥ 0, and +Jy +√̺0 +, Jyy, Jt, Jyt ∈ L∞(0, T; L2(R)), +√̺0v, √̺0v2, vy, vyy +√̺0 +, √̺0vt ∈ L∞(0, T; L2(R)), +vyyy, vyt ∈ L2(0, T; L2(R)), +√̺0ϑ, √̺0ϑy, √̺0ϑyy, ̺ +3 +2 +0 ϑt ∈ L∞(0, T; L2(R)), +ϑy ∈ L2(0, T; H2(R)), +̺0ϑt, ̺0ϑyt ∈ L2(0, T; L2(R)), +Gt, +�Gy +̺0 +� +y +∈ L2(0, T; L2(R)), +for any positive time T, where G := µ vy +J − R ̺0 +J ϑ. +Remark 1.1. (i) Condition (H1) allows arbitrary algebraic and even exponential de- +cay rate of ̺0 at far fields. Indeed, one can check that functions of the forms +A +(1+y2)ℓ +and e−(1+y2)δ, with A, ℓ ∈ (0, ∞) and δ ∈ (0, 1 +2], satisfy (H1). Thus, Theorem 1.1 gen- +eralizes the global existence result in our previous work [35], where some assumptions +on slow decay at far fields on ̺0 are assumed. +(ii) Condition (1.16) is used only to construct suitable approximated initial data +for the corresponding initial boundary value problems (which are expected to converge +to the Cauchy problem), see Step 1 in the proof of Theorem 1.1. +The second theorem gives the immediate unboundedness of the specific entropy if +the algebraic decay rate of the initial density is greater than 2. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +7 +Theorem 1.2. Assume, in addition to the conditions in Theorem 1.1, that +(1 + |y|)ℓρ̺0(y) ≤ K2, +∀y ∈ R, +(H2) +for some positive constants ℓρ ∈ (2, ∞) and K2, and either ϑ0 is not identically zero +or v0 is not identically a constant. Let (J, v, ϑ) be a solution to system (1.11)–(1.13), +subject to (1.14), satisfying the properties stated in Theorem 1.1. Then, the specific +entropy s ̸∈ L∞(R × (0, T)), for any positive time T ∈ (0, ∞). +Remark 1.2. Theorem 1.2 reveals a completely different phenomenon from that in +[34–36], where the initial density decays no faster than O( 1 +y2) at far fields, so that the +entropy keeps uniformly bounded. While Theorem 1.2 shows that if the initial density +decays faster than O +� +1 +|y|ℓρ +� +, with ℓρ > 2, at far fields, then the entropy becomes not +uniformly bounded immediately after the initial time. Consequently, we have given a +complete answer to the problem concerning the propagation of uniform boundedness +of entropy for ideal gases in one dimension: the uniform boundedness of the entropy +for the ideal gases, in the presence of vacuum at the far fields only in one dimension, +can be propagated if and only if the algebraic decay rate of the initial density is not +greater than 2. In other words, the decay rate 2 of the initial density at the far fields +is sharp for the uniform boundedness of the entropy in one dimension. +The main ingredients of the proof of Theorem 1.2 are based on using some scaling +transform to transform the far field vacuum to an interior vacuum and applying a +Hopf type lemma for a class of linear degenerate elliptic equations with degeneracy +in the time variable and possible unbounded coefficients. The scaling transform for +the temperature to be used here is +f(y, t) := ϑ(y−β, t), +y ∈ (0, ∞), t ∈ [0, ∞), +for some suitably chosen β > 0. Similar transform can also be introduced for negative +y. Due to the continuity equation (1.6) and the assumption that the initial density +reaches vacuum only at the far fields, the density remains positive on any compact +interval for all positive time. Thus the equation (1.8) can be regarded a uniform +parabolic equation for θ on compact domains. Consequently, the temperature will +be positive on any finite interval for any positive time t by the strong maximum +principle, and thus f is positive for any positive y and t. By using the properties of ϑ +stated in Theorem 1.1, one can verify that 0 < f ∈ C2,1((0, ∞) × (0, ∞)). Assuming +by contradiction that the entropy is uniformly bounded, one can extend f by zero on +the positive time axis, such that 0 ≤ f ∈ C([0, ∞) × [0, ∞)) and reaches zero on the +positive time axis only. The temperature equation yields +a0ft − afyy + bfy + ˜cf ≥ 0, +in (0, ∞) × (0, ∞), +which motivates us to apply the Hopf type lema to f at the points on the positive +time axis. By choosing β suitably, one can verify that the coefficients a0 and ˜c are +uniformly bounded near the positive time axis; however, the coefficient b contains an + +8 +JINKAI LI AND ZHOUPING XIN +unbounded term involving 1 +y. Fortunately, such an unbounded term in b is of “right” +sign while the remaining term in b is uniformly bounded for suitably chosen β, so +that the Hopf type lemma still holds (see Lemma 4.2). Thus applying the Hopf type +lemma to f near the positive time axis leads to a quantitative asymptotic behavior +of the temperature at the far field. The contradiction comes from the fact that the +asymptotic behavior of the temperature derived from the Hopf type lemma is not +consistent with that derived from (H2) and the uniform boundedness of the entropy. +This inconsistency implies that the entropy can not be uniformly bounded and thus +Theorem 1.2 follows. +The third theorem gives the uniform positivity of the temperature and consequently +the asymptotic unboundedness of the entropy, which are sharper results than those +in Theorem 1.2, under the stronger assumption that the algebraic decay rate of the +initial density at the far field is greater than 4. +Theorem 1.3. Assume, in addition to the conditions in Theorem 1.1, that +(1 + |y|)4̺0(y) ≤ K3, +∀y ∈ R, +(H3) +for a positive constant K3, and either ϑ0 is not identically zero or v0 is not identically +a constant. +Let (J, v, ϑ) be a solution to system (1.11)–(1.13), subject to (1.14), +satisfying the properties stated in Theorem 1.1. +Then, the following statements hold: +(i) the temperature ϑ satisfies +inf +y∈R ϑ(y, t) > 0, +∀t ∈ (0, ∞); +(ii) the specific entropy s satisfies +R ≤ lim +|y|→∞ +s(y, t) +− log(̺0(y)) ≤ lim +|y|→∞ +s(y, t) +− log(̺0(y)) < ∞, +∀t ∈ (0, ∞). +In particular, s becomes unbounded immediately after the initial time, regardless of +whether it is uniformly bounded or not at the initial time. +Remark 1.3. It is an interesting question to show whether Theorem 1.3 still holds in +the case that the algebraic decay rate of ̺0 lies between 2 and 4. However, as already +shown in Theorem 1.2, in this case, though the uniform positivity of the temperature +is not clear, yet the specific entropy becomes not uniformly bounded in any positive +time. +Recall that the temperature is positive on any finite interval for any positive time +t. To obtain the positive lower bound for the temperature at any positive time, it +suffices to achieve this at far fields. To this end, similar as in the proof of Theorem +1.2, we apply some scaling technique to transform the far field vacuum to an interior +vacuum and take advantage of the Hopf type lemma. However, the scaling transform + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +9 +introduced before does not work here directly. Instead, we apply the Kelvin transform +to the temperature ϑ and denote by h the transformed temperature, that is, +h(y, t) = yϑ +�1 +y, t +� +, +∀y ̸= 0, t ∈ [0, ∞), +which satisfies a linear degenerate equation, with all coefficients being uniformly +bounded by the assumption (H3). By using the properties of ϑ stated in Theorem +1.1, one can verify that 0 ≤ h ∈ C2,1(Ω) ∩ C(Ω) and more importantly h(0, t) = 0, +where Ω = ((−∞, 0)∪(0, ∞))×(0, ∞). Note that different from the proof of Theorem +1.2, here the important property that h(0, t) = 0 holds without any condition on the +entropy. By the Hopf type lemma (Lemma 4.2) and applying the strong maximum +principle, we can derive that h behaves linearly near the origin at each positive time +and hence obtain the uniformly positive lower bound for the temperature near the far +fields. With the aid of the positive lower bound of the temperature, the asymptotic +unboundedness of the entropy follows from (1.10) as J has uniform positive lower +and upper bounds. +The rest of this paper is arranged as follows: in Section 2, we consider a carefully +designed initial-boundary value problem for the system (1.11)–(1.13) and establish a +series of a priori estimates on the solution independent of the length of the spatial +interval; in Section 3, we obtain the global existence of solutions to the Cauchy +problem and thus prove Theorem 1.1 by taking limit of the solutions obtained in +Section 2; Section 4 is devoted to the proof of Theorem 1.2; and finally, the proof of +Theorem 1.3 is given in Section 5. +Throughout this paper, C will denote a generic positive constant which may vary +from place to place. +2. Initial-boundary value problem and a priori estimates +Throughout this section, we consider the initial-boundary value problem to the +system (1.11)–(1.13), in (α, β) × (0, ∞), with −∞ < α < β < +∞, subject to the +initial-boundary conditions: +(J, v, ϑ)|t=0 = (1, v0, ϑ0), +(2.1) +(vy, ϑ)|y=α,β = (0, 0). +(2.2) +The following global well-posedness can be proved in the same way as in [30]. +Proposition 2.1. Let (̺0, v0, ϑ0) ∈ H2((α, β)) be given such that ̺0, ϑ0 ≥ 0 on (α, β) +and v′ +0(α) = v′ +0(β) = ϑ0(α) = ϑ0(β) = 0. Assume that +µv′′ +0 − R(̺0ϑ0)′ = √̺0g1, +κϑ′′ +0 + µ(v′ +0)2 − Rv′ +0̺0ϑ0 = √̺0g2, +for two functions g1, g2 ∈ L2((α, β)). + +10 +JINKAI LI AND ZHOUPING XIN +Then, there is a unique global solution (J, v, ϑ) to system (1.11)–(1.13), in (α, β)× +[0, ∞), subject to (2.1)–(2.2), satisfying inf(y,t)∈(α,β)×(0,T) J > 0, ϑ ≥ 0, and +J ∈ C([0, T]; H2((α, β))), +Jt ∈ L2(0, T; H2((α, β))), +v, ϑ ∈ C([0, T]; H2((α, β))) ∩ L2(0, T; H3((α, β))), +vt, ϑt ∈ L2(0, T; H1((α, β))), +for any T ∈ (0, ∞). +The rest of this section is devoted to deriving the a priori estimates, independent +of α and β, on the unique global solution (J, v, ϑ) stated in Proposition 2.1. Keeping +this in mind, in the rest of this section, we will always assume that (J, v, ϑ) is the +solution stated in Proposition 2.1. +Throughout this section, for simplicity of notations, the norms ∥ · ∥q and ∥ · ∥H1 +are the corresponding ones on the interval (α, β), that is, +∥ · ∥q := ∥ · ∥Lq((α,β)) +and +∥ · ∥H1 := ∥ · ∥H1((α,β)). +Denote +m0 := +� β +α +̺0dy, +E0 := +� β +α +̺0 +�v2 +0 +2 + cvϑ0 +� +dy. +Proposition 2.2. It holds that +� β +α +̺0 +�v2 +2 + cvϑ +� +dy ≤ E0. +Proof. Multiplying (1.12) with v, integrating over (α, β), and by the boundary con- +ditions, one gets by integration by parts that +1 +2 +d +dt +� β +α +̺0v2dy + µ +� β +α +|vy|2 +J +dy − +� β +α +vyπdy = 0. +(2.3) +Since ϑ ≥ 0 in (α, β) × (0, ∞), it is clear that ϑy(α, t) ≥ 0 and ϑy(β, t) ≤ 0, for any +t ∈ (0, ∞). As are result, integrating (1.13) over (α, β) and integration by parts yield +cv +d +dt +� β +α +̺0ϑdy + +� β +α +vyπdy ≤ µ +� β +α +|vy|2 +J dy. +(2.4) +Summing (2.3) with (2.4) and integrating with respect to t lead to the conclusion. +□ +Proposition 2.3. It holds that +e− 2 +µ +√2m0E0 ≤ J ≤ e +4 +µ +√2m0E0 +� +1 + R +µ +� t +0 +̺0ϑdτ +� +, +∀t ∈ (0, ∞). +Proof. Since vy|y=α = 0 and J|t=0 = 1, it follows from (1.11) that J|y=α = 1. Substi- +tuting (1.11) into (1.12) yields +̺0vt − µ(log J)yt + πy = 0, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +11 +from which, integrating over (0, t) and using J|t=0 = 1, one can get +̺0(v − v0) + +� t +0 +πyds = µ(log J)y. +Integrating this over (α, y) and noticing that J|y=α = 1 and π|y=α = R ̺0 +J ϑ|y=α = 0, +one gets +� y +α +̺0(v − v0)dz + +� t +0 +πds = µ log J, +which leads to +J = e +1 +µ( +� y +α ̺0(v−v0)dz+ +� t +0 πds). +(2.5) +It follows from Proposition 2.2 and the H¨older inequality that +� β +α +̺0(|v| + |v0|)dz +≤ +�� β +α +̺0dz +� 1 +2 ��� β +α +̺0v2dz +� 1 +2 ++ +�� β +α +̺0v2 +0dz +� 1 +2� +≤ +2 +� +2m0E0. +(2.6) +With the aid of (2.6) and since π ≥ 0, it follows from (2.5) that +J ≥ e− 1 +µ +� β +α ̺0(|v|+|v0|)dz ≥ e− 2 +µ +√2m0E0. +(2.7) +Rewrite (2.5) as Je− 1 +µ +� y +α ̺0(v−v0)dz = e +1 +µ +� t +0 πds. Thus +1 +µJπ exp +� +−1 +µ +� y +α +̺0(v − v0)dz +� += ∂t(e +1 +µ +� t +0 πds). +Hence, one gets by noticing Jπ = R̺0ϑ that +exp +�1 +µ +� t +0 +πds +� += 1 + R +µ +� t +0 +̺0ϑ exp +� +−1 +µ +� y +α +̺0(v − v0)dz +� +ds. +Substituting this into (2.5) and using (2.6) lead to +J += +e +1 +µ +� y +α ̺0(v−v0)dz +� +1 + R +µ +� t +0 +̺0ϑ exp +� +−1 +µ +� y +α +̺0(v − v0)dz +� +ds +� +≤ +e +4 +µ +√2m0E0 +� +1 + R +µ +� t +0 +̺0ϑds +� +. +Combining this with (2.7) yields the conclusion. +□ +In the rest of this section, we will always assumed that C is a general positive +constant depending only on R, cv, µ, κ, K1, T, and the upper bound of N0, but inde- +pendent of α and β with β − α ≥ 1, where +N0 := ∥̺0∥∞ +m0 +E0 + +���� +�√̺0v2 +0, v′ +0, v′′ +0, √̺0ϑ0, √̺0ϑ′ +0, √̺0ϑ′′ +0, G0, G′ +0 +√̺0 +����� +2 +. (2.8) + +12 +JINKAI LI AND ZHOUPING XIN +Proposition 2.4. It holds that +sup +0≤t≤T +∥(√̺0v2, √̺0ϑ)∥2 +2 + +� T +0 +� +∥√̺0ϑ∥2 +∞ + +���� +vvy +√ +J +���� +2 +2 ++ +���� +ϑy +√ +J +���� +2 +2 +� +dt ≤ C. +Proof. Set E = v2 +2 + cvϑ. Then, it follows from (1.12) and (1.13) that +̺0Et − κ +�ϑy +J +� +y += +� +µvvy +J − R̺0 +J ϑv +� +y . +Note that ϑy(α, t) ≥ 0 and ϑy(β, t) ≤ 0 due to the boundary condition ϑ|y=α,β = 0 +and the fact that ϑ ≥ 0 in (α, β) × (0, ∞). Multiplying the above equation with E +and integration by parts yield +1 +2 +d +dt∥√̺0E∥2 +2 + κcv +� β +α +|ϑy|2 +J +dy − κE ϑy +J +��� +β +y=α +≤ +− +� β +α +� +µvvy +J +− R̺0 +J ϑv +� +(vvy + cvϑy)dy − κ +� β +α +ϑy +J vvydy +≤ +κcv +2 +���� +ϑy +√ +J +���� +2 +2 ++ C +� β +α +1 +J +� +|vvy|2 + ̺2 +0v2ϑ2� +dy, +and thus, by the Cauchy inequality and that −κE ϑy +J +��� +β +y=α ≥ 0, it follows that +d +dt∥√̺0E∥2 +2 + κcv +���� +ϑy +√ +J +���� +2 +2 +≤ A1 +���� +vvy +√ +J +���� +2 +2 ++ A1 +� β +α +1 +J ̺2 +0v2ϑ2dy, +(2.9) +for a positive constant A1 depending only on κ, cv, µ, and R. Multiplying (1.12) with +4v3, using the boundary conditions, and integration by parts, one deduces +d +dt +� β +α +̺0v4dy + 12µ +���� +vvy +√ +J +���� +2 +2 += 12R +� β +α +1 +J v2vy̺0ϑdy +≤ 6µ +���� +vvy +√ +J +���� +2 +2 ++ C +� β +α +1 +J ̺2 +0v2ϑ2dy, +and thus, +d +dt +� β +α +̺0v4dy + 6µ +���� +vvy +√ +J +���� +2 +2 +≤ C +� β +α +1 +J ̺2 +0v2ϑ2dy. +(2.10) +Multiplying (2.10) with A1 +3µ and summing the resultant with (2.9) yield +d +dt +� +∥√̺0E∥2 +2 + A1 +3µ∥√̺0v2∥2 +2 +� ++ κcv +���� +ϑy +√ +J +���� +2 +2 ++ A1 +���� +vvy +√ +J +���� +2 +2 +≤ C +� β +α +1 +J ̺2 +0v2ϑ2dy, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +13 +from which, by Proposition 2.2 and Proposition 2.3, one gets +d +dt +� +∥√̺0E∥2 +2 + A1 +3µ∥√̺0v2∥2 +2 +� ++ κcv +���� +ϑy +√ +J +���� +2 +2 ++ A1 +���� +vvy +√ +J +���� +2 +2 +≤ C∥√̺0v∥2 +2∥√̺0ϑ∥2 +∞ ≤ C∥√̺0ϑ∥2 +∞. +(2.11) +Since ϑ|y=α = 0, it follows from Proposition 2.3, the H¨older and Young inequalities, +and (H1) that +̺0ϑ2 += +� y +α +(̺0ϑ2)ydz = +� y +α +(̺′ +0ϑ2 + 2̺0ϑϑy)dz +≤ +� β +α +� +K1̺0ϑ2 + 2̺0ϑ ϑy +√ +J +√ +J +� +dz +≤ +K1∥√̺0ϑ∥2 +2 + 2∥̺0ϑ∥ +1 +2 +1 ∥̺0∥ +1 +4∞∥√̺0ϑ∥ +1 +2∞ +���� +ϑy +√ +J +���� +2 +∥J∥ +1 +2∞ +≤ +K1∥√̺0ϑ∥2 +2 + C∥√̺0ϑ∥ +1 +2∞ +���� +ϑy +√ +J +���� +2 +� +1 + +� t +0 +∥̺0ϑ∥∞dτ +� 1 +2 +≤ +K1∥√̺0ϑ∥2 +2 + C∥√̺0ϑ∥ +1 +2∞ +���� +ϑy +√ +J +���� +2 +� +1 + +�� t +0 +∥̺0ϑ∥2 +∞dτ +� 1 +4� +≤ +1 +2 +� +∥√̺0ϑ∥2 +∞ + ǫ +���� +ϑy +√ +J +���� +2 +2 +� ++ C +� +1 + ∥√̺0ϑ∥2 +2 + +� t +0 +∥√̺0ϑ∥2 +∞dτ +� +, +and thus +∥√̺0ϑ∥2 +∞ ≤ ǫ +���� +ϑy +√ +J +���� +2 +2 ++ Cǫ +� +1 + ∥√̺0ϑ∥2 +2 + +� t +0 +∥√̺0ϑ∥2 +∞dτ +� +(2.12) +for any ǫ > 0. Choosing ǫ sufficiently small and plugging (2.12) into (2.11) yield +d +dt +� +∥√̺0E∥2 +2 + A1 +3µ∥√̺0v2∥2 +2 +� ++ A1 +���� +vvy +√ +J +���� +2 +2 ++ κcv +2 +���� +ϑy +√ +J +���� +2 +2 +≤ C +� +1 + ∥√̺0ϑ∥2 +2 + +� t +0 +∥√̺0ϑ∥2 +∞dτ +� +. +(2.13) +Combining (2.12) with (2.13) leads to +d +dt +� +∥√̺0E∥2 +2 + A1 +3µ∥√̺0v2∥2 +2 + +� t +0 +∥√̺0ϑ∥2 +∞dτ +� ++ A1 +���� +vvy +√ +J +���� +2 +2 ++ κ +2 +���� +ϑy +√ +J +���� +2 +2 +≤ C +� +1 + ∥√̺0E∥2 +2 + +� t +0 +∥√̺0ϑ∥2 +∞dτ +� +, + +14 +JINKAI LI AND ZHOUPING XIN +which, together with the Gr¨onwall inequality, implies that +sup +0≤t≤T +∥√̺0E∥2 +2 + +� T +0 +� +∥√̺0ϑ∥2 +∞ + +���� +vvy +√ +J +���� +2 +2 ++ +���� +ϑy +√ +J +���� +2 +2 +� +dt ≤ C. +This completes the proof of the conclusion. +□ +Corollary 2.1. There are two positive constants C and C, such that +C ≤ J ≤ C +on (α, β) × (0, T), +� T +0 +∥vy∥2 +2dt ≤ C. +Proof. The lower bound of J follows directly from Proposition 2.3 while the upper +bound of J follows from combining Proposition 2.3 and Proposition 2.4. Testing +(1.12) with v and integrating by parts yield +1 +2 +d +dt∥√̺0v∥2 +2 + µ +���� +vy +√ +J +���� +2 +2 += +R +� β +α +̺0 +J ϑvydy +≤ C +���� +vy +√ +J +���� +2 +∥√̺0ϑ∥2 +≤ +µ +2 +���� +vy +√ +J +���� +2 +2 ++ C∥√̺0ϑ∥2 +2, +where the lower bound of J was used, and thus +d +dt∥√̺0v∥2 +2 + µ +���� +vy +√ +J +���� +2 +2 +≤ C∥√̺0ϑ∥2 +2. +The second conclusion follows from this, the upper bound of J just proved, and +Proposition 2.4. +□ +In the rest of this section, we always assume that β − α ≥ 1. We will use the +following elementary inequality. +Lemma 2.1. It holds that +∥f∥Lp((α,β)) ≤ C(∥f∥L2((α,β)) + ∥f∥ +1 +2+ 1 +p +L2((α,β))∥f ′∥ +1 +2− 1 +p +L2((α,β))), +p ∈ [2, ∞], +for any f ∈ H1((α, β)), and for a positive constant C depending only on p. +Proof. This can be proved by scaling the corresponding inequality in (α, β) to that +in (0, 1), applying the Gagliardo-Nirenberg inequality for functions in H1((0, 1)), and +using the condition β −α ≥ 1. Since the proof is straightforward, and thus is omitted +here. +□ +Let G be the effective viscous flux, i.e., +G := µvy +J − π = µvy +J − R̺0ϑ +J . + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +15 +Then, it holds that +Gt − µ +J +�Gy +̺0 +� +y += −κ(γ − 1) +J +�ϑy +J +� +y +− γ vy +J G +(2.14) +and +G|y=α,β = 0. +(2.15) +Proposition 2.5. It holds that +sup +0≤t≤T +∥G∥2 +2 + +� T +0 +����� +Gy +√̺0 +���� +2 +2 ++ ∥G∥4 +∞ +� +dt ≤ C(1 + ∥G0∥2 +2). +Proof. Testing (2.14) with JG, using (1.11), (2.15), Lemma 2.1, Corollary 2.1, and +the Young inequality, one obtains +1 +2 +d +dt∥ +√ +JG∥2 +2 + µ +���� +Gy +√̺0 +���� +2 +2 += +κ(γ − 1) +� β +α +ϑyGy +J +dy + +�1 +2 − γ +� � β +α +vyG2dy +≤ +C +� +∥ϑy∥2 +���� +Gy +√̺0 +���� +2 ++ ∥vy∥2∥G∥2 +4 +� +≤ +C +� +∥ϑy∥2 +���� +Gy +√̺0 +���� +2 ++ ∥vy∥2 +� +∥G∥2 +2 + ∥G∥ +3 +2 +2 ∥Gy∥ +1 +2 +2 +�� +≤ +µ +2 +���� +Gy +√̺0 +���� +2 +2 ++ C[∥ϑy∥2 +2 + (1 + ∥vy∥2 +2)∥G∥2 +2], +that is, +d +dt∥ +√ +JG∥2 +2 + µ +���� +Gy +√̺0 +���� +2 +2 +≤ C[∥ϑy∥2 +2 + (1 + ∥vy∥2 +2)∥G∥2 +2]. +Thanks to this and the Gr¨onwall inequality, the desired conclusion, except the es- +timate on +� T +0 ∥G∥4 +∞dt, follows from Proposition 2.4 and Corollary 2.1. While the +estimate for +� T +0 ∥G∥4 +∞dt follows from Corollary 2.1, Lemma 2.1, and the estimate +just proved. +□ +Proposition 2.6. It holds that +sup +0≤t≤T +����� +Jy +√̺0 +���� +2 +2 ++ ∥vy∥2 +2 + ∥Jt∥2 +2 +� ++ +� T +0 +� +∥√̺0vt∥2 +2 + +���� +vyy +√̺0 +���� +2 +2 +� +dt ≤ C. + +16 +JINKAI LI AND ZHOUPING XIN +Proof. Note that vy = 1 +µ(JG + R̺0ϑ) and √̺0vt = +Gy +√̺0. It follows from Proposition +2.4, Proposition 2.5, and Corollary 2.1 that +sup +0≤t≤T +∥vy∥2 +2 + +� T +0 +∥√̺0vt∥2 +2dt ≤ C, +which by (1.11) implies +sup +0≤t≤T +∥Jt∥2 +2 ≤ C. +Direct calculations yield +Jyt = 1 +µ(JGy + JyG + R̺′ +0ϑ + R̺0ϑy). +Taking the inner product of the above with Jy +̺0 , one obtains from Proposition 2.4, +Corollary 2.1, and (H1) that +1 +2 +d +dt +���� +Jy +√̺0 +���� +2 +2 +≤ +C +� β +α +� +|J| +���� +Gy +√̺0 +���� + +���� +Jy +√̺0 +���� |G| + √̺0ϑ + √̺0|ϑy| +� |Jy| +√̺0 +dy +≤ +C +����� +Gy +√̺0 +���� +2 ++ ∥G∥∞ +���� +Jy +√̺0 +���� +2 ++ ∥√̺0ϑ∥2 + ∥ϑy∥2 +� ���� +Jy +√̺0 +���� +2 +≤ +C(1 + ∥G∥∞) +���� +Jy +√̺0 +���� +2 +2 ++ C +� +1 + ∥ϑy∥2 +2 + +���� +Gy +√̺0 +���� +2 +2 +� +, +which, together with the Gr¨onwall inequality, Proposition 2.4, Corollary 2.1, and +Proposition 2.5, yields +sup +0≤t≤T +���� +Jy +√̺0 +���� +2 +2 +≤ C. +(2.16) +Since +vyy = 1 +µ(JyG + JGy + R̺′ +0ϑ + R̺0ϑy), +(2.17) +it follows from (2.16), Corollary 2.1, Propositions 2.4, Proposition 2.5, and (H1) that +� T +0 +���� +vyy +√̺0 +���� +2 +2 +dt ≤ C +� T +0 +����� +Jy +√̺0 +���� +2 +2 +∥G∥2 +∞ + +���� +� Gy +√̺0 +, √̺0ϑ, ϑy +����� +2 +2 +� +dt ≤ C. +This completes the proof. +□ +Proposition 2.7. It holds that +sup +0≤t≤T +∥√̺0ϑy∥2 +2 + +� T +0 +(∥̺0ϑt∥2 +2 + ∥ϑyy∥2 +2)dt ≤ C. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +17 +Proof. Rewrite (1.13) as +cv̺0ϑt − κ +�ϑy +J +� +y += vyG. +(2.18) +Note that ϑt|y=α,β = 0. Taking the inner product of the above equation with ̺0ϑt +yields +κ +� β +α +ϑy +J (̺0ϑyt + ̺′ +0ϑt)dy + cv∥̺0ϑt∥2 +2 = +� β +α +vyG̺0ϑtdy. +(2.19) +It follows from (1.11) that +� β +α +ϑy +J ̺0ϑytdy = 1 +2 +d +dt +���� +�̺0 +J ϑy +���� +2 +2 ++ 1 +2 +� β +α +vy +J2̺0|ϑy|2dy. +Substituting this into (2.19) and using (H1) and Corollary 2.1, one gets +κ +2 +d +dt +���� +�̺0 +J ϑy +���� +2 +2 ++ cv∥̺0ϑt∥2 +2 += +� β +α +� +vyG̺0ϑt − κ +2 +vy +J2̺0|ϑy|2 − κϑy +J ̺′ +0ϑt +� +dy +≤ +� β +α +� +|vy||G|̺0|ϑt| + κ +2 +|vy| +J2 ̺0|ϑy|2 + κK1 +|ϑy| +J ̺0|ϑt| +� +dy +≤ +cv +2 ∥̺0ϑt∥2 +2 + C(∥G∥2 +∞∥vy∥2 +2 + ∥vy∥∞∥√̺0ϑy∥2 +2 + ∥ϑy∥2 +2), +which implies +κ d +dt +���� +�̺0 +J ϑy +���� +2 +2 ++ cv∥̺0ϑt∥2 +2 +≤ +C +� +∥G∥2 +∞∥vy∥2 +2 + (∥G∥∞ + ∥̺0ϑ∥∞)∥√̺0ϑy∥2 +2 + ∥ϑy∥2 +2 +� +. +It follows from this, the Gr¨onwall inequality, Propositions 2.4–2.6, and Corollary 2.1 +that +sup +0≤t≤T +∥√̺0ϑy∥2 +2 + +� T +0 +∥̺0ϑt∥2 +2dt +≤ CeC � T +0 (∥G∥∞+∥̺0ϑ∥∞)dt +� +∥√̺0ϑ′ +0∥2 +2 + +� T +0 +(∥G∥2 +∞∥vy∥2 +2 + ∥ϑy∥2 +2)dt +� +≤ C. +(2.20) +Direct calculations and using (2.18) yield +κϑyy = κ +�ϑy +J +� +y +J + κϑy +J Jy = J(cv̺0ϑt − vyG) + κϑy +J Jy. + +18 +JINKAI LI AND ZHOUPING XIN +It follows from this, (2.20), Propositions 2.5–2.6, Corollary 2.1, and Lemma 2.1 that +� T +0 +∥ϑyy∥2 +2dt +≤ +C +� T +0 +� +∥̺0ϑt∥2 +2 + ∥vy∥2 +2∥G∥2 +∞ + ∥ϑy∥2 +∞∥Jy∥2 +2 +� +dt +≤ +C + C +� T +0 +∥ϑy∥2 +∞dt ≤ C + C +� T +0 +∥ϑy∥2(∥ϑy∥2 + ∥ϑyy∥2)dt +≤ +1 +2 +� T +0 +∥ϑyy∥2 +2dt + C, +and thus +� T +0 ∥ϑyy∥2 +2dt ≤ C. This completes the proof. +□ +Proposition 2.8. It holds that +sup +0≤t≤T +���� +Gy +√̺0 +���� +2 +2 ++ +� T +0 + +∥Gt∥2 +2 + +����� +�Gy +̺0 +� +y +����� +2 +2 + + dt ≤ C +� +1 + +���� +G′ +0 +√̺0 +���� +2 +2 +� +. +Proof. Combining (2.14) with (2.18) yields +Gt − µ +J +�Gy +̺0 +� +y += +−R +J ̺0ϑt − vy +J G. +Note that Gt|y=α,β = 0. Multiplying the above with JGt, integrating by parts, and +using Corollary 2.1 yield +µ +2 +d +dt +���� +Gy +√̺0 +���� +2 +2 ++ ∥ +√ +JGt∥2 +2 = − +� β +α +(R̺0ϑt + vyG) Gtdy +≤ 1 +2∥ +√ +JGt∥2 +2 + C(∥̺0ϑt∥2 +2 + ∥vy∥2 +2∥G∥2 +∞), +from which, by Propositions 2.5–2.7, the conclusion follows. +□ +Proposition 2.9. It holds that +sup +0≤t≤T +����̺ +3 +2 +0 ϑt +��� +2 +2 + ∥√̺0ϑyy∥2 +2 +� ++ +� T +0 +∥̺0ϑyt∥2 +2dt ≤ C. +Proof. Note that vy = 1 +µ(JG + R̺0ϑ) and +vyt = 1 +µ(JGt + vyG + R̺0ϑt). +(2.21) +It follows from (1.11), (2.18), and direct calculations that +cv̺0ϑtt − κ +�ϑyt +J +� +y += −κ +�vyϑy +J2 +� +y ++ vy +µ G2 + 1 +µ(2JG + R̺0ϑ)Gt + R +µ ̺0ϑtG. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +19 +Note that ϑt|y=α,β = 0. Multiplying the above equation with ̺2 +0ϑt and integrating by +parts yield +cv +2 +d +dt +���̺ +3 +2 +0 ϑt +��� +2 +2 + κ +� β +α +ϑyt +J (̺2 +0ϑyt + 2̺0̺′ +0ϑt)dy += +1 +µ +� β +α +[vyG2 + (2JG + R̺0ϑ)Gt + R̺0ϑtG]̺2 +0ϑtdy ++κ +� β +α +vyϑy +J2 (̺2 +0ϑyt + 2̺0̺′ +0ϑt)dy. +Then, by Corollary 2.1 and (H1), one deduces +cv +2 +d +dt∥̺ +3 +2 +0 ϑt∥2 +2 + κ +���� +̺0 +√ +J +ϑyt +���� +2 +2 +≤ +C +� β +α +[̺2 +0|ϑt||ϑyt| + |vy||ϑy|(̺2 +0|ϑyt| + ̺2 +0|ϑt|)]dy ++C +� β +α +[|vy|G2 + (|G| + ̺0ϑ)|Gt| + ̺0|ϑt||G|]̺2 +0|ϑt|dy +≤ +κ +2 +���� +̺0 +√ +J +ϑyt +���� +2 +2 ++ C(∥̺0ϑt∥2 +2 + ∥vy∥2 +∞∥̺0ϑy∥2 +2) + C∥G∥2 +∞(∥vy∥2 +2 + ∥̺2 +0ϑt∥2 +2) ++C∥Gt∥2 +2 + C(∥G∥2 +∞ + ∥̺0ϑ∥2 +∞)∥̺2 +0ϑt∥2 +2 + C∥G∥∞∥̺ +3 +2 +0 ϑt∥2 +2, +from which, by Propositions 2.6–2.7 and vy = 1 +µ(JG + R̺0ϑ), one obtains +cv +d +dt∥̺ +3 +2 +0 ϑt∥2 +2 + κ +���� +̺0 +√ +J +ϑyt +���� +2 +2 +≤ +C(∥G∥2 +∞ + ∥̺0ϑ∥2 +∞ + 1)∥̺ +3 +2 +0 ϑt∥2 +2 + C(∥vy∥2 +∞ + ∥G∥2 +∞ + ∥Gt∥2 +2 + ∥̺0ϑt∥2 +2) +≤ +C(∥G∥2 +∞ + ∥̺0ϑ∥2 +∞ + 1)∥̺ +3 +2 +0 ϑt∥2 +2 + C(∥G∥2 +∞ + ∥̺0ϑ∥2 +∞ + ∥Gt∥2 +2 + ∥̺0ϑt∥2 +2). +Applying the Gr¨onwall inequality to the above, one can get by Propositions 2.4–2.5 +and 2.7–2.8, and Corollary 2.1 that +sup +0≤t≤T +∥̺ +3 +2 +0 ϑt∥2 +2 + +� T +0 +∥̺0ϑyt∥2 +2dt +≤ +CeC � T +0 (∥G∥2 +∞+∥̺0ϑ∥2 +∞)dt ��̺ +3 +2 +0 ϑt +��2 +2 +��� +t=0 ++CeC � T +0 (∥G∥2 +∞+∥̺0ϑ∥2 +∞)dt +� T +0 +(∥G∥2 +∞ + ∥̺0ϑ∥2 +∞ + ∥Gt∥2 +2 + ∥̺0ϑt∥2 +2)dt +≤ +C(1 + ∥√̺0ϑ′′ +0∥2 +2 + ∥√̺0v′ +0G0∥2 +2), +(2.22) + +20 +JINKAI LI AND ZHOUPING XIN +where the fact that ̺ +3 +2 +0 ϑt|t=0 = +√̺0 +cv (κϑ′′ +0 + v′ +0G0) has been used, which follows from +(2.18). Therefore, noticing that Lemma 2.1 implies +∥√̺0v′ +0G0∥2 +2 ≤ C∥v′ +0∥2 +∞∥G0∥2 +2 ≤ C∥v′ +0∥2 +H1∥G0∥2 +2, +one gets from (2.22) that +sup +0≤t≤T +∥̺ +3 +2 +0 ϑt∥2 +2 + +� T +0 +∥̺0ϑyt∥2 +2dt ≤ C(1 + ∥√̺0ϑ′′ +0∥2 +2). +(2.23) +Note that +ϑyy = J +�ϑy +J +� +y ++ ϑy +J Jy = 1 +κ(cv̺0ϑt − vyG)J + 1 +J ϑyJy. +It follows from this, (2.23), Proposition 2.6, and Corollary 2.1 that +∥√̺0ϑyy∥2 +2 +≤ +C(∥̺ +3 +2 +0 ϑt∥2 +2 + ∥vy∥2 +2∥G∥2 +∞ + ∥√̺0ϑy∥2 +∞∥Jy∥2 +2) +≤ +C(1 + ∥G∥2 +∞ + ∥√̺0ϑy∥2 +∞). +(2.24) +It remains to estimate ∥G∥2 +∞ and ∥√̺0ϑy∥2 +∞ as follows. Note that Lemma 2.1, Propo- +sition 2.5, and Proposition 2.8 imply that +∥G∥2 +∞ ≤ C∥G∥2(∥G∥2 + ∥Gy∥2) ≤ C. +(2.25) +By Lemma 2.1 and (H1), and Proposition 2.7, it holds that +∥√̺0ϑy∥2 +∞ +≤ +C∥√̺0ϑy∥2 +� +∥√̺0ϑy∥2 + ∥√̺0ϑyy∥2 + +���� +̺′ +0 +√̺0 +ϑy +���� +2 +� +≤ +C(1 + ∥√̺0ϑyy∥2). +(2.26) +Plugging (2.25) and (2.26) into (2.24) and using the Cauchy inequality yield +∥√̺0ϑyy∥2 +2 ≤ C(1 + ∥√̺0ϑyy∥2) ≤ ∥√̺0ϑyy∥2 +2 +2 ++ C, +which gives ∥√̺0ϑyy∥2 +2 ≤ C. This completes the proof. +□ +Proposition 2.10. It holds that +sup +0≤t≤T +� +∥√̺0vt∥2 +2 + +���� +vyy +√̺0 +���� +2 +2 +� ++ +� T +0 +(∥vyt∥2 +2 + ∥vyyy∥2 +2 + ∥Jyy∥2 +2)dt ≤ C. +Proof. The estimate for √̺0vt follows directly from Proposition 2.8 since √̺0vt = +Gy +√̺0. +It follows from (H1), (2.17), (2.21), (2.25), Corollary 2.1, and Propositions +2.4–2.8 that +���� +vyy +√̺0 +���� +2 +2 +≤ C +����� +Jy +√̺0 +���� +2 +2 +∥G∥2 +∞ + +���� +Gy +√̺0 +���� +2 +2 ++ ∥√̺0ϑy∥2 +2 + ∥√̺0ϑ∥2 +2 +� +≤ C, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +21 +� T +0 +∥vyt∥2 +2dt ≤ C +� T +0 +(∥vy∥2 +2∥G∥2 +∞ + ∥Gt∥2 +2 + ∥̺0ϑt∥2 +2)dt ≤ C. +Noticing that +vyyy = 1 +µ(JyyG + 2JyGy + JGyy + R̺′′ +0ϑ + 2R̺′ +0ϑy + R̺0ϑyy). +one can get from (H1), (2.25), Corollary 2.1, and Propositions 2.4–2.5 and 2.7–2.8 +that +� t +0 +∥vyyy∥2 +2dτ +≤ +C +� t +0 +(∥Jyy∥2 +2∥G∥2 +∞ + ∥Jy∥2 +∞∥Gy∥2 +2 + ∥Gyy∥2 +2 ++∥̺0ϑ∥2 +2 + ∥̺0ϑy∥2 +2 + ∥̺0ϑyy∥2 +2)dτ +≤ +C +� t +0 +(∥Jyy∥2 +2 + ∥Jy∥2 +∞ + ∥Gyy∥2 +2)dτ + C, +(2.27) +where ∥G∥2 +∞ ≤ C(∥G∥2 +2 + ∥Gy∥2 +2) guaranteed by Lemma 2.1 wa used. Next, ∥Jy∥2 +∞ +and ∥Gyy∥2 +2 can be estimated as follows. Lemma 2.1 and Proposition 2.6 imply that +∥Jy∥2 +∞ ≤ C(∥Jy∥2 +2 + ∥Jy∥2∥Jyy∥2) ≤ C(1 + ∥Jyy∥2 +2). +(2.28) +While (H1) and Proposition 2.8 yield +� T +0 +∥Gyy∥2 +2dt +≤ +� T +0 +������̺0 +�Gy +̺0 +� +y +����� +2 ++ +����̺′ +0 +Gy +̺0 +���� +2 +�2 +dt +≤ +C +� T +0 + + +����� +�Gy +̺0 +� +y +����� +2 +2 ++ ∥Gy∥2 +2 + + dt ≤ C. +It follows from this, (2.27), and (2.28) that +� t +0 +∥vyyy∥2 +2dτ ≤ C +� +1 + +� t +0 +∥Jyy∥2 +2dτ +� +. +(2.29) +Since Jyy = +� t +0 vyyydτ, one has +� t +0 +∥Jyy∥2 +2dτ ≤ +� t +0 +���� +� τ +0 +vyyydτ ′ +���� +2 +2 +dτ ≤ C +� t +0 +�� τ +0 +∥vyyy∥2 +2dτ ′ +� +dτ. +(2.30) +Plugging this into (2.29) leads to +� t +0 +∥vyyy∥2 +2dτ ≤ C + C +� t +0 +�� τ +0 +∥vyyy∥2 +2dτ ′ +� +dτ, +which implies +� T +0 ∥vyyy∥2 +2dt ≤ CeT ≤ C by the Gr¨onwall inequality. This, together +with (2.30), shows that +� t +0 ∥Jyy∥2 +2dτ ≤ C. This completes the proof. +□ + +22 +JINKAI LI AND ZHOUPING XIN +Proposition 2.11. It holds that +sup +0≤t≤T +(∥Jyy∥2 +2 + ∥Jyt∥2 +2) ≤ C. +Proof. This follows from Proposition 2.10 by using Jyy = +� t +0 vyyydτ and Jyt = vyy. +□ +Proposition 2.12. It holds that +� T +0 +∥ϑyyy∥2 +2dt ≤ C. +Proof. By Lemma 2.1 and Propositions 2.5, 2.6, 2.8, 2.10, and 2.11, one has +∥Jy∥∞ + ∥Jyy∥2 + ∥vy∥∞ + ∥G∥∞ ≤ C. +(2.31) +It follows from (2.18) that +ϑyyy += +cv +κ (̺′ +0Jϑt + 2̺0Jyϑt + ̺0Jϑyt) + ϑy +J Jyy +−1 +κ(2JyvyG + JvyyG + JvyGy). +Then, by Corollary 2.1, (H1), (2.31), and Proposition 2.11, one deduces +� T +0 +∥ϑyyy∥2 +2dt +≤ +C +� T +0 +(∥̺0ϑt∥2 +2 + ∥Jy∥2 +∞∥̺0ϑt∥2 +2 + ∥̺0ϑyt∥2 +2 + ∥ϑy∥2 +∞∥Jyy∥2 +2 ++∥Jy∥2 +∞∥vy∥2 +2∥G∥2 +∞ + ∥vyy∥2 +2∥G∥2 +∞ + ∥vy∥2 +∞∥Gy∥2 +2)dt +≤ +C +� T +0 +(∥̺0ϑt∥2 +2 + ∥̺0ϑyt∥2 +2 + ∥ϑy∥2 +2 ++∥ϑyy∥2 +2 + ∥vy∥2 +2 + ∥vyy∥2 +2 + ∥Gy∥2 +2)dt, +where ∥ϑy∥2 +∞ ≤ C(∥ϑy∥2 +2 + ∥ϑyy∥2 +2) guaranteed by Lemma 2.1 was used, from which, +by Corollary 2.1 and Propositions 2.4–2.10, it follows +� T +0 ∥ϑyyy∥2 +2dt ≤ C. This proves +the conclusion. +□ +As a consequence of Propositions 2.2–2.12 and Corollary 2.1, one has: +Corollary 2.2. Let (J, v, ϑ) be the unique global solution stated in Proposition 2.1 +to system (1.11)–(1.13), subject to (2.1)–(2.2), and N0 be given by (2.8). Then, for +any T ∈ [0, ∞), it holds that +inf +(α,β)×(0,T) J ≥ CT, +sup +0≤t≤T +���� +� Jy +√̺0 +, Jyy, Jt, Jyt +����� +2 +L2((α,β)) +≤ CT, +sup +0≤t≤T +���� +�√̺0v, √̺0v2, vy, vyy +√̺0 +, √̺0vt +����� +2 +L2((α,β)) ++ +� T +0 +∥(vyyy, vyt)∥2 +L2((α,β))dt ≤ CT, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +23 +sup +0≤t≤T +� +∥̺0ϑ∥L1((α,β)) + +��� +�√̺0ϑ, √̺0ϑy, √̺0ϑyy, ̺ +3 +2 +0 ϑt +���� +2 +L2((α,β)) +� ++ +� T +0 +� +∥ϑy∥2 +H2((α,β)) + ∥(̺0ϑt, ̺0ϑyt)∥2 +L2((α,β)) +� +dt ≤ CT, +sup +0≤t≤T +���� +Gy +√̺0 +���� +2 +L2((α,β)) ++ +� T +0 + + +����� +�Gy +̺0 +� +y +����� +2 +L2((α,β)) ++ ∥Gt∥2 +L2((α,β)) + + dt ≤ CT, +where CT and CT are positive constants depending only on R, cv, µ, κ, K1, T, and the +upper bound of N0, but independent of α and β with β − α ≥ 1. +3. Global existence of solutions: proof of Theorem 1.1 +Proof of Theorem 1.1. The proof is given in three steps as follows. +Step 1. Approximations of the initial data. By the assumption (1.16), there +are two sequences {αn}∞ +n=1 and {βn}∞ +n=1, with limn→∞ αn = −∞ and limn→∞ βn = ∞, +and a positive constant M0, such that +����� +v′ +0(αn) +� +̺0(αn) +����� + +����� +v′ +0(βn) +� +̺0(βn) +����� ≤ M0, +∀n ≥ 1. +(3.1) +Set In = (αn − 1, βn + 1). For each n, choose 0 ≤ χn ∈ C∞ +0 (In), satisfying +χ ≡ 1 on [αn, βn], +0 ≤ χn ≤ 1 and |χ′ +n| + |χ′′ +n| ≤ C0 on In, +(3.2) +for a positive constant C0 independent of n. Define v0n and ϑ0n as +ϑ0n = ϑ0χn, +and +v0n = + + + +v0(αn) + 2 +πv′ +0(αn) sin +�π +2(y − αn) +� +, +y ∈ [αn − 1, αn], +v0(y), +y ∈ [αn, βn], +v0(βn) + 2 +πv′ +0(βn) sin +�π +2(y − βn) +� +, +y ∈ [βn, βn + 1]. +It can be checked easily that +v′ +0n(αn − 1) = v′ +0n(βn + 1) = ϑ0n(αn − 1) = ϑ0n(βn + 1) = 0. +(3.3) +Noticing that +v0n(αn) = v0(αn), +v′ +0n(αn) = v′ +0(αn), +v0n(βn) = v0(βn), +v′ +0n(βn) = v′ +0(βn), +and since v0 ∈ H2 +loc(R) and 0 ≤ ϑ0 ∈ H2 +loc(R), one has +v0n ∈ H2(In), +0 ≤ ϑ0n ∈ H2(In). +(3.4) +Due to 0 ≤ χn ≤ 1, it is clear that +∥√̺0ϑ0n∥L2(In) ≤ ∥√̺0ϑ0∥2. +(3.5) + +24 +JINKAI LI AND ZHOUPING XIN +For any y ∈ [αn − 1, αn), the definition of v0n implies that +|v0n(y) − v0(y)| ≤ |v0(αn) − v0(y)| + 2 +π|v′ +0(αn)| ≤ 2∥v′ +0∥∞. +Similarly, it holds that |v0n(y) − v0(y)| ≤ 2∥v′ +0∥∞, for any y ∈ (βn, βn + 1]. As a +result, one has +|v0n(y) − v0(y)| ≤ 2∥v′ +0∥∞, +∀y ∈ In. +(3.6) +Hence +∥√̺0v0n∥L2(In) +≤ +∥√̺0(v0n − v0)∥L2(In) + ∥√̺0v0∥L2(In) += +2∥v′ +0∥∞∥̺0∥ +1 +2 +1 + ∥√̺0v0∥2, +(3.7) +and +∥√̺0|v0n|2∥L2(In) +≤ +2 +��√̺0(|v0|2 + |v0 − v0n|2) +�� +L2(In) +≤ +2∥√̺0|v0|2∥2 + 8∥v′ +0∥2 +∞∥̺0∥ +1 +2 +1 . +(3.8) +It follows from (3.2) and direct calculations that +∥√̺0ϑ′ +0n∥L2(In) +≤ +∥√̺0ϑ′ +0∥2 + C0∥√̺0ϑ0∥2, +(3.9) +∥√̺0ϑ′′ +0n∥L2(In) +≤ +∥√̺0ϑ′′ +0∥2 + 2C0(∥√̺0ϑ′ +0∥2 + ∥√̺0ϑ0∥2). +(3.10) +By direct calculations, one gets by the Sobolev inequality that +∥v′ +0n∥H1(In) +≤ +∥v′ +0∥H1 + C(|v′ +0(αn)| + |v′ +0(βn)|) +≤ +∥v′ +0∥H1((αn,βn)) + ∥v′ +0∥H1((αn−1,αn)∪(βn,βn+1)) +≤ +C∥v′ +0∥2 +H1, +(3.11) +for a positive constant C independent of n. +Set G0n = µv′ +0n − R̺0ϑ0n. Combining (3.5) with (3.11) leads to +∥G0n∥L2(In) +≤ +µ∥v′ +0n∥L2(In) + R∥̺0∥ +1 +2∞∥√̺0ϑ0n∥L2(In) +≤ +C(∥v′ +0∥H1 + ∥̺0∥ +1 +2∞∥√̺0ϑ0∥2), +(3.12) +for a positive constant C independent of n. +For y ∈ (βn, βn + 1), one has +̺0(βn) +̺0(y) += +1 + +� y +βn +k(z)̺0(βn) +̺0(z) dz, +where k(z) = −̺′ +0(z) +̺0(z). +(3.13) +By (H1), it holds that |k(z)| ≤ K1, for any z ∈ R. Set +f(y) = 1 + +� y +βn +k(z)̺0(βn) +̺0(z) dz, +∀y ∈ (βn, βn + 1). + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +25 +Then, it follows from (3.13) that +f ′(y) = k(y)̺0(βn) +̺0(y) = k(y)f(y), +and thus +f(y) = e +� y +βn k(z)dzf(βn) = e +� y +βn k(z)dz ≤ eK1, +∀y ∈ (βn, βn + 1). +It follows from this and (3.13) that +̺0(βn) +̺0(y) = f(y) ≤ eK1, +∀y ∈ (βn, βn + 1). +(3.14) +Similarly, one has +̺0(αn) +̺0(y) ≤ eK1, +∀y ∈ (αn − 1, αn). +(3.15) +Recall that G0n = µv′ +0n − R̺0ϑ0n. Then, direct calculations yield +G′ +0n +√̺0 += + + + + + + + +−π +2µ v′ +0(αn) +√̺0 sin +�π +2(y − αn) +� +− R +�√̺0ϑ′ +0n + +̺′ +0 +√̺0ϑ0n +� +, +y ∈ (αn − 1, αn), +G′ +0 +√̺0, +y ∈ (αn, βn), +−π +2µ v′ +0(βn) +√̺0 sin +� π +2(y − βn) +� +− R +�√̺0ϑ′ +0n + +̺′ +0 +√̺0ϑ0n +� +, +y ∈ (βn, βn + 1). +It follows from (3.1) and (3.14)–(3.15) that +����� +v′ +0(αn) +� +̺0(y) +����� + +����� +v′ +0(βn) +� +̺0(y) +����� += +����� +v′ +0(αn) +� +̺0(αn) +� +̺0(αn) +̺0(y) +����� + +����� +v′ +0(βn) +� +̺0(βn) +� +̺0(βn) +̺0(y) +����� +≤ +2M0e +K1 +2 , +∀y ∈ (αn − 1, αn) ∪ (βn, βn + 1). +This together with (H1) yields +����� +G′ +0n(y) +� +̺0(y) +����� ≤ πµM0e +K1 +2 + R (√̺0|ϑ′ +0n| + K1 +√̺0ϑ0n) , +for any y ∈ (αn − 1, αn) ∪ (βn, βn + 1). Due to this and that G′ +0n +√̺0 = +G′ +0 +√̺0 on (αn, βn), +it follows from (3.9) that +���� +G′ +0n +√̺0 +���� +L2(In) +≤ +���� +G′ +0 +√̺0 +���� +2 ++ 2µπM0eK1/2 + R +� +∥√̺0ϑ′ +0n∥L2(In) + K1∥√̺0ϑ0n∥L2(In) +� +≤ +���� +G′ +0 +√̺0 +���� +2 ++ C (∥√̺0ϑ′ +0∥2 + ∥√̺0ϑ0∥2 + 1) , +(3.16) +for a positive constant C independent of n. +Step 2. Solutions to the system in In × (0, ∞) and a priori estimates. + +26 +JINKAI LI AND ZHOUPING XIN +For each positive integer n, let (v0n, ϑ0n) be the initial data constructed as before. +Consider the initial-boundary value problem to the system (1.11)–(1.13) in (αn − +1, βn + 1) × (0, ∞), subject to +(J, v, ϑ)|t=0 = (1, v0n, ϑ0n), +(vy, ϑ)|y=an−1,βn+1 = (0, 0). +(3.17) +Thanks to (3.3) and (3.4), and noticing that infy∈In ̺0 > 0, one can verify that the +initial datum (v0n, ϑ0n) satisfies all the assumptions in Proposition 2.1, for each fixed +n. Thus, there is a unique global strong solution (Jn, vn, ϑn) to (1.11)–(1.13) with +(3.17). Moreover, due to (3.5), (3.7)–(3.12), and (3.16), it follows from Corollary 2.2 +that +inf +In×(0,T) Jn ≥ CT , +ϑn(y, t) ≥ 0, +(3.18) +sup +0≤t≤T +���� +�∂yJn +√̺0 +, ∂2 +yJn, ∂tJn, ∂ytJn +����� +2 +L2(In) +≤ CT, +(3.19) +sup +0≤t≤T +���� +�√̺0vn, √̺0v2 +n, ∂yvn, ∂2 +yvn +√̺0 +, √̺0∂tvn +����� +2 +L2(In) ++ +� T +0 +∥(∂3 +yvn, ∂2 +ytvn)∥2 +L2(In)dt ≤ CT, +(3.20) +sup +0≤t≤T +� +∥̺0ϑn∥L1(In) + +��� +�√̺0ϑn, √̺0∂yϑn, √̺0∂2 +yϑn, ̺ +3 +2 +0 ∂tϑn +���� +2 +L2(In) +� ++ +� T +0 +� +∥∂yϑn∥2 +H2(In) + ∥(̺0∂tϑn, ̺0∂ytϑn)∥2 +L2(In) +� +dt ≤ CT, +(3.21) +and +sup +0≤t≤T +���� +∂yGn +√̺0 +���� +2 +L2(In) ++ +� T +0 + + +����� +�∂yGn +̺0 +� +y +����� +2 +L2(In) ++ ∥∂tGn∥2 +L2(In) + + dt ≤ CT, +(3.22) +for any positive time T, where Gn = µ ∂yvn +Jn − R ̺0 +Jnϑn, and CT and CT are positive +constants independent of n. +Step 3. Convergence and existence. +Thanks to the a priori estimates (3.18)–(3.22) and inf(−k,k) ̺0(y) > 0 for any k ∈ N, +the following estimate holds +∥(Jn, vn, ϑn)∥L∞(0,T;H2((−k,k))) + ∥(vn, ϑn)∥L2(0,T;H3((−k,k))) + ∥∂tJn∥L∞(0,T;H1((−k,k))) ++ ∥(∂tvn, ∂tϑn)∥L∞(0,T;L2((−k,k)))∩L2(0,T;H1((−k,k))) ≤ Ck,T, +∀k ∈ N, +for a positive constant Ck,T independent of n. Due to this and the Cantor’s diagonal +argument, there is a subsequence, still denoted by (Jn, vn, ϑn), and (J, v, ϑ), such that +(Jn, vn, ϑn) +∗⇀ (J, v, ϑ), +in L∞(0, T; H2((−k, k))), +(3.23) + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +27 +(vn, ϑn) ⇀ (v, ϑ), +in L2(0, T; H3((−k, k))), +(3.24) +∂tJn +∗⇀ Jt, +in L∞(0, T; H1((−k, k))), +(3.25) +(∂tvn, ∂tϑn) +∗⇀ (vt, ϑt), +in L∞(0, T; L2((−k, k))), +(3.26) +(∂tvn, ∂tϑn) ⇀ (vt, ϑt), +in L2(0, T; H1((−k, k))), +(3.27) +for any k ∈ N. Moreover, since H3((−k, k)) ֒→֒→ C2([−k, k]) and H2((−k, k)) ֒→֒→ +C1([−k, k]), it follows from the Aubin–Lions lemma that +(Jn, vn, ϑn) → (J, v, ϑ), +in C([0, T]; C1([−k, k])), +(3.28) +(vn, ϑn) → (v, ϑ), +in L2(0, T; C2([−k, k])), +(3.29) +for any k ∈ N. Thanks to these and by (3.18), one has +inf +(y,t)∈R×(0,T) J(y, t) ≥ CT, +1 +Jn +→ 1 +J in C([0, T]; C1([−k, k])), +(3.30) +for any k ∈ N. +Thanks to (3.23)–(3.30) and noticing that (v0n, ϑ0n) → (v0, ϑ0) in H2((−L, L)) for +any L > 0, one can take the limit as n → ∞ to show that (J, v, ϑ) is a solution to the +Cauchy problem to the system (1.11)–(1.13) subject to (J, v, ϑ)|t=0 = (1, v0, ϑ0). The +desired regularities of (J, v, ϑ) stated in Theorem 1.1 follow from the a priori estimates +(3.18)–(3.22) and convergence (3.23)–(3.29) by the weakly lower semi-continuity of +norms. This proves Theorem 1.1. +□ +4. A Hopf type lemma and unboundedness of the entropy +In this section, we prove the unboundedness of the entropy immediately after +the initial time, i.e. Theorem 1.2. As stated in the Introduction, this is based on +some suitable scaling transform and a Hopf type lemma for a class of general linear +degenerate equations. So, we first establish a Hopf type lemma in the first subsection +and then present the proof of Theorem 1.2 in the second subsection. The Hopf type +lemma has its own independent interests and will also be applied to prove the uniform +positivity of the temperature in the next section. +4.1. A Hopf type lemma. Since the results in this subsection hold in any di- +mension, we use the following notations. +Denote by x = (x1, x2, · · · , xn) and t +the spatial and time variables respectively and P = (x, t) a point in Rn+1. +For +P0 = (x0, t0) ∈ Rn+1 and r > 0, denote +Br(P0) := +� +(x, t) ∈ Rn+1���|x − x0|2 + (t − t0)2 < r2� +. +Let (aij)n×n, a0, b = (b1, b2, · · · , bn), and c be given functions satisfying suitable +properties to be specified later. Consider the operator +L ϕ = −aij∂ijϕ + a0∂tϕ + b · ∇ϕ + cϕ. + +28 +JINKAI LI AND ZHOUPING XIN +Note that here a0 is not required to have fixed sign and this linear operator can be +regarded only as a linear degenerate elliptic operator in the space and time variables +with degeneracy occurring in the time direction. +Lemma 4.1. Let O be a domain in Rn+1. Assume that aij, a0, b, and c are finitely +valued functions in O with c ≥ 0, and the matrix (aij)n×n is nonnegative definite in +O. Then, for any ϕ ∈ C2,1(O) ∩ C(O), satisfying +L ϕ > 0 in O, +and +ϕ|∂O ≥ 0, +it holds that ϕ > 0 in O. Here C2,1(O) denotes the space of all function f satisfying +f, ∂tf, ∇f, ∇2f ∈ C(O). +Proof. First, we claim that ϕ ≥ 0 in O. Otherwise, since ϕ ≥ 0 on ∂O and ϕ ∈ C(O), +there is P0 ∈ O, such that ϕ(P0) = minO ϕ < 0. Since ϕ ∈ C2,1(O), it is clear that +∂tϕ(P0) = ∇ϕ(P0) = 0 and ∇2ϕ(P0) is nonnegative definite. As a result +(L ϕ)(P0) = −aij(P0)∂ijϕ(P0) + c(P0)ϕ(P0) ≤ 0, +which contradicts to the assumption. Therefore, the claim holds. Next, we show +that ϕ > 0 in O. Otherwise, there is P ∗ +0 ∈ O, such that ϕ(P ∗ +0 ) = 0. Then, ϕ(P ∗ +0 ) = +minO ϕ = 0, from which, similar as before, one has (L ϕ)(P ∗ +0 ) ≤ 0, contradicting to +the assumption. Thus, ϕ > 0 in O, which proves the conclusion. +□ +Lemma 4.2 (Hopf type lemma). Given P0 = (x0, t0), r > 0, P∗ = (x∗, t∗) ∈ ∂Br(P0), +x∗ ̸= x0, and set P ∗ +0 = (x∗ +0, t∗ +0), with x∗ +0 = x0+x∗ +2 +and t∗ +0 = t0+t∗ +2 +. Assume that there +are positive constants λ, Λ, δ∗, and C∗, with δ∗ < |x0−x∗| +4 +, such that + + + +λ|ξ|2 ≤ aij(x, t)ξiξj ≤ Λ|ξ|2, +∀ξ ∈ Rn, +(t − t∗ +0)a0(x, t) + (x − x∗ +0) · b(x, t) ≥ −C∗, +0 ≤ c(x, t) +� +|x − x∗|2 + (t − t∗)2 ≤ C∗, +∀(x, t) ∈ B r +2(P ∗ +0 ) ∩ Bδ∗(P∗). +Let ϕ ∈ C2,1(Br(P0)) ∩ C(Br(P0)) satisfy +L ϕ ≥ 0, +ϕ > ϕ(P∗), +in B r +2(P ∗ +0 ) ∩ Bδ∗(P∗), +ϕ(P∗) ≤ 0. +Then, it holds that +lim +ℓ→0+ +ϕ(P∗) − ϕ(P∗ − ℓn∗) +ℓ +< 0, +where n∗ = P∗−P0 +r +is the unit outward normal vector to ∂Br(P0) at P∗. +Proof. Set +D = B r +2(P ∗ +0 ) ∩ Bδ∗(P∗). +It suffices to consider the case that ϕ(P∗) = 0. Otherwise, one may consider Φ := +ϕ − ϕ(P∗), which reduces to the case considered, due to +L Φ = L ϕ − L (ϕ(P∗)) = L ϕ − cϕ(P∗) ≥ L ϕ ≥ 0 +in D, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +29 +as ϕ(P∗) ≤ 0 and c ≥ 0 in D. It is clear that B r +2(P ∗ +0 ) ⊂ Br(P0). By assumption, it +holds that +ϕ(P) > ϕ(P∗) = 0, +∀P ∈ D \ {P∗}. +(4.1) +Define +φ(x, t) = e−ζ(|x−x∗ +0|2+(t−t∗ +0)2) − e− r2 +4 ζ = e−ζ|P −P ∗ +0 |2 − e− r2 +4 ζ, +where ζ > 0 is a constant to be determined. Then, it follows from direct calculations +that +L φ = − e−ζ|P −P ∗ +0 |2� +4(x − x∗ +0)TA(x − x∗ +0)ζ2 − 2trAζ ++ 2((t − t∗ +0)a0 + (x − x∗ +0) · b)ζ + c +� +eζ(|P −P ∗ +0 |2− r2 +4 ) − 1 +� � +, +(4.2) +where A = (aij)n×n and trA = aii. Note that the assumptions imply +4(x − x∗ +0)TA(x − x∗ +0)ζ2 − 2trAζ + 2((t − t∗ +0)a0 + (x − x∗ +0) · b)ζ +≥4λ|x − x∗ +0|2ζ2 − 2nΛζ − 2C∗ζ ≥ λ +4|x0 − x∗|2ζ2 − (2nΛ + 2C∗)ζ, +(4.3) +for any (x, t) ∈ D, due to trA ≤ nΛ and +|x − x∗ +0| ≥ |x∗ +0 − x∗| − |x∗ − x| ≥ |x0 − x∗| +2 +− δ∗ ≥ |x0 − x∗| +4 +, +∀(x, t) ∈ D. +Note that |P − P ∗ +0 | < r +2 for any P ∈ D. It follows from the mean value theorem and +the triangular inequality that +���eζ(|P −P ∗ +0 |2− r2 +4 ) − 1 +��� = eτζ(|P −P ∗ +0 |2− r2 +4 ) +����|P − P ∗ +0 |2 − r2 +4 +���� ζ +≤ +���|P − P ∗ +0 | − r +2 +��� +���|P − P ∗ +0 | + r +2 +��� ζ ≤ rζ +��|P − P ∗ +0 | − |P ∗ +0 − P∗| +�� ≤ rζ|P − P∗|, +for any P ∈ D, where τ ∈ (0, 1). This, together with the assumptions, yields +���c +� +eζ(|P −P ∗ +0 |2− r2 +4 ) − 1 +���� ≤ cr|P − P∗|ζ ≤ C∗rζ, +∀P ∈ D. +(4.4) +Combining (4.3) with (4.4) leads to +4(x − x∗ +0)TA(x − x∗ +0)ζ2 − 2trAζ ++2((t − t∗ +0)a0 + (x − x∗ +0) · b)ζ + c +� +eζ(|P −P ∗ +0 |2− r2 +4 ) − 1 +� +≥ +λ +4|x0 − x∗|2ζ2 − (2nΛ + 2C∗ + rC∗)ζ > 0, +∀P ∈ D, +(4.5) +if ζ > ζ0 := 8nΛ+8c∗+4rC∗ +λ|x0−x∗|2 +. Choose ζ = 2ζ0. Then, it follows from (4.2) and (4.5) that +L φ < 0 +in D. +(4.6) + +30 +JINKAI LI AND ZHOUPING XIN +It follows from (4.1) that +ϕ ≥ 0 = φ on ∂B r +2(P ∗ +0 ) ∩ Bδ∗(P∗), +inf +∂Bδ∗(P∗)∩B r +2 (P ∗ +0 ) ϕ > 0. +Therefore, for ε > 0 sufficiently small, it follows from the assumptions and (4.6) that +L ϕ ≥ 0 > L (εφ) in D, +ϕ ≥ εφ on ∂D, +and thus +L (ϕ − εφ) > 0 in D, +ϕ − εφ ≥ 0 on ∂D. +With the aid of this, noticing that ϕ − εφ ∈ C2,1(D) ∩ C(D), and applying Lemma +4.1, one gets +ϕ > εφ +in D. +Therefore, for ℓ > 0 sufficiently small, one has +ϕ(P∗) − ϕ(P∗ − ℓn∗) = −ϕ(P∗ − ℓn∗) < −εφ(P∗ − ℓn∗) = ε(φ(P∗) − φ(P∗ − ℓn∗)) +and thus +lim +ℓ→0+ +ϕ(P∗) − ϕ(P∗ − ℓn∗) +ℓ +≤ +ε lim +ℓ→0+ +φ(P∗) − φ(P∗ − ℓn∗) +ℓ += +ε∂n∗φ(P∗) = −εζre− r2 +4 ζ < 0. +This proves the conclusion. +□ +As a direct consequence of Lemma 4.2, the following corollary holds. +Corollary 4.1. Given P0 = (x0, t0), r > 0, P∗ = (x∗, t∗) ∈ ∂Br(P0), x∗ ̸= x0. +Assume that a0, b, c ∈ L∞(Br(P0)), c ≥ 0 in Br(P0), and +λ|ξ|2 ≤ aij(x, t)ξiξj ≤ Λ|ξ|2, +∀ξ ∈ Rn, (x, t) ∈ Br(P0), +for some positive constants λ and Λ. Let ϕ ∈ C2,1(Br(P0)) ∩ C(Br(P0)) satisfy +L ϕ ≥ 0, +ϕ > ϕ(P∗), +in Br(P0), +ϕ(P∗) ≤ 0. +Then, it holds that +lim +ℓ→0+ +ϕ(P∗) − ϕ(P∗ − ℓn∗) +ℓ +< 0, +where n∗ = P∗−P0 +r +is the unit outward normal vector to ∂Br(P0) at P∗. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +31 +4.2. Unboundedness of the entropy. This subsection is devoted to proving The- +orem 1.2. We start with the following embedding lemma, which is used to verify the +H¨older regularity of Jy required in the proof of Theorem 1.2. +Lemma 4.3. Let L > 0 be a positive number. Then, the following embedding in- +equality holds +∥f∥C +1 +2 , 1 +4 ([−L,L]×[0,T]) ≤ C(∥f∥L∞(0,T;H1((−L,L))) + ∥∂tf∥L2(0,T;L2((−L,L)))), +for any function f ∈ L∞(0, T; H1((−L, L))) such that ∂tf ∈ L2(0, T; L2((−L, L))), +where C is an absolute positive constant. +Proof. For any t, τ ∈ [0, T], one deduces by Lemma 2.1, the Minkovski, H¨older, and +Cauchy inequalities that +∥f(·, t) − f(·, τ)∥L∞((−L,L)) +≤ +C∥f(·, t) − f(·, τ)∥ +1 +2 +L2((−L,L))∥f(·, t) − f(·, τ)∥ +1 +2 +H1((−L,L)) +≤ +C∥f∥ +1 +2 +L∞(0,T;H1((−L,L))) +���� +� t +τ +∂tf(·, s)ds +���� +1 +2 +L2((−L,L)) +≤ +C∥f∥ +1 +2 +L∞(0,T;H1((−L,L))) +�� t +τ +∥∂tf∥L2((−L,L))ds +� 1 +2 +≤ +C(∥f∥L∞(0,T;H1((−L,L))) + ∥∂tf∥L2(0,T;L2((−L,L))))|t − τ| +1 +4, +for an absolute positive constant C. For any x, y ∈ [−L, L] and t ∈ [0, T], it follows +from the H¨older inequality that +|f(x, t) − f(y, t)| ≤ +���� +� y +x +∂xf(z, t)dz +���� +≤ +���� +� L +−L +|∂xf|2dx +���� +1 +2 +|y − x| +1 +2 ≤ ∥f∥L∞(0,T;H1((−L,L)))|y − x| +1 +2. +Therefore, for any x, y ∈ [−L, L] and t, τ ∈ [0, T], it holds that +|f(x, t) − f(y, τ)| ≤ |f(x, t) − f(y, t)| + |f(y, t) − f(y, τ)| +≤ +C(∥f∥L∞(0,T;H1((−L,L))) + ∥∂tf∥L2(0,T;L2((−L,L))))(|t − τ| +1 +4 + |y − x| +1 +2). +This leads to the conclusion. +□ +We are now ready to prove Theorem 1.2. +Proof of Theorem 1.2. The proof is dived into five steps as follows. +Step 1. Regularities and pointwise positivity of ϑ. For L > 0, denote by +W 2,1 +2 ((−L, L)×(0, T)) the space of all functions f ∈ L2(0, T; H2((−L, L))) satisfying + +32 +JINKAI LI AND ZHOUPING XIN +∂tf ∈ L2(0, T; L2((−L, L))). Recall the embedding that W 2,1 +2 ([−L, L] × (0, T)) ֒→ +C +1 +2, 1 +4([−L, L] × [0, T]) (Theorem 4.1 of [55]). Note that +(vy, ϑy) ∈ W 2,1 +2 ((−L, L) × (0, T)), +J, Jy ∈ L∞(0, T; H1((−L, L))), +and Jt, Jyt ∈ L∞(0, T; L2((−L, L))). Hence, it follows from the Sobolev embedding +and Lemma 4.3 that +(J, Jy, vy, ϑy) ∈ C +1 +2, 1 +4([−L, L] × [0, T]), +∀L > 0. +(4.7) +Rewrite (1.13) as +cv̺0ϑt − κ +J ϑyy + κJy +J2ϑy + R̺0 +vy +J ϑ = µ +J |vy|2. +(4.8) +Since J is uniformly positive on R×(0, T), it can be checked that all the coefficients in +(4.8), i.e., ̺0, 1 +J , Jy +J2, ̺0 +vy +J , and |vy|2 +J , belong to C +1 +2 , 1 +4([−L, L] × [0, T]). Thanks to these +and the fact that ̺0(y) > 0 for all y ∈ R, it follows from the classic Schauder theory +on interior regularities for uniform parabolic equations that ϑ ∈ C2+ 1 +2 ,1+ 1 +4((−L, L) × +(0, T)). On the other hand, by the embedding theorem, it follows from the regularities +of ϑ that ϑ ∈ C([−L, L] × [0, T]). Therefore, it holds that +ϑ ∈ C2,1((−L, L) × (0, T)) ∩ C([−L, L] × [0, T]). +Note that ϑ ̸≡ 0 on R × (0, T). Otherwise, noticing that ϑ ∈ C([0, T]; L2(−L, L)) +for any L > 0, one has ϑ0 ≡ 0; furthermore, it follows from (1.13) that vy ≡ 0 +on R × (0, T), from which, since v ∈ C([0, T]; L2((−L, L))) for any L > 0, one has +v0 ≡ Const. This contradicts to the assumptions. +Therefore, one has ϑ ̸≡ 0 on +R ×(0, T) and ϑ ≥ 0. Thanks to this and by the strong maximum principle, one gets +0 < ϑ ∈ C2,1(R × (0, T)) ∩ C(R × [0, T]). +(4.9) +Step 2. Asymptotic behavior of Jy. Note that +Jyt = 1 +µ(GJy + JGy + R̺′ +0ϑ + R̺0ϑy) +which implies +Jy = 1 +µ +� t +0 +e +1 +µ +� t +s Gdτ(JGy + R̺′ +0ϑ + R̺0ϑy)ds. +Therefore���� +Jy(y, t) +̺0(y) +���� ≤ 1 +µe +1 +µ +� t +0 ∥G∥∞dτ +� t +0 +� +J +���� +Gy +̺0 +���� + R +���� +̺′ +0 +̺0 +���� ϑ + R|ϑy| +� +ds. +(4.10) +For any y ≥ 0, it follows that +���� +Gy(y, t) +̺0(y) +���� += +����� +� 1 +0 +Gy +̺0 +dz + +� 1 +0 +� y +z +�Gy +̺0 +� +y +(z′, t)dz′dz +����� + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +33 +≤ +� 1 +0 +���� +Gy +̺0 +���� dz + +� y+1 +0 +����� +�Gy +̺0 +� +y +����� dz +≤ +���� +Gy +√̺0 +���� +2 +(t) + +� +y + 1 +����� +�Gy +̺0 +� +y +����� +2 +(t) +and +ϑ(y, t) += +� 1 +0 +ϑ(z, t)dz + +� 1 +0 +� y +z +ϑy(z′, t)dz′dz +≤ +� 1 +0 +√̺0ϑ +√̺0 +dz + +� y+1 +0 +|ϑy|dz ≤ ∥√̺0ϑ∥2(t) +√δ0 ++ +� +y + 1∥ϑy∥2(t),(4.11) +where δ0 := inf[−1,1] ̺0 > 0. Similar estimates hold also for y < 0 and thus it holds +for any y ∈ R that +���� +Gy(y, t) +̺0(y) +���� ≤ +���� +Gy +√̺0 +���� +2 +(t) + +� +|y| + 1 +����� +�Gy +̺0 +� +y +����� +2 +(t) +(4.12) +and +ϑ(y, t) ≤ ∥√̺0ϑ∥2(t) +√δ0 ++ +� +|y| + 1∥ϑy∥2(t). +(4.13) +Substituting (4.12)–(4.13) into (4.10) and using (H1), one can get by the H¨older +and Sobolev inequalities that +���� +Jy(y, t) +̺0(y) +���� +≤ +CeC � t +0 ∥G∥∞ds +� t +0 +� ���� +Gy +√̺0 +���� +2 ++ +� +|y| + 1 +����� +�Gy +̺0 +� +y +����� +2 +� +ds ++CeC +� t +0 ∥G∥∞ds +� t +0 +� +∥√̺0ϑ∥2 + ∥ϑy∥2 +� +|y| + 1 + ∥ϑy∥H1 +� +ds +≤ +C +√ +t +� +|y| + 1 + + +� t +0 + + +����� +� +Gy +√̺0 +, +�Gy +̺0 +� +y +������ +2 +2 ++ ∥ϑy∥2 +H1 + + ds + + +1 +2 ++Ct +� +|y| + 1∥√̺0ϑ∥L∞(0,T;L2) +≤ +C1 +� +|y| + 1, +that is, +���� +Jy(y, t) +̺0(y) +���� ≤ C1 +� +|y| + 1, +∀y ∈ R, t ∈ [0, T], +(4.14) +where the regularities of (ϑ, G) have been used. +Step 3. +A scaling transform. +Let T > 0 be any arbitrary given constant. +Assume by contradiction that s ∈ L∞(R×(0, T)). Since ϑ = A +Re +s +cv �̺0 +J +�γ−1 and J has + +34 +JINKAI LI AND ZHOUPING XIN +uniform positive lower and upper bounds, it follows that +0 ≤ ϑ(y, t) ≤ CT̺γ−1 +0 +(y), +∀(y, t) ∈ R × [0, T]. +(4.15) +Let β > 0 to be determined later and introduce a scaling transform as +f(y, t) := ϑ(y−β, t), +y ∈ (0, ∞), t ≥ 0. +Then, direct calculations yield +ϑ(y, t) = f(y− 1 +β , t), +ϑt(y, t) = ft(y− 1 +β , t), +ϑy(y, t) = − 1 +β y−(1+ 1 +β )fy(y− 1 +β , t), +ϑyy(y, t) = 1 +β2y−(2+ 2 +β )fyy(y− 1 +β , t) + β + 1 +β2 y−(2+ 1 +β )fy(y− 1 +β , t), +for any (y, t) ∈ (0, ∞) × (0, ∞). Besides, one deduces from (4.8) that +cv̺0(y−β)y−(2+2β)J(y−β, t)ft(y, t) − κ +β2fyy(y, t) +− +�κ(β + 1) +β2 +y−1 + κ +βy−(1+β)Jy(y−β, t) +J(y−β, t) +� +fy(y, t) ++R̺0(y−β)y−(2β+2)vy(y−β, t)f(y, t) ≥ 0, +(4.16) +for all (y, t) ∈ (0, ∞) × (0, ∞). +Step 4. Verifying conditions of Hopf type lemma. Let MT be a positive +constant to be determined later and define +F(y, t) = e−MT tf(y, t), +y ∈ (0, ∞), t ∈ [0, ∞). +Due to (4.9), it is clear that +0 < F ∈ C2,1((0, ∞) × (0, ∞)) ∩ C((0, ∞) × [0, ∞)). +Moreover, it follows from (4.15) and (H2) that +F(y, t) += +e−MT tϑ(y−β, t) ≤ CTe−MT t̺γ−1 +0 +(y−β) +≤ +CTKγ−1 +2 +e−MT t(1 + y−β)−(γ−1)ℓρ ≤ CTKγ−1 +2 +e−MT ty(γ−1)βℓρ, +for an y ∈ (0, ∞) and t ∈ [0, ∞). Thus, one can define F(0, t) = 0 for t ∈ [0, ∞), +such that F is well defined on [0, ∞) × [0, ∞), satisfying +� +F ∈ C2,1((0, ∞) × (0, ∞)) ∩ C([0, ∞) × [0, ∞)), +F > 0 in (0, ∞) × (0, ∞), +F(0, t) = 0, +∀t ∈ [0, ∞). +(4.17) +It follows from (4.16) that +a0(y, t)Ft(y, t) − aFyy(y, t) + b(y, t)Fy(y, t) + c(y, t)F(y, t) ≥ 0, +(4.18) +in (0, ∞) × (0, ∞), where +a0(y, t) += +cv̺0(y−β)y−(2+2β)J(y−β, t), +a = κ +β2, + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +35 +b(y, t) += +− +�κ(β + 1) +β2 +y−1 + κ +β y−(1+β)Jy(y−β, t) +J(y−β, t) +� +, +c(y, t) += +̺0(y−β)y−(2+2β)� +cvMT J(y−β, t) + Rvy(y−β, t) +� +. +Take arbitrary t0 ∈ (0, T), 0 < y0 < min{ 1 +2, t0}, and set +P0 = (y0, t0), +r = y0, +P∗ = (0, t0) =: (y∗, t∗), +δ∗ = y0 +8 , +P ∗ +0 = +�y0 +2 , t0 +� +=: (y∗ +0, t∗ +0), +D = Bδ∗(P∗) ∩ B r +2(P ∗ +0 ). +Then, +P∗ ∈ ∂Br(P0), +D = B y0 +8 ((0, t0)) ∩ B y0 +2 (( y0 +2 , t0)). +For any (y, t) ∈ D, due to 0 < y < y0 +8 < +1 +16 and t0 +2 < t < 3 +2t0, one deduces +(t − t∗ +0)a0(y, t) + (y − y∗ +0)b(y, t) += +cv(t − t0)̺0(y−β)y−(2+2β)J(y−β, t) +− +� +y − y0 +2 +� �κ(β + 1) +β2 +y−1 + κ +βy−(1+β)Jy(y−β, t) +J(y−β, t) +� +≥ +−cvt0̺0(y−β)y−(2+2β)J(y−β, t) − κy0 +2β y−(1+β)|Jy(y−β, t)| +J(y−β, t) +≥ +−cvt0jT̺0(y−β)y−(2+2β) − +κ +jTβ y−(1+β)|Jy(y−β, t)|, +(4.19) +where +jT := +sup +(y,t)∈R×[0,T] +J(y, t), +jT := +inf +(y,t)∈R×[0,T] J(y, t). +(4.20) +Set MT := +R∥vy∥L∞(R×(0,T )) +cvjT +. Then, +cvMT J(y−β, t) + Rvy(y−β, t) ≥ cvMT jT − R∥vy∥L∞(R×(0,T)) = 0 +and +cvMTJ(y−β, t) + Rvy(y−β, t) +≤ +cvMT jT + R∥vy∥L∞(R×(0,T)) += +cvMT (jT + jT). +Thus, for any (y, t) ∈ D, since +� +|y − y∗|2 + |t − t∗|2 ≤ δ∗ ≤ +1 +16, it holds that +0 ≤ c(y, t) +� +|y − y∗|2 + |t − t∗|2 ≤ cvMT (jT + jT)̺0(y−β)y−(2+2β). +(4.21) +For any (y, t) ∈ D, since 0 < y < 1, it follows from (4.14) and (H2) that +̺0(y−β)y−(2+2β) ≤ K2(1 + y−β)−ℓρy−(2+2β) ≤ K2y(ℓρ−2)β−2 ≤ K2 +(4.22) +and +y−(1+β)|Jy(y−β, t)| +≤ +C1y−(1+β)̺0(y−β) +� +1 + y−β ≤ C1K2y−(1+β)(1 + y−β)−ℓρ+ 1 +2 + +36 +JINKAI LI AND ZHOUPING XIN +≤ +C1K2y(ℓρ− 3 +2 )β−1 ≤ C1K2, +(4.23) +as long as β ≥ max +� +2 +ℓρ−2, +2 +2ℓρ−3 +� += +2 +ℓρ−2. +Due to (4.22) and (4.23), it follows from (4.19) and (4.21) that +(t − t∗ +0)a0(y, t) + (y − y∗ +0)b(y, t) ≥ −(C1 + 1)K2 +� +cvt0jT + +κ +jTβ +� +, +(4.24) +0 ≤ c(y, t) +� +|y − y∗|2 + |t − t∗|2 ≤ cvMT(jT + jT)K2, +(4.25) +for any (y, t) ∈ D, as long as β ≥ +2 +ℓρ−2. +Step 5. Unboundedness of entropy. Choose +β = β0 := max +� +2 +(γ − 1)ℓρ +, +2 +ℓρ − 2 +� +. +Due to (4.17), (4.18), (4.24), and (4.25), it follows from Lemma 4.2 that +lim +ℓ→0+ +F(P∗) − F(P∗ − n∗ℓ) +ℓ += − lim +ℓ→0+ +F(P∗ − n∗ℓ) +ℓ += −2ε2, +for some positive constant ε2, where we recall P∗ = (0, t0), n∗ = P∗−P0 +r += (−1, 0), and +thus P∗ − n∗ℓ = (ℓ, t0). Thus, there is a positive number ℓ0, such that +F(y, t0) = e−MT t0ϑ(y−β0, t0) ≥ ε2y, +∀y ∈ (0, ℓ0), +that is +ϑ(y, t0) ≥ ε2eMT t0y− 1 +β0 , +∀y ∈ +� +ℓ +− 1 +β0 +0 +, ∞ +� +. +On the other hand, it follows from (4.15) and (H2) that +ϑ(y, t) ≤ CTKγ−1 +2 +(1 + y)−ℓρ(γ−1) ≤ CTKγ−1 +2 +y−ℓρ(γ−1), +∀y > 0. +Combing the previous two inequalities leads to +y(γ−1)ℓρ− 1 +β0 ≤ CTKγ−1 +2 +ε−1 +2 e−MT t0, +∀y ∈ (ℓ +− 1 +β0 +0 +, ∞), +which is impossible when y → ∞, as (γ − 1)ℓρ − 1 +β0 ≥ γ−1 +2 ℓρ > 0. This contradiction +leads to the desired conclusion that s ̸∈ L∞(R × (0, T)). +□ +5. Uniform positivity of ϑ and asymptotic unboundedness of s +In this section, we prove the uniform positivity of the temperature and asymptotic +unboundedness of the entropy, under the condition that the initial density decays at +the far field not slower than O( 1 +x4), which yields the proof of Theorem 1.3. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +37 +Proof of Theorem 1.3. We need only to prove (i), while the conclusion (ii) follows +from (i), (1.10), and (4.13), as J has uniformly positive lower and upper bounds at +each time t ∈ (0, ∞). +Let h be the Kelvin transform of ϑ defined as +h(y, t) = yϑ +�1 +y, t +� +, +∀y ̸= 0, t ∈ [0, T]. +(5.1) +Then (4.9) implies +h ∈ C2,1((R+ ∪ R−) × (0, T)) ∩ C((R+ ∪ R−) × [0, T]). +(5.2) +Note that +ϑ(y, t) = yh +�1 +y, t +� +, +ϑt(y, t) = yht +�1 +y, t +� +, +ϑy(y, t) = h +�1 +y, t +� +− 1 +yhy +�1 +y, t +� +, +ϑyy(y, t) = 1 +y3hyy +�1 +y, t +� +, +for any y ̸= 0 and t ∈ (0, T). It follows from these and (4.8) that +cv̺0 +�1 +y +� 1 +y4ht (y, t) − +κ +J +� +1 +y, t +�hyy (y, t) − κ +Jy +� +1 +y, t +� +J2 +� +1 +y, t +� 1 +y2hy (y, t) ++ + +R +vy +� +1 +y, t +� +J +� +1 +y, t +� +̺0 +� +1 +y +� +y4 ++ κ +Jy +� +1 +y, t +� +J2 +� +1 +y, t +� 1 +y3 + + h(y, t) = µ +���vy +� +1 +y, t +���� +2 +y3J +� +1 +y, t +� , +for y ̸= 0. Define a0, a, b, and ˜c as +a0 := cv̺0 +�1 +y +� 1 +y4, +a := +κ +J +� +1 +y, t +�, +b := −κ +Jy +� +1 +y, t +� +J2 +� +1 +y, t +� 1 +y2, +˜c := R +vy +� +1 +y, t +� +J +� +1 +y, t +� +̺0 +� +1 +y +� +y4 ++ κ +Jy +� +1 +y, t +� +J2 +� +1 +y, t +� 1 +y3, +∀y ̸= 0, t ∈ [0, T]. +Then, it holds that +� +a0ht − ahyy + bhy + ˜ch ≥ 0, +in Q+ +T , +a0ht − ahyy + bhy + ˜ch ≤ 0, +in Q− +T , +(5.3) +where +Q+ +T := R+ × (0, T), +Q− +T := R− × (0, T). +Properties of a0, a, b, and ˜c are analyzed as follows. It follows from (4.7) and the +regularities of ̺0 and J that +a0 ∈ C(R+ ∪ R−), +a, b, ˜c ∈ C(Q+ +T ∪ Q− +T ). +(5.4) + +38 +JINKAI LI AND ZHOUPING XIN +For a0, it follows from (H3) that +0 ≤ a0(y) ≤ +cvK3 +(|y| + 1)4 ≤ cvK3, +∀y ̸= 0. +(5.5) +For a, it holds that +λT ≤ a(y, t) ≤ ΛT, +∀y ̸= 0, t ∈ [0, T], +(5.6) +where λT = +κ +jT and ΛT = +κ +jT , with jT and jT given by (4.20). +It follows from (4.14) and (H3) that +|Jy(y, t)| ≤ C1̺0(y) +� +|y| + 1 ≤ C1K3(|y| + 1)− 7 +2, +∀y ∈ R, t ∈ [0, T]. +(5.7) +This implies that +|b(y, t)| ≤ κ +j2 +T +1 +y2 +����Jy +�1 +y, t +����� ≤ κ +j2 +T +1 +y2C1K3 +� 1 +|y| + 1 +�− 7 +2 +≤ C1K3κ +j2 +T +, +(5.8) +for any y ̸= 0 and t ∈ [0, T]. By (H3) and (5.7), one deduces +|˜c(y, t)| +≤ +R +jT +1 +y4 +K3 +� +1 + +1 +|y| +�4∥vy∥∞(t) + κ +j2 +T +1 +|y|3C1K3 +� +1 + 1 +|y| +�− 7 +2 +≤ +RK3 +jT +∥vy∥∞(t) + κ +j2 +T +C1K3 ≤ C(∥vy∥L∞(0,T;H1) + 1), +(5.9) +for any y ̸= 0 and t ∈ [0, T]. +Set +NT = 2 +cv +� +R +jT +∥vy∥L∞(R×(0,T)) + +√ +2κC1 +j2 +T +� +and define +H(y, t) = e−NT th(y, t), +∀y ̸= 0, t ∈ [0, T]. +(5.10) +Due to (5.2), it is clear that +H ∈ C2,1((R+ ∪ R−) × (0, T)) ∩ C((R+ ∪ R−) × [0, T]). +(5.11) +Since ϑ > 0, it follows from (5.1) and (5.10) that +H > 0 in Q+ +T +and +H < 0 in Q− +T . +(5.12) +By (4.13) and recalling the definitions of h and H, one deduces +|H(y, t)| +≤ +|y|ϑ +�1 +y, t +� +≤ C(∥√̺0ϑ∥L∞(0,T;L2) + ∥ϑy∥L∞(0,T;L2))|y| +� +1 + 1 +|y| +≤ +C +� +y2 + |y|, +∀y ̸= 0, t ∈ [0, T]. + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +39 +Thanks to this, it holds that +lim +(y,τ)→(0,t) H(y, t) = 0, +∀t ∈ [0, T]. +(5.13) +It follows from direct calculations and (5.3) that +� +a0Ht − aHyy + bHy + cH ≥ 0, +in Q+ +T , +a0Ht − aHyy + bHy + cH ≤ 0, +in Q− +T , +(5.14) +where +c(y, t) +:= +˜c(y, t) + NTa0(y, t) += + +cvNT + R +vy +� +1 +y, t +� +J +� +1 +y, t +� + +κ +J2 +� +1 +y, t +� +Jy +� +1 +y, t +� +̺0 +� +1 +y +� y + + ̺0 +�1 +y +� 1 +y4. +For any y ∈ [−1, 0) ∩ (0, 1] and t ∈ [0, T], it follows from (4.14) that +c(y, t) +≥ +� +cvNT − R +jT +∥vy∥∞(t) − κ +j2 +T +|y|C1 +� +1 + 1 +|y| +� +̺0 +�1 +y +� 1 +y4 += +� +cvNT − R +jT +∥vy∥∞(t) − κ +j2 +T +C1 +� +|y|2 + |y| +� +̺0 +�1 +y +� 1 +y4 +≥ +� +cvNT − R +jT +∥vy∥∞(t) − +√ +2κC1 +j2 +T +� +̺0 +�1 +y +� 1 +y4 +≥ +� +R +jT +∥vy∥∞(t) + +√ +2κC1 +j2 +T +� +̺0 +�1 +y +� 1 +y4 +and thus +c(y, t) ≥ 0, +∀y ∈ [−1, 0) ∪ (0, 1], t ∈ [0, T]. +(5.15) +Define +�H(y, t) = + + + +H(y, t), +if y > 0, +0, +if y = 0, +−H(y, t), +if y < 0, +(5.16) +for all t ∈ [0, T]. Denote +Ω− := (−1, 0) × (0, T), +Ω+ := (0, 1) × (0, T), +Ω := Ω+ ∪ Ω−, +Γ := {0} × [0, T]. +Then, it follows from (5.11)–(5.14) that +�H ∈ C2,1(Ω) ∩ C(Ω), +�H > 0 in Ω, +�H|Γ = 0, +(5.17) +L �H = a0 �Ht − a �Hyy + b �Hy + c �H ≥ 0 +in Ω, +(5.18) + +40 +JINKAI LI AND ZHOUPING XIN +with a0, a, b, and c satisfying +� +a0 ∈ C((−1, 1) \ {0}) ∩ L∞((−1, 1) \ {0}), +a, b, c ∈ C(Ω) ∩ L∞(Ω), +λT ≤ a ≤ ΛT, +c ≥ 0 in Ω, +(5.19) +which follows from (5.4)–(5.6), (5.8)–(5.9), and (5.15). +For arbitrary t0 ∈ (0, T), set +P∗ = (0, t0), +P0 = (y0, t0), +r = y0 = min +�1 +2, t0, T − t0 +� +. +Then, it is clear that n∗ := P∗−P0 +r += (−1, 0). Let Br be the space-time ball of radius +r and centered at P0. Thanks to (5.17), (5.18), and (5.19), it is clear that �H satisfies +all the conditions in Corollary 4.1, and thus Corollary 4.1 implies +lim +ℓ→0+ +�H(P∗) − �H(P∗ − ℓn∗) +ℓ += lim +ℓ→0+ +− �H(ℓ, t0) +ℓ += −2ε0, +with a positive constant ε0. Thus, limℓ→0+ +� +H(ℓ,t0) +ℓ += 2ε0, which yields that +�H(y, t0) ≥ ε0y, +∀y ∈ (0, ℓ0), +(5.20) +for some positive constant ℓ0. Then, by the definition of �H, one derives +�H(y, t0) = e−NT t0h(y, t0) = e−NT t0yϑ +�1 +y, t0 +� +≥ ε0y, +∀y ∈ (0, ℓ0) +(5.21) +and thus, +ϑ(y, t0) ≥ ε0eNT t0 ≥ ε0, +∀y ∈ +� 1 +ℓ0 +, ∞ +� +. +(5.22) +Similarly, there are positive constants ε1 and ℓ1 such that +ϑ(y, t0) ≥ ε1, +∀y ∈ +� +−∞, − 1 +ℓ1 +� +. +Combining this with (5.22) and recalling that 0 < ϑ ∈ C(R × [0, T]), one has +inf +y∈R ϑ(y, t0) = min + + +ε0, ϑ1, +inf +y∈ +� +− 1 +ℓ1 , 1 +ℓ0 +� ϑ(y, t0) + + + > 0. +This yields the desired conclusion, and the proof of Theorem 1.3 is completed. +□ + +UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE +41 +Acknowledgments +This work was supported by the Key Project of National Natural Science Foun- +dation of China (Grant No. 12131010) and Guangdong Basic and Applied Basic +Research Foundation (Grant No. 2020B1515310002). The work of J. L. was also sup- +ported by the National Natural Science Foundation of China (Grants No. 11971009 +and 11871005) and by the Guangdong Basic and Applied Basic Research Foundation +(Grants No. 2019A1515011621 and 2020B1515310005). The work of Z.X. was also +supported by the Zheng Ge Ru Foundation and by the Hong Kong RGC Earmarked +Research Grants (Grants No. CUHK-14301421, CUHK-14300917, CUHK-14300819, +and CUHK-14302819). +References +[1] Bresch, D.; Jabin, P.-E: Global existence of weak solutions for compressible +Navier-Stokes equations: thermodynamically unstable pressure and anisotropic +viscous stress tensor, Ann. Math., (2) 188 (2018), no. 2, 577–684. +[2] Chen, G.-Q.; Hoff, D.; Trivisa, K.: Global solutions of the compressible Navier- +Stokes equations with large discontinuous initial data, Comm. 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Notes, 63 +(1998), 736–746. +(Jinkai Li) South China Research Center for Applied Mathematics and Interdisci- +plinary Studies, School of Mathematical Sciences, South China Normal University, +Guangzhou 510631, China +Email address: jklimath@m.scnu.edu.cn; jklimath@gmail.com +(Zhouping Xin) The Institute of Mathematical Sciences, The Chinese University of +Hong Kong, Hong Kong, China +Email address: zpxin@ims.cuhk.edu.hk + diff --git a/W9AyT4oBgHgl3EQfvPl0/content/tmp_files/load_file.txt b/W9AyT4oBgHgl3EQfvPl0/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ce6df8db5f790589497eecb908f1d2c6159d0d0 --- /dev/null +++ b/W9AyT4oBgHgl3EQfvPl0/content/tmp_files/load_file.txt @@ -0,0 +1,1579 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf,len=1578 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='00627v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='AP] 2 Jan 2023 INSTANTANEOUS UNBOUNDEDNESS OF THE ENTROPY AND UNIFORM POSITIVITY OF THE TEMPERATURE FOR THE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH FAST DECAY DENSITY JINKAI LI AND ZHOUPING XIN Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This paper concerns the physical behaviors of any solutions to the one dimensional compressible Navier-Stokes equations for viscous and heat conductive gases with constant viscosities and heat conductivity for fast decaying density at far fields only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' First, it is shown that the specific entropy becomes not uniformly bounded immediately after the initial time, as long as the initial density decays to vacuum at the far field at the rate not slower than O � 1 |x|ℓρ � with ℓρ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Further- more, for faster decaying initial density, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', ℓρ ≥ 4, a sharper result is discovered that the absolute temperature becomes uniformly positive at each positive time, no matter whether it is uniformly positive or not initially, and consequently the cor- responding entropy behaves as O(− log(̺0(x))) at each positive time, independent of the boundedness of the initial entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Such phenomena are in sharp contrast to the case with slowly decaying initial density of the rate no faster than O( 1 x2 ), for which our previous works [34–36] show that the uniform boundedness of the entropy can be propagated for all positive time and thus the temperature decays to zero at the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' These give a complete answer to the problem concerning the propagation of uniform boundedness of the entropy for the heat conductive ideal gases and, in particular, show that the algebraic decay rate 2 of the initial density at the far field is sharp for the uniform boundedness of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The tools to prove our main results are based on some scaling transforms, including the Kelvin transform, and a Hopf type lemma for a class of degenerate equations with possible unbounded coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Introduction The compressible Navier–Stokes equations for the ideal viscous and heat conductive gases read as ∂tρ + div (ρu) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) ρ(∂tu + (u · ∇)u) − µ∆u − (µ + λ)∇div u + ∇p = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) Date: January 2, 2023.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 2010 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 35A01, 35B45, 35Q86, 76D03, 76D09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Uniform positivity of temperature;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' immediately unboundedness of en- tropy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' global classic solution;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' compressible Navier–Stokes equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' fast decay density;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Kelvin transform;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Hopf type lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 1 2 JINKAI LI AND ZHOUPING XIN cvρ(∂tθ + u · ∇θ) + pdiv u − κ∆θ = Q(∇u), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) where the unknowns ρ ≥ 0, u ∈ RN, with N the spatial dimension, θ ≥ 0, and p = Rρθ, respectively, represent the density, velocity, temperature, and pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Here, R and cv are positive constants, µ and λ are the viscous coefficients, both assumed to be constants and satisfy the physical constraints µ > 0 and 2µ+Nλ > 0, κ is the heat conductive coefficient, assumed to be a positive constant, and Q(∇u) is a quadratic term of ∇u given as Q(∇u) = µ 2|∇u + (∇u)T|2 + λ(div u)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By the Gibbs equation θDs = De + pD( 1 ρ), where s is the specific entropy and e = cvθ is the specific internal energy, it holds that p = Ae s cv ργ for some positive constant A, where γ − 1 = R cv .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It is clear that γ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In terms of ρ and θ, the specific entropy s can be expressed as s = cv � log R A + log θ − (γ − 1) log ρ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) satisfying ρ(∂ts + u · ∇s) − κ cv ∆s = κ(γ − 1)div �∇ρ ρ � + 1 θ � Q(∇u) + κ|∇θ|2 θ � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) in the region where both ρ and θ are positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As the governing system in the gas dynamics, the compressible Navier–Stokes equations have been studied extensively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' One of the central concepts in the mathe- matical theory for the compressible Navier–Stokes equations is the vacuum, which, if occurs, means that the density vanishes at either some interior points or on the boundary or at the far fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Indeed, the possible presence of vacuum is one of the main difficulties in the theory of global well-posedness of general solutions to the compressible Navier–Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) for the entropy is highly degenerate and singular near the vacuum, it is even more difficult to analyze the dynamic behavior of the entropy in the presence of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to this, most of the mathematical theories developed in the existing literatures on the compress- ible Navier–Stokes equations in the presence of vacuum are for system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) regardless of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' There are extensive literatures on the mathematical studies concerning the com- pressible Navier–Stokes equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In the one-dimensional case, the corre- sponding theory is satisfactory and in particular the global well-posedness has been known for long time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In the absence of vacuum, for which the information of the entropy follows from that of the density and the temperature directly by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4), the global well-posedness of strong solutions was established by Kazhikov–Shelukin [24] and Kazhikov [25], which were later extended in the setting of weak solutions, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', [2, 23, 58, 59];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' large time behavior of solutions with general initial data was UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 3 proved by Li–Liang [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In the presence of vacuum, but without considering the en- tropy, the corresponding global well-posedness were established by the first author of this paper in [29, 30], for both heat conductive and non-heat conductive ideal gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As shown by Hoff–Smoller [17], for the one-dimensional compressible Navier–Stokes equations, no vacuum can be formed later in finite time from non-vacuum initial data, while such a result remains open in the multidimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In the multi-dimensional case, the mathematical theory for the compressible Navier– Stokes equations is less complete than that in the one-dimensional case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The break- through for the global existence of finite energy weak solutions with general initial data and possible vacuum, to the isentropic compressible Navier–Stokes equations, was achieved by Lions [37, 38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The results of Lions [37, 38] were later improved by Feireisl–Novotn´y–Petzeltov´a [12], Jiang–Zhang [22], and more recently Bresch– Jabin [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For the full compressible Navier–Stokes equations, the global existence of variational weak solutions was proved by Feireisl [14], under some assumptions on the equations of states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The uniqueness of weak solutions is still a challenging open problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' If the initial datum is suitably regular, then the compressible Navier–Stokes equations admit a unique local strong or classic solution, see [21, 39, 46, 48, 50, 51, 53] for the case in the absence of vacuum, and [5–7, 15, 18, 31, 49] for the case in the pres- ence of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, the corresponding global existence with general initial data may not be expected, due to the recent finite time blow up results by Merle–Rapha’el– Rodnianski–Szeftel [44, 45], where for the three-dimensional isentropic compressible Navier–Stokes equations with spherical symmetry, regular solutions with finite time singularities are constructed for a class of initial data with far field vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Indeed, up to now, global strong or classical solutions are established only under some ad- ditional conditions on the initial data: the case with small perturbed initial data around non-vacuum equilibriums was achieved by Matsumura–Nishida [40–43], and later developed in many works, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', [3, 4, 8–11, 16, 26, 47, 52];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' while the case with initial data of small energy but allowing large oscillations and vacuum was proved by Huang–Li–Xin [20] and Li–Xin [33] for the isentropic system, and later generalized to the full system in [19, 28, 54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It is worth pointing out that there are some significant differences in the mathe- matical theories for the compressible Navier–Stokes equations between the vacuum and non-vacuum cases and new phenomena may occur depending on the locations and states of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In the non-vacuum case, the solutions can be establish in both the homogeneous and inhomogeneous spaces depending on the properties of the ini- tial data, and the solution spaces guarantee the uniform boundedness of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, these may fail in general in the presence of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Indeed, in the case that the density has compact support, the solution can be established in the homo- geneous spaces, see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', [5–7, 15, 18, 20, 31], but not in the inhomogeneous spaces, see Li–Wang–Xin [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Further more, the blowup results of Xin [56] and Xin–Yan [57] imply that the global solutions established in [19, 28, 54] must have unbounded entropy, if initially there is an isolated mass group surrounded by the vacuum region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 4 JINKAI LI AND ZHOUPING XIN However, it is somewhat surprising that if the initial density vanishes only at far fields with a rate no more than O( 1 |x|2), then, as for the non-vacuum case, the solutions can be established in both the homogeneous and inhomogeneous spaces, and the entropy can be uniformly bounded, see the recent works by the authors [34–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It should be noted that since system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) is already closed, one can indeed establish self-contained mathematical theories for it, as already developed in the pre- vious works mentioned above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, since the second law of the thermodynamics is not taken in to account, these theories are insufficient from the physical point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, some new theories are needed to provide information for the entropy in the presence of vacuum to meet the physical requirements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, due to the lack of the expression and high singularity and degeneracy of the governing equation for the entropy near the vacuum region, in spite of its importance, the mathematical analysis of the entropy for the viscous compressible fluids in the presence of vac- uum was rarely carried out before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Mathematical studies towards this direction has been initiated in our previous works [34, 35] and further developed in [36], where the propagation of the uniform boundedness of the entropy and the inhomogeneous Sobolev regularities was achieved for the compressible Navier–Stokes equations, with or without heat conductivities, in the presence of vacuum at the far fields, under the crucial condition that the initial density decays to vacuum at the rate no faster than O( 1 |x|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In this paper, we continue our studies on the dynamic behavior of the entropy in the presence of vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Different from the cases considered in [34–36], where the density decays slowly to the vacuum at far fields, in the current paper, we investigate the case with fast decaying density at the far fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For simplicity, we study the one-dimensional case in the current paper while leave the multi-dimensional case as future works.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It will be shown in this paper that, in sharp contrast to the cases with slowly decaying density in [34–36], the uniform boundedness of the entropy can not be propagated by the compressible Navier–Stokes equations for viscous and heat conductive ideal gases with constant viscosities and heat conductivities, if the initial density decays faster than the order O( 1 |x|ℓρ ) at the far fields with ℓρ > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since the uniform boundedness of the entropy has already been established in [34–36] if the decay rate is less than O( 1 |x|2), our results in this paper reveal that the decay rate 2 of the initial density at the far field is sharp for the uniform boundedness of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Surprisingly, in case that the initial density decays faster than the order O( 1 x4), some sharper results can be achieved: the temperature is uniformly positive immediately after the initial time, for any general nonnegative (not identically zero) initial tem- perature, and, as a result, the entropy tends to infinity at the order O(− log(̺0(x))) at any positive time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Consider the Cauchy problem to the one-dimensional compressible Navier–Stokes equations for viscous and heat conductive ideal gases ρt + (ρu)x = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 5 ρ(ut + uux) − µuxx + px = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) cvρ(θt + uθx) + pux − κθxx = µ(ux)2, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) where p = Rρθ, subject to the initial condition (ρ, u, θ)|t=0 = (ρ0, u0, θ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) The main results of this paper will be stated and proved in the Lagrangian co- ordinates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' however, since the velocity of the solutions obtained in this paper have Lipschitz regularities in the spatial variable, the results can be transformed back to those in the Eulerian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Define the coordinate transform between the Lagrangian coordinate y and the Eulerian coordinate x as x = η(y, t) satisfying � ∂tη(y, t) = u(η(y, t), t), η(y, 0) = y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set ̺(y, t) := ρ(η(y, t), t), v(y, t) := u(η(y, t), t), ϑ(y, t) := θ(η(y, t), t), and J := J(y, t) = ηy(y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it holds that Jt = vy, J|t=0 ≡ 1, J̺ = ̺0, with ̺0 := ρ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' We still use s to denote the specific entropy in the Lagrangian coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) that s(y, t) = cv � log R A + log ϑ(y, t) − (γ − 1) log ̺0(y) + (γ − 1) log J(y, t) � , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) for any y ∈ R and t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, in the Lagrangian coordinates, the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) becomes Jt = vy, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) ̺0vt − µ �vy J � y + πy = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) cv̺0ϑt + vyπ − κ �ϑy J � y = µ|vy|2 J , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) where π = R ̺0 J ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The initial data can be taken as (J, v, ϑ)|t=0 = (1, v0, ϑ0), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) where v0 = u0 and ϑ0 = θ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The following conventions will be used throughout this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For 1 ≤ q ≤ ∞ and positive integer m, Lq = Lq(R) and W 1,q = W m,q(R) denote the standard Lebesgue and Sobolev spaces, respectively, and Hm = W m,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For simplicity, Lq and Hm denote 6 JINKAI LI AND ZHOUPING XIN also their N product spaces (Lq)N and (Hm)N, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' ∥u∥q is the Lq norm of u, and ∥(f1, f2, · · · , fn)∥X is the sum �N i=1 ∥fi∥X or the equivalent norm ��N i=1 ∥fi∥2 X � 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The main results of this paper are the following three theorems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The first one yields the global existence of a solution to the Cauchy problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), subject to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let the initial density ̺0 be given such that 0 < ̺0 ∈ L1(R)∩W 2,∞(R) and |̺′ 0| + |̺′′ 0| ≤ K1̺0 on R, (H1) for a positive constant K1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume that (v0, ϑ0) satisfies ϑ0 ≥ 0 on R and (√̺0v0, √̺0v2 0, v′ 0, v′′ 0, √̺0ϑ0, √̺0ϑ′ 0, √̺0ϑ′′ 0) ∈ L2(R), G′ 0 √̺0 ∈ L2(R), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) lim y→−∞ |v′ 0(y)| � ̺0(y) + lim y→+∞ |v′ 0(y)| � ̺0(y) < +∞, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) where G0 := µv′ 0 − R̺0ϑ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, there is a global solution (J, v, ϑ) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), subject to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14), satis- fying inf(y,t)∈R×(0,T) J > 0, θ ≥ 0, and Jy √̺0 , Jyy, Jt, Jyt ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), √̺0v, √̺0v2, vy, vyy √̺0 , √̺0vt ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), vyyy, vyt ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), √̺0ϑ, √̺0ϑy, √̺0ϑyy, ̺ 3 2 0 ϑt ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), ϑy ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2(R)), ̺0ϑt, ̺0ϑyt ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), Gt, �Gy ̺0 � y ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(R)), for any positive time T, where G := µ vy J − R ̺0 J ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (i) Condition (H1) allows arbitrary algebraic and even exponential de- cay rate of ̺0 at far fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Indeed, one can check that functions of the forms A (1+y2)ℓ and e−(1+y2)δ, with A, ℓ ∈ (0, ∞) and δ ∈ (0, 1 2], satisfy (H1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 gen- eralizes the global existence result in our previous work [35], where some assumptions on slow decay at far fields on ̺0 are assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (ii) Condition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) is used only to construct suitable approximated initial data for the corresponding initial boundary value problems (which are expected to converge to the Cauchy problem), see Step 1 in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The second theorem gives the immediate unboundedness of the specific entropy if the algebraic decay rate of the initial density is greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 7 Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume, in addition to the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, that (1 + |y|)ℓρ̺0(y) ≤ K2, ∀y ∈ R, (H2) for some positive constants ℓρ ∈ (2, ∞) and K2, and either ϑ0 is not identically zero or v0 is not identically a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let (J, v, ϑ) be a solution to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), subject to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14), satisfying the properties stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, the specific entropy s ̸∈ L∞(R × (0, T)), for any positive time T ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 reveals a completely different phenomenon from that in [34–36], where the initial density decays no faster than O( 1 y2) at far fields, so that the entropy keeps uniformly bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' While Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 shows that if the initial density decays faster than O � 1 |y|ℓρ � , with ℓρ > 2, at far fields, then the entropy becomes not uniformly bounded immediately after the initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Consequently, we have given a complete answer to the problem concerning the propagation of uniform boundedness of entropy for ideal gases in one dimension: the uniform boundedness of the entropy for the ideal gases, in the presence of vacuum at the far fields only in one dimension, can be propagated if and only if the algebraic decay rate of the initial density is not greater than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In other words, the decay rate 2 of the initial density at the far fields is sharp for the uniform boundedness of the entropy in one dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The main ingredients of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 are based on using some scaling transform to transform the far field vacuum to an interior vacuum and applying a Hopf type lemma for a class of linear degenerate elliptic equations with degeneracy in the time variable and possible unbounded coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The scaling transform for the temperature to be used here is f(y, t) := ϑ(y−β, t), y ∈ (0, ∞), t ∈ [0, ∞), for some suitably chosen β > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Similar transform can also be introduced for negative y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to the continuity equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) and the assumption that the initial density reaches vacuum only at the far fields, the density remains positive on any compact interval for all positive time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus the equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) can be regarded a uniform parabolic equation for θ on compact domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Consequently, the temperature will be positive on any finite interval for any positive time t by the strong maximum principle, and thus f is positive for any positive y and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By using the properties of ϑ stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, one can verify that 0 < f ∈ C2,1((0, ∞) × (0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assuming by contradiction that the entropy is uniformly bounded, one can extend f by zero on the positive time axis, such that 0 ≤ f ∈ C([0, ∞) × [0, ∞)) and reaches zero on the positive time axis only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The temperature equation yields a0ft − afyy + bfy + ˜cf ≥ 0, in (0, ∞) × (0, ∞), which motivates us to apply the Hopf type lema to f at the points on the positive time axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By choosing β suitably, one can verify that the coefficients a0 and ˜c are uniformly bounded near the positive time axis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' however, the coefficient b contains an 8 JINKAI LI AND ZHOUPING XIN unbounded term involving 1 y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Fortunately, such an unbounded term in b is of “right” sign while the remaining term in b is uniformly bounded for suitably chosen β, so that the Hopf type lemma still holds (see Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus applying the Hopf type lemma to f near the positive time axis leads to a quantitative asymptotic behavior of the temperature at the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The contradiction comes from the fact that the asymptotic behavior of the temperature derived from the Hopf type lemma is not consistent with that derived from (H2) and the uniform boundedness of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This inconsistency implies that the entropy can not be uniformly bounded and thus Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The third theorem gives the uniform positivity of the temperature and consequently the asymptotic unboundedness of the entropy, which are sharper results than those in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2, under the stronger assumption that the algebraic decay rate of the initial density at the far field is greater than 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume, in addition to the conditions in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, that (1 + |y|)4̺0(y) ≤ K3, ∀y ∈ R, (H3) for a positive constant K3, and either ϑ0 is not identically zero or v0 is not identically a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let (J, v, ϑ) be a solution to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), subject to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14), satisfying the properties stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, the following statements hold: (i) the temperature ϑ satisfies inf y∈R ϑ(y, t) > 0, ∀t ∈ (0, ∞);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (ii) the specific entropy s satisfies R ≤ lim |y|→∞ s(y, t) − log(̺0(y)) ≤ lim |y|→∞ s(y, t) − log(̺0(y)) < ∞, ∀t ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' In particular, s becomes unbounded immediately after the initial time, regardless of whether it is uniformly bounded or not at the initial time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It is an interesting question to show whether Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 still holds in the case that the algebraic decay rate of ̺0 lies between 2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, as already shown in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2, in this case, though the uniform positivity of the temperature is not clear, yet the specific entropy becomes not uniformly bounded in any positive time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Recall that the temperature is positive on any finite interval for any positive time t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' To obtain the positive lower bound for the temperature at any positive time, it suffices to achieve this at far fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' To this end, similar as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2, we apply some scaling technique to transform the far field vacuum to an interior vacuum and take advantage of the Hopf type lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' However, the scaling transform UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 9 introduced before does not work here directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Instead, we apply the Kelvin transform to the temperature ϑ and denote by h the transformed temperature, that is, h(y, t) = yϑ �1 y, t � , ∀y ̸= 0, t ∈ [0, ∞), which satisfies a linear degenerate equation, with all coefficients being uniformly bounded by the assumption (H3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By using the properties of ϑ stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, one can verify that 0 ≤ h ∈ C2,1(Ω) ∩ C(Ω) and more importantly h(0, t) = 0, where Ω = ((−∞, 0)∪(0, ∞))×(0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that different from the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2, here the important property that h(0, t) = 0 holds without any condition on the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By the Hopf type lemma (Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) and applying the strong maximum principle, we can derive that h behaves linearly near the origin at each positive time and hence obtain the uniformly positive lower bound for the temperature near the far fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' With the aid of the positive lower bound of the temperature, the asymptotic unboundedness of the entropy follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) as J has uniform positive lower and upper bounds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The rest of this paper is arranged as follows: in Section 2, we consider a carefully designed initial-boundary value problem for the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) and establish a series of a priori estimates on the solution independent of the length of the spatial interval;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' in Section 3, we obtain the global existence of solutions to the Cauchy problem and thus prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 by taking limit of the solutions obtained in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Section 4 is devoted to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' and finally, the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 is given in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Throughout this paper, C will denote a generic positive constant which may vary from place to place.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Initial-boundary value problem and a priori estimates Throughout this section, we consider the initial-boundary value problem to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), in (α, β) × (0, ∞), with −∞ < α < β < +∞, subject to the initial-boundary conditions: (J, v, ϑ)|t=0 = (1, v0, ϑ0), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) (vy, ϑ)|y=α,β = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) The following global well-posedness can be proved in the same way as in [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let (̺0, v0, ϑ0) ∈ H2((α, β)) be given such that ̺0, ϑ0 ≥ 0 on (α, β) and v′ 0(α) = v′ 0(β) = ϑ0(α) = ϑ0(β) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume that µv′′ 0 − R(̺0ϑ0)′ = √̺0g1, κϑ′′ 0 + µ(v′ 0)2 − Rv′ 0̺0ϑ0 = √̺0g2, for two functions g1, g2 ∈ L2((α, β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 10 JINKAI LI AND ZHOUPING XIN Then, there is a unique global solution (J, v, ϑ) to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), in (α, β)× [0, ∞), subject to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2), satisfying inf(y,t)∈(α,β)×(0,T) J > 0, ϑ ≥ 0, and J ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2((α, β))), Jt ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2((α, β))), v, ϑ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2((α, β))) ∩ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H3((α, β))), vt, ϑt ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H1((α, β))), for any T ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The rest of this section is devoted to deriving the a priori estimates, independent of α and β, on the unique global solution (J, v, ϑ) stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Keeping this in mind, in the rest of this section, we will always assume that (J, v, ϑ) is the solution stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Throughout this section, for simplicity of notations, the norms ∥ · ∥q and ∥ · ∥H1 are the corresponding ones on the interval (α, β), that is, ∥ · ∥q := ∥ · ∥Lq((α,β)) and ∥ · ∥H1 := ∥ · ∥H1((α,β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Denote m0 := � β α ̺0dy, E0 := � β α ̺0 �v2 0 2 + cvϑ0 � dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that � β α ̺0 �v2 2 + cvϑ � dy ≤ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) with v, integrating over (α, β), and by the boundary con- ditions, one gets by integration by parts that 1 2 d dt � β α ̺0v2dy + µ � β α |vy|2 J dy − � β α vyπdy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) Since ϑ ≥ 0 in (α, β) × (0, ∞), it is clear that ϑy(α, t) ≥ 0 and ϑy(β, t) ≤ 0, for any t ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As are result, integrating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) over (α, β) and integration by parts yield cv d dt � β α ̺0ϑdy + � β α vyπdy ≤ µ � β α |vy|2 J dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) Summing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) and integrating with respect to t lead to the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that e− 2 µ √2m0E0 ≤ J ≤ e 4 µ √2m0E0 � 1 + R µ � t 0 ̺0ϑdτ � , ∀t ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since vy|y=α = 0 and J|t=0 = 1, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) that J|y=α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Substi- tuting (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) into (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) yields ̺0vt − µ(log J)yt + πy = 0, UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 11 from which, integrating over (0, t) and using J|t=0 = 1, one can get ̺0(v − v0) + � t 0 πyds = µ(log J)y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Integrating this over (α, y) and noticing that J|y=α = 1 and π|y=α = R ̺0 J ϑ|y=α = 0, one gets � y α ̺0(v − v0)dz + � t 0 πds = µ log J, which leads to J = e 1 µ( � y α ̺0(v−v0)dz+ � t 0 πds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 and the H¨older inequality that � β α ̺0(|v| + |v0|)dz ≤ �� β α ̺0dz � 1 2 ��� β α ̺0v2dz � 1 2 + �� β α ̺0v2 0dz � 1 2� ≤ 2 � 2m0E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) With the aid of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) and since π ≥ 0, it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) that J ≥ e− 1 µ � β α ̺0(|v|+|v0|)dz ≥ e− 2 µ √2m0E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) Rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) as Je− 1 µ � y α ̺0(v−v0)dz = e 1 µ � t 0 πds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus 1 µJπ exp � −1 µ � y α ̺0(v − v0)dz � = ∂t(e 1 µ � t 0 πds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Hence, one gets by noticing Jπ = R̺0ϑ that exp �1 µ � t 0 πds � = 1 + R µ � t 0 ̺0ϑ exp � −1 µ � y α ̺0(v − v0)dz � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Substituting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) lead to J = e 1 µ � y α ̺0(v−v0)dz � 1 + R µ � t 0 ̺0ϑ exp � −1 µ � y α ̺0(v − v0)dz � ds � ≤ e 4 µ √2m0E0 � 1 + R µ � t 0 ̺0ϑds � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Combining this with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) yields the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ In the rest of this section, we will always assumed that C is a general positive constant depending only on R, cv, µ, κ, K1, T, and the upper bound of N0, but inde- pendent of α and β with β − α ≥ 1, where N0 := ∥̺0∥∞ +m0 +E0 + ���� �√̺0v2 0, v′ 0, v′′ 0, √̺0ϑ0, √̺0ϑ′ 0, √̺0ϑ′′ 0, G0, G′ 0 √̺0 ����� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) 12 JINKAI LI AND ZHOUPING XIN Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ∥(√̺0v2, √̺0ϑ)∥2 2 + � T 0 � ∥√̺0ϑ∥2 ∞ + ���� vvy √ J ���� 2 2 + ���� ϑy √ J ���� 2 2 � dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set E = v2 2 + cvϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) that ̺0Et − κ �ϑy J � y = � µvvy J − R̺0 J ϑv � y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that ϑy(α, t) ≥ 0 and ϑy(β, t) ≤ 0 due to the boundary condition ϑ|y=α,β = 0 and the fact that ϑ ≥ 0 in (α, β) × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Multiplying the above equation with E and integration by parts yield 1 2 d dt∥√̺0E∥2 2 + κcv � β α |ϑy|2 J dy − κE ϑy J ��� β y=α ≤ − � β α � µvvy J − R̺0 J ϑv � (vvy + cvϑy)dy − κ � β α ϑy J vvydy ≤ κcv 2 ���� ϑy √ J ���� 2 2 + C � β α 1 J � |vvy|2 + ̺2 0v2ϑ2� dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' and thus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' by the Cauchy inequality and that −κE ϑy J ��� β y=α ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' it follows that d dt∥√̺0E∥2 2 + κcv ���� ϑy √ J ���� 2 2 ≤ A1 ���� vvy √ J ���� 2 2 + A1 � β α 1 J ̺2 0v2ϑ2dy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) for a positive constant A1 depending only on κ, cv, µ, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Multiplying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) with 4v3, using the boundary conditions, and integration by parts, one deduces d dt � β α ̺0v4dy + 12µ ���� vvy √ J ���� 2 2 = 12R � β α 1 J v2vy̺0ϑdy ≤ 6µ ���� vvy √ J ���� 2 2 + C � β α 1 J ̺2 0v2ϑ2dy, and thus, d dt � β α ̺0v4dy + 6µ ���� vvy √ J ���� 2 2 ≤ C � β α 1 J ̺2 0v2ϑ2dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) Multiplying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) with A1 3µ and summing the resultant with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) yield d dt � ∥√̺0E∥2 2 + A1 3µ∥√̺0v2∥2 2 � + κcv ���� ϑy √ J ���� 2 2 + A1 ���� vvy √ J ���� 2 2 ≤ C � β α 1 J ̺2 0v2ϑ2dy, UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 13 from which, by Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3, one gets d dt � ∥√̺0E∥2 2 + A1 3µ∥√̺0v2∥2 2 � + κcv ���� ϑy √ J ���� 2 2 + A1 ���� vvy √ J ���� 2 2 ≤ C∥√̺0v∥2 2∥√̺0ϑ∥2 ∞ ≤ C∥√̺0ϑ∥2 ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) Since ϑ|y=α = 0, it follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' the H¨older and Young inequalities,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='and (H1) that ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='̺0ϑ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='(̺0ϑ2)ydz = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� y ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='(̺′ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='0ϑ2 + 2̺0ϑϑy)dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� β ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='α ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='K1̺0ϑ2 + 2̺0ϑ ϑy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='dz ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='K1∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 + 2∥̺0ϑ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ∥̺0∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4∞∥√̺0ϑ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='ϑy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∥J∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='K1∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 + C∥√̺0ϑ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='ϑy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∥̺0ϑ∥∞dτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='K1∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 + C∥√̺0ϑ∥ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2∞ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='ϑy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='�� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∥̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∞dτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='≤ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∞ + ǫ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='ϑy ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='√ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='J ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='���� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='+ C ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 + ∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� t ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∥√̺0ϑ∥2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='∞dτ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=',' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' and thus ∥√̺0ϑ∥2 ∞ ≤ ǫ ���� ϑy √ J ���� 2 2 + Cǫ � 1 + ∥√̺0ϑ∥2 2 + � t 0 ∥√̺0ϑ∥2 ∞dτ � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) for any ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Choosing ǫ sufficiently small and plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) yield d dt � ∥√̺0E∥2 2 + A1 3µ∥√̺0v2∥2 2 � + A1 ���� vvy √ J ���� 2 2 + κcv 2 ���� ϑy √ J ���� 2 2 ≤ C � 1 + ∥√̺0ϑ∥2 2 + � t 0 ∥√̺0ϑ∥2 ∞dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) leads to d dt � ∥√̺0E∥2 2 + A1 3µ∥√̺0v2∥2 2 + � t 0 ∥√̺0ϑ∥2 ∞dτ � + A1 ���� vvy √ J ���� 2 2 + κ 2 ���� ϑy √ J ���� 2 2 ≤ C � 1 + ∥√̺0E∥2 2 + � t 0 ∥√̺0ϑ∥2 ∞dτ � , 14 JINKAI LI AND ZHOUPING XIN which, together with the Gr¨onwall inequality, implies that sup 0≤t≤T ∥√̺0E∥2 2 + � T 0 � ∥√̺0ϑ∥2 ∞ + ���� vvy √ J ���� 2 2 + ���� ϑy √ J ���� 2 2 � dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This completes the proof of the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' There are two positive constants C and C, such that C ≤ J ≤ C on (α, β) × (0, T), � T 0 ∥vy∥2 2dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The lower bound of J follows directly from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 while the upper bound of J follows from combining Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Testing (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) with v and integrating by parts yield 1 2 d dt∥√̺0v∥2 2 + µ ���� vy √ J ���� 2 2 = R � β α ̺0 J ϑvydy ≤ C ���� vy √ J ���� 2 ∥√̺0ϑ∥2 ≤ µ 2 ���� vy √ J ���� 2 2 + C∥√̺0ϑ∥2 2, where the lower bound of J was used, and thus d dt∥√̺0v∥2 2 + µ ���� vy √ J ���� 2 2 ≤ C∥√̺0ϑ∥2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The second conclusion follows from this, the upper bound of J just proved, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ In the rest of this section, we always assume that β − α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' We will use the following elementary inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that ∥f∥Lp((α,β)) ≤ C(∥f∥L2((α,β)) + ∥f∥ 1 2+ 1 p L2((α,β))∥f ′∥ 1 2− 1 p L2((α,β))), p ∈ [2, ∞], for any f ∈ H1((α, β)), and for a positive constant C depending only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This can be proved by scaling the corresponding inequality in (α, β) to that in (0, 1), applying the Gagliardo-Nirenberg inequality for functions in H1((0, 1)), and using the condition β −α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since the proof is straightforward, and thus is omitted here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Let G be the effective viscous flux, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', G := µvy J − π = µvy J − R̺0ϑ J .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 15 Then, it holds that Gt − µ J �Gy ̺0 � y = −κ(γ − 1) J �ϑy J � y − γ vy J G (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) and G|y=α,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ∥G∥2 2 + � T 0 ����� Gy √̺0 ���� 2 2 + ∥G∥4 ∞ � dt ≤ C(1 + ∥G0∥2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Testing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) with JG, using (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15), Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and the Young inequality, one obtains 1 2 d dt∥ √ JG∥2 2 + µ ���� Gy √̺0 ���� 2 2 = κ(γ − 1) � β α ϑyGy J dy + �1 2 − γ � � β α vyG2dy ≤ C � ∥ϑy∥2 ���� Gy √̺0 ���� 2 + ∥vy∥2∥G∥2 4 � ≤ C � ∥ϑy∥2 ���� Gy √̺0 ���� 2 + ∥vy∥2 � ∥G∥2 2 + ∥G∥ 3 2 2 ∥Gy∥ 1 2 2 �� ≤ µ 2 ���� Gy √̺0 ���� 2 2 + C[∥ϑy∥2 2 + (1 + ∥vy∥2 2)∥G∥2 2], that is, d dt∥ √ JG∥2 2 + µ ���� Gy √̺0 ���� 2 2 ≤ C[∥ϑy∥2 2 + (1 + ∥vy∥2 2)∥G∥2 2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to this and the Gr¨onwall inequality, the desired conclusion, except the es- timate on � T 0 ∥G∥4 ∞dt, follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' While the estimate for � T 0 ∥G∥4 ∞dt follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and the estimate just proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ����� Jy √̺0 ���� 2 2 + ∥vy∥2 2 + ∥Jt∥2 2 � + � T 0 � ∥√̺0vt∥2 2 + ���� vyy √̺0 ���� 2 2 � dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 16 JINKAI LI AND ZHOUPING XIN Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that vy = 1 µ(JG + R̺0ϑ) and √̺0vt = Gy √̺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5, and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 that sup 0≤t≤T ∥vy∥2 2 + � T 0 ∥√̺0vt∥2 2dt ≤ C, which by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) implies sup 0≤t≤T ∥Jt∥2 2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Direct calculations yield Jyt = 1 µ(JGy + JyG + R̺′ 0ϑ + R̺0ϑy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Taking the inner product of the above with Jy ̺0 , one obtains from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and (H1) that 1 2 d dt ���� Jy √̺0 ���� 2 2 ≤ C � β α � |J| ���� Gy √̺0 ���� + ���� Jy √̺0 ���� |G| + √̺0ϑ + √̺0|ϑy| � |Jy| √̺0 dy ≤ C ����� Gy √̺0 ���� 2 + ∥G∥∞ ���� Jy √̺0 ���� 2 + ∥√̺0ϑ∥2 + ∥ϑy∥2 � ���� Jy √̺0 ���� 2 ≤ C(1 + ∥G∥∞) ���� Jy √̺0 ���� 2 2 + C � 1 + ∥ϑy∥2 2 + ���� Gy √̺0 ���� 2 2 � , which, together with the Gr¨onwall inequality, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5, yields sup 0≤t≤T ���� Jy √̺0 ���� 2 2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) Since vyy = 1 µ(JyG + JGy + R̺′ 0ϑ + R̺0ϑy), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17) it follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5, and (H1) that � T 0 ���� vyy √̺0 ���� 2 2 dt ≤ C � T 0 ����� Jy √̺0 ���� 2 2 ∥G∥2 ∞ + ���� � Gy √̺0 , √̺0ϑ, ϑy ����� 2 2 � dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ∥√̺0ϑy∥2 2 + � T 0 (∥̺0ϑt∥2 2 + ∥ϑyy∥2 2)dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 17 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) as cv̺0ϑt − κ �ϑy J � y = vyG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) Note that ϑt|y=α,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Taking the inner product of the above equation with ̺0ϑt yields κ � β α ϑy J (̺0ϑyt + ̺′ 0ϑt)dy + cv∥̺0ϑt∥2 2 = � β α vyG̺0ϑtdy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) It follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) that � β α ϑy J ̺0ϑytdy = 1 2 d dt ���� �̺0 J ϑy ���� 2 2 + 1 2 � β α vy J2̺0|ϑy|2dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Substituting this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) and using (H1) and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, one gets κ 2 d dt ���� �̺0 J ϑy ���� 2 2 + cv∥̺0ϑt∥2 2 = � β α � vyG̺0ϑt − κ 2 vy J2̺0|ϑy|2 − κϑy J ̺′ 0ϑt � dy ≤ � β α � |vy||G|̺0|ϑt| + κ 2 |vy| J2 ̺0|ϑy|2 + κK1 |ϑy| J ̺0|ϑt| � dy ≤ cv 2 ∥̺0ϑt∥2 2 + C(∥G∥2 ∞∥vy∥2 2 + ∥vy∥∞∥√̺0ϑy∥2 2 + ∥ϑy∥2 2), which implies κ d dt ���� �̺0 J ϑy ���� 2 2 + cv∥̺0ϑt∥2 2 ≤ C � ∥G∥2 ∞∥vy∥2 2 + (∥G∥∞ + ∥̺0ϑ∥∞)∥√̺0ϑy∥2 2 + ∥ϑy∥2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from this, the Gr¨onwall inequality, Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6, and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 that sup 0≤t≤T ∥√̺0ϑy∥2 2 + � T 0 ∥̺0ϑt∥2 2dt ≤ CeC � T 0 (∥G∥∞+∥̺0ϑ∥∞)dt � ∥√̺0ϑ′ 0∥2 2 + � T 0 (∥G∥2 ∞∥vy∥2 2 + ∥ϑy∥2 2)dt � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20) Direct calculations and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) yield κϑyy = κ �ϑy J � y J + κϑy J Jy = J(cv̺0ϑt − vyG) + κϑy J Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 18 JINKAI LI AND ZHOUPING XIN It follows from this, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20), Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 that � T 0 ∥ϑyy∥2 2dt ≤ C � T 0 � ∥̺0ϑt∥2 2 + ∥vy∥2 2∥G∥2 ∞ + ∥ϑy∥2 ∞∥Jy∥2 2 � dt ≤ C + C � T 0 ∥ϑy∥2 ∞dt ≤ C + C � T 0 ∥ϑy∥2(∥ϑy∥2 + ∥ϑyy∥2)dt ≤ 1 2 � T 0 ∥ϑyy∥2 2dt + C, and thus � T 0 ∥ϑyy∥2 2dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ���� Gy √̺0 ���� 2 2 + � T 0 \uf8eb \uf8ed∥Gt∥2 2 + ����� �Gy ̺0 � y ����� 2 2 \uf8f6 \uf8f8 dt ≤ C � 1 + ���� G′ 0 √̺0 ���� 2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) yields Gt − µ J �Gy ̺0 � y = −R J ̺0ϑt − vy J G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that Gt|y=α,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Multiplying the above with JGt, integrating by parts, and using Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 yield µ 2 d dt ���� Gy √̺0 ���� 2 2 + ∥ √ JGt∥2 2 = − � β α (R̺0ϑt + vyG) Gtdy ≤ 1 2∥ √ JGt∥2 2 + C(∥̺0ϑt∥2 2 + ∥vy∥2 2∥G∥2 ∞), from which, by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7, the conclusion follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T ����̺ 3 2 0 ϑt ��� 2 2 + ∥√̺0ϑyy∥2 2 � + � T 0 ∥̺0ϑyt∥2 2dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that vy = 1 µ(JG + R̺0ϑ) and vyt = 1 µ(JGt + vyG + R̺0ϑt).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21) It follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18), and direct calculations that cv̺0ϑtt − κ �ϑyt J � y = −κ �vyϑy J2 � y + vy µ G2 + 1 µ(2JG + R̺0ϑ)Gt + R µ ̺0ϑtG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 19 Note that ϑt|y=α,β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Multiplying the above equation with ̺2 0ϑt and integrating by parts yield cv 2 d dt ���̺ 3 2 0 ϑt ��� 2 2 + κ � β α ϑyt J (̺2 0ϑyt + 2̺0̺′ 0ϑt)dy = 1 µ � β α [vyG2 + (2JG + R̺0ϑ)Gt + R̺0ϑtG]̺2 0ϑtdy +κ � β α vyϑy J2 (̺2 0ϑyt + 2̺0̺′ 0ϑt)dy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 and (H1), one deduces cv 2 d dt∥̺ 3 2 0 ϑt∥2 2 + κ ���� ̺0 √ J ϑyt ���� 2 2 ≤ C � β α [̺2 0|ϑt||ϑyt| + |vy||ϑy|(̺2 0|ϑyt| + ̺2 0|ϑt|)]dy +C � β α [|vy|G2 + (|G| + ̺0ϑ)|Gt| + ̺0|ϑt||G|]̺2 0|ϑt|dy ≤ κ 2 ���� ̺0 √ J ϑyt ���� 2 2 + C(∥̺0ϑt∥2 2 + ∥vy∥2 ∞∥̺0ϑy∥2 2) + C∥G∥2 ∞(∥vy∥2 2 + ∥̺2 0ϑt∥2 2) +C∥Gt∥2 2 + C(∥G∥2 ∞ + ∥̺0ϑ∥2 ∞)∥̺2 0ϑt∥2 2 + C∥G∥∞∥̺ 3 2 0 ϑt∥2 2, from which, by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7 and vy = 1 µ(JG + R̺0ϑ), one obtains cv d dt∥̺ 3 2 0 ϑt∥2 2 + κ ���� ̺0 √ J ϑyt ���� 2 2 ≤ C(∥G∥2 ∞ + ∥̺0ϑ∥2 ∞ + 1)∥̺ 3 2 0 ϑt∥2 2 + C(∥vy∥2 ∞ + ∥G∥2 ∞ + ∥Gt∥2 2 + ∥̺0ϑt∥2 2) ≤ C(∥G∥2 ∞ + ∥̺0ϑ∥2 ∞ + 1)∥̺ 3 2 0 ϑt∥2 2 + C(∥G∥2 ∞ + ∥̺0ϑ∥2 ∞ + ∥Gt∥2 2 + ∥̺0ϑt∥2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Applying the Gr¨onwall inequality to the above, one can get by Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8, and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 that sup 0≤t≤T ∥̺ 3 2 0 ϑt∥2 2 + � T 0 ∥̺0ϑyt∥2 2dt ≤ CeC � T 0 (∥G∥2 ∞+∥̺0ϑ∥2 ∞)dt ��̺ 3 2 0 ϑt ��2 2 ��� t=0 +CeC � T 0 (∥G∥2 ∞+∥̺0ϑ∥2 ∞)dt � T 0 (∥G∥2 ∞ + ∥̺0ϑ∥2 ∞ + ∥Gt∥2 2 + ∥̺0ϑt∥2 2)dt ≤ C(1 + ∥√̺0ϑ′′ 0∥2 2 + ∥√̺0v′ 0G0∥2 2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) 20 JINKAI LI AND ZHOUPING XIN where the fact that ̺ 3 2 0 ϑt|t=0 = √̺0 cv (κϑ′′ 0 + v′ 0G0) has been used, which follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, noticing that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 implies ∥√̺0v′ 0G0∥2 2 ≤ C∥v′ 0∥2 ∞∥G0∥2 2 ≤ C∥v′ 0∥2 H1∥G0∥2 2, one gets from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) that sup 0≤t≤T ∥̺ 3 2 0 ϑt∥2 2 + � T 0 ∥̺0ϑyt∥2 2dt ≤ C(1 + ∥√̺0ϑ′′ 0∥2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23) Note that ϑyy = J �ϑy J � y + ϑy J Jy = 1 κ(cv̺0ϑt − vyG)J + 1 J ϑyJy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from this, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23), Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6, and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 that ∥√̺0ϑyy∥2 2 ≤ C(∥̺ 3 2 0 ϑt∥2 2 + ∥vy∥2 2∥G∥2 ∞ + ∥√̺0ϑy∥2 ∞∥Jy∥2 2) ≤ C(1 + ∥G∥2 ∞ + ∥√̺0ϑy∥2 ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='24) It remains to estimate ∥G∥2 ∞ and ∥√̺0ϑy∥2 ∞ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, Propo- sition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5, and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8 imply that ∥G∥2 ∞ ≤ C∥G∥2(∥G∥2 + ∥Gy∥2) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25) By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 and (H1), and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7, it holds that ∥√̺0ϑy∥2 ∞ ≤ C∥√̺0ϑy∥2 � ∥√̺0ϑy∥2 + ∥√̺0ϑyy∥2 + ���� ̺′ 0 √̺0 ϑy ���� 2 � ≤ C(1 + ∥√̺0ϑyy∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='26) Plugging (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='26) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='24) and using the Cauchy inequality yield ∥√̺0ϑyy∥2 2 ≤ C(1 + ∥√̺0ϑyy∥2) ≤ ∥√̺0ϑyy∥2 2 2 + C, which gives ∥√̺0ϑyy∥2 2 ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T � ∥√̺0vt∥2 2 + ���� vyy √̺0 ���� 2 2 � + � T 0 (∥vyt∥2 2 + ∥vyyy∥2 2 + ∥Jyy∥2 2)dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The estimate for √̺0vt follows directly from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8 since √̺0vt = Gy √̺0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from (H1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8 that ���� vyy √̺0 ���� 2 2 ≤ C ����� Jy √̺0 ���� 2 2 ∥G∥2 ∞ + ���� Gy √̺0 ���� 2 2 + ∥√̺0ϑy∥2 2 + ∥√̺0ϑ∥2 2 � ≤ C, UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 21 � T 0 ∥vyt∥2 2dt ≤ C � T 0 (∥vy∥2 2∥G∥2 ∞ + ∥Gt∥2 2 + ∥̺0ϑt∥2 2)dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Noticing that vyyy = 1 µ(JyyG + 2JyGy + JGyy + R̺′′ 0ϑ + 2R̺′ 0ϑy + R̺0ϑyy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' one can get from (H1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25), Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8 that � t 0 ∥vyyy∥2 2dτ ≤ C � t 0 (∥Jyy∥2 2∥G∥2 ∞ + ∥Jy∥2 ∞∥Gy∥2 2 + ∥Gyy∥2 2 +∥̺0ϑ∥2 2 + ∥̺0ϑy∥2 2 + ∥̺0ϑyy∥2 2)dτ ≤ C � t 0 (∥Jyy∥2 2 + ∥Jy∥2 ∞ + ∥Gyy∥2 2)dτ + C, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='27) where ∥G∥2 ∞ ≤ C(∥G∥2 2 + ∥Gy∥2 2) guaranteed by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 wa used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Next, ∥Jy∥2 ∞ and ∥Gyy∥2 2 can be estimated as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6 imply that ∥Jy∥2 ∞ ≤ C(∥Jy∥2 2 + ∥Jy∥2∥Jyy∥2) ≤ C(1 + ∥Jyy∥2 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='28) While (H1) and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8 yield � T 0 ∥Gyy∥2 2dt ≤ � T 0 ������̺0 �Gy ̺0 � y ����� 2 + ����̺′ 0 Gy ̺0 ���� 2 �2 dt ≤ C � T 0 \uf8eb \uf8ed ����� �Gy ̺0 � y ����� 2 2 + ∥Gy∥2 2 \uf8f6 \uf8f8 dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from this, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='27), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='28) that � t 0 ∥vyyy∥2 2dτ ≤ C � 1 + � t 0 ∥Jyy∥2 2dτ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='29) Since Jyy = � t 0 vyyydτ, one has � t 0 ∥Jyy∥2 2dτ ≤ � t 0 ���� � τ 0 vyyydτ ′ ���� 2 2 dτ ≤ C � t 0 �� τ 0 ∥vyyy∥2 2dτ ′ � dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='30) Plugging this into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='29) leads to � t 0 ∥vyyy∥2 2dτ ≤ C + C � t 0 �� τ 0 ∥vyyy∥2 2dτ ′ � dτ, which implies � T 0 ∥vyyy∥2 2dt ≤ CeT ≤ C by the Gr¨onwall inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This, together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='30), shows that � t 0 ∥Jyy∥2 2dτ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ 22 JINKAI LI AND ZHOUPING XIN Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that sup 0≤t≤T (∥Jyy∥2 2 + ∥Jyt∥2 2) ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This follows from Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10 by using Jyy = � t 0 vyyydτ and Jyt = vyy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It holds that � T 0 ∥ϑyyy∥2 2dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 and Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11, one has ∥Jy∥∞ + ∥Jyy∥2 + ∥vy∥∞ + ∥G∥∞ ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='31) It follows from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) that ϑyyy = cv κ (̺′ 0Jϑt + 2̺0Jyϑt + ̺0Jϑyt) + ϑy J Jyy −1 κ(2JyvyG + JvyyG + JvyGy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, (H1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='31), and Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11, one deduces � T 0 ∥ϑyyy∥2 2dt ≤ C � T 0 (∥̺0ϑt∥2 2 + ∥Jy∥2 ∞∥̺0ϑt∥2 2 + ∥̺0ϑyt∥2 2 + ∥ϑy∥2 ∞∥Jyy∥2 2 +∥Jy∥2 ∞∥vy∥2 2∥G∥2 ∞ + ∥vyy∥2 2∥G∥2 ∞ + ∥vy∥2 ∞∥Gy∥2 2)dt ≤ C � T 0 (∥̺0ϑt∥2 2 + ∥̺0ϑyt∥2 2 + ∥ϑy∥2 2 +∥ϑyy∥2 2 + ∥vy∥2 2 + ∥vyy∥2 2 + ∥Gy∥2 2)dt, where ∥ϑy∥2 ∞ ≤ C(∥ϑy∥2 2 + ∥ϑyy∥2 2) guaranteed by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 was used, from which, by Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 and Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10, it follows � T 0 ∥ϑyyy∥2 2dt ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This proves the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ As a consequence of Propositions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2–2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12 and Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, one has: Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let (J, v, ϑ) be the unique global solution stated in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 to system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), subject to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2), and N0 be given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' for any T ∈ [0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' ∞),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' it holds that inf (α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)×(0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='T) J ≥ CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' sup 0≤t≤T ���� � Jy √̺0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Jyy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Jt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Jyt ����� 2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) ≤ CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' sup 0≤t≤T ���� �√̺0v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' √̺0v2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' vy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' vyy √̺0 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' √̺0vt ����� 2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) + � T 0 ∥(vyyy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' vyt)∥2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β))dt ≤ CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 23 sup 0≤t≤T � ∥̺0ϑ∥L1((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) + ��� �√̺0ϑ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' √̺0ϑy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' √̺0ϑyy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' ̺ 3 2 0 ϑt ���� 2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) � + � T 0 � ∥ϑy∥2 H2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) + ∥(̺0ϑt,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' ̺0ϑyt)∥2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) � dt ≤ CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' sup 0≤t≤T ���� Gy √̺0 ���� 2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) + � T 0 \uf8eb \uf8ed ����� �Gy ̺0 � y ����� 2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) + ∥Gt∥2 L2((α,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='β)) \uf8f6 \uf8f8 dt ≤ CT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' where CT and CT are positive constants depending only on R,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' cv,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' µ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' κ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' K1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' and the upper bound of N0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' but independent of α and β with β − α ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Global existence of solutions: proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The proof is given in three steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Approximations of the initial data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By the assumption (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16), there are two sequences {αn}∞ n=1 and {βn}∞ n=1, with limn→∞ αn = −∞ and limn→∞ βn = ∞, and a positive constant M0, such that ����� v′ 0(αn) � ̺0(αn) ����� + ����� v′ 0(βn) � ̺0(βn) ����� ≤ M0, ∀n ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) Set In = (αn − 1, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For each n, choose 0 ≤ χn ∈ C∞ 0 (In), satisfying χ ≡ 1 on [αn, βn], 0 ≤ χn ≤ 1 and |χ′ n| + |χ′′ n| ≤ C0 on In, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) for a positive constant C0 independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Define v0n and ϑ0n as ϑ0n = ϑ0χn, and v0n = \uf8f1 \uf8f2 \uf8f3 v0(αn) + 2 πv′ 0(αn) sin �π 2(y − αn) � , y ∈ [αn − 1, αn], v0(y), y ∈ [αn, βn], v0(βn) + 2 πv′ 0(βn) sin �π 2(y − βn) � , y ∈ [βn, βn + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It can be checked easily that v′ 0n(αn − 1) = v′ 0n(βn + 1) = ϑ0n(αn − 1) = ϑ0n(βn + 1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) Noticing that v0n(αn) = v0(αn), v′ 0n(αn) = v′ 0(αn), v0n(βn) = v0(βn), v′ 0n(βn) = v′ 0(βn), and since v0 ∈ H2 loc(R) and 0 ≤ ϑ0 ∈ H2 loc(R), one has v0n ∈ H2(In), 0 ≤ ϑ0n ∈ H2(In).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) Due to 0 ≤ χn ≤ 1, it is clear that ∥√̺0ϑ0n∥L2(In) ≤ ∥√̺0ϑ0∥2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) 24 JINKAI LI AND ZHOUPING XIN For any y ∈ [αn − 1, αn), the definition of v0n implies that |v0n(y) − v0(y)| ≤ |v0(αn) − v0(y)| + 2 π|v′ 0(αn)| ≤ 2∥v′ 0∥∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Similarly, it holds that |v0n(y) − v0(y)| ≤ 2∥v′ 0∥∞, for any y ∈ (βn, βn + 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As a result, one has |v0n(y) − v0(y)| ≤ 2∥v′ 0∥∞, ∀y ∈ In.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) Hence ∥√̺0v0n∥L2(In) ≤ ∥√̺0(v0n − v0)∥L2(In) + ∥√̺0v0∥L2(In) = 2∥v′ 0∥∞∥̺0∥ 1 2 1 + ∥√̺0v0∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) and ∥√̺0|v0n|2∥L2(In) ≤ 2 ��√̺0(|v0|2 + |v0 − v0n|2) �� L2(In) ≤ 2∥√̺0|v0|2∥2 + 8∥v′ 0∥2 ∞∥̺0∥ 1 2 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) and direct calculations that ∥√̺0ϑ′ 0n∥L2(In) ≤ ∥√̺0ϑ′ 0∥2 + C0∥√̺0ϑ0∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) ∥√̺0ϑ′′ 0n∥L2(In) ≤ ∥√̺0ϑ′′ 0∥2 + 2C0(∥√̺0ϑ′ 0∥2 + ∥√̺0ϑ0∥2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) By direct calculations, one gets by the Sobolev inequality that ∥v′ 0n∥H1(In) ≤ ∥v′ 0∥H1 + C(|v′ 0(αn)| + |v′ 0(βn)|) ≤ ∥v′ 0∥H1((αn,βn)) + ∥v′ 0∥H1((αn−1,αn)∪(βn,βn+1)) ≤ C∥v′ 0∥2 H1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) for a positive constant C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set G0n = µv′ 0n − R̺0ϑ0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) leads to ∥G0n∥L2(In) ≤ µ∥v′ 0n∥L2(In) + R∥̺0∥ 1 2∞∥√̺0ϑ0n∥L2(In) ≤ C(∥v′ 0∥H1 + ∥̺0∥ 1 2∞∥√̺0ϑ0∥2), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) for a positive constant C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For y ∈ (βn, βn + 1), one has ̺0(βn) ̺0(y) = 1 + � y βn k(z)̺0(βn) ̺0(z) dz, where k(z) = −̺′ 0(z) ̺0(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) By (H1), it holds that |k(z)| ≤ K1, for any z ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set f(y) = 1 + � y βn k(z)̺0(βn) ̺0(z) dz, ∀y ∈ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 25 Then, it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) that f ′(y) = k(y)̺0(βn) ̺0(y) = k(y)f(y), and thus f(y) = e � y βn k(z)dzf(βn) = e � y βn k(z)dz ≤ eK1, ∀y ∈ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from this and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) that ̺0(βn) ̺0(y) = f(y) ≤ eK1, ∀y ∈ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) Similarly, one has ̺0(αn) ̺0(y) ≤ eK1, ∀y ∈ (αn − 1, αn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) Recall that G0n = µv′ 0n − R̺0ϑ0n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, direct calculations yield G′ 0n √̺0 = \uf8f1 \uf8f4 \uf8f4 \uf8f2 \uf8f4 \uf8f4 \uf8f3 −π 2µ v′ 0(αn) √̺0 sin �π 2(y − αn) � − R �√̺0ϑ′ 0n + ̺′ 0 √̺0ϑ0n � , y ∈ (αn − 1, αn), G′ 0 √̺0, y ∈ (αn, βn), −π 2µ v′ 0(βn) √̺0 sin � π 2(y − βn) � − R �√̺0ϑ′ 0n + ̺′ 0 √̺0ϑ0n � , y ∈ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) that ����� v′ 0(αn) � ̺0(y) ����� + ����� v′ 0(βn) � ̺0(y) ����� = ����� v′ 0(αn) � ̺0(αn) � ̺0(αn) ̺0(y) ����� + ����� v′ 0(βn) � ̺0(βn) � ̺0(βn) ̺0(y) ����� ≤ 2M0e K1 2 , ∀y ∈ (αn − 1, αn) ∪ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This together with (H1) yields ����� G′ 0n(y) � ̺0(y) ����� ≤ πµM0e K1 2 + R (√̺0|ϑ′ 0n| + K1 √̺0ϑ0n) , for any y ∈ (αn − 1, αn) ∪ (βn, βn + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to this and that G′ 0n √̺0 = G′ 0 √̺0 on (αn, βn), it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) that ���� G′ 0n √̺0 ���� L2(In) ≤ ���� G′ 0 √̺0 ���� 2 + 2µπM0eK1/2 + R � ∥√̺0ϑ′ 0n∥L2(In) + K1∥√̺0ϑ0n∥L2(In) � ≤ ���� G′ 0 √̺0 ���� 2 + C (∥√̺0ϑ′ 0∥2 + ∥√̺0ϑ0∥2 + 1) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) for a positive constant C independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Solutions to the system in In × (0, ∞) and a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 26 JINKAI LI AND ZHOUPING XIN For each positive integer n, let (v0n, ϑ0n) be the initial data constructed as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Consider the initial-boundary value problem to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) in (αn − 1, βn + 1) × (0, ∞), subject to (J, v, ϑ)|t=0 = (1, v0n, ϑ0n), (vy, ϑ)|y=an−1,βn+1 = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17) Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4), and noticing that infy∈In ̺0 > 0, one can verify that the initial datum (v0n, ϑ0n) satisfies all the assumptions in Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, for each fixed n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, there is a unique global strong solution (Jn, vn, ϑn) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Moreover, due to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16), it follows from Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 that inf In×(0,T) Jn ≥ CT , ϑn(y, t) ≥ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) sup 0≤t≤T ���� �∂yJn √̺0 , ∂2 yJn, ∂tJn, ∂ytJn ����� 2 L2(In) ≤ CT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) sup 0≤t≤T ���� �√̺0vn, √̺0v2 n, ∂yvn, ∂2 yvn √̺0 , √̺0∂tvn ����� 2 L2(In) + � T 0 ∥(∂3 yvn, ∂2 ytvn)∥2 L2(In)dt ≤ CT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20) sup 0≤t≤T � ∥̺0ϑn∥L1(In) + ��� �√̺0ϑn, √̺0∂yϑn, √̺0∂2 yϑn, ̺ 3 2 0 ∂tϑn ���� 2 L2(In) � + � T 0 � ∥∂yϑn∥2 H2(In) + ∥(̺0∂tϑn, ̺0∂ytϑn)∥2 L2(In) � dt ≤ CT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21) and sup 0≤t≤T ���� ∂yGn √̺0 ���� 2 L2(In) + � T 0 \uf8eb \uf8ed ����� �∂yGn ̺0 � y ����� 2 L2(In) + ∥∂tGn∥2 L2(In) \uf8f6 \uf8f8 dt ≤ CT, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) for any positive time T, where Gn = µ ∂yvn Jn − R ̺0 Jnϑn, and CT and CT are positive constants independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Convergence and existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to the a priori estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) and inf(−k,k) ̺0(y) > 0 for any k ∈ N, the following estimate holds ∥(Jn, vn, ϑn)∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H2((−k,k))) + ∥(vn, ϑn)∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H3((−k,k))) + ∥∂tJn∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−k,k))) + ∥(∂tvn, ∂tϑn)∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2((−k,k)))∩L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−k,k))) ≤ Ck,T, ∀k ∈ N, for a positive constant Ck,T independent of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to this and the Cantor’s diagonal argument, there is a subsequence, still denoted by (Jn, vn, ϑn), and (J, v, ϑ), such that (Jn, vn, ϑn) ∗⇀ (J, v, ϑ), in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2((−k, k))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23) UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 27 (vn, ϑn) ⇀ (v, ϑ), in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H3((−k, k))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='24) ∂tJn ∗⇀ Jt, in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H1((−k, k))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25) (∂tvn, ∂tϑn) ∗⇀ (vt, ϑt), in L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2((−k, k))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='26) (∂tvn, ∂tϑn) ⇀ (vt, ϑt), in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H1((−k, k))), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='27) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Moreover, since H3((−k, k)) ֒→֒→ C2([−k, k]) and H2((−k, k)) ֒→֒→ C1([−k, k]), it follows from the Aubin–Lions lemma that (Jn, vn, ϑn) → (J, v, ϑ), in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' C1([−k, k])), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='28) (vn, ϑn) → (v, ϑ), in L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' C2([−k, k])), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='29) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to these and by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18), one has inf (y,t)∈R×(0,T) J(y, t) ≥ CT, 1 Jn → 1 J in C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' C1([−k, k])), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='30) for any k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='30) and noticing that (v0n, ϑ0n) → (v0, ϑ0) in H2((−L, L)) for any L > 0, one can take the limit as n → ∞ to show that (J, v, ϑ) is a solution to the Cauchy problem to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) subject to (J, v, ϑ)|t=0 = (1, v0, ϑ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The desired regularities of (J, v, ϑ) stated in Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 follow from the a priori estimates (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) and convergence (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23)–(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='29) by the weakly lower semi-continuity of norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This proves Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' A Hopf type lemma and unboundedness of the entropy In this section, we prove the unboundedness of the entropy immediately after the initial time, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As stated in the Introduction, this is based on some suitable scaling transform and a Hopf type lemma for a class of general linear degenerate equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' So, we first establish a Hopf type lemma in the first subsection and then present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 in the second subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The Hopf type lemma has its own independent interests and will also be applied to prove the uniform positivity of the temperature in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' A Hopf type lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since the results in this subsection hold in any di- mension, we use the following notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Denote by x = (x1, x2, · · · , xn) and t the spatial and time variables respectively and P = (x, t) a point in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For P0 = (x0, t0) ∈ Rn+1 and r > 0, denote Br(P0) := � (x, t) ∈ Rn+1���|x − x0|2 + (t − t0)2 < r2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let (aij)n×n, a0, b = (b1, b2, · · · , bn), and c be given functions satisfying suitable properties to be specified later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Consider the operator L ϕ = −aij∂ijϕ + a0∂tϕ + b · ∇ϕ + cϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 28 JINKAI LI AND ZHOUPING XIN Note that here a0 is not required to have fixed sign and this linear operator can be regarded only as a linear degenerate elliptic operator in the space and time variables with degeneracy occurring in the time direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let O be a domain in Rn+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume that aij, a0, b, and c are finitely valued functions in O with c ≥ 0, and the matrix (aij)n×n is nonnegative definite in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, for any ϕ ∈ C2,1(O) ∩ C(O), satisfying L ϕ > 0 in O, and ϕ|∂O ≥ 0, it holds that ϕ > 0 in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Here C2,1(O) denotes the space of all function f satisfying f, ∂tf, ∇f, ∇2f ∈ C(O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' First, we claim that ϕ ≥ 0 in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Otherwise, since ϕ ≥ 0 on ∂O and ϕ ∈ C(O), there is P0 ∈ O, such that ϕ(P0) = minO ϕ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since ϕ ∈ C2,1(O), it is clear that ∂tϕ(P0) = ∇ϕ(P0) = 0 and ∇2ϕ(P0) is nonnegative definite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' As a result (L ϕ)(P0) = −aij(P0)∂ijϕ(P0) + c(P0)ϕ(P0) ≤ 0, which contradicts to the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, the claim holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Next, we show that ϕ > 0 in O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Otherwise, there is P ∗ 0 ∈ O, such that ϕ(P ∗ 0 ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, ϕ(P ∗ 0 ) = minO ϕ = 0, from which, similar as before, one has (L ϕ)(P ∗ 0 ) ≤ 0, contradicting to the assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, ϕ > 0 in O, which proves the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 (Hopf type lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Given P0 = (x0, t0), r > 0, P∗ = (x∗, t∗) ∈ ∂Br(P0), x∗ ̸= x0, and set P ∗ 0 = (x∗ 0, t∗ 0), with x∗ 0 = x0+x∗ 2 and t∗ 0 = t0+t∗ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume that there are positive constants λ, Λ, δ∗, and C∗, with δ∗ < |x0−x∗| 4 , such that \uf8f1 \uf8f2 \uf8f3 λ|ξ|2 ≤ aij(x, t)ξiξj ≤ Λ|ξ|2, ∀ξ ∈ Rn, (t − t∗ 0)a0(x, t) + (x − x∗ 0) · b(x, t) ≥ −C∗, 0 ≤ c(x, t) � |x − x∗|2 + (t − t∗)2 ≤ C∗, ∀(x, t) ∈ B r 2(P ∗ 0 ) ∩ Bδ∗(P∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let ϕ ∈ C2,1(Br(P0)) ∩ C(Br(P0)) satisfy L ϕ ≥ 0, ϕ > ϕ(P∗), in B r 2(P ∗ 0 ) ∩ Bδ∗(P∗), ϕ(P∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it holds that lim ℓ→0+ ϕ(P∗) − ϕ(P∗ − ℓn∗) ℓ < 0, where n∗ = P∗−P0 r is the unit outward normal vector to ∂Br(P0) at P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set D = B r 2(P ∗ 0 ) ∩ Bδ∗(P∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It suffices to consider the case that ϕ(P∗) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Otherwise, one may consider Φ := ϕ − ϕ(P∗), which reduces to the case considered, due to L Φ = L ϕ − L (ϕ(P∗)) = L ϕ − cϕ(P∗) ≥ L ϕ ≥ 0 in D, UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 29 as ϕ(P∗) ≤ 0 and c ≥ 0 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It is clear that B r 2(P ∗ 0 ) ⊂ Br(P0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By assumption, it holds that ϕ(P) > ϕ(P∗) = 0, ∀P ∈ D \\ {P∗}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) Define φ(x, t) = e−ζ(|x−x∗ 0|2+(t−t∗ 0)2) − e− r2 4 ζ = e−ζ|P −P ∗ 0 |2 − e− r2 4 ζ, where ζ > 0 is a constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it follows from direct calculations that L φ = − e−ζ|P −P ∗ 0 |2� 4(x − x∗ 0)TA(x − x∗ 0)ζ2 − 2trAζ + 2((t − t∗ 0)a0 + (x − x∗ 0) · b)ζ + c � eζ(|P −P ∗ 0 |2− r2 4 ) − 1 � � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) where A = (aij)n×n and trA = aii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that the assumptions imply 4(x − x∗ 0)TA(x − x∗ 0)ζ2 − 2trAζ + 2((t − t∗ 0)a0 + (x − x∗ 0) · b)ζ ≥4λ|x − x∗ 0|2ζ2 − 2nΛζ − 2C∗ζ ≥ λ 4|x0 − x∗|2ζ2 − (2nΛ + 2C∗)ζ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) for any (x, t) ∈ D, due to trA ≤ nΛ and |x − x∗ 0| ≥ |x∗ 0 − x∗| − |x∗ − x| ≥ |x0 − x∗| 2 − δ∗ ≥ |x0 − x∗| 4 , ∀(x, t) ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that |P − P ∗ 0 | < r 2 for any P ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from the mean value theorem and the triangular inequality that ���eζ(|P −P ∗ 0 |2− r2 4 ) − 1 ��� = eτζ(|P −P ∗ 0 |2− r2 4 ) ����|P − P ∗ 0 |2 − r2 4 ���� ζ ≤ ���|P − P ∗ 0 | − r 2 ��� ���|P − P ∗ 0 | + r 2 ��� ζ ≤ rζ ��|P − P ∗ 0 | − |P ∗ 0 − P∗| �� ≤ rζ|P − P∗|, for any P ∈ D, where τ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This, together with the assumptions, yields ���c � eζ(|P −P ∗ 0 |2− r2 4 ) − 1 ���� ≤ cr|P − P∗|ζ ≤ C∗rζ, ∀P ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) Combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) leads to 4(x − x∗ 0)TA(x − x∗ 0)ζ2 − 2trAζ +2((t − t∗ 0)a0 + (x − x∗ 0) · b)ζ + c � eζ(|P −P ∗ 0 |2− r2 4 ) − 1 � ≥ λ 4|x0 − x∗|2ζ2 − (2nΛ + 2C∗ + rC∗)ζ > 0, ∀P ∈ D, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) if ζ > ζ0 := 8nΛ+8c∗+4rC∗ λ|x0−x∗|2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Choose ζ = 2ζ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) that L φ < 0 in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) 30 JINKAI LI AND ZHOUPING XIN It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) that ϕ ≥ 0 = φ on ∂B r 2(P ∗ 0 ) ∩ Bδ∗(P∗), inf ∂Bδ∗(P∗)∩B r 2 (P ∗ 0 ) ϕ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, for ε > 0 sufficiently small, it follows from the assumptions and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) that L ϕ ≥ 0 > L (εφ) in D, ϕ ≥ εφ on ∂D, and thus L (ϕ − εφ) > 0 in D, ϕ − εφ ≥ 0 on ∂D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' With the aid of this, noticing that ϕ − εφ ∈ C2,1(D) ∩ C(D), and applying Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, one gets ϕ > εφ in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, for ℓ > 0 sufficiently small, one has ϕ(P∗) − ϕ(P∗ − ℓn∗) = −ϕ(P∗ − ℓn∗) < −εφ(P∗ − ℓn∗) = ε(φ(P∗) − φ(P∗ − ℓn∗)) and thus lim ℓ→0+ ϕ(P∗) − ϕ(P∗ − ℓn∗) ℓ ≤ ε lim ℓ→0+ φ(P∗) − φ(P∗ − ℓn∗) ℓ = ε∂n∗φ(P∗) = −εζre− r2 4 ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This proves the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ As a direct consequence of Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2, the following corollary holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Given P0 = (x0, t0), r > 0, P∗ = (x∗, t∗) ∈ ∂Br(P0), x∗ ̸= x0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume that a0, b, c ∈ L∞(Br(P0)), c ≥ 0 in Br(P0), and λ|ξ|2 ≤ aij(x, t)ξiξj ≤ Λ|ξ|2, ∀ξ ∈ Rn, (x, t) ∈ Br(P0), for some positive constants λ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let ϕ ∈ C2,1(Br(P0)) ∩ C(Br(P0)) satisfy L ϕ ≥ 0, ϕ > ϕ(P∗), in Br(P0), ϕ(P∗) ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it holds that lim ℓ→0+ ϕ(P∗) − ϕ(P∗ − ℓn∗) ℓ < 0, where n∗ = P∗−P0 r is the unit outward normal vector to ∂Br(P0) at P∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 31 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Unboundedness of the entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This subsection is devoted to proving The- orem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' We start with the following embedding lemma, which is used to verify the H¨older regularity of Jy required in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let L > 0 be a positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, the following embedding in- equality holds ∥f∥C 1 2 , 1 4 ([−L,L]×[0,T]) ≤ C(∥f∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L))) + ∥∂tf∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2((−L,L)))), for any function f ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H1((−L, L))) such that ∂tf ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2((−L, L))), where C is an absolute positive constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For any t, τ ∈ [0, T], one deduces by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, the Minkovski, H¨older, and Cauchy inequalities that ∥f(·, t) − f(·, τ)∥L∞((−L,L)) ≤ C∥f(·, t) − f(·, τ)∥ 1 2 L2((−L,L))∥f(·, t) − f(·, τ)∥ 1 2 H1((−L,L)) ≤ C∥f∥ 1 2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L))) ���� � t τ ∂tf(·, s)ds ���� 1 2 L2((−L,L)) ≤ C∥f∥ 1 2 L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L))) �� t τ ∥∂tf∥L2((−L,L))ds � 1 2 ≤ C(∥f∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L))) + ∥∂tf∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2((−L,L))))|t − τ| 1 4, for an absolute positive constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For any x, y ∈ [−L, L] and t ∈ [0, T], it follows from the H¨older inequality that |f(x, t) − f(y, t)| ≤ ���� � y x ∂xf(z, t)dz ���� ≤ ���� � L −L |∂xf|2dx ���� 1 2 |y − x| 1 2 ≤ ∥f∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L)))|y − x| 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, for any x, y ∈ [−L, L] and t, τ ∈ [0, T], it holds that |f(x, t) − f(y, τ)| ≤ |f(x, t) − f(y, t)| + |f(y, t) − f(y, τ)| ≤ C(∥f∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1((−L,L))) + ∥∂tf∥L2(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2((−L,L))))(|t − τ| 1 4 + |y − x| 1 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This leads to the conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ We are now ready to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The proof is dived into five steps as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Regularities and pointwise positivity of ϑ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For L > 0, denote by W 2,1 2 ((−L, L)×(0, T)) the space of all functions f ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H2((−L, L))) satisfying 32 JINKAI LI AND ZHOUPING XIN ∂tf ∈ L2(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2((−L, L))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Recall the embedding that W 2,1 2 ([−L, L] × (0, T)) ֒→ C 1 2, 1 4([−L, L] × [0, T]) (Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 of [55]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that (vy, ϑy) ∈ W 2,1 2 ((−L, L) × (0, T)), J, Jy ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' H1((−L, L))), and Jt, Jyt ∈ L∞(0, T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2((−L, L))).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Hence, it follows from the Sobolev embedding and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 that (J, Jy, vy, ϑy) ∈ C 1 2, 1 4([−L, L] × [0, T]), ∀L > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) Rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) as cv̺0ϑt − κ J ϑyy + κJy J2ϑy + R̺0 vy J ϑ = µ J |vy|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) Since J is uniformly positive on R×(0, T), it can be checked that all the coefficients in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=', ̺0, 1 J , Jy J2, ̺0 vy J , and |vy|2 J , belong to C 1 2 , 1 4([−L, L] × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to these and the fact that ̺0(y) > 0 for all y ∈ R, it follows from the classic Schauder theory on interior regularities for uniform parabolic equations that ϑ ∈ C2+ 1 2 ,1+ 1 4((−L, L) × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' On the other hand, by the embedding theorem, it follows from the regularities of ϑ that ϑ ∈ C([−L, L] × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, it holds that ϑ ∈ C2,1((−L, L) × (0, T)) ∩ C([−L, L] × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that ϑ ̸≡ 0 on R × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Otherwise, noticing that ϑ ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2(−L, L)) for any L > 0, one has ϑ0 ≡ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' furthermore, it follows from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) that vy ≡ 0 on R × (0, T), from which, since v ∈ C([0, T];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L2((−L, L))) for any L > 0, one has v0 ≡ Const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This contradicts to the assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore, one has ϑ ̸≡ 0 on R ×(0, T) and ϑ ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to this and by the strong maximum principle, one gets 0 < ϑ ∈ C2,1(R × (0, T)) ∩ C(R × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Asymptotic behavior of Jy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Note that Jyt = 1 µ(GJy + JGy + R̺′ 0ϑ + R̺0ϑy) which implies Jy = 1 µ � t 0 e 1 µ � t s Gdτ(JGy + R̺′ 0ϑ + R̺0ϑy)ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Therefore���� Jy(y, t) ̺0(y) ���� ≤ 1 µe 1 µ � t 0 ∥G∥∞dτ � t 0 � J ���� Gy ̺0 ���� + R ���� ̺′ 0 ̺0 ���� ϑ + R|ϑy| � ds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) For any y ≥ 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' it follows that ���� Gy(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' t) ̺0(y) ���� = ����� � 1 0 Gy ̺0 dz + � 1 0 � y z �Gy ̺0 � y (z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' t)dz′dz ����� UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 33 ≤ � 1 0 ���� Gy ̺0 ���� dz + � y+1 0 ����� �Gy ̺0 � y ����� dz ≤ ���� Gy √̺0 ���� 2 (t) + � y + 1 ����� �Gy ̺0 � y ����� 2 (t) and ϑ(y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' t) = � 1 0 ϑ(z,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' t)dz + � 1 0 � y z ϑy(z′,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' t)dz′dz ≤ � 1 0 √̺0ϑ √̺0 dz + � y+1 0 |ϑy|dz ≤ ∥√̺0ϑ∥2(t) √δ0 + � y + 1∥ϑy∥2(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) where δ0 := inf[−1,1] ̺0 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Similar estimates hold also for y < 0 and thus it holds for any y ∈ R that ���� Gy(y, t) ̺0(y) ���� ≤ ���� Gy √̺0 ���� 2 (t) + � |y| + 1 ����� �Gy ̺0 � y ����� 2 (t) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) and ϑ(y, t) ≤ ∥√̺0ϑ∥2(t) √δ0 + � |y| + 1∥ϑy∥2(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) Substituting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12)–(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) and using (H1), one can get by the H¨older and Sobolev inequalities that ���� Jy(y, t) ̺0(y) ���� ≤ CeC � t 0 ∥G∥∞ds � t 0 � ���� Gy √̺0 ���� 2 + � |y| + 1 ����� �Gy ̺0 � y ����� 2 � ds +CeC � t 0 ∥G∥∞ds � t 0 � ∥√̺0ϑ∥2 + ∥ϑy∥2 � |y| + 1 + ∥ϑy∥H1 � ds ≤ C √ t � |y| + 1 \uf8ee \uf8f0 � t 0 \uf8eb \uf8ed ����� � Gy √̺0 , �Gy ̺0 � y ������ 2 2 + ∥ϑy∥2 H1 \uf8f6 \uf8f8 ds \uf8f9 \uf8fb 1 2 +Ct � |y| + 1∥√̺0ϑ∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2) ≤ C1 � |y| + 1, that is, ���� Jy(y, t) ̺0(y) ���� ≤ C1 � |y| + 1, ∀y ∈ R, t ∈ [0, T], (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) where the regularities of (ϑ, G) have been used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' A scaling transform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let T > 0 be any arbitrary given constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Assume by contradiction that s ∈ L∞(R×(0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Since ϑ = A Re s cv �̺0 J �γ−1 and J has 34 JINKAI LI AND ZHOUPING XIN uniform positive lower and upper bounds, it follows that 0 ≤ ϑ(y, t) ≤ CT̺γ−1 0 (y), ∀(y, t) ∈ R × [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) Let β > 0 to be determined later and introduce a scaling transform as f(y, t) := ϑ(y−β, t), y ∈ (0, ∞), t ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, direct calculations yield ϑ(y, t) = f(y− 1 β , t), ϑt(y, t) = ft(y− 1 β , t), ϑy(y, t) = − 1 β y−(1+ 1 β )fy(y− 1 β , t), ϑyy(y, t) = 1 β2y−(2+ 2 β )fyy(y− 1 β , t) + β + 1 β2 y−(2+ 1 β )fy(y− 1 β , t), for any (y, t) ∈ (0, ∞) × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Besides, one deduces from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) that cv̺0(y−β)y−(2+2β)J(y−β, t)ft(y, t) − κ β2fyy(y, t) − �κ(β + 1) β2 y−1 + κ βy−(1+β)Jy(y−β, t) J(y−β, t) � fy(y, t) +R̺0(y−β)y−(2β+2)vy(y−β, t)f(y, t) ≥ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) for all (y, t) ∈ (0, ∞) × (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Verifying conditions of Hopf type lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let MT be a positive constant to be determined later and define F(y, t) = e−MT tf(y, t), y ∈ (0, ∞), t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9), it is clear that 0 < F ∈ C2,1((0, ∞) × (0, ∞)) ∩ C((0, ∞) × [0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Moreover, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) and (H2) that F(y, t) = e−MT tϑ(y−β, t) ≤ CTe−MT t̺γ−1 0 (y−β) ≤ CTKγ−1 2 e−MT t(1 + y−β)−(γ−1)ℓρ ≤ CTKγ−1 2 e−MT ty(γ−1)βℓρ, for an y ∈ (0, ∞) and t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, one can define F(0, t) = 0 for t ∈ [0, ∞), such that F is well defined on [0, ∞) × [0, ∞), satisfying � F ∈ C2,1((0, ∞) × (0, ∞)) ∩ C([0, ∞) × [0, ∞)), F > 0 in (0, ∞) × (0, ∞), F(0, t) = 0, ∀t ∈ [0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17) It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) that a0(y, t)Ft(y, t) − aFyy(y, t) + b(y, t)Fy(y, t) + c(y, t)F(y, t) ≥ 0, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) in (0, ∞) × (0, ∞), where a0(y, t) = cv̺0(y−β)y−(2+2β)J(y−β, t), a = κ β2, UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 35 b(y, t) = − �κ(β + 1) β2 y−1 + κ β y−(1+β)Jy(y−β, t) J(y−β, t) � , c(y, t) = ̺0(y−β)y−(2+2β)� cvMT J(y−β, t) + Rvy(y−β, t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Take arbitrary t0 ∈ (0, T), 0 < y0 < min{ 1 2, t0}, and set P0 = (y0, t0), r = y0, P∗ = (0, t0) =: (y∗, t∗), δ∗ = y0 8 , P ∗ 0 = �y0 2 , t0 � =: (y∗ 0, t∗ 0), D = Bδ∗(P∗) ∩ B r 2(P ∗ 0 ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, P∗ ∈ ∂Br(P0), D = B y0 8 ((0, t0)) ∩ B y0 2 (( y0 2 , t0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For any (y, t) ∈ D, due to 0 < y < y0 8 < 1 16 and t0 2 < t < 3 2t0, one deduces (t − t∗ 0)a0(y, t) + (y − y∗ 0)b(y, t) = cv(t − t0)̺0(y−β)y−(2+2β)J(y−β, t) − � y − y0 2 � �κ(β + 1) β2 y−1 + κ βy−(1+β)Jy(y−β, t) J(y−β, t) � ≥ −cvt0̺0(y−β)y−(2+2β)J(y−β, t) − κy0 2β y−(1+β)|Jy(y−β, t)| J(y−β, t) ≥ −cvt0jT̺0(y−β)y−(2+2β) − κ jTβ y−(1+β)|Jy(y−β, t)|, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) where jT := sup (y,t)∈R×[0,T] J(y, t), jT := inf (y,t)∈R×[0,T] J(y, t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20) Set MT := R∥vy∥L∞(R×(0,T )) cvjT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, cvMT J(y−β, t) + Rvy(y−β, t) ≥ cvMT jT − R∥vy∥L∞(R×(0,T)) = 0 and cvMTJ(y−β, t) + Rvy(y−β, t) ≤ cvMT jT + R∥vy∥L∞(R×(0,T)) = cvMT (jT + jT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, for any (y, t) ∈ D, since � |y − y∗|2 + |t − t∗|2 ≤ δ∗ ≤ 1 16, it holds that 0 ≤ c(y, t) � |y − y∗|2 + |t − t∗|2 ≤ cvMT (jT + jT)̺0(y−β)y−(2+2β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21) For any (y, t) ∈ D, since 0 < y < 1, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) and (H2) that ̺0(y−β)y−(2+2β) ≤ K2(1 + y−β)−ℓρy−(2+2β) ≤ K2y(ℓρ−2)β−2 ≤ K2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) and y−(1+β)|Jy(y−β, t)| ≤ C1y−(1+β)̺0(y−β) � 1 + y−β ≤ C1K2y−(1+β)(1 + y−β)−ℓρ+ 1 2 36 JINKAI LI AND ZHOUPING XIN ≤ C1K2y(ℓρ− 3 2 )β−1 ≤ C1K2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23) as long as β ≥ max � 2 ℓρ−2, 2 2ℓρ−3 � = 2 ℓρ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='23), it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21) that (t − t∗ 0)a0(y, t) + (y − y∗ 0)b(y, t) ≥ −(C1 + 1)K2 � cvt0jT + κ jTβ � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='24) 0 ≤ c(y, t) � |y − y∗|2 + |t − t∗|2 ≤ cvMT(jT + jT)K2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25) for any (y, t) ∈ D, as long as β ≥ 2 ℓρ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Step 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Unboundedness of entropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Choose β = β0 := max � 2 (γ − 1)ℓρ , 2 ℓρ − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Due to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='24), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='25), it follows from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2 that lim ℓ→0+ F(P∗) − F(P∗ − n∗ℓ) ℓ = − lim ℓ→0+ F(P∗ − n∗ℓ) ℓ = −2ε2, for some positive constant ε2, where we recall P∗ = (0, t0), n∗ = P∗−P0 r = (−1, 0), and thus P∗ − n∗ℓ = (ℓ, t0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, there is a positive number ℓ0, such that F(y, t0) = e−MT t0ϑ(y−β0, t0) ≥ ε2y, ∀y ∈ (0, ℓ0), that is ϑ(y, t0) ≥ ε2eMT t0y− 1 β0 , ∀y ∈ � ℓ − 1 β0 0 , ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' On the other hand, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) and (H2) that ϑ(y, t) ≤ CTKγ−1 2 (1 + y)−ℓρ(γ−1) ≤ CTKγ−1 2 y−ℓρ(γ−1), ∀y > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Combing the previous two inequalities leads to y(γ−1)ℓρ− 1 β0 ≤ CTKγ−1 2 ε−1 2 e−MT t0, ∀y ∈ (ℓ − 1 β0 0 , ∞), which is impossible when y → ∞, as (γ − 1)ℓρ − 1 β0 ≥ γ−1 2 ℓρ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This contradiction leads to the desired conclusion that s ̸∈ L∞(R × (0, T)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Uniform positivity of ϑ and asymptotic unboundedness of s In this section, we prove the uniform positivity of the temperature and asymptotic unboundedness of the entropy, under the condition that the initial density decays at the far field not slower than O( 1 x4), which yields the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 37 Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' We need only to prove (i), while the conclusion (ii) follows from (i), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13), as J has uniformly positive lower and upper bounds at each time t ∈ (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let h be the Kelvin transform of ϑ defined as h(y, t) = yϑ �1 y, t � , ∀y ̸= 0, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) Then (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) implies h ∈ C2,1((R+ ∪ R−) × (0, T)) ∩ C((R+ ∪ R−) × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2) Note that ϑ(y, t) = yh �1 y, t � , ϑt(y, t) = yht �1 y, t � , ϑy(y, t) = h �1 y, t � − 1 yhy �1 y, t � , ϑyy(y, t) = 1 y3hyy �1 y, t � , for any y ̸= 0 and t ∈ (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from these and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) that cv̺0 �1 y � 1 y4ht (y, t) − κ J � 1 y, t �hyy (y, t) − κ Jy � 1 y, t � J2 � 1 y, t � 1 y2hy (y, t) + \uf8eb \uf8edR vy � 1 y, t � J � 1 y, t � ̺0 � 1 y � y4 + κ Jy � 1 y, t � J2 � 1 y, t � 1 y3 \uf8f6 \uf8f8 h(y, t) = µ ���vy � 1 y, t ���� 2 y3J � 1 y, t � , for y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Define a0, a, b, and ˜c as a0 := cv̺0 �1 y � 1 y4, a := κ J � 1 y, t �, b := −κ Jy � 1 y, t � J2 � 1 y, t � 1 y2, ˜c := R vy � 1 y, t � J � 1 y, t � ̺0 � 1 y � y4 + κ Jy � 1 y, t � J2 � 1 y, t � 1 y3, ∀y ̸= 0, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it holds that � a0ht − ahyy + bhy + ˜ch ≥ 0, in Q+ T , a0ht − ahyy + bhy + ˜ch ≤ 0, in Q− T , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) where Q+ T := R+ × (0, T), Q− T := R− × (0, T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Properties of a0, a, b, and ˜c are analyzed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) and the regularities of ̺0 and J that a0 ∈ C(R+ ∪ R−), a, b, ˜c ∈ C(Q+ T ∪ Q− T ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4) 38 JINKAI LI AND ZHOUPING XIN For a0, it follows from (H3) that 0 ≤ a0(y) ≤ cvK3 (|y| + 1)4 ≤ cvK3, ∀y ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='5) For a, it holds that λT ≤ a(y, t) ≤ ΛT, ∀y ̸= 0, t ∈ [0, T], (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6) where λT = κ jT and ΛT = κ jT , with jT and jT given by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' It follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) and (H3) that |Jy(y, t)| ≤ C1̺0(y) � |y| + 1 ≤ C1K3(|y| + 1)− 7 2, ∀y ∈ R, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7) This implies that |b(y, t)| ≤ κ j2 T 1 y2 ����Jy �1 y, t ����� ≤ κ j2 T 1 y2C1K3 � 1 |y| + 1 �− 7 2 ≤ C1K3κ j2 T , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8) for any y ̸= 0 and t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' By (H3) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='7), one deduces |˜c(y, t)| ≤ R jT 1 y4 K3 � 1 + 1 |y| �4∥vy∥∞(t) + κ j2 T 1 |y|3C1K3 � 1 + 1 |y| �− 7 2 ≤ RK3 jT ∥vy∥∞(t) + κ j2 T C1K3 ≤ C(∥vy∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='H1) + 1), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9) for any y ̸= 0 and t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Set NT = 2 cv � R jT ∥vy∥L∞(R×(0,T)) + √ 2κC1 j2 T � and define H(y, t) = e−NT th(y, t), ∀y ̸= 0, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) Due to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='2), it is clear that H ∈ C2,1((R+ ∪ R−) × (0, T)) ∩ C((R+ ∪ R−) × [0, T]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11) Since ϑ > 0, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='10) that H > 0 in Q+ T and H < 0 in Q− T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='12) By (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) and recalling the definitions of h and H, one deduces |H(y, t)| ≤ |y|ϑ �1 y, t � ≤ C(∥√̺0ϑ∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2) + ∥ϑy∥L∞(0,T;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='L2))|y| � 1 + 1 |y| ≤ C � y2 + |y|, ∀y ̸= 0, t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 39 Thanks to this, it holds that lim (y,τ)→(0,t) H(y, t) = 0, ∀t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='13) It follows from direct calculations and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3) that � a0Ht − aHyy + bHy + cH ≥ 0, in Q+ T , a0Ht − aHyy + bHy + cH ≤ 0, in Q− T , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) where c(y, t) := ˜c(y, t) + NTa0(y, t) = \uf8eb \uf8edcvNT + R vy � 1 y, t � J � 1 y, t � + κ J2 � 1 y, t � Jy � 1 y, t � ̺0 � 1 y � y \uf8f6 \uf8f8 ̺0 �1 y � 1 y4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For any y ∈ [−1, 0) ∩ (0, 1] and t ∈ [0, T], it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) that c(y, t) ≥ � cvNT − R jT ∥vy∥∞(t) − κ j2 T |y|C1 � 1 + 1 |y| � ̺0 �1 y � 1 y4 = � cvNT − R jT ∥vy∥∞(t) − κ j2 T C1 � |y|2 + |y| � ̺0 �1 y � 1 y4 ≥ � cvNT − R jT ∥vy∥∞(t) − √ 2κC1 j2 T � ̺0 �1 y � 1 y4 ≥ � R jT ∥vy∥∞(t) + √ 2κC1 j2 T � ̺0 �1 y � 1 y4 and thus c(y, t) ≥ 0, ∀y ∈ [−1, 0) ∪ (0, 1], t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15) Define �H(y, t) = \uf8f1 \uf8f2 \uf8f3 H(y, t), if y > 0, 0, if y = 0, −H(y, t), if y < 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='16) for all t ∈ [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Denote Ω− := (−1, 0) × (0, T), Ω+ := (0, 1) × (0, T), Ω := Ω+ ∪ Ω−, Γ := {0} × [0, T].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='11)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='14) that �H ∈ C2,1(Ω) ∩ C(Ω), �H > 0 in Ω, �H|Γ = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17) L �H = a0 �Ht − a �Hyy + b �Hy + c �H ≥ 0 in Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18) 40 JINKAI LI AND ZHOUPING XIN with a0, a, b, and c satisfying � a0 ∈ C((−1, 1) \\ {0}) ∩ L∞((−1, 1) \\ {0}), a, b, c ∈ C(Ω) ∩ L∞(Ω), λT ≤ a ≤ ΛT, c ≥ 0 in Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19) which follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='4)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='6), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='8)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='9), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' For arbitrary t0 ∈ (0, T), set P∗ = (0, t0), P0 = (y0, t0), r = y0 = min �1 2, t0, T − t0 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, it is clear that n∗ := P∗−P0 r = (−1, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Let Br be the space-time ball of radius r and centered at P0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thanks to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='17), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='18), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='19), it is clear that �H satisfies all the conditions in Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1, and thus Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='1 implies lim ℓ→0+ �H(P∗) − �H(P∗ − ℓn∗) ℓ = lim ℓ→0+ − �H(ℓ, t0) ℓ = −2ε0, with a positive constant ε0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Thus, limℓ→0+ � H(ℓ,t0) ℓ = 2ε0, which yields that �H(y, t0) ≥ ε0y, ∀y ∈ (0, ℓ0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='20) for some positive constant ℓ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Then, by the definition of �H, one derives �H(y, t0) = e−NT t0h(y, t0) = e−NT t0yϑ �1 y, t0 � ≥ ε0y, ∀y ∈ (0, ℓ0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='21) and thus, ϑ(y, t0) ≥ ε0eNT t0 ≥ ε0, ∀y ∈ � 1 ℓ0 , ∞ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) Similarly, there are positive constants ε1 and ℓ1 such that ϑ(y, t0) ≥ ε1, ∀y ∈ � −∞, − 1 ℓ1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' Combining this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='22) and recalling that 0 < ϑ ∈ C(R × [0, T]), one has inf y∈R ϑ(y, t0) = min \uf8f1 \uf8f2 \uf8f3ε0, ϑ1, inf y∈ � − 1 ℓ1 , 1 ℓ0 � ϑ(y, t0) \uf8fc \uf8fd \uf8fe > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' This yields the desired conclusion, and the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='3 is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' □ UNBOUNDEDNESS OF ENTROPY AND UNIFORM POSITIVITY OF TEMPERATURE 41 Acknowledgments This work was supported by the Key Project of National Natural Science Foun- dation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 12131010) and Guangdong Basic and Applied Basic Research Foundation (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 2020B1515310002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The work of J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' was also sup- ported by the National Natural Science Foundation of China (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 11971009 and 11871005) and by the Guangdong Basic and Applied Basic Research Foundation (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' 2019A1515011621 and 2020B1515310005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' The work of Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' was also supported by the Zheng Ge Ru Foundation and by the Hong Kong RGC Earmarked Research Grants (Grants No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' CUHK-14301421, CUHK-14300917, CUHK-14300819, and CUHK-14302819).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' References [1] Bresch, D.' metadata={'source': 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+page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='cn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content=' jklimath@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='com (Zhouping Xin) The Institute of Mathematical Sciences, The Chinese University of Hong Kong, Hong Kong, China Email address: zpxin@ims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='cuhk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} +page_content='hk' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/W9AyT4oBgHgl3EQfvPl0/content/2301.00627v1.pdf'} diff --git a/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/2301.00876v1.pdf.txt b/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/2301.00876v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d71faa71e336b6a77c392bc18e7d0220cc4dc15f --- /dev/null +++ b/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/2301.00876v1.pdf.txt @@ -0,0 +1,1437 @@ +MAUD: An Expert-Annotated Legal NLP Dataset for +Merger Agreement Understanding +Steven H. Wang1∗ +Antoine Scardigli1 +Leonard Tang2 +Wei Chen3 +Dimitry Levkin3 +Anya Chen4 +Spencer Ball5 +Thomas Woodside6 +Oliver Zhang7 +Dan Hendrycks8 +1ETH Z¨urich 2Harvard University 3The Atticus Project 4The Nueva School +5University of Wisconsin, Madison 6Yale University 7Stanford University 8UC Berkeley +Abstract +Reading comprehension of legal text can be +a particularly challenging task due to the +length and complexity of legal clauses and a +shortage of expert-annotated datasets. To ad- +dress this challenge, we introduce the Merger +Agreement Understanding Dataset (MAUD), +an expert-annotated reading comprehension +dataset based on the American Bar Associa- +tion’s 2021 Public Target Deal Points Study, +with over 39,000 examples and over 47,000 +total annotations. Our fine-tuned Transformer +baselines show promising results, with models +performing well above random on most ques- +tions. However, on a large subset of questions, +there is still room for significant improvement. +As the only expert-annotated merger agree- +ment dataset, MAUD is valuable as a bench- +mark for both the legal profession and the NLP +community. +1 +Introduction +While pretrained Transformers (Devlin et al., 2019; +Brown et al., 2020) have surpassed humans on read- +ing comprehension tasks such as SQuAD 2.0 (Ra- +jpurkar et al., 2018a) and SuperGLUE (Wang et al., +2019), their accuracy in understanding real-world +specialized legal texts remains underexplored. +Reading comprehension of legal text can be a +particularly challenging natural language process- +ing (NLP) task due to the length and complexity of +legal clauses and the difficulty of collecting expert- +annotated datasets. To help address this challenge, +we introduce the Merger Agreement Understanding +Dataset (MAUD), a legal reading comprehension +dataset curated under the supervision of highly spe- +cialized mergers-and-acquisitions (M&A) lawyers +and used in the American Bar Association’s 2021 +Public Target Deal Points Study (“ABA Study”). +The dataset and code for MAUD can be found at +github.com/TheAtticusProject/maud. +∗Correspondence to stewang@student.ethz.edu +Public target company acquisitions are the most +prominent business transactions, valued at hun- +dreds of billions of dollars each year. Merger agree- +ments are the legal documents that enable these +acquisitions, and key clauses in these merger agree- +ments are called “deal points.” +Lawyers working on the ABA Study perform +contract review on merger agreements. In general, +contract review is a two-step process. First, lawyers +extract key legal clauses from the contract (an entity +extraction task). Second, they interpret the meaning +of these legal clauses (a reading comprehension +task). In the ABA Study, the lawyers extract deal +points from merger agreements, and for each deal +point they answer a set of standardized multiple- +choice questions. +Models trained on MAUD’s expert-annotated +data can learn to answer 92 reading comprehen- +sion questions from the 2021 ABA Study, given +extracted deal point text from merger agreements. +By answering these questions, models interpret +the meaning of specialized legal language and cat- +egorize the different agreements being made by +companies in the contract. +Entity extraction and reading comprehension are +both important and challenging tasks in legal con- +tract review. A large-scale expert-annotated entity +extraction benchmark for contract review is already +available in Hendrycks et al. (2021b). However, to +the best of our knowledge, there is no large-scale +expert-annotated reading comprehension dataset +for contract review or any other legal task in the +English language. Therefore in this short paper, +we focus on the legal reading comprehension task. +(Appendix A.12 presents a preliminary benchmark +for the extraction task for interested researchers.) +Annotating MAUD was a collective effort of +over 10,000 hours by law students and experienced +lawyers. Prior to labeling, each law student at- +tended 70-100 hours of training, including lectures +and workshops from experienced M&A lawyers. +arXiv:2301.00876v1 [cs.CL] 2 Jan 2023 + +Outputs +No +Business and operation of Target +Ability to consummate transaction +Deal Point Answers: +☑ +☑ +☐ +☐ +☐ +☐ +Deal Point Question: FLS (MAE) applies to +Deal Point Category: Material Adverse Effect +Deal Point Text: +"Material Adverse Effect" means, with respect to any Person, +any event, change, circumstance, occurence or effect that (i) has, +or would have, a material adverse effect on the business... +Inputs +Deal Point Question: Stock Deal: Fixed Ratio v. Fixed Value +Deal Point Category: Type of Consideration +Deal Point Text: +...the right to receive an amount in cash equal to $37.00 (such +amount of cash, as may be adjusted pursuant to Section 3.01(3), +is hereinafter referred to as the "Merger Consideration")... +Model +Mixed Cash/Stock +All Cash +All Stock +Deal Point Answers: +☑ +☐ +☐ +☐ +☐ +☐ +Figure 1: MAUD contains 39,000+ examples for 92 different reading comprehension questions about merger +agreements. Given a deal point question and deal point text, a model learns to predict the correct answer(s) from a +list of possible answers standardized by the 2021 ABA Study. The deal point texts above are truncated for display. +Each annotation was labelled by three law student +annotators, and these labels were verified by an +experienced lawyer. See Appendix A.11 for more +information on the annotation process. We estimate +the pecuniary value of MAUD to be over $5 million +using a prevailing rate of $500 per hour in M&A +legal fees. +2 +Related Work +Due to the high costs of contract review and the +specialized skills it requires, understanding legal +text has proven to be a ripe area for NLP research. +Legal Entity Extraction. +One area of contract +review research focuses on legal entity extraction +and document segmentation. Chalkidis et al. (2017) +introduce a dataset for extracting basic information +from contracts, with follow-up modeling work us- +ing RNNs (Chalkidis et al., 2018) and Transform- +ers (Chalkidis et al., 2020). Lippi et al. (2019) +introduce a small expert-annotated dataset for iden- +tifying “unfair” clauses in 50 online terms of ser- +vices. Tuggener et al. (2020) introduce a semi- +automatically constructed dataset of legal contracts +for entity extraction. Leivaditi et al. (2020) intro- +duce an expert-annotated dataset of 2960 annota- +tions for 179 lease agreements. Hendrycks et al. +(2021b) introduce CUAD, an expert-annotated con- +tract review dataset containing 13,010 annotations +for 150 legal contracts. Unlike CUAD, which is a +entity extraction task for 16 different types of con- +tracts, MAUD is a multiple-choice reading compre- +hension task focusing on merger agreements. +Reading Comprehension for Legal NLP. +Ko- +reeda and Manning (2021) introduce a crowd- +worker-annotated dataset containing 7191 Natural +Language Inference questions about spans of non- +disclosure agreements. Hendrycks et al. (2021a) +propose a question-answering dataset sourced from +freely available online materials, containing ques- +tions (including legal exam questions) from dozens +of specialized areas. Zheng et al. (2021) present +a multiple-choice reading comprehension dataset +with 53,317 annotations automatically extracted +from US case law citations. Duan et al. (2019) +present a Chinese-language legal reading com- +prehension dataset, with about 50,000 expert- +generated annotations of Chinese judicial rulings. +In our work we present a legal reading comprehen- +sion dataset with 47,457 expert-generated annota- +tions about merger agreements. To the best of our +knowledge, MAUD is the only English-language +legal reading comprehension dataset that is both +large-scale and expert-annotated. +3 +MAUD: A Legal NLP Dataset for +Merger Agreement Understanding +MAUD consists of 47,457 annotations based on le- +gal text extracted from 151 English-language pub- +lic merger agreements. MAUD’s merger agree- +ments were sourced from the EDGAR system main- +tained by the U.S. Securities and Exchange Com- +mission. +Terminology. +Deal points are legal clauses that +define when and how the parties in a merger agree- +ment are obligated to complete an acquisition. We +refer to the text of these clauses (extracted by an- +notators from merger agreements) as deal point +texts. One or more predefined deal point questions +can be asked about each deal point text. Each deal +point question can be answered by one or more pre- + +Deal Point Category +Main +Dataset +Rare Answers +Dataset +Abridged +Dataset +All +Datasets +Conditions to Closing +3,411 +298 +4,052 +7,761 +Deal Protection and Related Provisions +6,491 +2,280 +5,937 +14,708 +General Information +152 +17 +173 +342 +Knowledge +388 +23 +258 +669 +Material Adverse Effect +8,816 +871 +3,273 +12,960 +Operating and Efforts Covenant +1,216 +191 +1,054 +2,461 +Remedies +149 +0 +181 +330 +All Categories +19,407 +3,680 +14,928 +39,231 +Table 1: Number of MAUD examples contained in each dataset by category. Each example is a question-answer +pair corresponding to an extracted deal point text. +defined deal point answers. Deal point questions +and texts are grouped into mutually exclusive deal +point categories. +Deal Points in MAUD. +The deal points in +MAUD are standardized by the 2021 ABA Study. +For the 2021 ABA Study, the American Bar Asso- +ciation appointed an M&A attorney to design 130 +deal point questions and 7 deal point categories re- +flecting recent legal developments and deal trends +of interest. +Of the 130 different deal point questions in the +2021 ABA Study, 92 are represented in MAUD. +MAUD contains 8,226 unique deal point text anno- +tations and 39,231 question-answer annotations (i.e. +examples), for a total of 47,457 annotations. There +are seven different deal point categories in MAUD: +Conditions to Closing, Deal Protection and Re- +lated Provisions, General Information, Knowledge, +Material Adverse Effect, Operating and Efforts +Covenant, and Remedies. +Task. +MAUD is a multiple-choice reading com- +prehension task. The model predicts the correct +deal point answer from a predefined list of possible +answers associated with each question. (See Figure +1 for an example). Several deal point questions in +the ABA Study are in fact multilabel questions, but +for uniformity we cast all multilabel questions as +binary multiple-choice questions. This increases +the effective number of questions from 92 to 144. +3.1 +MAUD Datasets and Splits +MAUD contains three datasets (main, abridged, +and rare answers) corresponding to three methods +of generating examples. See Table 2 for the number +of examples contained in each dataset. +Main Dataset. +The main dataset contains 20,623 +examples with original deal point text extracted +from 151 merger agreements by expert annotators. +Abridged Dataset. +The abridged dataset con- +tains 14,928 examples with deal point text extracted +from 94 of the 151 merger agreements included +in the main dataset. In the abridged dataset, deal +point texts are abridged to delete portions of legal +text in the main dataset that are not pertinent to +the deal point question. Because many texts con- +tain answers to multiple questions, we provide the +abridged data to guide a model to recognize the +most pertinent text. Appendix A.8 compares the +difficulty of main and abridged test examples. +Rare +Answers +Dataset. +The +rare +answers +dataset contains 3,680 examples that have rare an- +swers to a question. Legal experts made small edits +to texts in the main dataset to create deal points +with rare answers. See Appendix A.11 for an ex- +ample edit. We introduced the rare answers dataset +to ameliorate imbalanced answer distributions in +the main dataset. In particular, some answers in +the main dataset appear in fewer than 3 contracts, +making a train-dev-test split impossible. +train +dev +test +overall +main +13,256 +3,471 +3,896 +20,623 +abridged +9,647 +2,526 +2,755 +14,928 +rare +2,924 +756 +0 +3,680 +overall +25,827 +6,753 +6,651 +39,231 +Table 2: The number of examples in MAUD, grouped +by splits (train, dev, test) and by dataset (main, +abridged, rare answers). +Train, Dev, and Test Splits. +We construct the +train-dev-test split as follows. We reserve a random +20% of the combined main and abridged datasets +as the test split. The remaining main and abridged +examples are combined with the rare answers data, +and then split 80%-20% to form the train and dev + +Deal Point Category +Random +BERT +RoBERTa +LegalBERT +DeBERTa +BigBird +Conditions to Closing +20.4% +41.7% +41.6% +32.0% +48.2% +46.6% +Deal Protections +17.2% +53.8% +57.1% +58.6% +57.9% +58.0% +General Information +23.4% +85.7% +81.7% +82.0% +87.2% +81.2% +Knowledge +18.8% +75.6% +81.4% +71.6% +80.9% +81.0% +Material Adverse Effect +14.5% +44.0% +47.7% +49.8% +48.8% +50.9% +Operating and Efforts Cov. +22.0% +84.8% +85.7% +89.0% +86.9% +86.6% +Remedies +10.9% +88.2% +94.3% +100% +96.6% +95.0% +Overall +16.8% +52.6% +55.5% +55.9% +57.1% +57.8% +Table 3: Single-task AUPR scores for each deal point category and fine-tuned model. Each category score is +calculated as the mean minority-class AUPR over all questions in the category and over three runs. The overall +score is the mean AUPR score over all questions (not the mean over categories). See Appendix A.10 for category +descriptions. +Deal Point Category +Random +RoBERTa +LegalBERT +DeBERTa +Conditions to Closing +20.4% +40.3% +46.2% +46.2% +Deal Protections +17.2% +48.6% +53.6% +53.0% +General Information +23.4% +80.2% +74.8% +67.7% +Knowledge +18.8% +68.3% +73.0% +71.8% +Material Adverse Effect +14.5% +48.3% +50.7% +47.8% +Operating and Efforts Cov. +22.0% +80.3% +87.3% +74.2% +Remedies +10.9% +51.0% +83.6% +77.9% +Overall +16.8% +51.4% +55.8% +53.0% +Table 4: Multi-task AUPR scores for each deal point category and fine-tuned model. +splits. All splits are stratified by deal point question- +answer pairs. +To avoid data leakage due to main dataset and +abridged dataset examples having overlapping text +and the same answer, we always split the main +examples first and then place abridged examples +from the same contract in the same split. +4 +Experiments +4.1 +Setup +Metrics. +Because many questions have an imbal- +anced answer distribution, we use area under the +precision-recall curve (AUPR) as our primary met- +ric. For every question, we calculate the minority- +class AUPR score for each answer and then average +to get a mean AUPR score for the question. Then +we average over all question scores to get an overall +AUPR score for a model. +For example, consider a deal point question +Q, with three possible answers: A1, A2, and +A3, which have 50, 10, and 10 test examples re- +spectively. For the unique question-answer pair +(Q, A1), we first binarize all answers as A1 or +¬A1. The minority binarized answer is ¬A1, with +20 examples, and so the AUPR score for (Q, A1) +is calculated using positive class ¬A1. To get the +AUPR score for question Q, we average the AUPR +scores for (Q, A1), (Q, A2), and (Q, A3). +Models. +We fine-tune both single-task and multi- +task pretrained language models on MAUD using +the Transformers library (Wolf et al., 2020). +In the single-task setting, we evaluate the perfor- +mance of fine-tuned BERT-base (110M params), +RoBERTa-base (125M params), +LegalBERT- +base (110M params), DeBERTa-v3-base (184M +params), and BigBird-base (127M params). +In the multi-task setting, we evaluate RoBERTa- +base, LegalBERT-base, and DeBERTa-v3-base. +BERT (Devlin et al., 2019) is a bidirectional +Transformer that established state-of-the-art perfor- +mance on many NLP tasks. LegalBERT (Chalkidis +et al., 2020) pretrains BERT on a legal corpus. +RoBERTa (Liu et al., 2019) improves on BERT, +using the same architecture, but pretraining on an +order of magnitude more data. DeBERTa (He et al., +2020) improves upon RoBERTa by using a disen- +tangled attention mechanism and more parameters. +27.6% of the unique deal point texts in MAUD +and 50.0% of texts across all examples are longer +than 512 RoBERTa-base tokens, motivating our +evaluation of BigBird-base. BigBird (Zaheer et al., +2020) is initialized with RoBERTa and trained on +longer input sequences up to 4,096 tokens using +a sparse attention pattern that scales linearly with + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +Recall +0.1 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +Precision +MAUD Multi-Task Precision Recall Curve +LegalBERT +DeBERTa +RoBERTa +Random +Figure 2: Precision-recall curves for multi-task models, +averaged over all MAUD questions. +the number of input tokens. No deal point texts in +MAUD have more than 4,096 tokens. +Training. +We +fine-tune +models +using +the +AdamW optimizer (Loshchilov and Hutter, 2018) +and oversample to give every answer equal +proportion. +The learning rate and number of +updates were chosen by grid search. We trained +our final models on the combined training and +development splits, averaging test AUPR scores +over three runs. +See Appendix A.3 for more +training details. +4.2 +Results +Our fine-tuned models achieved high AUPR scores +in the Remedies, General Information, and Oper- +ating & Efforts Covenant categories, but scored +lower on other categories, particularly Deal Protec- +tions & Related Provisions (best single-task AUPR +58.6%), Conditions to Closing (48.2%), and Ma- +terial Adverse Effect (50.9%). Our results indi- +cate that there is substantial room for improvement +on these three hardest categories, which have the +longest text lengths (see Table 9) and which attor- +neys also find to be the most difficult to review. See +Tables 3 and 4 for full results. +Generally, larger and newer models had higher +mean performance on MAUD. In the single-task +setting, DeBERTa achieved an overall score of +57.1% AUPR, compared with 55.5% for RoBERTa +and 52.6% for BERT. BigBird achieved the high- +est score of 57.8% AUPR, slightly outperforming +DeBERTa. +Effect of Pretraining on Legal Corpus. +In the +single-task setting, LegalBERT outperforms BERT +103 +103.5 +104 +104.5 +Number of Training Examples +30 +35 +40 +45 +50 +55 +60 +AUPR +MAUD Performance vs. Dataset Size +Figure 3: RoBERTa-base AUPR as a function of the +number of training examples, highlighting the value of +our dataset’s size. AUPR is averaged over three runs. +and slightly outperforms RoBERTa, which have +the same model architecture but are not specialized +for law. In the multi-task setting, LegalBERT also +outperforms DeBERTa. The strong performance of +LegalBERT suggests that that pretraining on legal +data is helpful for MAUD. +Single-Task versus Multi-Task Performance. +RoBERTa and DeBERTa multi-task models per- +formed worse than their single-task counterparts by +about 4% AUPR. However, for LegalBERT these +models had approximately the same performance. +Dataset Size Ablation. +We trained single-task +RoBERTa models on random subsets of MAUD +training data to evaluate the effect of dataset size +on performance (see Figure 3). We found that +RoBERTa models trained on all training examples +had an overall AUPR score 7.3% higher than those +trained on a 50% subset of the dataset and 23.7% +higher than models trained on only a 5% subset. +5 +Conclusion +MAUD is a large-scale expert-annotated dataset +which facilitates NLP research on a specialized +merger agreement review task, based on the Amer- +ican Bar Association’s Public Target Deal Point +Study. MAUD can accelerate research towards +specialized legal tasks like merger agreement re- +view, while also serving as a benchmark for as- +sessing NLP models in legal text understanding. +Fine-tuned Transformer baselines exhibit strong +performance on some deal point categories, but +there is significant room for improvement on the +three hardest categories. + +6 +Ethics Statement +6.1 +Data Collection +Our data was created by volunteer annotators from +a non-profit legal organization, who joined the or- +ganization in order to create this dataset. None of +our annotators were compensated monetarily for +their time. Among our 36 annotators, 20 were male +and 16 were female. 33 annotators are based in the +United States and 3 annotators are based in Europe. +6.2 +Societal Impact +Advances in ML contract review, including merger +agreement review, can reduce the costs of and in- +crease the availability of legal services to busi- +nesses and individuals. In coming years, M&A +attorneys would likely benefit from having auxil- +iary analysis provided by ML models. +6.3 +Limitations +MAUD enables research on models that can auto- +mate a specialized labelling task in the ABA Study, +but does not target the other task performed in the +ABA Study, which is the extraction of deal point +texts from merger agreements. +We reserve this task for future work. For re- +searchers interested in the deal point extraction +task, we also release the 152 original contract texts +and span annotations. Details on the span annota- +tions and a preliminary baseline can be found in +Appendix A.12. +The 152 merger agreements in MAUD involve +the acquisitions of most but not all of the U.S. pub- +lic target companies exceeding $200 million in +value that were closed in 2021. Merger agreements +for private companies or public companies that do +not exceed $200 million in value are not included, +and consequently models trained on MAUD may +be less performant for deal point texts extracted for +these merger agreements. +The deal point questions and the list of prede- +fined deal point answers to each question were +created by experienced M&A attorneys and stan- +dardized by the ABA, but they do not represent all +of the deal points that are important in a merger +agreement. 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Advances in Neural Information +Processing Systems, 33:17283–17297. +Lucia Zheng, Neel Guha, Brandon R Anderson, Peter +Henderson, and Daniel E Ho. 2021. When Does Pre- +training Help? Assessing Self-Supervised Learning +for Law and the CaseHOLD Dataset. In Proceed- +ings of the Eighteenth International Conference on +Artificial Intelligence and Law, pages 159–168. + +contract name +category +text +question +answer +contract 93 +Material Adverse Effect +“Company Material Adverse Effect” +shall mean any state of facts, change, +condition, occurrence, effect, event, ... +FLS (MAE) Standard-Answer +��“Would” (reasonably) be expected to +�“Could” (reasonably) be expected to +�Other forward-looking standard +contract 102 +General Information +(i) each share of Company Common +Stock (including each share of +Company Common Stock described ... +Type of Consideration +��All Cash +�All Stock +�Mixed Cash/Stock +contract 77 +Conditions to Closing +Section 3.1 Organization, Standing +and Power. Section 3.2 +Capital Stock. (b) All +outstanding shares of capital stock +and other voting securities or ... +Accuracy of +Fundamental Target +R&Ws-Types of R&Ws +��Capitalization-Other +��Authority +��Approval +... +�Other +Table 5: MAUD contains three CSV files corresponding to the train, dev, and test splits of the dataset. We illustrate +some example rows in the table above, using a subset of the CSV columns. For full details on the dataset’s format, +we refer the reader to the MAUD Data Sheet or the dataset README. +A +Appendix +A.1 +Licensing +MAUD is licensed under CC-BY 4.0. +A.2 +Original Merger Agreement Texts +The 152 original merger agreement texts are avail- +able as text files in the supplementary data. +A.3 +Training details +Models. +The BERT, RoBERTA, LegalBERT, +DeBERTa-v3, and BigBird pretrained language +models that we use in our experiments are avail- +able on HuggingFace Hub as bert-base-cased, +roberta-base, +nlpaueb/legal-bert-base-uncased, +microsoft/deberta-v3-base, and google/bigbird- +roberta-base. +Single- and Multi-Task Settings. +In the single- +task setting, we train an individual model for every +MAUD question. In the multi-task setting, we train +a single model that answers all questions. +Training. +We fine-tune both single-task and +multi-task models using the AdamW optimizer +with weight decay 0.01. We oversample to give +every answer equal proportion. For all models ex- +cept BigBird we truncate deal point texts to 512 +tokens. +To reduce computational costs while fine-tuning +BigBird models, we set the maximum input se- +quence length to the minimum required to en- +compass all deal point texts associated with the +model’s deal point question. In particular, this +means that questions whose longest deal point text +has fewer than 704 tokens were fine-tuned with +full attention rather than sparse attention, because +google/bigbird-roberta-base requires a sequence +length of 704 or higher for sparse attention. +Multi-Task +Training. +For multitask experi- +ments we attached 144 classification heads to each +model, one for each question (including each mul- +tilabel binary question) in the dataset. +For every question we maintain a different shuf- +fled queue of training examples. Each training +batch fed to the classifier consists of 16 training ex- +amples drawn in round-robin order from the ques- +tion queues. +Single-Task Grid Search. +For BERT, RoBERTa, +LegalBERT, and DeBERTa-v3 experiments, in- +cluding the RoBERTa dataset size ablation experi- +ment,we used batch size 16. We grid-searched over +learning rates {1 × 10−5, 3 × 10−5, 1 × 10−4} +and number of updates {100, 200, 300, 400}. +For +BigBird +experiments +we +used +batch +size 8. +We grid-searched over learning rates +{1 × 10−5, 1 × 10−4} and number of updates +{200, 400, 600, 800}. +Validation AUPR scores were averaged over 3 +runs. +Multi-Task Grid Search. +For all models, we +used batch size 16, grid-searched over learning +rates {1 × 10−5, 3 × 10−5, 1 × 10−4} and num- +ber of epochs {1, 2, 3, 4, 5, 6}. Validation AUPR +scores were averaged over 3 runs. +Infrastructure and Computational Costs. +We +trained BERT and RoBERTa experiments in paral- +lel on A5000 GPUs, using about 12GB of GPU +memory. Three runs of fine-tuning models for +every question with 400 updates took about one +GPU-day per learning rate setting. +We trained DeBERTa-v3 experiments in parallel +on A4000 GPUs, using about 20GB of GPU mem- +ory. Three runs of fine-tuning models for every +question with 400 updates took about two GPU- + +Data Subset +Random +RoBERTa +LegalBERT +DeBERTa +Abridged +22.1% +67.5% +73.3% +64.3% +Main +23.1% +65.1% +70.0% +63.5% +Table 6: Mean AUPR scores for multi-task models, computed separately for main test examples and for abridged +test examples. For a fair comparison, we compute these scores only over questions that have any abridged examples. +As expected, abridged subscores are higher than main subscores. +days per learning rate setting. +We trained BigBird experiments in parallel on +A4000, A5000, and A100 GPUs, choosing the min- +imum GPU size required to accomodate the GPU +usage of the model, which varied with the maximi- +mum deal point text length. The experiments with +the longest deal point text lengths required about +75 GB of GPU memory. Three runs of fine-tuning +models for every question with 800 updates took 4 +to 5 GPU-days per learning rate setting. +Multi-task models were trained on a A100 GPU. +3 runs of 6 epochs of training took about 6 GPU- +hours per learning rate setting and model. +A.4 +Best-Performing Hyperparameters +For brevity we present the over 300 combinations +of best hyperparameters as CSV files in the supple- +mentary materials. +A.5 +Evaluation Variability +We find that the average overall AUPR over three +runs for our models can vary by 1-2%. +A.6 +Example Annotations in the Datasets +Table 5 shows the dataset structure as well as a few +example annotations. +A.7 +Other Dataset Statistics +Table 9 shows the percentage of deal point texts +that are longer than 512 tokens and the number of +deal point questions in each category. +A.8 +Main and Abridged Example Subscores +In Table 6 we report the mean multi-task AUPR +scores over main and abridged examples separately. +As expected, the abridged subscore is higher +than the main subscore for all three models. +A.9 +F1 Scores +Tables 7 and 8 show the average test F1 micro and +macro scores for multi-task models. +A.10 +Category Descriptions +We describe the seven categories of deal points +found in our dataset. +1. General Information. This category includes +the type of consideration and the deal structure +of an acquisition. +2. Conditions to Closing. This category spec- +ifies the conditions upon the satisfaction of +which a party is obligated to close the acqui- +sition. These conditions include the accuracy +of a target company’s representations and war- +ranties, compliance with a target company’s +covenants, absence of certain litigation, ab- +sence of exercise of appraisal or dissenters +rights, absence of material adverse effect on +the target company. +3. Material Adverse Effect. This category in- +cludes a number of questions based on the +Material Adverse Effect definition. Material +Adverse Effect defines what types of event +constitutes a material adverse effect on the +target company that would allow the buyer to, +among other things, terminate the agreement. +4. Knowledge. This category includes several +questions based on the definition of Knowl- +edge. Knowledge defines the standard and +scope of knowledge of the individuals making +representations on behalf of the target compa- +nies. +5. Deal Protection and Related Provisions. This +category describes the circumstances where a +target company’s board is permitted to change +its recommendation or terminate the merger +agreement in order to fulfill its fiduciary obli- +gations. +6. Operating and Efforts Covenants. This cate- +gory includes requirements for a party to take +or not to take specified actions between the +signing of the merger agreement and closing +of the acquisition. The types of covenants + +Deal Point Category +RoBERTa +LegalBERT +DeBERTa +Conditions to Closing +66.0% +66.5% +64.0% +Deal Protections +65.5% +67.1% +65.3% +General Information +85.5% +86.0% +82.8% +Knowledge +88.7% +87.9% +87.9% +Material Adverse Effect +79.7% +81.0% +76.6% +Operating and Efforts Cov. +87.5% +89.9% +83.0% +Remedies +90.6% +97.4% +94.3% +Overall +74.8% +76.1% +72.8% +Table 7: F1 micro scores for each deal point category and multi-task model. +Deal Point Category +RoBERTa +LegalBERT +DeBERTa +Conditions to Closing +41.9% +41.9% +44.0% +Deal Protections +50.9% +52.8% +50.7% +General Information +76.3% +68.4% +61.1% +Knowledge +76.5% +75.4% +75.3% +Material Adverse Effect +61.6% +62.3% +60.1% +Operating and Efforts Cov. +79.1% +82.6% +73.2% +Remedies +68.6% +91.7% +84.7% +Overall +53.3% +59.7% +57.3% +Table 8: F1 macro scores for each deal point category and multi-task model. +Deal Point Category +Deal Point +Questions +Percent +Long Texts +Conditions to Closing +9 +43.9% +Deal Protection and Related Provisions +31 +21.7% +General Information +1 +5.6% +Knowledge +3 +16.7% +Material Adverse Effect +39 +99.0% +Operating and Efforts Covenant +8 +2.1% +Remedies +1 +0.0% +All Categories +92 +50.0% +Table 9: Number of deal point questions and long text proportions by category. “Percent Long Texts” refers to the +proportion of annotations with deal point texts longer than 512 tokens when using a roberta-base tokenizer. Condi- +tions to Closing, Deal Protection and Related Provisions, and Material Adverse Effect have the largest proportion +of long texts. + +include obligation to conduct business in the +ordinary course of business and to use reason- +able efforts to secure antitrust approval. +7. Remedies. This category describes whether a +party has the right to specific performance. +A.11 +Labeling Process +MAUD is a collective effort of over 10,000 hours +by law students, experienced lawyers, and machine +learning researchers. Prior to labeling, each law +student attended 70-100 hours of training that in- +cluded live and recorded lectures by experienced +M&A lawyers and passing multiple quizzes. Law +students also read an instructions handbook. +Our volunteer annotators are experienced +lawyers and law students who are part of a non- +profit legal organization. None of the volunteers +were compensated monetarily for their time. +Annotator Demographics. +Among our 36 an- +notators, 20 were male and 16 were female. 33 +annotators are based in the United States and 3 +annotators are based in Europe. +Data Verification. +The law students who anno- +tated MAUD worked in teams of three. Each anno- +tation in the main and abridged datasets was first +annotated collectively, by a consensus established +by a law student team. This annotation includes +both the text extraction and the deal point answers. +When the team of law students could not reach an +agreement on an annotation, they escalated to an +experienced M&A lawyer. Finally, every law stu- +dent annotation was reviewed by an experienced +M&A lawyer for accuracy. +We unfortunately did not retain the records nec- +essary for us to calculate inter-annotator agreement +metrics. However, lawyers reviewing the student +annotations report that they agreed about 80% of +the time with student annotations. +Main and Abridged Datasets. +To create the +main dataset and the abridged dataset, the law stu- +dents conducted manual review and labeling of the +merger agreements uploaded in eBrevia, an elec- +tronic contract review tool. On a periodic basis, the +law students exported the annotations into reports, +and sent them to experienced lawyers for a quality +check. The lawyers reviewed the reports or the +labeled contracts in eBrevia, provided comments, +and addressed student questions. +Where needed, reviewing lawyers escalated +questions to a panel of 3-5 expert lawyers for dis- +cussions and reached consensus. Students or the +lawyers made changes in eBrevia accordingly. +Rare Answers Dataset. +To create the rare an- +swers dataset, legal experts copied an example +from the main dataset and minimally edited the +deal point text to create an example with a rare +answer. These edits were then reviewed by an ex- +perienced attorney to ensure accuracy. +For example, the deal point question “Limita- +tions on Antitrust Efforts” originally had very few +examples of “Dollar-based standard” deal point an- +swer. To create examples with this rare answer, the +annotators changed phrases in the deal point text +similar to “no obligation to divest or take other +actions” with language implying a dollar-based +standard, such as “Remedy Action or Remedy Ac- +tions with assets which generated in the aggregate +an amount of revenues that is in excess of USD +50,000,000.” +Final Annotation Formatting. +We exported the +final annotations as three CSV files corresponding +to the main, abridged, and rare answers datasets. +For example rows in the dataset, see Table 5. +A.12 +Extraction Dataset and Baseline +We also release a deal point extraction dataset +in SQuADv2 JSON format (Rajpurkar et al., +2018b), a format commonly used for Extrac- +tive Question Answering datasets. +The extrac- +tion dataset and the training and evaluation code +for a roberta-base baseline can be found at +github.com/TheAtticusProject/maud-extraction. +The annotated contract spans in this extraction +dataset correspond to deal point texts in the pri- +mary reading comprehension task’s main dataset. +The deal point texts in the main dataset are often +noncontiguous, and therefore the number of spans +exceeds the number of deal point texts. +Deal Point Types. +In MAUD’s primary reading +comprehension task, each example contains an ex- +tracted deal point text, a deal point question that +can be asked of this text, and a deal point category. +Another field in each example is the deal point +type. There are 22 deal point types, and each deal +point type belongs to exactly one deal point cate- +gory. The same set of questions are asked of all +deal points with the same type. + +Deal Point Type +Train +Dev +Test +Absence of Litigation Closing Condition +25 +6 +2 +Accuracy of Target R&W Closing Condition +1120 +153 +161 +Agreement provides for matching rights in connection with COR +469 +67 +53 +Agreement provides for matching rights in connection with FTR +424 +56 +46 +Breach of Meeting Covenant +102 +14 +6 +Breach of No Shop +269 +28 +35 +Compliance with Covenant Closing Condition +231 +32 +26 +FTR Triggers +241 +45 +27 +Fiduciary exception to COR convent +454 +58 +50 +Fiduciary exception: Board determination (no-shop) +276 +43 +35 +General Antitrust Efforts Standard +181 +27 +22 +Intervening Event Definition +114 +15 +12 +Knowledge Definition +121 +16 +16 +Limitations on FTR Exercise +385 +47 +45 +MAE Definition +367 +59 +40 +Negative interim operating convenant +169 +25 +22 +No-Shop +302 +45 +39 +Ordinary course covenant +167 +28 +26 +Specific Performance +134 +18 +20 +Superior Offer Definition +183 +30 +23 +Tail Period & Acquisition Proposal Details +360 +47 +40 +Type of Consideration +217 +26 +21 +All Types +6311 +885 +767 +Table 10: Span counts in the different splits of the MAUD extraction dataset, grouped by deal point type. +In MAUD’s extraction task, the Extractive QA +model is asked to identify spans of text inside the +full contract text that were annotated by legal ex- +perts as belonging to a particular deal point type. +See Table 10 for a list of deal point types and the +number of examples of each type. +Extraction Data Formatting. +For every deal +point type and every contract, we format the ques- +tion as follows: “Highlight the parts of the text (if +any) related to “” that should +be reviewed by a lawyer.” +Extraction Dataset Splits. +We build train, dev, +and test splits by splitting the 152 contracts in a +80-10-10 ratio. +Metrics. +MAUD’s extraction task is very sim- +ilar to the contract review extraction task from +Hendrycks et al. (2021b). In both tasks there is +a large imbalance between the number of negative +examples and positive examples in each contract. +Consequently, we use area under the precision re- +call curve (AUPR), averaged over three runs, as +our primary metric. Following Hendrycks et al. +(2021b), we consider a candidate span from our +model to be a match for a lawyer-annotated span if +the Jaccard similarity index is at least 50%. +Training Setup. +Since the most spans in the con- +tract text are negative examples, we oversample +positive examples to create a balanced training +dataset. +We fine-tune a roberta-base model on the com- +bined train and dev datasets using an A100 GPU. +We use Adam optimizer (Kingma and Ba, 2014) +and batch size 40. We use validation AUPR to +select the best learning rate from {1 × 10−5, 3 × +10−5, 1 × 10−4} and the best number of training +epochs from {4, 6, 8}. The best learning rate was +1 × 10−4 and the best number of epochs was 4. We +average our test AUPR score over three runs. +Results. +Our RoBERTa model has an AUPR +score of 19.7%. This is far lower than the 42.6% +baseline AUPR score achieved by RoBERTa in +Hendrycks et al. (2021b), suggesting that our con- +tract review extraction task is much more challeng- +ing. + diff --git a/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/load_file.txt b/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..78acb9cc83110bd2b07c1184109a2c8a160d415b --- /dev/null +++ b/YtAyT4oBgHgl3EQf9fqe/content/tmp_files/load_file.txt @@ -0,0 +1,793 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf,len=792 +page_content='MAUD: An Expert-Annotated Legal NLP Dataset for Merger Agreement Understanding Steven H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Wang1∗ Antoine Scardigli1 Leonard Tang2 Wei Chen3 Dimitry Levkin3 Anya Chen4 Spencer Ball5 Thomas Woodside6 Oliver Zhang7 Dan Hendrycks8 1ETH Z¨urich 2Harvard University 3The Atticus Project 4The Nueva School 5University of Wisconsin, Madison 6Yale University 7Stanford University 8UC Berkeley Abstract Reading comprehension of legal text can be a particularly challenging task due to the length and complexity of legal clauses and a shortage of expert-annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To ad- dress this challenge, we introduce the Merger Agreement Understanding Dataset (MAUD), an expert-annotated reading comprehension dataset based on the American Bar Associa- tion’s 2021 Public Target Deal Points Study, with over 39,000 examples and over 47,000 total annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Our fine-tuned Transformer baselines show promising results, with models performing well above random on most ques- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' However, on a large subset of questions, there is still room for significant improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' As the only expert-annotated merger agree- ment dataset, MAUD is valuable as a bench- mark for both the legal profession and the NLP community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 1 Introduction While pretrained Transformers (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Brown et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020) have surpassed humans on read- ing comprehension tasks such as SQuAD 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0 (Ra- jpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2018a) and SuperGLUE (Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2019), their accuracy in understanding real-world specialized legal texts remains underexplored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Reading comprehension of legal text can be a particularly challenging natural language process- ing (NLP) task due to the length and complexity of legal clauses and the difficulty of collecting expert- annotated datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To help address this challenge, we introduce the Merger Agreement Understanding Dataset (MAUD), a legal reading comprehension dataset curated under the supervision of highly spe- cialized mergers-and-acquisitions (M&A) lawyers and used in the American Bar Association’s 2021 Public Target Deal Points Study (“ABA Study”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The dataset and code for MAUD can be found at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='com/TheAtticusProject/maud.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' ∗Correspondence to stewang@student.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='ethz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='edu Public target company acquisitions are the most prominent business transactions, valued at hun- dreds of billions of dollars each year.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Merger agree- ments are the legal documents that enable these acquisitions, and key clauses in these merger agree- ments are called “deal points.” Lawyers working on the ABA Study perform contract review on merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In general, contract review is a two-step process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' First, lawyers extract key legal clauses from the contract (an entity extraction task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Second, they interpret the meaning of these legal clauses (a reading comprehension task).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the ABA Study, the lawyers extract deal points from merger agreements, and for each deal point they answer a set of standardized multiple- choice questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Models trained on MAUD’s expert-annotated data can learn to answer 92 reading comprehen- sion questions from the 2021 ABA Study, given extracted deal point text from merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' By answering these questions, models interpret the meaning of specialized legal language and cat- egorize the different agreements being made by companies in the contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Entity extraction and reading comprehension are both important and challenging tasks in legal con- tract review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A large-scale expert-annotated entity extraction benchmark for contract review is already available in Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' However, to the best of our knowledge, there is no large-scale expert-annotated reading comprehension dataset for contract review or any other legal task in the English language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Therefore in this short paper, we focus on the legal reading comprehension task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='12 presents a preliminary benchmark for the extraction task for interested researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=') Annotating MAUD was a collective effort of over 10,000 hours by law students and experienced lawyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Prior to labeling, each law student at- tended 70-100 hours of training, including lectures and workshops from experienced M&A lawyers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='00876v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='CL] 2 Jan 2023 Outputs No Business and operation of Target Ability to consummate transaction Deal Point Answers: ☑ ☑ ☐ ☐ ☐ ☐ Deal Point Question: FLS (MAE) applies to Deal Point Category: Material Adverse Effect Deal Point Text: "Material Adverse Effect" means, with respect to any Person, any event, change, circumstance, occurence or effect that (i) has, or would have, a material adverse effect on the business.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Inputs Deal Point Question: Stock Deal: Fixed Ratio v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Fixed Value Deal Point Category: Type of Consideration Deal Point Text: .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='the right to receive an amount in cash equal to $37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='00 (such amount of cash, as may be adjusted pursuant to Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='01(3), is hereinafter referred to as the "Merger Consideration").' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Model Mixed Cash/Stock All Cash All Stock Deal Point Answers: ☑ ☐ ☐ ☐ ☐ ☐ Figure 1: MAUD contains 39,000+ examples for 92 different reading comprehension questions about merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Given a deal point question and deal point text, a model learns to predict the correct answer(s) from a list of possible answers standardized by the 2021 ABA Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The deal point texts above are truncated for display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each annotation was labelled by three law student annotators, and these labels were verified by an experienced lawyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='11 for more information on the annotation process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We estimate the pecuniary value of MAUD to be over $5 million using a prevailing rate of $500 per hour in M&A legal fees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 2 Related Work Due to the high costs of contract review and the specialized skills it requires, understanding legal text has proven to be a ripe area for NLP research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Legal Entity Extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' One area of contract review research focuses on legal entity extraction and document segmentation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2017) introduce a dataset for extracting basic information from contracts, with follow-up modeling work us- ing RNNs (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2018) and Transform- ers (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Lippi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2019) introduce a small expert-annotated dataset for iden- tifying “unfair” clauses in 50 online terms of ser- vices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Tuggener et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2020) introduce a semi- automatically constructed dataset of legal contracts for entity extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Leivaditi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2020) intro- duce an expert-annotated dataset of 2960 annota- tions for 179 lease agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021b) introduce CUAD, an expert-annotated con- tract review dataset containing 13,010 annotations for 150 legal contracts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Unlike CUAD, which is a entity extraction task for 16 different types of con- tracts, MAUD is a multiple-choice reading compre- hension task focusing on merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Reading Comprehension for Legal NLP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Ko- reeda and Manning (2021) introduce a crowd- worker-annotated dataset containing 7191 Natural Language Inference questions about spans of non- disclosure agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021a) propose a question-answering dataset sourced from freely available online materials, containing ques- tions (including legal exam questions) from dozens of specialized areas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Zheng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021) present a multiple-choice reading comprehension dataset with 53,317 annotations automatically extracted from US case law citations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Duan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2019) present a Chinese-language legal reading com- prehension dataset, with about 50,000 expert- generated annotations of Chinese judicial rulings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In our work we present a legal reading comprehen- sion dataset with 47,457 expert-generated annota- tions about merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To the best of our knowledge, MAUD is the only English-language legal reading comprehension dataset that is both large-scale and expert-annotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 3 MAUD: A Legal NLP Dataset for Merger Agreement Understanding MAUD consists of 47,457 annotations based on le- gal text extracted from 151 English-language pub- lic merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD’s merger agree- ments were sourced from the EDGAR system main- tained by the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Securities and Exchange Com- mission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Terminology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal points are legal clauses that define when and how the parties in a merger agree- ment are obligated to complete an acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We refer to the text of these clauses (extracted by an- notators from merger agreements) as deal point texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' One or more predefined deal point questions can be asked about each deal point text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each deal point question can be answered by one or more pre- Deal Point Category Main Dataset Rare Answers Dataset Abridged Dataset All Datasets Conditions to Closing 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='411 298 4,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='052 7,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='761 Deal Protection and Related Provisions 6,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='491 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='280 5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='937 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='708 General Information 152 17 173 342 Knowledge 388 23 258 669 Material Adverse Effect 8,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='816 871 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='273 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='960 Operating and Efforts Covenant 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='216 191 1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='054 2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='461 Remedies 149 0 181 330 All Categories 19,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='407 3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='680 14,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='928 39,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='231 Table 1: Number of MAUD examples contained in each dataset by category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each example is a question-answer pair corresponding to an extracted deal point text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' defined deal point answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal point questions and texts are grouped into mutually exclusive deal point categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Points in MAUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The deal points in MAUD are standardized by the 2021 ABA Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For the 2021 ABA Study, the American Bar Asso- ciation appointed an M&A attorney to design 130 deal point questions and 7 deal point categories re- flecting recent legal developments and deal trends of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Of the 130 different deal point questions in the 2021 ABA Study, 92 are represented in MAUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD contains 8,226 unique deal point text anno- tations and 39,231 question-answer annotations (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' examples), for a total of 47,457 annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' There are seven different deal point categories in MAUD: Conditions to Closing, Deal Protection and Re- lated Provisions, General Information, Knowledge, Material Adverse Effect, Operating and Efforts Covenant, and Remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD is a multiple-choice reading com- prehension task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The model predicts the correct deal point answer from a predefined list of possible answers associated with each question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (See Figure 1 for an example).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Several deal point questions in the ABA Study are in fact multilabel questions, but for uniformity we cast all multilabel questions as binary multiple-choice questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This increases the effective number of questions from 92 to 144.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 MAUD Datasets and Splits MAUD contains three datasets (main, abridged, and rare answers) corresponding to three methods of generating examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Table 2 for the number of examples contained in each dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Main Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The main dataset contains 20,623 examples with original deal point text extracted from 151 merger agreements by expert annotators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Abridged Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The abridged dataset con- tains 14,928 examples with deal point text extracted from 94 of the 151 merger agreements included in the main dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the abridged dataset, deal point texts are abridged to delete portions of legal text in the main dataset that are not pertinent to the deal point question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Because many texts con- tain answers to multiple questions, we provide the abridged data to guide a model to recognize the most pertinent text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8 compares the difficulty of main and abridged test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Rare Answers Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The rare answers dataset contains 3,680 examples that have rare an- swers to a question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Legal experts made small edits to texts in the main dataset to create deal points with rare answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='11 for an ex- ample edit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We introduced the rare answers dataset to ameliorate imbalanced answer distributions in the main dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In particular, some answers in the main dataset appear in fewer than 3 contracts, making a train-dev-test split impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' train dev test overall main 13,256 3,471 3,896 20,623 abridged 9,647 2,526 2,755 14,928 rare 2,924 756 0 3,680 overall 25,827 6,753 6,651 39,231 Table 2: The number of examples in MAUD, grouped by splits (train, dev, test) and by dataset (main, abridged, rare answers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Train, Dev, and Test Splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We construct the train-dev-test split as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We reserve a random 20% of the combined main and abridged datasets as the test split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The remaining main and abridged examples are combined with the rare answers data, and then split 80%-20% to form the train and dev Deal Point Category Random BERT RoBERTa LegalBERT DeBERTa BigBird Conditions to Closing 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% Deal Protections 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% General Information 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% Knowledge 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Material Adverse Effect 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% Operating and Efforts Cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% Remedies 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 100% 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Overall 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% Table 3: Single-task AUPR scores for each deal point category and fine-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each category score is calculated as the mean minority-class AUPR over all questions in the category and over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The overall score is the mean AUPR score over all questions (not the mean over categories).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='10 for category descriptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Point Category Random RoBERTa LegalBERT DeBERTa Conditions to Closing 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% Deal Protections 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% General Information 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% Knowledge 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% Material Adverse Effect 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 47.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% Operating and Efforts Cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% Remedies 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 77.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% Overall 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Table 4: Multi-task AUPR scores for each deal point category and fine-tuned model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' All splits are stratified by deal point question- answer pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To avoid data leakage due to main dataset and abridged dataset examples having overlapping text and the same answer, we always split the main examples first and then place abridged examples from the same contract in the same split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 4 Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 Setup Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Because many questions have an imbal- anced answer distribution, we use area under the precision-recall curve (AUPR) as our primary met- ric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For every question, we calculate the minority- class AUPR score for each answer and then average to get a mean AUPR score for the question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Then we average over all question scores to get an overall AUPR score for a model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For example, consider a deal point question Q, with three possible answers: A1, A2, and A3, which have 50, 10, and 10 test examples re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For the unique question-answer pair (Q, A1), we first binarize all answers as A1 or ¬A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The minority binarized answer is ¬A1, with 20 examples, and so the AUPR score for (Q, A1) is calculated using positive class ¬A1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To get the AUPR score for question Q, we average the AUPR scores for (Q, A1), (Q, A2), and (Q, A3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We fine-tune both single-task and multi- task pretrained language models on MAUD using the Transformers library (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the single-task setting, we evaluate the perfor- mance of fine-tuned BERT-base (110M params), RoBERTa-base (125M params), LegalBERT- base (110M params), DeBERTa-v3-base (184M params), and BigBird-base (127M params).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the multi-task setting, we evaluate RoBERTa- base, LegalBERT-base, and DeBERTa-v3-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' BERT (Devlin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2019) is a bidirectional Transformer that established state-of-the-art perfor- mance on many NLP tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' LegalBERT (Chalkidis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020) pretrains BERT on a legal corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2019) improves on BERT, using the same architecture, but pretraining on an order of magnitude more data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' DeBERTa (He et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020) improves upon RoBERTa by using a disen- tangled attention mechanism and more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% of the unique deal point texts in MAUD and 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% of texts across all examples are longer than 512 RoBERTa-base tokens, motivating our evaluation of BigBird-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' BigBird (Zaheer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2020) is initialized with RoBERTa and trained on longer input sequences up to 4,096 tokens using a sparse attention pattern that scales linearly with 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0 Recall 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0 Precision MAUD Multi-Task Precision Recall Curve LegalBERT DeBERTa RoBERTa Random Figure 2: Precision-recall curves for multi-task models, averaged over all MAUD questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' the number of input tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' No deal point texts in MAUD have more than 4,096 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We fine-tune models using the AdamW optimizer (Loshchilov and Hutter, 2018) and oversample to give every answer equal proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The learning rate and number of updates were chosen by grid search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We trained our final models on the combined training and development splits, averaging test AUPR scores over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3 for more training details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 Results Our fine-tuned models achieved high AUPR scores in the Remedies, General Information, and Oper- ating & Efforts Covenant categories, but scored lower on other categories, particularly Deal Protec- tions & Related Provisions (best single-task AUPR 58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6%), Conditions to Closing (48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2%), and Ma- terial Adverse Effect (50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Our results indi- cate that there is substantial room for improvement on these three hardest categories, which have the longest text lengths (see Table 9) and which attor- neys also find to be the most difficult to review.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Tables 3 and 4 for full results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Generally, larger and newer models had higher mean performance on MAUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the single-task setting, DeBERTa achieved an overall score of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% AUPR, compared with 55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% for RoBERTa and 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% for BERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' BigBird achieved the high- est score of 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% AUPR, slightly outperforming DeBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Effect of Pretraining on Legal Corpus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the single-task setting, LegalBERT outperforms BERT 103 103.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5 104 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5 Number of Training Examples 30 35 40 45 50 55 60 AUPR MAUD Performance vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Dataset Size Figure 3: RoBERTa-base AUPR as a function of the number of training examples, highlighting the value of our dataset’s size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' AUPR is averaged over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' and slightly outperforms RoBERTa, which have the same model architecture but are not specialized for law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the multi-task setting, LegalBERT also outperforms DeBERTa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The strong performance of LegalBERT suggests that that pretraining on legal data is helpful for MAUD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Single-Task versus Multi-Task Performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' RoBERTa and DeBERTa multi-task models per- formed worse than their single-task counterparts by about 4% AUPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' However, for LegalBERT these models had approximately the same performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Dataset Size Ablation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We trained single-task RoBERTa models on random subsets of MAUD training data to evaluate the effect of dataset size on performance (see Figure 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We found that RoBERTa models trained on all training examples had an overall AUPR score 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% higher than those trained on a 50% subset of the dataset and 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% higher than models trained on only a 5% subset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 5 Conclusion MAUD is a large-scale expert-annotated dataset which facilitates NLP research on a specialized merger agreement review task, based on the Amer- ican Bar Association’s Public Target Deal Point Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD can accelerate research towards specialized legal tasks like merger agreement re- view, while also serving as a benchmark for as- sessing NLP models in legal text understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Fine-tuned Transformer baselines exhibit strong performance on some deal point categories, but there is significant room for improvement on the three hardest categories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 6 Ethics Statement 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 Data Collection Our data was created by volunteer annotators from a non-profit legal organization, who joined the or- ganization in order to create this dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' None of our annotators were compensated monetarily for their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Among our 36 annotators, 20 were male and 16 were female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 33 annotators are based in the United States and 3 annotators are based in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 Societal Impact Advances in ML contract review, including merger agreement review, can reduce the costs of and in- crease the availability of legal services to busi- nesses and individuals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In coming years, M&A attorneys would likely benefit from having auxil- iary analysis provided by ML models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3 Limitations MAUD enables research on models that can auto- mate a specialized labelling task in the ABA Study, but does not target the other task performed in the ABA Study, which is the extraction of deal point texts from merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We reserve this task for future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For re- searchers interested in the deal point extraction task, we also release the 152 original contract texts and span annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Details on the span annota- tions and a preliminary baseline can be found in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The 152 merger agreements in MAUD involve the acquisitions of most but not all of the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' pub- lic target companies exceeding $200 million in value that were closed in 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Merger agreements for private companies or public companies that do not exceed $200 million in value are not included, and consequently models trained on MAUD may be less performant for deal point texts extracted for these merger agreements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The deal point questions and the list of prede- fined deal point answers to each question were created by experienced M&A attorneys and stan- dardized by the ABA, but they do not represent all of the deal points that are important in a merger agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD should not be used as the sole source for developing AI tools for merger agree- ment review and 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' When Does Pre- training Help?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Assessing Self-Supervised Learning for Law and the CaseHOLD Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In Proceed- ings of the Eighteenth International Conference on Artificial Intelligence and Law, pages 159–168.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' contract name category text question answer contract 93 Material Adverse Effect “Company Material Adverse Effect” shall mean any state of facts, change, condition, occurrence, effect, event, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' FLS (MAE) Standard-Answer ��“Would” (reasonably) be expected to �“Could” (reasonably) be expected to �Other forward-looking standard contract 102 General Information (i) each share of Company Common Stock (including each share of Company Common Stock described .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Type of Consideration ��All Cash �All Stock �Mixed Cash/Stock contract 77 Conditions to Closing Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 Organization, Standing and Power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 Capital Stock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (b) All outstanding shares of capital stock and other voting securities or .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Accuracy of Fundamental Target R&Ws-Types of R&Ws ��Capitalization-Other ��Authority ��Approval .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' �Other Table 5: MAUD contains three CSV files corresponding to the train, dev, and test splits of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We illustrate some example rows in the table above, using a subset of the CSV columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For full details on the dataset’s format, we refer the reader to the MAUD Data Sheet or the dataset README.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1 Licensing MAUD is licensed under CC-BY 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 Original Merger Agreement Texts The 152 original merger agreement texts are avail- able as text files in the supplementary data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3 Training details Models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The BERT, RoBERTA, LegalBERT, DeBERTa-v3, and BigBird pretrained language models that we use in our experiments are avail- able on HuggingFace Hub as bert-base-cased, roberta-base, nlpaueb/legal-bert-base-uncased, microsoft/deberta-v3-base, and google/bigbird- roberta-base.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Single- and Multi-Task Settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the single- task setting, we train an individual model for every MAUD question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In the multi-task setting, we train a single model that answers all questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We fine-tune both single-task and multi-task models using the AdamW optimizer with weight decay 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We oversample to give every answer equal proportion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For all models ex- cept BigBird we truncate deal point texts to 512 tokens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To reduce computational costs while fine-tuning BigBird models, we set the maximum input se- quence length to the minimum required to en- compass all deal point texts associated with the model’s deal point question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In particular, this means that questions whose longest deal point text has fewer than 704 tokens were fine-tuned with full attention rather than sparse attention, because google/bigbird-roberta-base requires a sequence length of 704 or higher for sparse attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Multi-Task Training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For multitask experi- ments we attached 144 classification heads to each model, one for each question (including each mul- tilabel binary question) in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For every question we maintain a different shuf- fled queue of training examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each training batch fed to the classifier consists of 16 training ex- amples drawn in round-robin order from the ques- tion queues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Single-Task Grid Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For BERT, RoBERTa, LegalBERT, and DeBERTa-v3 experiments, in- cluding the RoBERTa dataset size ablation experi- ment,we used batch size 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We grid-searched over learning rates {1 × 10−5, 3 × 10−5, 1 × 10−4} and number of updates {100, 200, 300, 400}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For BigBird experiments we used batch size 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We grid-searched over learning rates {1 × 10−5, 1 × 10−4} and number of updates {200, 400, 600, 800}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Validation AUPR scores were averaged over 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Multi-Task Grid Search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For all models, we used batch size 16, grid-searched over learning rates {1 × 10−5, 3 × 10−5, 1 × 10−4} and num- ber of epochs {1, 2, 3, 4, 5, 6}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Validation AUPR scores were averaged over 3 runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Infrastructure and Computational Costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We trained BERT and RoBERTa experiments in paral- lel on A5000 GPUs, using about 12GB of GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Three runs of fine-tuning models for every question with 400 updates took about one GPU-day per learning rate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We trained DeBERTa-v3 experiments in parallel on A4000 GPUs, using about 20GB of GPU mem- ory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Three runs of fine-tuning models for every question with 400 updates took about two GPU- Data Subset Random RoBERTa LegalBERT DeBERTa Abridged 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% Main 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 70.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% Table 6: Mean AUPR scores for multi-task models, computed separately for main test examples and for abridged test examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For a fair comparison, we compute these scores only over questions that have any abridged examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' As expected, abridged subscores are higher than main subscores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' days per learning rate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We trained BigBird experiments in parallel on A4000, A5000, and A100 GPUs, choosing the min- imum GPU size required to accomodate the GPU usage of the model, which varied with the maximi- mum deal point text length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The experiments with the longest deal point text lengths required about 75 GB of GPU memory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Three runs of fine-tuning models for every question with 800 updates took 4 to 5 GPU-days per learning rate setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Multi-task models were trained on a A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 3 runs of 6 epochs of training took about 6 GPU- hours per learning rate setting and model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4 Best-Performing Hyperparameters For brevity we present the over 300 combinations of best hyperparameters as CSV files in the supple- mentary materials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5 Evaluation Variability We find that the average overall AUPR over three runs for our models can vary by 1-2%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6 Example Annotations in the Datasets Table 5 shows the dataset structure as well as a few example annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7 Other Dataset Statistics Table 9 shows the percentage of deal point texts that are longer than 512 tokens and the number of deal point questions in each category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8 Main and Abridged Example Subscores In Table 6 we report the mean multi-task AUPR scores over main and abridged examples separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' As expected, the abridged subscore is higher than the main subscore for all three models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9 F1 Scores Tables 7 and 8 show the average test F1 micro and macro scores for multi-task models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='10 Category Descriptions We describe the seven categories of deal points found in our dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' General Information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category includes the type of consideration and the deal structure of an acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Conditions to Closing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category spec- ifies the conditions upon the satisfaction of which a party is obligated to close the acqui- sition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' These conditions include the accuracy of a target company’s representations and war- ranties, compliance with a target company’s covenants, absence of certain litigation, ab- sence of exercise of appraisal or dissenters rights, absence of material adverse effect on the target company.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Material Adverse Effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category in- cludes a number of questions based on the Material Adverse Effect definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Material Adverse Effect defines what types of event constitutes a material adverse effect on the target company that would allow the buyer to, among other things, terminate the agreement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category includes several questions based on the definition of Knowl- edge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Knowledge defines the standard and scope of knowledge of the individuals making representations on behalf of the target compa- nies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Protection and Related Provisions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category describes the circumstances where a target company’s board is permitted to change its recommendation or terminate the merger agreement in order to fulfill its fiduciary obli- gations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Operating and Efforts Covenants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This cate- gory includes requirements for a party to take or not to take specified actions between the signing of the merger agreement and closing of the acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The types of covenants Deal Point Category RoBERTa LegalBERT DeBERTa Conditions to Closing 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 64.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Deal Protections 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% General Information 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% Knowledge 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% Material Adverse Effect 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% Operating and Efforts Cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Remedies 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% Overall 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 72.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% Table 7: F1 micro scores for each deal point category and multi-task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Point Category RoBERTa LegalBERT DeBERTa Conditions to Closing 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Deal Protections 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% 52.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='8% 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% General Information 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% Knowledge 76.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='5% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='4% 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% Material Adverse Effect 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% Operating and Efforts Cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 79.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2% Remedies 68.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 84.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% Overall 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% 59.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% 57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='3% Table 8: F1 macro scores for each deal point category and multi-task model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Point Category Deal Point Questions Percent Long Texts Conditions to Closing 9 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='9% Deal Protection and Related Provisions 31 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% General Information 1 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% Knowledge 3 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7% Material Adverse Effect 39 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Operating and Efforts Covenant 8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1% Remedies 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% All Categories 92 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='0% Table 9: Number of deal point questions and long text proportions by category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' “Percent Long Texts” refers to the proportion of annotations with deal point texts longer than 512 tokens when using a roberta-base tokenizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Condi- tions to Closing, Deal Protection and Related Provisions, and Material Adverse Effect have the largest proportion of long texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' include obligation to conduct business in the ordinary course of business and to use reason- able efforts to secure antitrust approval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Remedies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This category describes whether a party has the right to specific performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='11 Labeling Process MAUD is a collective effort of over 10,000 hours by law students, experienced lawyers, and machine learning researchers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Prior to labeling, each law student attended 70-100 hours of training that in- cluded live and recorded lectures by experienced M&A lawyers and passing multiple quizzes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Law students also read an instructions handbook.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Our volunteer annotators are experienced lawyers and law students who are part of a non- profit legal organization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' None of the volunteers were compensated monetarily for their time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Annotator Demographics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Among our 36 an- notators, 20 were male and 16 were female.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' 33 annotators are based in the United States and 3 annotators are based in Europe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Data Verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The law students who anno- tated MAUD worked in teams of three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Each anno- tation in the main and abridged datasets was first annotated collectively, by a consensus established by a law student team.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This annotation includes both the text extraction and the deal point answers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' When the team of law students could not reach an agreement on an annotation, they escalated to an experienced M&A lawyer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Finally, every law stu- dent annotation was reviewed by an experienced M&A lawyer for accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We unfortunately did not retain the records nec- essary for us to calculate inter-annotator agreement metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' However, lawyers reviewing the student annotations report that they agreed about 80% of the time with student annotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Main and Abridged Datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To create the main dataset and the abridged dataset, the law stu- dents conducted manual review and labeling of the merger agreements uploaded in eBrevia, an elec- tronic contract review tool.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' On a periodic basis, the law students exported the annotations into reports, and sent them to experienced lawyers for a quality check.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The lawyers reviewed the reports or the labeled contracts in eBrevia, provided comments, and addressed student questions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Where needed, reviewing lawyers escalated questions to a panel of 3-5 expert lawyers for dis- cussions and reached consensus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Students or the lawyers made changes in eBrevia accordingly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Rare Answers Dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To create the rare an- swers dataset, legal experts copied an example from the main dataset and minimally edited the deal point text to create an example with a rare answer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' These edits were then reviewed by an ex- perienced attorney to ensure accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For example, the deal point question “Limita- tions on Antitrust Efforts” originally had very few examples of “Dollar-based standard” deal point an- swer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' To create examples with this rare answer, the annotators changed phrases in the deal point text similar to “no obligation to divest or take other actions” with language implying a dollar-based standard, such as “Remedy Action or Remedy Ac- tions with assets which generated in the aggregate an amount of revenues that is in excess of USD 50,000,000.” Final Annotation Formatting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We exported the final annotations as three CSV files corresponding to the main, abridged, and rare answers datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For example rows in the dataset, see Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='12 Extraction Dataset and Baseline We also release a deal point extraction dataset in SQuADv2 JSON format (Rajpurkar et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=', 2018b), a format commonly used for Extrac- tive Question Answering datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The extrac- tion dataset and the training and evaluation code for a roberta-base baseline can be found at github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='com/TheAtticusProject/maud-extraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The annotated contract spans in this extraction dataset correspond to deal point texts in the pri- mary reading comprehension task’s main dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The deal point texts in the main dataset are often noncontiguous, and therefore the number of spans exceeds the number of deal point texts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Deal Point Types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In MAUD’s primary reading comprehension task, each example contains an ex- tracted deal point text, a deal point question that can be asked of this text, and a deal point category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Another field in each example is the deal point type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' There are 22 deal point types, and each deal point type belongs to exactly one deal point cate- gory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The same set of questions are asked of all deal points with the same type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Deal Point Type ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Train ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Dev ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Test ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Absence of Litigation Closing Condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Accuracy of Target R&W Closing Condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='1120 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='153 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='161 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Agreement provides for matching rights in connection with COR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='469 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='67 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='53 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Agreement provides for matching rights in connection with FTR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='424 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='56 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='46 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Breach of Meeting Covenant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='102 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Breach of No Shop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='269 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Compliance with Covenant Closing Condition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='231 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='32 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='FTR Triggers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='241 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Fiduciary exception to COR convent ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='454 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='58 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Fiduciary exception: Board determination (no-shop) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='276 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='43 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='General Antitrust Efforts Standard ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='181 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='27 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Intervening Event Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='114 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Knowledge Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='121 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Limitations on FTR Exercise ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='385 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='MAE Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='367 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='59 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Negative interim operating convenant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='169 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='No-Shop ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='302 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='39 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Ordinary course covenant ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='167 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='28 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Specific Performance ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='134 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Superior Offer Definition ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='183 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='23 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Tail Period & Acquisition Proposal Details ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='360 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='47 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Type of Consideration ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='217 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='26 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='21 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='All Types ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6311 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='885 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='767 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='Table 10: Span counts in the different splits of the MAUD extraction dataset,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' grouped by deal point type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In MAUD’s extraction task, the Extractive QA model is asked to identify spans of text inside the full contract text that were annotated by legal ex- perts as belonging to a particular deal point type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' See Table 10 for a list of deal point types and the number of examples of each type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Extraction Data Formatting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' For every deal point type and every contract, we format the ques- tion as follows: “Highlight the parts of the text (if any) related to “” that should be reviewed by a lawyer.” Extraction Dataset Splits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We build train, dev, and test splits by splitting the 152 contracts in a 80-10-10 ratio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' MAUD’s extraction task is very sim- ilar to the contract review extraction task from Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' In both tasks there is a large imbalance between the number of negative examples and positive examples in each contract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Consequently, we use area under the precision re- call curve (AUPR), averaged over three runs, as our primary metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Following Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021b), we consider a candidate span from our model to be a match for a lawyer-annotated span if the Jaccard similarity index is at least 50%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Training Setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Since the most spans in the con- tract text are negative examples, we oversample positive examples to create a balanced training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We fine-tune a roberta-base model on the com- bined train and dev datasets using an A100 GPU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We use Adam optimizer (Kingma and Ba, 2014) and batch size 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We use validation AUPR to select the best learning rate from {1 × 10−5, 3 × 10−5, 1 × 10−4} and the best number of training epochs from {4, 6, 8}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' The best learning rate was 1 × 10−4 and the best number of epochs was 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' We average our test AUPR score over three runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' Our RoBERTa model has an AUPR score of 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='7%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' This is far lower than the 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content='6% baseline AUPR score achieved by RoBERTa in Hendrycks et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} +page_content=' (2021b), suggesting that our con- tract review extraction task is much more challeng- ing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/YtAyT4oBgHgl3EQf9fqe/content/2301.00876v1.pdf'} diff --git a/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf b/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..fc9a11c4f97af40e263fd627524235ffaa4dc814 --- /dev/null +++ b/Z9E1T4oBgHgl3EQfcwRV/content/2301.03187v1.pdf @@ -0,0 +1,3 @@ +version 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We prove that traveling waves in viscous compressible liquids are a generic phenomenon. +The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, +viscous fluid, modeled by the free boundary compressible Navier-Stokes equations in dimension +n ⩾ 2. The bottom boundary of the fluid is flat and rigid, while the top is a moving free boundary. +A constant gravitational field acts normal to the flat bottom. We allow external forces to act in the +fluid’s bulk and external stresses to act on its free surface. These are posited to be in traveling wave +form, i.e. time-independent when viewed in a coordinate system moving at a constant, nontrivial +velocity parallel to the lower rigid boundary. +In the absence of such external sources of stress and force, the fluid system reverts to equilibrium, +which corresponds to a flat, quiescent fluid layer with vertically stratified density. In contrast, when +such sources of stress or force are present, the system admits traveling wave solutions. We establish +a small data well-posedness theory for this problem by proving that for every nontrivial traveling +wave speed there exists a nonempty open set of stress and forcing data that give rise to unique +traveling wave solutions, and that these solutions depend continuously on the data and the wave +speed. When n ⩾ 3 we prove this with surface tension accounted for at the free boundary, while in +the case n = 2 we prove this with or without surface tension. To the best of our knowledge, this +result constitutes the first general construction of traveling wave solutions to any free boundary +compressible fluid equations. +The traveling wave formulation of the equations is a quasilinear system of mixed type. The +interaction of the hyperbolic and elliptic parts leads to derivative loss in the linearizations of +the system. As such, we are compelled to construct solutions via an inverse function theorem of +Nash-Moser type. Our well-posedness proof has a number of novelties and elements of broader +interest, including: a new Nash-Moser variant that works in Banach scales but guarantees minimal +regularity loss in the existence as well as continuity of the local inverse; a host of results about +steady transport equations and their elliptic regularizations; new results about and uses of a scale of +anisotropic Sobolev spaces suited for constructing traveling waves; and a robust, streamlined, and +flexible approach for constructing solutions to our family of linearized free boundary problems. +2020 Mathematics Subject Classification. Primary 35Q30, 35R35, 35C07; Secondary 47J07, 76N06, 76N30. +Key words and phrases. Free boundary compressible Navier-Stokes, traveling waves, Nash-Moser inverse function +theorem. +N. Stevenson was supported by an NSF Graduate Research Fellowship. +I. Tice was supported by an NSF Grant (DMS #2204912). +1 +arXiv:2301.00773v1 [math.AP] 2 Jan 2023 + +2 +NOAH STEVENSON AND IAN TICE +Contents +1. +Introduction +3 +1.1. +Dynamics in Eulerian coordinates and stratified equilibria +3 +1.2. +Traveling waves and the role of the stress and forces +5 +1.3. +Previous work +7 +1.4. +Enthalpy and flattened reformulations +9 +1.5. +Statement of main result +11 +1.6. +Summary of strategy and layout of paper +14 +1.7. +Conventions of notation +21 +2. +A variation on the Nash-Moser inverse function theorem +22 +2.1. +Tame structure abstraction +23 +2.2. +Mapping hypotheses and statement of the inverse function theorem +30 +2.3. +Local surjectivity and injectivity +32 +2.4. +Proof of the inverse function theorem +42 +2.5. +Refinements +45 +3. +Nonlinear analysis of traveling free boundary compressible Navier-Stokes +51 +3.1. +Banach scales for the traveling wave problem +51 +3.2. +Analysis of atomic nonlinearities +53 +3.3. +Smooth tameness of the nonlinear operator +58 +3.4. +Derivative splitting +63 +3.5. +Prelude to linear analysis +70 +4. +Analysis of steady transport equations and their regularizations +74 +4.1. +Preliminary tame estimates +74 +4.2. +Some results on steady transport equations and elliptic regularizations +79 +5. +Analysis of weak solutions to the principal part linear equations +85 +5.1. +Estimates +85 +5.2. +Existence of solutions to the regularization +93 +6. +Analysis of strong solutions to the linearization +99 +6.1. +Analysis of tangential derivatives +99 +6.2. +Analysis of normal systems +104 +6.3. +Estimates and existence for the principal part +115 +6.4. +Synthesis of linear analysis +117 +7. +Conclusion +119 +7.1. +Abstract construction +119 +7.2. +PDE construction +120 +Appendix A. +Standard Sobolev space tools +122 +A.1. +Extension operators +122 +A.2. +Korn’s inequalities +123 +A.3. +Refined interpolation of Sobolev spaces +123 +Appendix B. +Some nonstandard function spaces +125 +B.1. +The anisotropic Sobolev spaces +125 +B.2. +Adapted Sobolev spaces +130 +Appendix C. +A selection of PDE tools +132 +C.1. +The divergence boundary value problem +132 +C.2. +Elliptic theory tools +135 +C.3. +Dissipation calculation for traveling compressible Navier-Stokes +138 +Appendix D. +Fine tools for nonlinear analysis +142 +D.1. +Smoothness of superposition nonlinearities +142 +D.2. +Tools for tame estimates +147 +Notation Index +154 +References +155 + +COMPRESSIBLE TRAVELING WAVES +3 +1. Introduction +The study of traveling wave solutions to the free boundary problems of fluid mechanics has been +of fundamental interest in mathematics for nearly two centuries. During this time, tremendous +progress has been made in the analysis of such solutions for incompressible fluids. Throughout +most of this period, the primary focus was inviscid, irrotational fluids; only in the past two decades +has progress been made on models that account for more robust phenomena such as vorticity and +viscosity. However, to the best of our knowledge, there are no rigorous results in the literature that +account for the fundamental fluid mechanical effect of compressibility. It is this effect that we aim +to study in the present paper. +All real fluids experience some degree of compressibility and viscosity, even if small, so it is +physically important to verify that traveling waves remain a generic phenomenon when these effects +are accounted for. From a mathematical perspective, the development of the compressible viscous +theory also opens the door to studying incompressible or inviscid limits, which may then shed light +on the zoo of incompressible inviscid solutions that have been constructed in the literature. We +emphasize that in this work we only study compressible fluids for which the density does not vanish +at the free boundary; these are often referred to as compressible liquids in the literature. +The rest of the introduction proceeds as follows. In Section 1.1 we formulate the dynamical +equations for a compressible viscous fluid with free boundary and identify the equilibrium solutions, +which correspond to stratified layers of quiescent fluid. Section 1.2 is concerned with the traveling +wave ansatz and a discussion of the role played by external stresses and forces. Previous work +on traveling waves is discussed in Section 1.3. Further reformulations of the equations, made in +the interest of identifying ‘good unknowns,’ are recorded in Section 1.4. Our main results are +stated in Section 1.5 along with some discussion of their implications. Section 1.6 contains a +high-level summary of the difficulties in the proof and our strategies for overcoming them. Finally, +Section 1.7 records the notational conventions we employ throughout the paper. We emphasize +that for convenience we have included a Notation Index at the end of the paper, just before the +references. +1.1. Dynamics in Eulerian coordinates and stratified equilibria. We begin by formulating +the free boundary compressible Navier-Stokes equations, which govern the dynamics of a layer +of viscous, compressible, barotropic (isentropic) fluid. First we must set some notation. We let +n ∈ N \ {0, 1} denote the spatial dimension; the cases n ∈ {2, 3} are the physically relevant ones, but +our analysis works in general dimension. The parameter b ∈ R+ designates the equilibrium depth of +the fluid. If η : Rn−1 → R is a continuous function satisfying η + b > 0, then we define the open set +Ω[η] = +� +(x, y) ∈ Rn−1 × R : 0 < y < b + η (x) +� +(1.1.1) +as well as the interfacial sets +Σ[η] = +� +(x, y) ∈ Rn−1 × R : y = b + η (x) +� +and Σ0 = Rn−1 × {0}. +(1.1.2) +Note that ∂Ω[η] = Σ[η] ⊔ Σ0. See Figure 1 for a depiction of these sets. Throughout the paper we +will extensively use the shorthand notation +Ω = Ω[0] = Rn−1 × (0, b) and Σ = Σ[0] = Rn−1 × {b}. +(1.1.3) +The fluid is assumed to occupy a semi-infinite layer of finite depth that changes in time; more +precisely, at time t the fluid occupies the set Ω[ζ (t, ·)] ⊂ Rn for an unknown continuous free surface +function ζ (t, ·) : Rn−1 → R satisfying ζ (t, ·) + b > 0. The upper boundary of this set, Σ[ζ (t, ·)], +is called the free boundary, while the lower boundary Σ0 is referred to as the fixed boundary. +The fluid is described by its velocity vector field w (t, ·) : Ω[ζ (t, ·)] → Rn and its scalar density +τ (t, ·) : Ω[ζ (t, ·)] → R+. Associated to w and τ are two crucial fluid mechanical quantities: the +pressure and the viscous stress tensor. The pressure within the fluid is given by P(τ), where +P ∈ C∞ (R+; R) is a given pressure law that is strictly increasing and satisfies P ′ > 0. The + +4 +NOAH STEVENSON AND IAN TICE +b + η +Σ[η] +Σ0 +Ω[η] +Figure 1. Depiction of the domain Ω[η] and its boundary, Σ[η] and Σ0, when n = 3. +assumption that the pressure depends only on the density is what makes the fluid barotropic; this +can be viewed as a consequence of assuming the fluid flow is isentropic, i.e. entropy remains constant. +The viscous stress tensor within the fluid is the symmetric tensor +Sτw = µ (τ) +� +∇w + ∇wt − 2 +n∇ · wI +� ++ λ (τ) (∇ · w)I = µ(τ)D0w + λ(τ)(∇ · w)I, +(1.1.4) +where the shear and bulk viscosity coefficients are µ, λ ∈ C∞ (R+; [0, ∞)). The non-negativity +of the viscosity coefficients is a requirement of the Clausius–Duhem inequality from continuum +thermodynamics, but we will place more assumptions on these below. Note that the specific forms +of the functions P, µ, and λ can be thought of as characterizing the specific material comprising +the fluid. +The dynamics of ζ, w, and τ are then coupled to the various forces and stresses acting on the +fluid through the free boundary compressible Navier-Stokes equations: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂tτ + ∇ · (τw) = 0 +in Ω[ζ (t, ·)] +τ (∂tw + w · ∇w) + ∇(P (τ)) − ∇ · Sτw = −gτen + τG + F +in Ω[ζ (t, ·)] +−(P (τ) − Sτw)νζ + Pextνζ − ςH (ζ) νζ = Tνζ +on Σ[ζ (t, ·)] +∂tζ + w · (∇∥ζ, −1) = 0 +on Σ[ζ (t, ·)] +w = 0 +on Σ0. +(1.1.5) +Note that in the above we have written ∇∥ = (∂1, . . . , ∂n−1) to refer to the ‘tangential gradient’. +We now enumerate these forces and stresses. The term −gτen is the gravitational force acting on +the fluid, with gravitational strength g > 0 and unit vector en = (0, . . . , 1) ∈ Rn perpendicular to +Σ0. The vector fields F (t, ·) , G(t, ·) : Ω[ζ (t, ·)] → Rn are the applied bulk and specific bulk forces, +respectively. The parameter Pext ∈ R is a constant external pressure, T (t, ·) : Σ[ζ (t, ·)] → Rn×n is +the applied surface stress, and νζ is the outward unit normal to the surface Σ[ζ (t, ·)]. We note that +in continuum mechanics it is usually the case that T (t, ·) is symmetric, but this condition plays no +role in our analysis, so have allowed for the most general case. The mean curvature operator is +H (ζ) = ∇∥ · ((1 + +��∇∥ζ +��2)−1/2∇∥ζ), +(1.1.6) +and the parameter ς ⩾ 0 is called the coefficient of surface tension. For technical reasons that +will be discussed later, we make the following assumptions about the viscosity coefficients and the +coefficient of surface tension:� +µ > 0, λ ⩾ 0 in R+, and ς > 0 +if n ⩾ 3 +µ > 0, λ > 0 in R+, and ς ⩾ 0 +if n = 2. +(1.1.7) +The first equation in (1.1.5) is the continuity equation, which asserts conservation of mass. The +next is the momentum equation, and it dictates a Newtonian balance of forces in the fluid bulk. +After this is the dynamic boundary condition, which enforces a balance of stresses acting on the +free surface. The penultimate equation is the kinematic boundary condition, which determines how + +COMPRESSIBLE TRAVELING WAVES +5 +the free surface evolves according to the fluid velocity. The final equation in (1.1.5) is simply the +no-slip boundary condition for the velocity on the rigid bottom. For a more thorough introduction +to the compressible Navier-Stokes equations, including their derivation, we refer to the books of +Wehausen and Laitone [104], Feireisl [33], Lions [66], Novotn´y and Straˇskraba [83], Gurtin, Fried, +and Anand [38], and Plotnikov and Soko�lowski [88]. +The compressible Navier-Stokes system (1.1.5) admits a vertically stratified equilibrium solution, +provided that the barotropic pressure law P, the external depth b, the external pressure Pext, and +the gravitational field strength g satisfy some compatibility conditions. Indeed, suppose that the +fluid experiences no external forces or stresses, i.e. F = G = 0 and T = 0, and that the fluid is +quiescent and occupies a flat slab of depth b, i.e. w = 0 and ζ = 0. Finally, suppose that ∂tτ = 0; +then (1.1.5) reduces to τ(t, x, y) = ϱ(y), where ϱ : [0, b] → R+ is a smooth function solving the +Cauchy problem +� +(P ◦ ϱ)′ = −gϱ +in (0, b) +P ◦ ϱ(b) = Pext. +(1.1.8) +In order to guarantee that a solution exists, we henceforth assume that the following pair of +compatibility conditions are satisfied (these conditions are actually necessary and sufficient): +Pext ∈ P(R+) and (0, ∞] ∋ +� ∞ +P −1(Pext) +t−1P ′(t) dt > gb. +(1.1.9) +With (1.1.9) in hand, we can conveniently solve (1.1.8) by introducing the enthalpy H : R+ → R, +which is the smooth increasing function defined via +H(s) = −gb + +� s +P −1(Pext) +t−1P ′(t) dt. +(1.1.10) +Note that since P ′ > 0 we have that H′ > 0 as well. We can also calculate the image H(R+) = +(Hmin, Hmax) ⊆ R, where +Hmin = −gb− +� P −1(Pext) +0 +t−1P ′(t) dt < −gb and Hmax = −gb+ +� ∞ +P −1(Pext) +t−1P ′(t) dt > 0. (1.1.11) +Since we now know that H : R+ → (Hmin, Hmax) is a smooth diffeomorphism and [−gb, 0] ⊆ H(R+), +we may realize ϱ as the smooth decreasing function defined by +ϱ(y) = H−1(−gy) for y ∈ [0, b], +(1.1.12) +which is the unique solution to (1.1.8) in light of the construction of H. The equilibrium density ϱ +will play a crucial role in our subsequent analysis. As a concrete example, if the pressure satisfies +the well-known polytropic law P(t) = Ktα for α ⩾ 1 and K ∈ R+, then +ϱ(y) = +� +PextK−1 exp(gK−1(b − y)) +if α = 1 +((PextK−1)(α−1)/α + (α − 1)α−1K−1g(b − y))1/(α−1) +if α > 1. +(1.1.13) +1.2. Traveling waves and the role of the stress and forces. The main thrust of this paper +is the study of traveling wave solutions to the system (1.1.5). These are solutions that are time- +independent when viewed in an inertial coordinate system obtained from the above Eulerian +coordinates through a Galilean transformation. In order for time-independence to hold, the moving +coordinate system must travel at a constant velocity parallel to Σ0. Without loss of generality (we +can always apply a rigid rotation that fixes the vector en to change coordinates), we may assume +that the traveling coordinate system moves at constant velocity γe1 for a wave speed γ ∈ R+. +In the new coordinates, the stationary free boundary is described by η (x − γte1) = ζ (t, x) for a +new unknown free surface function η : Rn−1 → (−b, ∞), which then determines the fluid domain +Ω[η] and the free boundary Σ[η]. We then posit that the other quantities are also in traveling + +6 +NOAH STEVENSON AND IAN TICE +wave form: γv (x − tγe1, y) = w (t, x, y), σ(x − γte1, y) = τ(t, x, y), F (x − tγe1, y) = F (t, x, y), +G (x − tγe1, y) = G (t, x, y), and T (x − tγe1, y) = T (t, x, y), where v : Ω[η] → Rn, σ : Ω[η] → R, +F, G : Ω[η] → Rn, and T : Σ[η] → Rn×n define the stationary velocity field, density, external and +specific forces, and external stresses, respectively. Under these assumptions, (1.1.5) is equivalent to +the following traveling compressible Navier-Stokes system for unknowns (σ, v, η) and data (T , G, F): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +−∂1σ + ∇ · (σv) = 0 +in Ω[η] +γ2σ (v − e1) · ∇v + ∇(P (σ)) − γ∇ · Sσv = −gσen + σG + F +in Ω[η] +−(P (σ) − γSσv)νη + Pextνη − ςH (η) νη = T νη +on Σ[η] +−∂1η + v · (∇∥η, −1) = 0 +on Σ[η] +v = 0 +on Σ0. +(1.2.1) +Note that in changing unknowns, we have rescaled the velocity vector by γ. This has the effect of +nondimensionalizing the vector field v and, more importantly, removing the γ-dependence from the +continuity equation and the kinematic boundary condition. +With the traveling wave system (1.2.1) formulated, we turn to a discussion of the role played +by the surface stress, specific bulk force, and bulk force data triple, (T , G, F). We have chosen to +study this general form of (T , G, F) in order to allow these to model a variety of physical effects. +As a specific example, the bulk force term G can be thought of as a localized perturbation of the +gravitational field caused by a massive object translating above the fluid (a primitive model of the +ocean-moon system). Similarly, a simple example of the surface stress occurs when T = −ϕIn×n for +a given scalar function ϕ : Rn → R; in this configuration, ϕ can be viewed as a spatially localized +source of pressure translating above the fluid. See Figure 2 for depictions of the free surface for this +latter case of applied stress. +If (T , G, F) = 0, then it is a simple matter to verify that the stratified equilibrium solution, +v = 0, η = 0, and σ = ϱ (defined by (1.1.12)), provides a solution to (1.2.1) with any value of γ. +This suggests that we should seek solutions to (1.2.1) as perturbations of this stratified equilibrium. +An elementary formal calculation (we will state and prove a rigorous version later after a further +reformulation of the problem: see Appendix C.3 and, in particular Corollary C.12) reveals that if +σ − ϱ, v, and η are in a Sobolev-type framework, then +� +Ω[η] +µ(σ) +2 +|D0v|2 + λ(σ)|∇ · v|2 = +� +Ω[η] +(σG + F) · v + +� +Σ[η] +T νη · v. +(1.2.2) +The physical interpretation of this identity is that if a traveling wave solution exists, then the power +supplied by the forces and stress (the right side of (1.2.2)) must be in exact balance with the energy +dissipation rate due to viscosity (the left side of (1.2.2)). +The identity (1.2.2) reveals even more if we assume there are no applied forces or stress, i.e. +(T , G, F) = 0. Without a source of external power, (1.2.2) requires that the left integral vanishes, +and so the assumptions (1.1.7) together with the Korn inequality (see Propositions A.3 and A.4 for +Korn in Ω, but similar results hold in Ω[η] if η is sufficiently regular) imply that v = 0. In turn, the +momentum equation implies that ∇(P(σ)) = −gσen and hence σ = H−1(−gidRn · en + c) for some +constant c, but we must have c = 0 since σ − ϱ vanishes at infinity. The normal part of the dynamic +boundary condition then requires +Pext − P ◦ H−1(−g(b + η)) − ς∇∥ · ((1 + |∇∥η|2)−1/2∇∥η) = 0, +(1.2.3) +and by multiplying this equation by η and integrating by parts in the mean curvature term, we +deduce that +� +Rn−1(Pext − P ◦ H−1(−g(b + η)))η ⩽ 0. +(1.2.4) + +COMPRESSIBLE TRAVELING WAVES +7 +Since Pext − P ◦ H−1(−g(b + ·)) is a strictly increasing function vanishing at zero, the integrand +in (1.2.4) is nonnegative and must thus vanish pointwise. Hence, η = 0, and we have deduced that +(σ, v, η) = (ϱ, 0, 0) when (T , G, F) = 0. +The above formal computation suggests that the dissipative nature of viscosity prohibits the +existence of nontrivial traveling wave solutions (in Sobolev-type spaces) without applied stress or +forcing. This shows that the triple (T , G, F) plays an essential role in the study of traveling wave +solutions, as the stress and forcing data are necessary for solutions to exist. We emphasize, however, +that the above argument does not preclude the existence of nontrivial solutions with (T , G, F) = 0 +in non-Sobolev functional frameworks. +Now that the importance of the data triple in the traveling wave theory is evident, we can +roughly summarize our goal for the paper: we aim to prove that for every traveling wave speed +γ ∈ R+ the problem (1.2.1) admits a small-data well-posedness theory. That is, we aim to identify +a nontrivial open set, Uγ, of stress and forcing data in a Sobolev-type framework such that for every +(T , G, F) ∈ Uγ there exists a locally unique solution triple (σ, v, η) to (1.2.1), also in a Sobolev-type +framework, that depends continuously on (T , G, F). However, for technical reasons that we will +explain at the beginning of Section 1.4, the variables (σ, v, η) are not suitable for this task, and +we must introduce a further reformulation of (1.2.1) with a new set of ‘good unknowns’ in order +to achieve our goal. Interestingly, this new formulation has the added benefit of allowing us to +establish the continuity of solutions with respect to the wave speed γ as well. +1.3. Previous work. The time dependent free boundary compressible fluid equations of (1.1.5) +and their variants have received much attention in the literature. A full review is beyond the scope +of the paper, so we will settle for a brief survey of some results closely related to ours. +A significant portion of the literature on the dynamic problem concerns the inviscid analog +of (1.1.5), i.e. the free boundary compressible Euler equations. Lindblad [62, 63] proved local +well-posedness of the liquid droplet problem. +Jang and Masmoudi [49, 50] and Coutand and +Shkoller [20, 21] proved local well-posedness for the vacuum droplet problem. Well-posedness of the +liquid droplet problem with surface tension was studied by Coutand, Hole, and Shkoller [19] and by +Disconzi and Kukavica [28]. The incompressible limit for the liquid droplet problem was derived by +Lindblad and Luo [65] without surface tension and by Disconzi and Luo [29] with surface tension. +Trakhinin [102] and Luo and Zhang [69] proved local well-posedness for inviscid liquid layers of +infinite depth. +There are also a number of dynamical studies of the free boundary compressible Navier-Stokes +equations. Denisova proved local well-posedness for the compressible bubble in a compressible fluid +problem in Sobolev spaces [23] and in weighted H¨older spaces [25]. Denisova [24] also proved similar +results for the compressible bubble in an incompressible fluid problem. The viscous liquid droplet +problem was studied by Secchi and Valli [92], who proved local well-posedness with heat conduction, +Solonnikov and Tani [95], who proved local well-posedness with surface tension, and by Denisova +and Solonnikov [26], who developed a well-posedness theory with and without surface tension. The +well-posedness of the viscous liquid droplet problem has also been proved under various conditions +using maximal regularity techniques: see the work of Enomoto, von Below, and Shibata [32], Shibata +[94], and Burczak, Shibata, and Zaj¸aczkowski [13]. +The viscous literature also has several studies of layer geometries. Jin [53] and Jin and Padula [54] +proved local and global well-posedness for a layer of periodic barotropic fluid with surface tension. +Tanaka and Tani [100] gave local and global well-posedness results for layers of heat conducting +fluids. Jang, Tice, and Wang [51, 52] proved local and global well-posedness for multiple layers of +barotropic fluid. Huang and Luo [45] proved global existence for a layer of heat conducting fluid +without surface tension. +It is also possible to use compressible fluids with free boundaries as a simple model of stellar +structure, in which case the fluid is subject to the force of its own gravitational field. When gravity + +8 +NOAH STEVENSON AND IAN TICE +is assumed to be Newtonian rather than relativistic, this problem is called the Euler-Poisson system +for inviscid fluids and the Navier-Stokes-Poisson system for viscous ones. Makino [71] proved an +early local existence result for Euler-Poisson under special assumptions on the data. Makino [84] and +Matuˇsu-Neˇcasov´a, Okada, and Makino [76] studied the viscous problem with spherical symmetry +outside a solid inner-core. Jang [47] proved local existence for the Navier-Stokes-Poisson problem +with vacuum boundary. Luo, Xin, and Zeng [70] studied local well-posedness for radial solutions to +the inviscid problem with vacuum boundary. Ginsberg, Lindblad, and Luo [36] studied the local +well-posedness of a self gravitating compressible liquid. Jang and Hadˇzi´c [39] constructed global +expanding solutions for Euler-Poisson. The self-gravitating problem admits nontrivial radial steady +states known as Lane-Emden solutions (see, for instance, the book of Chandrasekhar [14]). More +recent work has rigorously constructed rotating solutions: for a variational approach we refer to +Auchmuty and Beals [4] and Li [61], and for a perturbative approach we refer to Jang and Makino +[48] and Strauss and Wu [99]. +In stark contrast to the above discussion, the compressible traveling problem (1.2.1) has, to +the best of our knowledge, not received any prior attention in the PDE literature either with or +without viscosity. Perhaps the closest work, though still rather distant, concerns the construction of +stationary (but not traveling) solutions to some free boundary problems for viscous compressible +fluids. Pileckas and Zaj¸aczkowski [87] found stationary solutions to a bounded viscous compressible +fluid droplet with surface tension under symmetry considerations. Jin and Padula [55] considered +steady flows of viscous compressible fluids in a bounded rigid container with partially free boundary. +The incompressible analogs of (1.2.1) have, however, received attention in the literature. The +incompressible and inviscid analog of (1.2.1), which is also known as the traveling water wave problem, +has received enormous attention in the mathematics literature for more than a century. We refer to +the surveys of Toland [101], Groves [37], Strauss [98], and Haziot, Hur, Strauss, Toland, Wahl´en, +Walsh, and Wheeler [41] and the references therein for a thorough review of this extensive literature. +On the other hand, progress on the traveling wave theory for the free boundary incompressible +Navier-Stokes equations only began quite recently. A small data well-posedness theory was first +developed by Leoni and Tice [60]. This result was subsequently generalized by Stevenson and +Tice [97] and Koganemaru and Tice [57] to multi-layered and inclined geometries, respectively. A +similar well-posedness and stability theory for traveling wave solutions to the one-phase Muskat +problem was developed by Nguyen and Tice [82]. Viscous traveling waves were also empirically +identified recently in experiments with a tube of air, translating uniformly above a wave tank, +blowing onto a single layer of viscous fluid. For details, we refer to the works of Akylas, Cho, Diorio, +and Duncan [18, 27], Masnadi and Duncan [74], and Park and Cho [85, 86]. +As we have already mentioned, there are only a few recent works in the literature that rigorously +treat traveling waves with viscosity and none that treat compressibility. Viscosity and compressibility +are important physical effects to account for in studying traveling waves because all real fluids +experience some degree of compressibility and viscosity, even if small. Within the applied literature +there are works that study the observable effects of compressibility in simplified fluid models. For +example, Longuet-Higgins [68] and Kadri [56] studied how the compressibility of water leads to +microseisms in the ocean, and Long and Morton [67] and Miesen, Kamp, and Sluijter [78, 79] used +asymptotic expansions and numerics to study the role of compressibility in atmospheric solitary +traveling waves. +Our proof of well-posedness for the traveling wave problem (1.2.1) uses a novel variation on +the Nash-Moser inverse function theorem. A literature review concerning our version is delayed +until Sections 1.6 and 2, but we will conclude this subsection with a sampling of interesting results +from the fluids PDE literature that have been proved with Nash-Moser. +Beale [7] used it to +prove the existence of steady water waves for the irrotational incompressible free boundary Euler +equations in two dimensions. Plotnikov and Toland [89] found time periodic standing water waves. +Lindblad [63, 64] used Nash-Moser to study the droplet problem for incompressible and compressible + +COMPRESSIBLE TRAVELING WAVES +9 +Euler. Chen and Wang [15] studied the existence and stability of compressible current-vortex sheets +in three-dimensional magnetohydrodynamics. Trakhinin [102] considered the infinite depth surface +wave problem for compressible Euler. Makino [72, 73] studied spherically symmetric motions of a +planet’s compressible atmosphere and the vacuum boundary problem for gaseous stars. Buffoni +and Wahl´en [12] used a version of Nash-Moser to produce steady three dimensional rotation flows +for incompressible Euler. Chen, Secchi, and Wang [16] studied relativistic vortex sheets in three- +dimensional Minkowski spacetime for compressible, relativistic Euler. Chen, Hu, Wang, Wang, and +Yuan [17] used Nash-Moser to study compressible vortex sheets in two-dimensional elastodynamics. +Trakhinin and Wang [103] studied the ideal compressible magnetohydrodynamic equations with +surface tension via Nash-Moser. +1.4. Enthalpy and flattened reformulations. The goal of this subsection is to further reformu- +late the traveling wave system (1.2.1) so that it is more convenient to analyze. The motivations +for this are three-fold: two common difficulties in free boundary problems and a third more subtle +issue specific to the problem at hand. The first issue is that we wish to establish a well-posedness +theory in Sobolev-type spaces. However, as we discussed at the end of Section 1.2, if (T , G, F) = 0 +then the solution reduces to the stratified equilibrium, but we cannot expect σ = ϱ to belong to +Sobolev-type spaces on sets of infinite measure. This suggests that we should rewrite (1.2.1) as +a perturbation of the equilibrium solution (ϱ, 0, 0). The second issue is that, even if we rewrite +in perturbed form, the resulting equations are still posed in an unknown domain Ω[η]. In order +to conveniently employ standard PDE toolboxes, it is advantageous to recast the system in a +fixed, known domain. The traveling wave formulation precludes the common choice of Lagrangian +coordinates, so we instead employ a flattening into the equilibrium domain Ω based only on the +free surface function. The third issue arises because the traveling wave structure ultimately forces +the free surface function η to belong to a scale of anisotropic Sobolev-type spaces (see Appendix +B.1). After the perturbation and flattened reformulations, η will end up appearing in various +nonlinearities in a form that, due to the strange properties of the anisotropic spaces, is quite difficult +or impossible to control in a Sobolev-type framework. Roughly speaking, our way around this +problem is to make a nonlinear change of unknown, shifting from the density to the perturbed +enthalpy h = H(σ) − H(ϱ). The unknown h turns out to serve as a sort of ‘good unknown’ in +that it recasts the worst nonlinearity, which comes from the term ∇(P(σ)) + gσen in (1.2.1), in the +form σ∇(H(σ) − H(ϱ)) = H−1(h + H(ϱ))∇h, which shifts the nonlinearity outside of the gradient +and permits simple Sobolev estimates. We will also employ a nonlinear change of the velocity in +order to similarly linearize the kinematic boundary condition. The cost of these changes is that the +continuity equation becomes more cumbersome, but fortunately we can handle its new form. +We now turn to the execution of these reformulations of (1.2.1). We start by switching to a +perturbed enthalpy reformulation by defining the new unknown h = H(σ) − H(ϱ). Note that σ +can be recovered from h via σ = H−1(h + H(ϱ)), provided h + H(ϱ) takes values in (Hmin, Hmax) +from (1.1.11), which will always hold for the solutions we construct. Then the perturbative enthalpy +reformulation of the traveling free boundary compressible Navier-Stokes equations is the following +system for (h, v, η) with data (T , G, F): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +−∂1(H−1(h + H(ϱ))) + ∇ · (H−1(h + H(ϱ))v) = 0 +in Ω[η] +γ2H−1(h + H(ϱ))(v − e1) · ∇v + H−1(h + H(ϱ))∇h − γ∇ · SH−1(h+H(ϱ))v += H−1(h + H(ϱ))G + F +in Ω[η] +−((P − Pext) ◦ H−1(h + H(ϱ)) − γSH−1(h+H(ϱ))v)νη − ςH (η)νη = T νη +on Σ[η] +−∂1η + v · (∇∥η, −1) = 0 +on Σ[η] +v = 0 +on Σ0. +(1.4.1) + +10 +NOAH STEVENSON AND IAN TICE +Next we turn our attention to reformulating (1.4.1) in the fixed domain Ω = Rn−1 × (0, b). We +first define a Poisson-like extension operator that takes functions defined on Σ and extends them to +functions defined on Ω. The auxiliary mapping E0 : Hs−1/2(Σ) → Hs(Ω) ∩ 0H1(Ω) (see (1.7.3)), for +s ∈ N+, is defined via ϕ �→ E0ϕ, where E0ϕ is the unique solution to the PDE +� +� +� +� +� +−∆E0ϕ = 0 +in Ω, +E0ϕ = ϕ +on Σ, +E0ϕ = 0 +on Σ0. +(1.4.2) +Our Poisson-like extension operator E is then defined for appropriate functions (see Lemma A.1) +η : Σ → R through the assignment +Eη(x, y) = y · b−1Π1 +Lη(x) + E0Π1 +Hη(x, y) for (x, y) ∈ Rn−1 × (0, b), +(1.4.3) +where the Fourier projectors are given by Π1 +Lη = F −1[1B(0,1)F[η]] and Π1 +Hη = η − Π1 +Lη. +Our flattening map is then built from E as follows. For η : Σ → R satisfying η > −b and belonging +to an appropriate function space (see (1.5.1) with s > 1 + d/2 and d = n − 1) we define Fη : Ω → Rn +via +Fη(x, y) = (x, y + Eη(x, y)) for (x, y) ∈ Ω = Rn−1 × (0, b). +(1.4.4) +We associate to Fη two crucial quantities: the Jacobian Jη : Ω → R+ and (when Jη is nowhere +vanishing) the geometry matrix Aη : Ω → Rn×n, which are defined via +Jη = det(∇Fη) = 1 + ∂nEη = ∂n(Fη · en) and Aη = (∇Fη))−t. +(1.4.5) +Then, provided that +Jη > 0 and Jη, 1/Jη ∈ L∞(Ω), +(1.4.6) +Fη(Ω) = Ω[η] and Fη is a bi-Lipschitz homeomorphism from Ω to Ω[η] such that its restriction to Ω +defines a smooth diffeomorphism to Ω[η]. Moreover, Fη(Σ) = Σ[η] and Fη is the identity on Σ0. +For our penultimate change of equations, we set h = h ◦ Fη and u = JηAt +ηv ◦ Fη to be our +new unknowns defined in the fixed domain Ω. Equations (1.4.1) then transform to the following +equivalent system for unknowns (h, u, η) with data (T , G, F): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (H−1(h + H ◦ ϱ(Fη · en))(u − JηAt +ηe1)) = 0 +in Ω +H−1(h + H ◦ ϱ(Fη · en))(γ2(A−t +η u/Jη − e1) · Aη∇(A−t +η u/Jη) + Aη∇h) +−γ(Aη∇) · SH−1(h+H◦ϱ(Fη·en)) +Aη +(A−t +η u/Jη) = H−1(h + H ◦ ϱ(Fη · en))G ◦ Fη ++F ◦ Fη +in Ω +−((P − Pext) ◦ H−1(h + H ◦ ϱ(Fη · en)) − γSH−1(h+H◦ϱ(Fη·en)) +Aη +(A−t +η u/Jη))Nη +−ςH (η)Nη = T ◦ FηNη +on Σ +u · en + ∂1η = 0 +on Σ +u = 0 +on Σ0, +(1.4.7) +where for τ = H−1(h + H ◦ ϱ(Fη · en)), M = Aη, and w = A−t +η u/Jη we have used the notation +Sτ +Mw = µ(τ) +� +∇wMt + M∇wt − 2 +n((M∇) · w)I +� ++ λ(τ)((M∇) · w)I, +(1.4.8) +and we also denote Nη = (−∇∥η, 1). +Note that in obtaining (1.4.7) we did not only flatten the velocity vector, we also multiplied by +the matrix JηAt +η. This has the following important consequences. First, we are able to maintain +the continuity equation in ‘perfect divergence’ form. Second, the kinematic boundary condition +transforms to a linear equation. These properties are indispensable in our analysis. +We have one more change of unknowns left to make before we reach the desired formulation of +the problem. The issue is that the formulation (1.4.7) will not quite let us ensure that h belongs to + +COMPRESSIBLE TRAVELING WAVES +11 +a standard Sobolev space. Rather than develop more nonstandard Sobolev theory, we circumvent +this issue with a final change of unknowns and equations by defining q = h − gη, and multiplying +the momentum equation by A−1 +η , which leads to considerable simplifications in our analysis. We are +then left with the following final equivalent form of our equations for the unknowns (q, u, η) with +data (T , G, F): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (σq,η(u − Mηe1)) = 0 +in Ω, +γ2σq,ηM−t +η (((u − Mηe1) · ∇)(M−1 +η u)) + σq,η∇(q + gη) +−γM−t +η (∇ · (Sσq,η +Aη (M−1 +η u)Mt +η)) = Jησq,ηM−t +η G ◦ Fη + JηM−t +η F ◦ Fη +in Ω, +−((P − Pext) ◦ σq,η − γSσq,η +Aη (M−1 +η u))Mt +ηen − ςH (η)Mt +ηen = T ◦ FηMt +ηen +on Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0, +(1.4.9) +where we have set +Mη = JηAt +η = +�(1 + ∂nEη)I(n−1)×(n−1) +0(n−1)×1 +−E(∇∥η) +1 +� +: Ω → Rn×n +(1.4.10) +and +σq,η = H−1(q + gη + H ◦ ϱ(Fη · en)) = H−1(−gidRn · en + q + g(I − E)η) : Ω → R. +(1.4.11) +As we discussed in Section 1.1, the original system (1.1.5) admits a stratified equilibrium solution. +This solution is still encoded in the new formulation (1.4.9) in the sense that for any γ ∈ R+, +(q, u, η) = (0, 0, 0) is a solution to (1.4.9) when there are no additional stress or forces present, i.e. +when (T , G, F) = (0, 0, 0). Furthermore, we show in Corollary C.13 that these trivial solutions are +unique among triples (q, u, η) satisfying +� +∥q∥2 +H4+⌊n/2⌋ + ∥u∥2 +H5+⌊n/2⌋ + ∥η∥2 +H11/2+⌊n/2⌋ < ρ, +(1.4.12) +where ρ is a constant depending only on the equilibrium depth b, gravity g, the pressure law P, +the viscosity coefficients µ and λ, and the dimension n. We emphasize, though, that this result is +recorded in this form for simplicity but could be improved. This reflects the fact that we only expect +traveling wave solutions to exist for viscous fluids if they are generated by stress and force. Solutions +to (1.4.9) also obey a balance of power and dissipation analogous to (1.2.2); this is recorded in +Corollary C.12. +1.5. Statement of main result. We now state our main results. To do so, we first need to +introduce a bit of notation to describe the functional framework. In our work, as in the previous +work of Leoni and Tice [60], Stevenson and Tice [97], and Koganemaru and Tice [57] on traveling +wave solutions to the incompressible analog of (1.2.1), and Nguyen and Tice [82] on traveling wave +solutions to the one-phase Muskat problem, the traveling wave structure forces the free surface +functions to belong to a scale of nonstandard anisotropic Sobolev spaces, the properties of which +end up playing a crucial role in the analysis. Therefore, in order to properly state our main theorem, +we first introduce these spaces. +For R ∋ s ⩾ 0 and d ∈ N+ we define the anisotropic Sobolev space +Hs(Rd) = {f ∈ S ∗(Rd; R) : F[f] ∈ L1 +loc(Rd; C), ∥f∥Hs < ∞}, +(1.5.1) +equipped with the norm +∥f∥Hs = +� � +Rd +� +|ξ|−2(ξ2 +1 + |ξ|4)1B(0,1)(ξ) + ⟨ξ⟩2s1Rd\B(0,1)(ξ) +� +|F[f](ξ)|2 dξ +�1/2 +. +(1.5.2) +We refer to Appendix B.1 for more information on these function spaces, but for the purposes of +stating our main theorem, we note here that Hs(Rd) is a Hilbert space and Hs(Rd) �→ Hs(Rd) �→ +Hs(Rd) + C∞ +0 (Rd), with equality in the first embedding if and only if d = 1. + +12 +NOAH STEVENSON AND IAN TICE +For the following statement of the main theorem, we set r = 10 + 2⌊n/2⌋ and for s ∈ N we define +the sets +Us = H1+s+r(Rn; Rn×n) × Hs+r(Rn; Rn) × Hs+r(Rn; Rn) × R+, +(1.5.3) +Vs = H1+s+r(Ω) × H2+s+r(Ω; Rn) × H5/2+s+r(Rn−1), +(1.5.4) +V 0 = {(q, u, η) ∈ V0 : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = 0}. +(1.5.5) +We can now state our main theorem, which establishes the well-posedness of the traveling wave +formulation of free boundary compressible Navier-Stokes equations (1.4.9). +Theorem 1 (Proved in Theorem 7.3). Assume that the parameters µ, λ, and ς satisfy (1.1.7) and +that P satisfies P ′ > 0 and (1.1.9). There exist a collection {V (γ)}γ∈R+ of open subsets of V 0 and +a nonincreasing sequence {Us}∞ +s=0 of open subsets of U0 such that the following hold. +(1) Nondegeneracy: We have that {0} × R+ ⊆ �∞ +s=0 Us and 0 ∈ � +γ∈R+ V (γ). +(2) Existence and uniqueness: For all tuples (T , G, F, γ) ∈ U0 of applied stress, specific bulk +force, bulk force, and wave speed, there exists a unique solution (q, u, η) ∈ V (γ) such that +the traveling wave reformulation for the free boundary compressible Navier-Stokes equations, +system (1.4.9), is classically satisfied with data (T , G, F) and wave speed γ. +(3) Regularity, given low norm smallness: If s ∈ N+ and (T , G, F, γ) ∈ Us ∩ Us, then the +corresponding solution satisfies (q, u, η) ∈ V (γ) ∩ Vs. +(4) Continuous dependence: For any s ∈ N, the solution map +Us ∩ Us ∋ (T , G, F, γ) �→ (q, u, η) ∈ Vs ∩ V 0 +(1.5.6) +is continuous with respect to the Us and Vs ∩ V 0 topologies. +(5) No vacuum formation: There exists positive constants c, C ∈ R+ such that for all (q, u, η) ∈ +� +γ∈R+ V (γ) we have that c ⩽ σq,η ⩽ C, where σq,η is defined in (1.4.11). +(6) Flattening map diffeomorphism: For any s ∈ N and (q, u, η) ∈ Vs ∩ � +γ∈R+ V (γ), we have +that the flattening map Fη from (1.4.4) is a smooth diffeomorphism from Ω to Ω[η] that +extends to a C12+⌊n/2⌋+s diffeomorphism from Ω to Ω[η]. +Before enumerating some corollaries, we pause for a few comments and remarks. First, we +emphasize that a high-level summary of our theorem is that traveling waves for the free boundary +compressible Navier-Stokes system are generic: they exist for all nontrivial wave speeds γ ∈ R+, and +for a fixed regularity index s ∈ N the set of stress-force-speed data, Us ∩ Us, is open. In particular, +for each fixed wave speed γ ∈ R+ and s ∈ N, the set of stress and force data for which we can +solve (1.4.9) is an open set containing the origin, so the existence of small-data solutions is a generic +phenomenon. The utility of working in Us ∩ Us is that it allows us to prove the joint continuity +of our solutions with respect to both the stress-forcing data (T , G, F) and the wave speed γ. In +fact, we can say a bit more: we show in Theorem 7.1 and Remark 7.2 that the solution map (1.5.6) +enjoys a certain form of continuous differentiability, and even higher regularity. Stating this precisely +entails carefully dealing with more issues related to derivative loss, so we have skipped this technical +point in the statement of Theorem 1 for the sake of brevity. +Second, it is worth highlighting that our theorem does slightly different things when n = 2 as +compared to when n ⩾ 3. In the case n ⩾ 3, the hypothesis (1.1.7) requires that the coefficient +of surface tension is positive, ς > 0, and our analysis crucially uses this and the ellipticity of the +mean curvature operator to gain regularity for η. When n = 2 we only require ς ⩾ 0, as in this +case there is another mechanism built into the equations for regularity gain, namely the equation +∂1η = −u · en on Σ, which is elliptic when n = 2 since then ∂1 is elliptic as a differential operator on +Σ ≃ R. When n ⩾ 3 we have to contend with the free surface function η belonging to the anisotropic +spaces H5/2+s(Rn−1), which are strictly larger than H5/2+s(Rn−1), but when n = 2 we have that +H5/2+s(R) = H5/2+s(R), and so our solutions live in standard Sobolev spaces. Interestingly, when +n = 2 there is one condition that must be stronger than when n ⩾ 3; indeed, (1.1.7) requires positive + +COMPRESSIBLE TRAVELING WAVES +13 +Figure 2. A depiction of three traveling wave free surfaces generated by the same +applied stress tensor T = −ϕI3×3, where ϕ ⩾ 0 is compactly supported, with +increasing wave speed γ moving from left to right, indicated by the red arrows. +bulk viscosity, λ > 0, when n = 2 and only λ ⩾ 0 when n ⩾ 3. This is ultimately related to technical +issues with the deviatoric Korn inequality; we refer to Remark A.5 for further details. +In spite of the above dimensional differences in the precise functional setting, the space V 0 ∩ Vs +from (1.5.4) and (1.5.5) always satisfies the embedding (thanks to Proposition B.1 and standard +Sobolev embeddings) +V 0 ∩ Vs �→ Cs+k +0 +(Ω) × Cs+k+1 +0 +(Ω; Rn) × Cs+k+2 +0 +(Rn−1) +(1.5.7) +for k = 10 + ⌊n/2⌋. Since our theorem guarantees the inclusion (q, u, η) ∈ V0 ∩ Vs, we see that our +solutions always decay to zero at infinity, which means that our solutions are what are known as +solitary waves in the parlance of the traveling wave literature. At the level of generality of our +well-posedness theory, there is not much more qualitative information that can be deduced about +our solutions. However, our result opens the door to more detailed qualitative studies given specific +forms of the stress-forcing data tuple (T , G, F). +We also point out that our techniques only allow us to construct traveling wave solutions (γ ∈ R+) +and not stationary (γ = 0) solutions. The strict sign condition on γ plays a key role in our analysis +by, in part, allowing us to use a nondegenerate norm on the collection of free surface functions. Our +selected functional framework simply does not work with γ = 0. We face a similar issue with the +gravitational constant in our analysis; indeed, we can only construct solutions in the case that g > 0. +We now turn our attention to two corollaries of Theorem 1. The first formalizes the above +discussion about the open set of stress-force data for a given wave speed γ ∈ R+. +Corollary 2 (Proved in Corollary 7.4). With r = 10 + 2⌊n/2⌋ as before, for each γ ∈ R+ there +exists a nonempty open set +(0, 0, 0) ∈ W (γ) ⊂ H1+r(Rn; Rn×n) × Hr(Rn; Rn) × Hr(Rn; Rn) +(1.5.8) +with the property that for all stress-force data tuples (T , G, F) ∈ W (γ) there exists a unique +(q, u, η) ∈ V (γ) (where the latter open set is from the statement of Theorem 1) such that system (1.4.9) +is satisfied with solution (q, u, η), wave speed γ, and data (T , G, F). +Although the natural formulation for the traveling wave problem from the perspective of well- +posedness is in a flattened domain as in (1.4.9), we can also switch our solutions back to the Eulerian +formulation (1.2.1). We record this in our second corollary. +Corollary 3 (Proved in Corollary 7.5). Each solution to the flattened perturbative enthalpy formula- +tion for the traveling wave problem for free boundary compressible Navier-Stokes, i.e system (1.4.9), +produced by Theorem 1 gives rise to a classical solution to the traveling Eulerian formulation of the +problem given by system (1.2.1). + +14 +NOAH STEVENSON AND IAN TICE +1.6. Summary of strategy and layout of paper. In this subsection we aim to summarize the +principal difficulties in proving Theorem 1 and our strategies for overcoming them. This will also +serve to outline the structure of the paper. +High level summary of difficulties. The boundary value problem (1.4.9) is posed in an +unbounded domain with infinite measure and non-compact boundary, the equations are quasilinear, +and there is no variational structure; as such, compactness, Fredholm, and variational techniques are +unavailable. This, along with our expectation of a robust linear theory, suggests that the construction +of solutions should proceed through perturbative techniques such as the implicit function theorem, +or more fundamentally, an iteration scheme based on some sort of linearization. Indeed, an implicit +function theorem strategy, based on the linearization of the equations around vanishing stress-force +data and trivial solution triple for a fixed arbitrary wave speed γ ∈ R+, proved successful in recent +work [57, 60, 97] on the incompressible version of (1.4.9), so it is enlightening to begin our discussion +by stating the corresponding linearization of (1.4.9): +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (ϱu) + g−1ϱ′∂1(q + gη) = g +in Ω, +−γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f +in Ω, +−(ϱq − γSϱu)en − ς∆∥ηen = k +on Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0, +(1.6.1) +where the linearized unknowns are still labeled (q, u, η) but the linearized data is now the triple +(g, f, k). +The reader familiar with the elliptic structure of the incompressible Stokes problem will recognize +a fundamental difficulty appearing already in the first two equations of (1.6.1): even if we ignore η +or view it as given, these two equations do not constitute an elliptic system for (q, u) in the sense of +Agmon, Douglis, and Nirenberg [3], due to the appearance of ∂1q in the first equation. Without +this elliptic structure to serve as a base for the analysis, it is not obvious that (1.6.1) will give rise +to an isomorphism between Banach spaces, a necessary ingredient for the perturbation strategy. +Remarkably, in spite of this ellipticity failure, we are able to show in Theorem 6.17 that the forward +linear map (q, u, η) �→ (g, f, k) defined by (1.6.1) actually does induce an isomorphism between the +Sobolev-type Hilbert spaces +0,0,0 +Xs and Ys for s ∈ N, as defined by (3.4.36) and (3.1.6), respectively. +Unpacking the details of the space +0,0,0 +Xs reveals the fundamental difficulties lurking in (1.4.9) and +motivates our overall strategy. +A cursory glance at the spaces +0,0,0 +Xs and Ys shows that the regularity count essentially matches +that of the incompressible problem if we formally identify q with the pressure in the incompressible +problem: in the domain space, if q gets 1 + s derivatives, then u gets 2 + s, and η gets 5/2 + s, while +in the codomain g gets 1 + s derivatives, f gets s, and k gets 1/2 + s. However, a closer inspection +reveals two crucial complications with these spaces and this counting scheme. The first, which was +already present in the analysis of the incompressible problem, is that the structure of the operators +hitting η in (1.6.1) only allows for the recovery of estimates of +∆∥η ∈ H1/2+s(Rn−1), ∇∥η ∈ Hs(Rn−1; Rn−1), and ∂1η ∈ ˙H−1(Rn−1), +(1.6.2) +where +˙H−1(Rn−1) is defined by (1.7.4), which is not enough to guarantee the inclusion η ∈ +H5/2+s(Rn−1) for general n. Instead, as we prove in Proposition B.4, these inclusions essentially +characterize the anisotropic inclusion η ∈ H5/2+s(Rn−1). The takeaway is that the anisotropic +spaces are inextricably linked to the traveling wave problem through the structure of the differential +operators (and also through the positivity g > 0 and γ > 0, which give us the latter two estimates +of (1.6.2)). The second complication is more severe and new to the compressible problem: knowing + +COMPRESSIBLE TRAVELING WAVES +15 +that q ∈ H1+s(Ω), u ∈ H2+s(Ω; Rn), and η ∈ H5/2+s(Rn−1) alone is not enough to guarantee +that g ∈ H1+s(Ω). With this count, the continuity equation in (1.6.1) only yields g ∈ Hs(Ω). To +achieve the higher regularity inclusion g ∈ H1+s(Ω) we have to build the extra condition that +∂1q ∈ H1+s(Ω) into the domain space +0,0,0 +Xs , and conversely, with g ∈ H1+s(Ω) we are able to recover +that ∂1q ∈ H1+s(Ω) through the linearized continuity equation. This is what the 0, 0, 0 adornment +actually indicates for the space +0,0,0 +Xs : the q elements in this space enjoy some ‘bonus partial regularity’ +whose precise form is dictated by the structure of the linearized operator around the trivial triple +(0, 0, 0). The bonus partial regularity can also be viewed as another manifestation of anisotropy in +the problem. +The fact that (1.6.1) induces an isomorphism +0,0,0 +Xs +∋ (q, u, η) �→ (g, f, k) ∈ Ys is certainly +encouraging, but in reality it exposes a much deeper complication with the nonlinear problem (1.4.9). +Indeed, if we attempt to formulate (1.4.9) as a nonlinear mapping problem on a space in which +(q, u, η) ∈ +0,0,0 +Xs , then we immediately see that the bonus partial regularity is lost by the nonlinearity. +In more concrete terms: any attempt to solve (1.4.9) through an iteration scheme based on the +isomorphism from (1.6.1) will suffer from derivative loss, rendering the scheme useless. +This isomorphism issue is actually more generic. We also prove in Theorem 6.17 that for any +appropriately small triple (q0, u0, η0), the linearization of (1.4.9) around (q0, u0, η0) for (T , G, F) = 0 +and γ ∈ R+ induces an isomorphism +q0,u0,η0 +Xs +∋ (q, u, η) �→ (g, f, k) ∈ Ys, where now the ‘adapted +space’ +q0,u0,η0 +Xs +encodes the bonus partial regularity vq0,u0,η0 · ∇q ∈ H1+s(Ω) for a vector field vq0,u0,η0 +that is determined by the linearization location (q0, u0, η0) (the field is collinear with e1 at (0, 0, 0)). +Once more, this can be viewed as a sort of anisotropy, but now it is clear that the favored direction +depends on the background triple (q0, u0, η0). These general adapted spaces face the same problem +described above: the bonus regularity is lost by the nonlinearity, leading to derivative loss in any +iteration scheme. The failure of the nonlinearity to preserve the adapted spaces can ultimately be +traced to the fact that the bonus regularity is not perturbative: we cannot use control of vq0,u0,η0 ·∇q +to say anything useful about vq1,u1,η1 · ∇q for general distinct triples (qi, ui, ηi) with i ∈ {0, 1}. In +other words, in general the adapted spaces +q0,u0,η0 +Xs +and +q1,u1,η1 +Xs +are inequivalent. +While these issues preclude the use of elementary perturbation techniques, they also reveal the +potential utility of more sophisticated Nash-Moser techniques. Indeed, for any appropriate triple +(q0, u0, η0), we have the natural inclusion Xs+1 �→ +q0,u0,η0 +Xs +, where the former space is defined by (3.1.3) +and encodes no location-specific bonus regularity. This suggests that we may pose the nonlinear +mapping from (1.4.9) on a scale of spaces, indexed by s, in which the codomain involves Ys but, to +compensate for the derivative loss, the domain scale is shifted and requires (q, u, η) ∈ Xs+1. The +above isomorphism results then suggest that the maps Ys ∋ (g, f, k) �→ (q, u, η) ∈ +q0,u0,η0 +Xs +�→ Xs will +allow us to construct the right and left inverse to the derivative of the nonlinear map, provided +we expand our view to scales of Banach spaces and accept the reality of derivative loss. This is +precisely the purview of the Nash-Moser technique [81, 80], which we have thus chosen as the engine +to prove Theorem 1. +Nash-Moser framework. Our goal in studying (1.4.9) is not just to show the existence and +uniqueness of solutions, but to establish a proper well-posedness theory that shows the solutions +depend continuously on the data in the optimal topology. To the best of our knowledge, the only +Nash-Moser inverse function theorems in the literature with this capability are the formulations +of Sergeraert [93] and Hamilton [40], which actually yield smoothness of the inverse map. The +Sergeraert and Hamilton Nash-Moser theorems work in the context of smooth tame maps between +Fr´echet spaces. Roughly speaking, one can think of tameness as a family of structured estimates + +16 +NOAH STEVENSON AND IAN TICE +associated to the derivatives of the maps; these bounds play an essential role in the use of a +modification of Newton’s method to overcome the derivative loss in proving surjectivity. The Fr´echet +spaces that serve as the domain and codomain of the nonlinear operator in these theorems can be +thought of as the intersection of all of the spaces in a Banach scale (like H∞(Rn) = � +k∈N Hk(Rn) +vis-`a-vis the Banach scale {Hk(Rn)}k∈N). The hypotheses of the Sergeraert and Hamilton Nash- +Moser variants require, among other things, a family of right inverses to the derivative mapping +into the domain Fr´echet space. Unfortunately, for reasons that are ultimately attributable to the +failure of hypoellipticity for the hyperbolic structure appearing in the traveling wave formulation of +the continuity equation, in our context we can only verify that in a given open neighborhood of the +trivial solution, the derivatives’ inverses only map into finite regularity spaces in the Banach scale, +and so we cannot satisfy the basic hypotheses of Sergeraert’s or Hamilton’s formulation. +There are Nash-Moser theorems in the literature that allow for this finite invertibility range. The +oldest we are aware of is found in the work of Schwartz [90, 91], but this is formulated only for a +very specific Banach scale, produces solutions in a suboptimal space, and has no mechanism for +regularity promotion. The version due to H¨ormander [42, 43, 44] works for general Banach scales +and produces solutions in the optimal or nearly-optimal space in an associated ‘weak Banach scale,’ +which in some cases coincides with the original (e.g. H¨older scales) but in general is slightly larger +(e.g. if the original is the Sobolev scale {Hk(Rn)}k∈N, then the weak scale is the larger Besov scale +{Bk +2,∞(Rn)}k∈N). This result also has no regularity promotion mechanism. The recent Nash-Moser +theorem of Baldi and Haus [5] produces solutions in the optimal space and also has a regularity +promotion mechanism. Famously, the Nash-Moser technique overcomes the problem of derivative loss +by employing a family of smoothing operators that satisfy a host of precise quantitative estimates. +Baldi and Haus achieve their significant improvement by placing more strenuous conditions on +these smoothing operators than in the other Nash-Moser formulations, which allows them to port +techniques from Littlewood-Paley theory and the paradifferential calculus into the abstract setting. +In the context of the function spaces we employ, it is relatively easy to construct smoothing +operators that satisfy the hypotheses of, say H¨ormander’s Nash-Moser theorem [43]. However, in +our context it is rather delicate to show that these operators satisfy the stronger hypotheses of +Baldi and Haus [5]. Rather than focus effort on this construction, which would mostly apply to +our specific problem, we have chosen to use a more general, abstract approach, which is a Banach +scale generalization of the notion of a tame Fr´echet space as defined in Hamilton [40], and which +we believe may be of use in other problems. Roughly speaking, the idea is to view a given scale of +Banach spaces, in which it is hard or impossible to construct the smoothing operators, as being a +retract of another Banach scale in which the smoothing operators are known to exist. In the setting +of Banach spaces, we can think of the retract property in terms of the former spaces being direct +summands, or complemented subspaces, of the latter. We emphasize that this idea does not pull +the smoothing operators back to the initial scale, but rather pushes the nonlinear map forward to +the larger scale, and so in some sense our method shifts the focus from constructing smoothing +operators to identifying the direct summand structure, which is more amenable to PDE techniques. +In light of the above discussion, we have opted to craft another version of the abstract Nash-Moser +inverse function theorem, synthesizing the desired elements of the Sergeraert, Hamilton, and Baldi +and Haus formulations, that is capable of working within our finite range of invertibility context, +produces solutions in the optimal space, provides a regularity promotion mechanism, and provides +some degree of regularity for the inverse map, in particular a continuity assertion in an optimally +strong norm. Our new version employs the direct summand method to sidestep the smoothing +operator construction. This new Nash-Moser formulation, the precise statement of which is given in +Theorems 2.21 and 2.24, is essential for our proof of Theorem 1, but we believe it is likely to be of +broader interest and applicability due to the flexibility of its hypotheses and improved conclusions. +We refer to Section 2 for further exposition. + +COMPRESSIBLE TRAVELING WAVES +17 +With our Nash-Moser variant in hand, our strategy for proving Theorem 1 is simple to state: +encode the conclusions of the theorem as properties of a nonlinear map associated the system (1.4.9) +that can be granted by our Nash-Moser inverse function theorem, and then verify the hypotheses of +Nash-Moser. These hypotheses, which are stated precisely in Definition 2.20 and Theorem 2.21, +are divided into two categories: nonlinear and linear. For the former, we need to verify that the +nonlinear operator satisfies certain differentiability conditions and that the derivatives obey certain +tame estimates. At the linear level we need to study the linearization of (1.4.9) around a generic +triple (q0, u0, η0) in an open neighborhood of zero, and in particular we need to construct the family +of left and right inverses to these derivatives and verify they also obey a set of tame estimates. +The diagram in Figure 3 represents the logical flow of dependencies for our strategy as it is +implemented in this paper. The gray boxes correspond to the abstract nonlinear analysis in Section 2, +which culminates with the inverse function theorem. The green boxes correspond to the main +features of our nonlinear analysis, which are found in Section 3. The blue boxes show our linear +analysis strategy, which is then executed in Sections 4, 5, and 6. Finally, at the bottom, we have +the red box representing our conclusion and final proof of Theorem 1, appearing in Section 7. +Well posed- +ness, Section 7 +Existence for +regularized principal +part, Section 5.2 +Linear analysis syn- +thesis, Section 6.4 +Refined Nash-Moser +inverse function +theorem, Section 2 +Smooth tameness +verification for +nonlinear operator, +Section 3.3 +Tame structure +abstraction, +Section 2.1 +Derivative loss +vector field, +Lemma 3.19 +Analysis of atomic +nonlinearities, +Section 3.2 +Spitting of +the derivative, +Section 3.4 +Analysis of +(regularized) +steady transport +equations, Section 4 +Estimates on +principal part +strong solutions, +Theorem 6.16 +Estimates for +regularized +principal part +strong solutions, +Theorem 6.16 +Estimates on +regularized principal +part weak solutions, +Proposition 5.4 +Estimates on +principal part +weak solutions, +Proposition 5.2 +Figure 3. Logical flow of the paper +Nonlinear analysis. Our abstract nonlinear analysis begins in Section 2 with the study of +the tame structure of differentiable maps between scales of Banach spaces, which form the basic + +18 +NOAH STEVENSON AND IAN TICE +framework of our Nash-Moser theorem. +In contrast to the Sergeraert [93] and Hamilton [40] +frameworks, we study tameness in a finite regularity context and on possibly finite Banach scales +rather than Fr´echet spaces. We develop a calculus of such maps, showing closure under various +operations such as sums, products, and compositions. These results turn out to be essential in our +subsequent verification of the ‘nonlinear hypotheses’ of our Nash-Moser inverse function theorem. +Once this is done, we then formulate and prove our version of the inverse function theorem. +In Section 3.1 we precisely define the nonlinear operator associated to the system (1.4.9), namely +Ψ defined by (3.3.1), and formulate the Banach scales that serve as its domain and codomain. +We verify, in Lemma 3.2, that the domain Banach scale is tame and that the codomain Banach +scale is a tame direct summand of the domain. Then we endeavor to check that Ψ is tamely twice +continuously differentiable and has order one derivative loss. Within a standard Sobolev framework +this could be accomplished, more or less, with off-the-shelf tools. Unfortunately, in our context the +free surface functions η belong to the anisotropic spaces H5/2+s(Rn−1), and their ubiquity in the +nonlinearities of (1.4.9) then requires a much more delicate and customized analysis. To make this +task less arduous, we first identify within Ψ a number of simpler ‘atomic’ nonlinearities that can be +analyzed separately. +For the most part, these atoms are handled via elementary high-low estimates from Appendix D.2, +combined with some results from the abstraction of tame structure. We emphasize that our choice +of the enthalpy formulation makes the atoms that comprise the momentum equation all of this form. +By contrast, the nonlinearity arising from the continuity equation in (1.4.9), namely +(q, u, η) �→ Ξ(q, u, η) = ∇ · (σq,η(u − Mηe1)), +(1.6.3) +is significantly more delicate and requires more sophisticated ideas. The principal difficulty here is +that our functional setting requires that +� b +0 +Ξ(q, u, η)(·, y) dy ∈ ˙H−1(Rn−1), +(1.6.4) +where ˙H−1(Rn−1) is defined by (1.7.4). At first glance, this inclusion appears to follow readily from +the identity +� b +0 +Ξ(q, u, η)(·, y) dy = (∇∥, 0) · +� b +0 +(σq,η(u − Mηe1) + ϱe1)(·, y) dy, +(1.6.5) +but a closer inspection reveals that, because of the anisotropic spaces, the integral argument of +the divergence on the right does not belong to L2(Rn−1) in general. To get around this problem +we employ the ‘Taylor expansion trick’ of Lemma 3.8 in conjunction with the subtle vector field +decomposition from Lemma 3.9. These rely crucially on various nontrivial algebraic properties of +the anisotropic spaces. +We then synthesize the analysis of the atomic nonlinearities with our analysis of tame structure +to complete the verification of most of the ‘nonlinear hypotheses’ of the inverse function theorem. +This is done in Theorem 3.17. +Principal part identification. We next pass through some applications of our nonlinear results +to set up the linear analysis in a simpler form. We identify the manifestation of the derivative loss at +the linear level as the vector field vq0,u0,η0 in the linearized continuity equation, as described above. +The relevant properties of this derivative loss vector field are enumerated in Lemma 3.19. This +understanding allows us to perform a ‘derivative splitting’ for the nonlinear operator Ψ of the form +DΨ = DΨprin + DΨrem, where the linear operator DΨprin is the ‘principal part’ and DΨrem is the +perturbative remainder. Associated to the principal part operator DΨprin is the following principal + +COMPRESSIBLE TRAVELING WAVES +19 +part PDE system for linearized unknowns (q, u, η) with given data (g, f, k) and wave speed γ ∈ R+: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (ϱu) + ∇ · (vq0,u0,η0(q + gη)) = g +in Ω, +−γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f +in Ω, +−(ϱq − γSϱu)en − ς∆∥ηen = k +on Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0. +(1.6.6) +This splitting provides two key benefits. First, the problem (1.6.6) is as close as possible to the +linearization around the trivial background, namely (1.6.1), while retaining the entirety of the +derivative loss information. Second, the remainder piece of the linearization has no derivative loss +and is effectively small so that it can be handled perturbatively. We refer to Propositions 3.21 +and 3.22 for the precise details of this derivative splitting. As a result of this careful splitting, we +effectively reduce most of our linear analysis to the study of this family of principal part linear +equations, (1.6.6). +Linear analysis. Now we discuss the strategy for the verification of the ‘linear hypotheses’ for +our Nash-Moser inverse function theorem. The majority of the work is devoted to studying the +principal part system (1.6.6) with vq0,u0,η0 obtained from a general background triple (q0, u0, η0) +in an open neighborhood of zero, and in particular showing that it induces the aforementioned +isomorphism +q0,u0,η0 +Xs +∋ (q, u, η) �→ (g, f, k) ∈ Ys and obeys related tame estimates. The fundamental +difficulty here, as in (1.6.1), is the lack of ellipticity in the base Stokes system for q and u; indeed, +one should view (1.6.6) as an unhappy marriage of elliptic (Lam´e and mean-curvature type) and +hyperbolic (steady transport type) operators whose individual regularity theories are incompatible +and appear not to combine without substantial difficulty. To deal with the difficulties inherent in +the system (1.6.6), it is convenient to initially decouple the problems of estimates and existence and +only combine them at the last moment. +The key to this strategy is the introduction of a regularizing term in (1.6.6) that makes the +elliptic parts interface better with the hyperbolic steady transport part. Unfortunately, the natural +technique of applying a smoothing operator to the steady transport term in the continuity equation +of (1.6.6) does not work well in Ω: operators that preserve the good elliptic energy structure +seemingly lack good commutators and so fail to give high regularity estimates, while operators that +have good commutators do not seem to respect the energy structure. We are thus led to employ an +elliptic regularization by replacing the system (1.6.6) with +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (ϱu) + τ∇ · (vq0,u0,η0(q + gη)) + N−1Lm(q + gη) = g +in Ω, +−γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f +in Ω, +−(ϱq − γSϱu)en − ς∆∥ηen = k +on Σ, +u · en + ∂1η = N−1(−∆∥)m−1/4η +on Σ, +u = 0 +on Σ0, +∂m +n q = · · · = ∂2m−1 +n +q = 0 +on ∂Ω, +(1.6.7) +where the regularization parameter is N−1 for N ∈ N+, Lm is the 2mth-order linear elliptic +differential operator +Lm = (−1)m +n +� +j=1 +∂2m +j +, +(1.6.8) +(−∆∥)m−1/4 is a standard fractional power of the Laplace operator on Σ ≃ Rn−1, m ∈ N+ is a +tunable regularity parameter, and τ ∈ [0, 1] is an operator homotopy parameter (τ = 1 corresponds +to a regularization of (1.6.6)). Note that the Neumann boundary conditions for q recorded in the + +20 +NOAH STEVENSON AND IAN TICE +final equation of (1.6.7) are new relative to (1.6.6); their presence is dictated by our introduction of +Lm, but the specific choice of the Neumann conditions plays a crucial role later. +The benefits of the elliptic regularization (1.6.7) are manifold. First, the domain space for +(1.6.7) is Xs +m,N, as defined by (3.5.7). Crucially, this space is independent of the background triple +(q0, u0, η0) but continuously embeds into the background-dependent space +q0,u0,η0 +Xs +. This makes it +an ideal setting for using arguments based on the method of continuity to extend the solvability +theory from τ = 0 to τ = 1, which is the form of the problem we actually care about. Second, +the regularization operators preserve the energy structure of the original problem (1.6.6) while +giving a relatively simple regularity gain mechanism. Third, and perhaps most important, they +are compatible with the derivation of N−independent estimates at high-regularity, which we will +employ to solve (1.6.6) in +q0,u0,η0 +Xs +via weak compactness arguments. Note that in doing so we will +always need m ⩾ 1 + s so that the artificial Neumann conditions for q in (1.6.7) do not pass to the +limit. Of course, regularization does not come without its downsides, and a good amount of work +is needed to deal with technical complications it introduces. We now turn to a somewhat more +detailed account of how we use (1.6.7) to complete the linear analysis. +We study a priori estimates for the principal part and its regularization in tandem; for the former +the goal is to develop the desired tame estimates, but for the latter the goal is high regularity +N−independent estimates with only a weaker form of tameness with respect to the background +tuple (q0, u0, η0). The starting point for the principal part is estimates for weak solutions, which +are proved in the usual manner modulo some minor technical complications due to some nonlinear +expressions involving members of anisotropic Sobolev spaces. See Proposition 5.2 for more details. +With the weak estimates in hand, we then turn our attention to high regularity estimates for strong +solutions. The idea here is to exploit the fact that we can apply tangential derivatives to the +system (1.6.6) without changing the basic structure of the equations; this allows us to employ the +weak solution estimates to get bounds on these tangential derivatives as in Theorem 6.5. After +we achieve this control of the tangential derivatives, it remains to recover control of the normal, +or vertical derivatives. Here we implement a version of the classic technique of Matsumura and +Nishida [75], originally developed for fixed domains with Dirichlet boundary conditions, to reveal +a subtle dissipative structure for the normal derivatives of the density. This, together with some +supplementary analysis of steady transport equations (see Proposition 4.4), provides an estimate +on the high norms of the solution in terms of the data and the tangentially differentiated solution +alone. This is recorded in the first conclusion of Theorem 6.15. Synthesizing, we then derive the first +conclusion of Theorem 6.16, which is the desired a priori estimates for the principal part equations. +The strategy for the a priori estimates of the regularized problem (1.6.7) is similar at a bird’s +eye view, although the specifics of the argument are rather distinct. We again start by proving +a priori estimates for weak solutions and then study the equations satisfied by the tangentially +differentiated solution: see Proposition 5.4 and the second conclusion of Theorem 6.5, respectively. +The argument used to derive estimates for the normal system, the second conclusion of Theorem 6.15, +now substantially deviates from that used for the principal part case because of the need to pass +through analysis of the regularized steady transport equations. It is here that the regularization Lm +serves as a liability rather than an asset. Indeed, deriving estimates with good dependence on N is +technically delicate and requires a very careful use of the bilinear form associated to Lm and the +precise Neumann boundary conditions imposed on q. The majority of Section 4 is devoted to this +regularized transport analysis. In the end, we obtain the uniform in N estimates as stated in the +second conclusion of Theorem 6.16. +The existence theory for the regularized problem (1.6.7) for all τ ∈ [0, 1] is developed by first +establishing the existence of weak solutions in Theorem 5.5. This is proved through the method +of continuity, based on the a priori estimates from Proposition 5.4 and an existence result for the +problem with τ = 0, which itself requires a further two-parameter regularization and compactness + +COMPRESSIBLE TRAVELING WAVES +21 +argument to establish. The weak solutions are then promoted to strong solutions in Corollary 5.6 +via elliptic regularity arguments. We emphasize that while this result shows that (1.6.7) induces an +isomorphism, it does not establish N−uniform estimates. The existence theory for the principal +part problem (1.6.6) is then established by way of the regularized existence theory, the N−uniform +a priori estimates, and another weak limiting argument. This is our Theorem 6.17. +For the conclusion of the linear analysis, we combine the work that allowed us to identify the +principal part equations, namely the splitting of the derivative, with the previously discussed +principal part analysis on estimates and existence. This is the synthesis of linear analysis result of +Theorem 6.18. +We conclude with a couple remarks on our linear strategy. First, we emphasize that our existence +strategy for (1.6.6) intentionally bypasses direct regularity promotion of weak solutions in favor +of high regularity inherited by weak limits of solutions to the regularized equations (1.6.7). This +is advantageous since the mixed elliptic-hyperbolic nature of (1.6.6) creates substantial technical +difficulties in attempting to implement the standard techniques for regularity promotion (e.g. finite +differences or mollification and commutators). The second feature we wish to highlight is that our +methods of handling the free surface unknown in the linear existence theory are significantly different +and simpler than in the previous work on the incompressible problem [60, 97, 57]. In these works the +free surface unknown is constructed in terms of the data alone via a pseudodifferential equation, the +symbol for which is inverted after a careful asymptotic analysis. In our work, we entirely circumvent +the use of these delicate pseudodifferential techniques in our existence theory, replacing them with +regularizations, a priori estimates, weak limits, and a new spatial characterization of the anisotropic +Sobolev spaces Hs(Rd). +Conclusion and appendices. Once we have completed the nonlinear and linear analysis, the +hypotheses of our Nash-Moser theorem are verified. We then invoke the result with a few minor +additional arguments in Theorems 7.1 and 7.3 to complete the proof of well-posedness. +The remainder of the paper consists of four appendices. These mostly consist of various analytical +and PDE tools that are customized or optimized for our particular needs in the paper. However, +some of the results there appear to be new and may be of independent interest for use in other +problems. Appendix A records tools related to standard Sobolev spaces. In contrast, Appendix +B is concerned with properties of the nonstandard spaces we employ in this paper, namely the +anisotropic spaces and the adapted spaces. Appendix C focuses on PDE tools, with an emphasis on +the specified divergence problem in various contexts. Appendix D contains a selection of nonlinear +analysis tools that are essential in our abstract tame analysis. +1.7. Conventions of notation. We recall that we have included a Notation Index at the end +of the paper, which catalogs the numerous operators, function spaces, and other symbols used +throughout. +We write N for the set of nonnegative integers, N+ = N \ {0}, and R+ = (0, ∞). Whenever +α ≲ β appears in a result, it means there is a constant C ∈ R+ depending only on the parameters +mentioned in the formulation of the result such that α ⩽ Cβ. To emphasize this dependence, we +will sometimes write α ≲a,...,b β to indicate the parameters a, . . . , b. We will also write α ≍ β to +mean α ≲ β and β ≲ α. We will use the bracket notation +⟨x1, . . . , xp⟩ = (1 + x2 +1 + · · · + x2 +p)1/2 +(1.7.1) +for p ∈ N+ and x1, . . . , xp ∈ R. Given x ∈ Rd, we will often abbreviate ⟨x⟩ = ⟨x1, . . . , xd⟩. If U and +V are some open sets we write U ⋐ V to mean that the closure U is compact and U ⊂ V . +We denote the gradient and its tangential counterpart by ∇ = (∂1, . . . , ∂n) and ∇∥ = (∂1, . . . , ∂n−1), +respectively. The divergence and tangential divergence operators are written ∇ · f = �n +j=1 ∂j(f · ej) +and (∇∥, 0) · f = �n−1 +k=1 ∂k(f · ek) for appropriate Rn-valued functions f. We will also use subscript +‘∥’ to indicate that a differential operator depends only on ∂1, . . . , ∂n−1, e.g. ∆∥ = �n−1 +ℓ=1 ∂2 +ℓ . + +22 +NOAH STEVENSON AND IAN TICE +If ∥·∥ is a norm on a product of normed spaces, X1 × · · · × Xq, we will typically write ∥x1, . . . , xq∥ +in place of ∥(x1, . . . , xq)∥, where xi ∈ Xi for i ∈ {1, . . . , q}. Given Λ ⊆ R, we say a decreasing +collection of Banach spaces {Xs}s∈Λ with norms {∥·∥Xs}s∈Λ is log-convex if for all s0, s1, s ∈ Λ such +that s0 < s1 and s = (1 − σ)s0 + σs1 for some σ ∈ [0, 1] we have that +∥x∥Xs ≲s0,s1,σ ∥x∥1−σ +s0 +∥x∥σ +s1 for all x ∈ Xs1. +(1.7.2) +Suppose that k ∈ N, U ⊆ Rd is open, and V is a finite dimensional real vector space. For +p ∈ [1, ∞], we write W k,p(U; V ) for the usual Lp-based Sobolev space of order k with functions +valued in V , and we write Hk(U; V ) = W k,2(U; V ). We write Ck +b (U; V ) = Ck(U; V ) ∩ W k,∞(U; V ), +endowed with the obvious norm. The space Ck +0 (U; V ) is the closure of C∞ +c (Rd; V ) in Ck +b (U; V ). +We also write C∞ +b (U; V ) = � +k∈N Ck +b (U; V ) and C∞ +0 (U; V ) = � +k∈N Ck +0 (U; V ). Similarly, we write +H∞(U; V ) = � +k∈N Hk(U; V ) and W ∞,∞(U; V ) = � +k∈N W k,∞(U; V ). Note that W ∞,∞(U; V ) = +C∞ +b (U; V ). +For S ∈ {Σ0, Σ}, we write TrS for the trace operator that maps appropriate functions defined on +Ω to functions on S. The following closed subspace of H1(Ω; Rn) is frequently used: +0H1(Ω; Rn) = {u ∈ H1(Ω; Rn) : TrΣ0(u) = 0}. +(1.7.3) +The Fourier transform, which we normalize to be unitary on L2, is denoted by F. We will utilize +the homogeneous Sobolev space of order −1, which is defined as +˙H−1(Rd) = {f ∈ S ∗(Rd; R) : F[f] ∈ L1 +loc(Rd \ {0}; C) and [f] ˙H−1 < ∞} +(1.7.4) +for the seminorm [f] ˙H−1 = ∥1Rd\{0}| · |−1F[f]∥L2. Finally, we emphasize that we frequently identify +Σ and Rn−1 in the canonical way when working with function spaces defined on Σ. +2. A variation on the Nash-Moser inverse function theorem +As we discussed in Section 1.6, our strategy for proving the well-posedness of the system (1.4.9) +is to employ a new version of the Nash-Moser inverse function theorem. The goal of this section is +to prove this theorem and develop its abstract framework, which will be employed generally in all of +our subsequent analysis. Before we begin, we provide a brief overview of the Nash-Moser strategy +and some variants of the theorem that have been developed in the literature. +The abstract setting of the classical inverse function theorem is C1 maps Ψ : U → F, where E +and F are Banach spaces and U ⊆ E is an open set, for which DΨ(u) ∈ L(E; F) is invertible for +some u ∈ U. The typical proof leverages the invertibility of DΨ(u) to prove the local bijectivity +of Ψ by way of the contraction mapping principle. In particular, surjectivity is established via a +Picard iteration scheme that crucially employs the map DΨ(u)−1. +The Nash-Moser approach aims to handle the case when DΨ(u) fails to be invertible but remains +‘nearly invertible’ in the sense that it admits a right inverse L(u), defined on F but only mapping +into some larger vector space E0 ⊃ E. This phenomenon is typically regarded as ‘derivative loss,’ +as in practice E0 often consists of functions, and the subspace E consists of functions of higher +regularity than those in E0. To fully take advantage of this, the Nash-Moser strategy generalizes the +setting of the standard inverse function theorem to maps between one-parameter scales of Banach +spaces, say {Es}s∈S and {F s}s∈S for some S ⊆ R, where roughly speaking one should think of the +parameter s as measuring the regularity of the elements of the space. The derivative loss is then +required to be uniform along the scale in the sense that DΨ maps from Es+µ to F s, for some µ > 0 +measuring the extent of the derivative loss, but its right inverse L (which is now required to exist +in the entirety of an appropriate open set) only maps from F s to Es. See Figure 4 for a diagram. +The profound idea of Nash [81], which was expanded upon by Moser [80], was to establish local +surjectivity not via Picard iteration, which is not available due to the derivative loss, but instead +with an iteration scheme, based on Newton’s method, that employs a family of smoothing operators +that increase the regularity along the Banach scales. The extreme speed of convergence of Newton’s + +COMPRESSIBLE TRAVELING WAVES +23 +method is needed to make the derivative loss and smoothing operators cooperate, and in order to +properly implement this idea various precise quantitative estimates are required for the map and +the smoothing operators. +The above approach is extremely flexible and customizable, and has thus become more of a +strategy than a specific theorem. Indeed, many variants of the Nash-Moser theorem have appeared +in the literature. Most of these are really local surjectivity theorems rather than full inverse +function theorems, which must establish injectivity as well as continuity or higher regularity of the +induced local inverse map. We are aware of a few exceptions in the literature. Sergeraert [93] and +Hamilton [40] prove smoothness of the inverse map by working in the more restrictive context of +smooth tame maps between Fr´echet spaces. Berti, Bolle, and Procesi [10] study an implicit function +theorem with parameters, and they prove continuous differentiability of the local solution map only +with respect to a finite dimensional space of parameters. Ekeland [30] and Ekeland and S´er´e [31] +prove an inverse function theorem and deduce some suboptimal Lipschitz estimates of the inverse +map. +Unfortunately, the well-posedness problem for (1.4.9) is not amenable to any of the above results, +so we have endeavored to develop a new Nash-Moser inverse function theorem that works well for +our problem, produces solutions in optimal spaces, provides a regularity promotion mechanism, +and provides an optimal continuity and even some higher regularity results for the inverse map. +Our approach is to synthesize ideas from Hamilton [40] with the iteration scheme of Baldi and +Haus [5], which is an improvement on the results of H¨ormander [42, 43, 44] that employs ideas from +Littlewood-Paley theory. +When compared to the standard inverse function theorem, the Nash-Moser variants have hypothe- +ses that are much more involved, but are similar at a high level. To guarantee a local inverse for a +nonlinear operator, one needs to check that: 1) the nonlinear operator is defined on tame scales of +Banach spaces; 2) the operator is tamely C2, with a fixed derivative loss; and, 3) the derivative +of the operator admits a tame family of inverses in an open neighborhood of a point. As all of +these hypotheses involve the adjective ‘tame,’ we devote Section 2.1 to defining tame structures +and developing a calculus of tame maps. Once this is done, we use Section 2.2 to state the precise +hypotheses and conclusions of our version of the Nash-Moser inverse function theorem. Afterward, +in Sections 2.3, 2.4, and 2.5 the theorem is then proved in several parts. For the reader looking +to take the inverse function theorem as a ‘black box’ and more readily proceed to the analysis of +the PDE (1.4.9), we suggest restricting to the abstract tame structure and the statement of our +Nash-Moser theorem, but initially skipping over Sections 2.3, 2.4, and 2.5. +2.1. Tame structure abstraction. In this subsection we are first concerned with tame mappings +between scales of Banach spaces. Our presentation here primarily inspired by Hamilton [40] and +Baldi and Haus [5]. Our initial concern is scales of Banach spaces, starting with some notation for +indexing them. +Definition 2.1 (Subsets of N). Given N ∈ N ∪ {∞} we define �N� ⊆ N to be the set �N� = +{n ∈ N : n ⩽ N}. +Now we define scales of Banach spaces. +Definition 2.2 (Banach scales). Let N ∈ N+ ∪ {∞} and let E = {Es}s∈�N� be a collection of +Banach spaces over a common field, either R or C. +(1) We say that E is a Banach scale if for each s ∈ �N − 1� we have the non-expansive inclusion +Es+1 �→ Es, i.e. ∥·∥Es ⩽ ∥·∥Es+1. +(2) If E is a Banach scale, then we define the scale’s terminal space to be EN = � +s∈�N� Es and +endow it with the Fr´echet topology induced by the collection of norms {∥·∥Es}s∈�N�. Note +that if N < ∞, then EN has the standard Banach topology from its norm. + +24 +NOAH STEVENSON AND IAN TICE +(3) If E is a Banach scale, then we write BEr(u, δ) ⊆ Er for the Er-open ball of radius δ > 0, +centered at u ∈ Er. +(4) If E is a Banach scale and EN is dense in Es for each s ∈ �N�, then we say that E is +terminally dense. +We note that finite scales of Banach spaces correspond to the case N ∈ N in this definition. Next +we record a quick remark regarding the product of Banach scales. +Remark 2.3 (Products of Banach scales). Suppose that Ei = {Es +i }s∈�N� is a Banach scale for +i ∈ {1, . . . , n}, each over the same field. Then F = �n +i=1 Ei = {�n +i=1 Es +i }s∈�N� is a Banach scale +when each �n +i=1 Es +i is endowed with the norm +∥u1, . . . , un∥�n +i=1 Es +i = +� +n +� +i=1 +∥ui∥2 +Es +i +�1/2 +. +(2.1.1) +We now consider tame mappings between Banach scales in our next definition. +Definition 2.4 (Tame maps). Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over the +same field, µ, r ∈ �N� with µ ⩽ r, U ⊆ Er be an open set, and P : U → F 0. +(1) We say that P satisfies tame estimates of order µ and base r (with respect to the Banach +scales E and F) if for all �N� ∋ s ⩾ r, there exists a constant Cs ∈ R+ such that for all +f ∈ U ∩ Es we have the inclusion P(f) ∈ F s−µ as well as the estimate +∥P(f)∥F s−µ ⩽ Cs⟨∥f∥Es⟩, +(2.1.2) +where ⟨·⟩ is defined by (1.7.1). +(2) We say that P is µ-tame with base r if for each f ∈ U, there exists an Er-open set V ⊆ U +such that f ∈ V and the restricted map P ↾ V : V → F 0 satisfies tame estimates of order µ +and base r. +(3) We say that P is strongly µ-tame with base r if on every Er-open and bounded subset V ⊆ U, +the restricted map P ↾ V satisfies tame estimates of order µ and base r. +(4) We say that P is (strongly) µ-tamely C0 with base r if P is (strongly) µ-tame with base r +and if for every �N� ∋ s ⩾ r we have that P : U ∩ Es → F s−µ is continuous as a map from +Es to F s−µ. The collections of such maps will be denoted by T 0 +µ,r(U, E; F) and sT 0 +µ,r(U, E; F), +respectively. +(5) For k ∈ N+ we say that P is (strongly) µ-tamely Ck with base r if for all �N� ∋ s ⩾ r the +map P : U ∩ Es → F s−µ is Ck and for all j ∈ {0, 1, . . . , k} the jth derivative map, thought +of as mapping DjP : U × �j +ℓ=1 Er → F 0, is (strongly) µ-tame with base r with respect to +the Banach scales E1+j and F. In other words, DjP ∈ T 0 +µ,r(U × �j +ℓ=1 Er, E1+j; F) or, in +the strong case, DjP ∈ sT 0 +µ,r(U × �j +ℓ=1 Er, E1+j; F). The collections of such maps will be +denoted by T k +µ,r(U, E; F) and sT k +µ,r(U, E; F), respectively. +(6) We denote the collections of (strongly) µ-tamely C∞ with base r maps by T ∞ +µ,r(U, E; F) = +� +k∈N T k +µ,r(U, E; F) and sT ∞ +µ,r(U, E; F) = � +k∈N sT k +µ,r(U, E; F). +(7) When U = Er we will often use the abbreviated notation T k +µ,r(E, F) and sT k +µ,r(E; F) in place +of T k +µ,r(Er, E; F) and sT k +µ,r(U, E; F), respectively. +The following result is a useful characterization of tameness when there is multilinear structure +present in the map. +Lemma 2.5 (Multilinearity and tame maps). Let k ∈ N+ and r, µ ∈ �N� with µ ⩽ r. +Let +F = {F s}s∈�N�, G = {Gs}s∈�N�, and Ej = {Es +j}s∈�N�, for j ∈ {1, . . . , k}, be Banach scales over +the same field. +Let U ⊆ Gr be an open set. +Then the following are equivalent for all maps +P : U × �k +j=1 Er +j → F 0 such that P(g, ·) is k-multilinear for all g ∈ U. + +COMPRESSIBLE TRAVELING WAVES +25 +(1) P ∈ T 0 +µ,r(U × �k +j=1 Er +j ; G × �k +j=1 Ej; F). +(2) The restriction of P to (U ∩ Gs) × �k +j=1 Es +j is continuous as a map into F s−µ for all +�N� ∋ s ⩾ r, and for all g ∈ U there exists a Gr-open subset V ⊆ U such that g ∈ V and +whenever �N� ∋ s ⩾ r, f ∈ V ∩ Gs, and hi ∈ Es +i for i ∈ {1, . . . , k} we have the estimate +∥P(f, h1, . . . , hk)∥F s−µ ≲ ⟨∥f∥Gs⟩ +k +� +i=1 +∥hi∥Er +i + +k +� +j=1 +∥hj∥Es +j +� +i̸=j +∥hi∥Er +i . +(2.1.3) +A similar equivalence holds for maps P ∈ sT 0 +µ,r(U ×�k +j=1 Er +j ; G×�k +j=1 Ej; F) if we change the space +in the first item to the space of strongly tame maps and we change the second item’s quantification +of V to ‘for all bounded Gr-open sets V ⊆ U’. +Proof. The second item implies the first by noting that if {hi}k +i=1 lies within a bounded subset of +�k +i=1 Er +i , then we immediately obtain the required tame estimates on P from inequality (2.1.3). +Now we look to the converse, fixing g ∈ U. The first item provides an open set �V ⊆ U × �k +j=1 Er +j +such that (g, 0, . . . , 0) ∈ �V and if (f, h1, . . . , hk) ∈ �V ∩ (Gs × Es +1 × · · · × Es +k) for �N� ∋ s ⩾ r, then +∥P(f, h1, . . . , hk)∥F s−µ ≲ ⟨∥f∥Gs, ∥h1, . . . , hk∥Es +1×···×Es +k⟩. +(2.1.4) +Now, since �V is open, there exists δ ∈ R+ such that Vδ = BGr(0, δ) × �k +j=1 BEr +j (0, δ) ⊆ �V . Hence, +if f ∈ BGr(0, δ) and h1, . . . , hk ∈ �k +j=1(Er +k \ {0}) are such that (g, h1, . . . , hk) ∈ Gs × Es +1 × · · · × Es +k, +then we have that +(f, δh1/∥h1∥Er +1, . . . , δhk/∥hk∥Er +k) ∈ Vδ +(2.1.5) +and so we can invoke estimate (2.1.4) and multiply through by δ−k∥h1∥Er +1 · · · · · ∥hk∥Er +k to acquire +the desired bound (2.1.3). A similar argument applies in the case of strongly tame maps. +□ +Remark 2.6. If P ∈ T 0 +µ,r(�k +j=1 Ej; F) is a k-multilinear mapping, then a simple modification of +the proof of Lemma 2.5 shows that P is actually strongly tame. Moreover, by multilinearity, we +have that P is automatically a smooth function whose derivatives are also multilinear maps. This +combines with the previous fact to show that actually P ∈ sT ∞ +µ,r(�k +j=1 Ej; F). +As a consequence of the previous lemma, we have structured estimates of the derivatives of tame +maps. +Corollary 2.7 (Derivative estimates on tame maps). Let E = {Es}s∈�N� and F = {F s}s∈�N� be +Banach scales over a common field. Let r, µ ∈ �N� with µ ⩽ r, U ⊆ Er be an open set, and +P ∈ T k +µ,r(U, E; F). Then for each f0 ∈ U, there exists an open set f0 ∈ V ⊆ U with the property +that for every �N� ∋ s ⩾ r there exists a constant Cs,V ∈ R+, depending on s and V , such that for +all f ∈ V ∩ Es and all g1, . . . , gk ∈ Es we have the estimate +∥DkP(f)[g1, . . . , gk]∥F s−µ ⩽ Cs,V +k +� +ℓ=1 +∥gℓ∥Es +� +j̸=ℓ +∥gj∥Er + Cs,V ⟨∥f∥Es⟩ +k +� +j=1 +∥gj∥Er. +(2.1.6) +Moreover, if P ∈ sT k +µ,r(U, E; F), then a similar assertion holds for every bounded and open subset +V ⊆ U. +Proof. This is a direct application of Lemma 2.5. +□ +Our next result studies the interaction of tame maps via composition. +Lemma 2.8 (Composition of tame maps). Let E = {Es}s∈�N�, F = {F s}s∈�N�, and G = {Gs}s∈�N� +be a triple of Banach scales over a common field (R or C), and let k ∈ N and µ, µ′, r, r′ ∈ �N� +be such that µ ⩽ r, µ′ ⩽ r′, r′ + µ ∈ �N�. +Suppose that U ⊆ Er, U ′ ⊆ F r′ are open sets, + +26 +NOAH STEVENSON AND IAN TICE +P ∈ T k +µ,r(U, E; F), and Q ∈ T k +µ′,r′(U ′, F; G). Set V = U ∩ Emax{r,r′+µ}. If P(V ) ⊆ U ′, then +Q ◦ P ∈ T k +µ+µ′,max{r,µ+r′}(V, E; G). A similar assertion holds for the composition of strongly tame +maps. +Proof. We proceed by induction on k ∈ N. Consider first the case that k = 0. Fix f0 ∈ V . Since +P(f0) ∈ U ′, we can appeal to the tameness of Q to obtain an open set P(f0) ∈ W ′ ⊆ U ′ such +that Q satisfies a tame estimate of order µ′ and base r′ in W ′. The map P is continuous and +hence P −1(W ′) ⊆ V is open and contains f0. In light of the tameness of P, there exists an open +set f0 ∈ W ⊆ V in which P satisfies a tame estimate of order µ and base r. Now, the open +set W ∩ P −1(W ′) ⊆ V contains f0 and is such that whenever �N� ∋ s ⩾ max{r, r′ + µ} and +f ∈ Es ∩ W ∩ P −1(W ′) we may estimate +∥Q ◦ P(f)∥Gs−(µ+µ′) ≲ ⟨∥P(f)∥F s−µ⟩ ≲ ⟨∥f∥Es⟩. +(2.1.7) +Hence, Q ◦ P is indeed (µ + µ′)-tamely C0 with base max{r, r′ + µ}. +Now suppose that for 1 ⩽ k ∈ N the result has been proved at the level k − 1, and P and Q +satisfy the hypotheses at level k. Applying the induction hypothesis handles the tameness of all +derivatives up to order k − 1, so it suffices to show that Dk(Q ◦ P) is (µ + µ′)-tame with base +max{r, r′ + µ}. Fix f0 ∈ V . For ℓ ∈ {0, 1, . . . , k} we have that DℓQ is µ-tame with base r and +hence there exists an open set P(f0) ∈ W ′ +ℓ ⊆ U ′ such that in W ′ +ℓ we have that DℓQ satisfies an +ℓ-multilinear tame estimate of order µ′ and base r′ (see Lemma 2.7). The map P is continuous and +hence �k +ℓ=1 P −1(W ′ +ℓ) is an open subset of V containing f0. By appealing to the tameness of P and +Lemma 2.7 again, for ℓ ∈ {0, 1, . . . , k} there exists an open set f0 ∈ Wℓ ⊆ V such that in Wℓ we +have that DℓP satisfies an ℓ-multilinear tame estimate of order µ and base r. +For each R > 0, the set Of0,R = �k +ℓ=1(Wℓ ∩ P −1(W ′ +ℓ)) × B(Emax{r,r′+µ})k(0, R) is an open V × +�k +p=1 Emax{r,r′+µ} containing (f0, 0, . . . , 0). For �N� ∋ s ⩾ max{r, r′ + µ} and (f, g1, . . . , gk) ∈ +Of0,R ∩ Es, we use the Fa`a di Bruno theorem (see, for instance, Section 2.4A in Abraham, Marsden, +and Ratiu [1]) to compute +Dk(Q ◦ P)(f)[g1, . . . , gk] += +� +σ∈Sk +k +� +ℓ=1 +� +j1+···+jℓ=k +cσ,ℓ,k,j1,...,jℓDℓQ ◦ P(f){Dj1P(f), . . . , DjℓP(f)}[gσ(1), . . . , gσ(k)], +(2.1.8) +where Sk denotes the permutation group on {1, . . . , k}, and c∗ denotes some combinatorial constant. +We will estimate the norm in Gs−(µ+µ′) of each term in the sum above. Fix ℓ ∈ {1, . . . , k} and +j1, . . . , jℓ ∈ {1, . . . , k} such that j1 + · · · + jℓ = k. +Thanks to Lemma 2.7 and the fact that +max1⩽i⩽k∥gi∥Er ⩽ R, we are free to estimate +� +� +� +� +� +� +� +∥Dj1P(f)[g1, . . . , gj1]∥F s−µ +... +∥DjℓP(f)[gjℓ−k+1, . . . , gk]∥F s−µ +≲R ⟨∥f, g1, . . . , gk∥Es×···×Es⟩. +(2.1.9) +By the same argument, +� +� +� +� +� +� +� +∥Dj1P(f)[g1, . . . , gj1]∥F r′ +... +∥DjℓP(f)[gjℓ−k+1, . . . , gk]∥F r′ +≲R ⟨∥f, g1, . . . , gk∥Emax{r′+µ}×···×E{r′+µ}⟩ ≲R 1. +(2.1.10) + +COMPRESSIBLE TRAVELING WAVES +27 +Thus, applying the estimate from Lemma 2.7 once more yields +∥DℓQ ◦ P(f){Dj1P(f)[g1, . . . , gj1], . . . , DjℓP(f)[gjℓ−k+1, . . . , gk]}∥Gs−(µ+µ′) +≲R ⟨∥f, g1, . . . , gk∥Es×···×Es⟩. +(2.1.11) +Then (2.1.8) and (2.1.11) combine to give us the estimate +∥Dk(Q ◦ P)(f)[g1, . . . gk]∥Gs−(µ+µ′) ≲R ⟨∥f, g1, . . . , gk∥Es×···×Es⟩, +(2.1.12) +which is the desired tame bound at level k. The result thus holds for all k ∈ N by induction. +□ +A convenient application of the previous result is that tame products of tame maps result in a +tame map. More precisely, we have the following. +Corollary 2.9 (Tame products of tame Ck maps). Let ℓ, k ∈ N with ℓ ⩾ 1. For j ∈ {1, . . . , ℓ} +let Ej = {Es +j}s∈�N� and Fj = {Es +j}s∈�N� be Banach scales over a common field, and let G = +{Gs}s∈�N� be a Banach scale over the same field. Let r1, . . . , rℓ ∈ �N�, µj ∈ �rj� for j ∈ {1, . . . , ℓ}, +r′ ∈ �N�, and µ′ ∈ �r′�. Finally, suppose that Uj ⊆ Erj +j +is open, Pj ∈ T k +µj,rj(Uj, Ej; Fj), and +B ∈ T 0 +µ′,r′(�ℓ +j=1 Fj; G) is k-multilinear. If we set +µ = µ′ + max{µ1, . . . , µℓ} and r = max{r1, . . . , rℓ, µ1 + r′, . . . , µℓ + r′} +(2.1.13) +and assume that r ∈ �N�, then the product map satisfies the inclusion +B(P1, . . . , Pℓ) ∈ T k +µ,r +� +ℓ� +j=1 +Uj ∩ Er +j , +ℓ� +j=1 +Ej; G +� +. +(2.1.14) +A similar assertion holds for products of strongly tame maps. +Proof. We first note that by Remark 2.6, B in fact belongs to sT ∞ +µ′,r′(�ℓ +j=1 Fj; G). The inclusion +(P1, . . . , Pℓ) ∈ T k +max{µ1,...,µℓ},max{r1,...,rℓ} +� +ℓ� +j=1 +Uj ∩ Emax{r1,...,rℓ} +j +, +ℓ� +j=1 +Ej; +ℓ� +j=1 +Fj +� +(2.1.15) +is clear, so the conclusion then follows from Lemma 2.8. +□ +We can also combine families of tame maps by integrating over parameters. Although more +general results hold, we will need only the following simple realization of this fact. +Lemma 2.10 (Integrals of one-parameter families of tame Ck maps). Let k ∈ N ∪ {∞}, E = +{Es}s∈�N� and F = {F s}s∈�N� be a pair of Banach scales over the same field, and let µ, r ∈ �N� +satisfy µ ⩽ r. Suppose that U ⊆ Er is an open set, and for all t ∈ [0, 1] let Pt ∈ T k +µ,r(U, E; F). +Suppose additionally that the defining inequalities for these tame estimates (as in the first item of +Definition 2.4 for the map and its derivatives) are satisfied uniformly in t ∈ [0, 1] and that for every +j ∈ {0, . . . , k}, �N� ∋ s ⩾ r, f ∈ U ∩ Es, and f1, . . . , fj ∈ Es the map +[0, 1] ∋ t �→ DjPt(f)[f1, . . . , fj] ∈ F s−µ +(2.1.16) +is continuous. Then the integral map +� 1 +0 Pt dt : U → F 0 given by f �→ +� 1 +0 Pt(f) dt is well-defined +and satisfies +� 1 +0 Pt dt ∈ T k +µ,r(U, E; F). A similar assertion holds for integrals of strongly tame maps. +Proof. The proof essentially amounts to checking definitions, so we will only sketch the argument. +Hypothesis (2.1.16) ensures that the map +� 1 +0 Pt dt is well-defined, while the assumed uniform tame +estimates ensure that the integrands are uniformly bounded with respect to t ∈ [0, 1]. The map is Ck +as a map from U ∩ Es to F s−µ thanks to (2.1.16) and standard dominated convergence arguments. +The fact that +� 1 +0 Pt dt obeys the defining inequalities to be µ-tamely Ck with base r follows from +uniformity in t and the fact that the norm of the integral is at most the integral of the norm. +□ + +28 +NOAH STEVENSON AND IAN TICE +The latter half of this subsection is concerned with various specialized classes of Banach scales. +The nicest classes of these are introduced in the subsequent definition, which closely follows Baldi +and Haus [5]. +Definition 2.11 (Smoothable and LP-smoothable Banach scales). Let E = {Es}s∈�N� be a Banach +scale. +(1) We say that E is smoothable if for each j ∈ N+ there exists a linear map Sj : E0 → EN +satisfying the following smoothing conditions for every u ∈ E0: +∥Sju∥Es ≲ ∥u∥Es for all s ∈ �N�, +(2.1.17) +∥Sju∥Et ≲ 2j(t−s) ∥Sju∥Es for all s, t ∈ �N� with s < t, +(2.1.18) +∥(I − Sj)u∥Es ≲ 2−j(t−s) ∥(I − Sj)u∥Et for all s, t ∈ �N� with s < t, +(2.1.19) +∥(Sj+1 − Sj)u∥Et ≲ 2j(t−s) ∥(Sj+1 − Sj)u∥Es for all s, t ∈ �N�, +(2.1.20) +where the implicit constants are independent of j and are increasing in s and t. +(2) We say that E is LP-smoothable if E is smoothable and the smoothing operators satisfy the +following Littlewood-Paley condition: for s ∈ �N� and u ∈ Es we have that +lim +j→∞∥(I − Sj)u∥Es = 0, +(2.1.21) +and there exists a constant A > 0, possibly depending on s, such that +A−1 ∥u∥Es ⩽ +� ∞ +� +j=0 +∥∆ju∥2 +Es +�1/2 +⩽ A ∥u∥Es , +(2.1.22) +where the operators {∆j}∞ +j=0 are defined via ∆j = Sj+1 − Sj with the convention that S0 = 0. +We now give some examples of LP-smoothable Banach scales. The first is trivial, but instructive. +Example 2.12 (A fixed Banach space). Suppose that X is a Banach space and N ∈ N ∪ {∞}. +Then the scale {X}s∈�N�, generated by X alone, is LP-smoothable in the sense of Definition 2.11, +provided we take Sj = I for all j ∈ N+. +We also have the less trivial example of Sobolev spaces on all of Euclidean space. +Example 2.13 (Sobolev spaces on Rd). Let V be a finite dimensional real vector space and +N ∈ N ∪ {∞}. The real Banach scale of Sobolev spaces {Hs(Rd; V )}s∈�N� is LP-smoothable for the +smoothing operators {Sj}∞ +j=0 given by Sj = 1B(0,2j)(∇/2πi). +Unfortunately, the (LP-)smoothable Banach scales are insufficiently general for our purposes. As +such, we introduce a broader class that captures Banach scales which are essentially closed and +complemented subspaces of LP-smoothable Banach scales. This is analogous to Hamilton’s notion +of a tame Fr´echet space, which is given in Definition 1.3.2 of [40]. +Definition 2.14 (Tame direct summands and tame Banach scales). Let E = {Es}s∈�N� be a Banach +scale. +(1) Suppose that F = {F s}s∈�N� is a Banach scale over the same field as E. We say that E is a +tame direct summand of F if there exist bounded linear maps λ : E0 → F 0 and ρ : F 0 → E0 +such that the following hold. +(a) ρλu = u for all u ∈ E0. +(b) For s ∈ �N� we have that λ(Es) ⊆ F s and ρ(F s) ⊆ Es, and the induced maps +λ : Es → F s and ρ : F s → Es are bounded and linear. +In this case, we say λ is the lifting map and ρ is the restriction map. +(2) We say that E is tame if there exists an LP-smoothable Banach scale F such that E is a +tame direct summand of F. + +COMPRESSIBLE TRAVELING WAVES +29 +Remark 2.15. We note that if E is a tame direct summand of F, then basic functional analysis +shows that λ(Es) ⊆ F s is a closed and complemented subspace of F s, and π = λ ◦ ρ : F s → F s +is bounded and linear projection onto λ(Es). Consequently, for each s ∈ �N� there exists a closed +subspace Gs ⊆ F s such that F s = λ(Es)⊕Gs. This is the motivation for calling E a direct summand +of F. Moreover, if E is a direct summand of F, then for each s ∈ N we have the equivalence +∥u∥Es ≍ ∥λu∥F s for u ∈ Es. +We have the following example of a tame Banach scale. +Example 2.16 (Sobolev spaces on domains). Let U ⊂ Rd be an open set that is a Stein extension +domain in the sense of Definition A.2, and let EU denote the associated extension operator. Let +V be a finite dimensional real vector space and N ∈ N ∪ {∞}. The real Banach scale of Sobolev +spaces {Hs(U; V )}s∈�N� is tame in the sense of Definition 2.14. Indeed, we realize this scale as +a tame direct summand of {Hs(Rd; V )}s∈�N�, which is LP-smoothable by Example 2.13, with the +lifting operator λ = EU and restriction operator given by standard restriction, ρ = RU. +We now record a result on products of (LP-)smoothable and tame Banach scales, the proof of +which is straightforward and thus omitted. +Lemma 2.17 (Products of Banach scales). The product of a finite family of (smoothable, LP- +smoothable, tame) Banach scales over a common field is again a (smoothable, LP-smoothable, tame) +Banach scale. +Consider now the following useful inequalities about smoothing in tame Banach scales. +Lemma 2.18 (Smoothing in tame Banach scales). Let E = {Es}s∈�N� be a tame Banach scale in +the sense of Definition 2.14. There exist a sequence of smoothing operators {Tj}∞ +j=0 ⊂ L(E0; EN) +and, for s ∈ �N�, sequences of seminorms {ms +j}∞ +j=0, {ns +j}∞ +j=0 on Es such that the following hold. +(1) For all g ∈ Es we have the estimates +∥(Tj+1 − Tj)g∥Es ≲ ms +j(g) and ∥(I − Tj)g∥Es ≲ ns +j(g), +(2.1.23) +where the implicit constants depend only on s. +(2) For s, t ∈ �N� we have that +� +ms +j ≍ 2j(s−t)mt +j +for all s, t, +ns +j ≲ 2j(s−t)nt +j +for all s ⩽ t, +(2.1.24) +with the implicit constants depending only on s and t. +(3) We have the equivalence +∥·∥2 +Es ≍ +∞ +� +j=0 +(ms +j)2 +(2.1.25) +with implicit constants depending only on s. +(4) For every g ∈ Es we have that ns +j(g) ≲ ∥g∥Es (with implicit constants depending only on s), +and ns +j(g) → 0 as j → ∞. +(5) E is terminally dense in the sense of the fourth item of Definition 2.2. +Proof. Let F be an LP-smoothable Banach scale witnessing the definition of tameness for E with +lifting λ ∈ L(E0; F 0), restriction ρ ∈ L(F 0; E0), and smoothing operators {Sj}∞ +j=0 ⊆ L(F 0; F N). +We define Tj = ρ ◦ Sj ◦ λ and for s ∈ �N� and g ∈ Es set +ms +j(g) = ∥(Sj+1 − Sj) ◦ λg∥F s and ns +j(g) = ∥(I − Sj) ◦ λg∥F s. +(2.1.26) +Then the first item follows from the boundedness of ρ. The second, third, and fourth items are +immediate consequences of Definition 2.11 and Remark 2.15. The fifth assertions follows from the +first and the fourth. +□ + +30 +NOAH STEVENSON AND IAN TICE +Next we give an interpolation result. +Lemma 2.19 (Log-convexity in tame Banach scales). Let E = {Es}s∈N be a tame Banach scale in +the sense of Definition 2.14 over either R or C. For r, s, t ∈ �N� with r < s < t we have that +∥u∥Es ≲ ∥u∥ +t−s +t−r +Er ∥u∥ +s−r +t−r +Et +(2.1.27) +for all u ∈ Et, where the implicit constant is increasing r, t. +Proof. The bound is trivial if u = 0, so assume u ̸= 0. Let F be an LP-smoothable Banach scale +witnessing the definition of tameness for E, with lifting, restriction, and smoothing operators λ, ρ, +and {Sj}∞ +j=0, respectively. By Remark 2.15, we have that ∥u∥Ep ≍ ∥λu∥F p for p ∈ {s, r, t}. Now we +invoke the properties of the smoothing operators from Definition 2.11 to see that for any j ∈ N +∥λu∥F s ⩽ ∥(I − Sj)λu∥F s + ∥Sjλu∥F s ≲ 2j(s−t)∥λu∥F t + 2j(s−r)∥λu∥F r +(2.1.28) +and hence ∥u∥Es ≲ 2j(s−t)∥u∥Et + 2j(s−r)∥u∥Er. Now, since 1 ⩽ ∥u∥Et/∥u∥Er, we can choose +j = ⌊log(∥u∥Et/∥u∥Er)/((t − r) log 2)⌋ ∈ N to obtain the desired inequality. +□ +2.2. Mapping hypotheses and statement of the inverse function theorem. We now intro- +duce a lengthy definition that records a number of conditions that must be placed on the nonlinear +map in our version of the Nash-Moser inverse function theorem. +Definition 2.20 (Mapping hypotheses). We say that a triple (E, F, Ψ) satisfies the RI mapping +hypotheses with parameters (µ, r, R) ∈ �N�3 if E = {Es}s∈�N� and F = {F s}s∈�N� are Banach scales +over a common field, 1 ⩽ µ ⩽ r < R < ∞, R + µ ∈ �N�, and there exists 0 < δr ∈ R such that +Ψ : BEr(0, δr) → F r−µ is a map satisfying the following. +(1) Ψ(0) = 0. +(2) µ-tamely C2: For every r − µ ⩽ s ∈ �N − µ� we have that Ψ : BEr(0, δr) ∩ Es+µ → F s is +C2, and for every u0 ∈ BEr(0, δr) ∩ Es+µ we have the tame estimate +��D2Ψ(u0)[v, w] +�� +F s ⩽ C1(s)(∥v∥Es+µ ∥w∥Er + ∥v∥Er ∥w∥Es+µ + ⟨∥u0∥Es+µ⟩ ∥v∥Er ∥w∥Er). (2.2.1) +Here the constant C1(s) is increasing in s. In other words, we have the inclusion Ψ ∈ +sT 2 +µ,r(BEr(0, δr), E; F) according to the notation from Definition 2.4. +(3) Derivative inversion: There exists δR ∈ R satisfying 0 < δR ⩽ δr such that for every +u0 ∈ BEr(0, δR) ∩ EN there exists a bounded linear operator L(u0) : F r → Er satisfying the +following three conditions. +(a) For every s ∈ N ∩ [r, R] we have that the restriction of L(u0) to F s defines a bounded +linear operator with values in Es, i.e. L(u0) ∈ L(F s; Es). +(b) DΨ(u0)L(u0)f = f for every f ∈ F r. +(c) We have the tame estimate +∥L(u0)f∥Es ⩽ C2(s)(∥f∥F s + ⟨∥u0∥Es+µ⟩ ∥f∥F r) +(2.2.2) +for every f ∈ F s and r ⩽ s ⩽ R, where again the constant C2(s) is increasing in s. +Here the use of the prefix RI- is meant to indicate that the maps L(u0) are only required to be right +inverses. We say that the triple (E, F, Ψ) satisfies the LRI mapping hypotheses if condition (b) in +the third item is augmented by the left-inverse condition (b′) : +L(u0)DΨ(u0)v = v for every v ∈ Er+µ. +(2.2.3) +See Figure 4 for a diagrammatic depiction of how L(u0) and DΨ(u0) interact with E and F. +To conclude this subsection, we state our version of the Nash-Moser inverse function theorem, +which is divided into two parts. + +COMPRESSIBLE TRAVELING WAVES +31 +F s+µ +Es+µ +L(u0) +F s +Es +L(u0) +DΨ(u0) +F s−µ +Es−µ +L(u0) +DΨ(u0) +F 0 +E0 +F N +EN +Figure 4. Commutative diagram arising from the LRI mapping hypotheses for +u0 ∈ EN ∩ BEr(0, δR) and r + µ ⩽ s ⩽ R − µ. The ‘�→’ are the inclusion maps. +Theorem 2.21 (Inverse function theorem). Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach +scales over the same field, and assume that E and F satisfy one of the following two conditions: +I: E and F are LP-smoothable (see Definition 2.11), or +II: E is tame and F is a tame direct summand of E (see Definition 2.14). +Assume the triple (E, F, Ψ) satisfies the LRI mapping hypotheses of Definition 2.20 with parameters +(µ, r, R) satisfying 2(r + µ) + 1 < (r + R)/2, and set β = 2(r + µ) + 1 ∈ �N�. Then there exist +ε, κ1, κ2 > 0 such that the following hold. +(1) Existence of local inverse: For every g ∈ BF β(0, ε) there exists a unique u ∈ BEβ(0, κ1ε) +such that Ψ(u) = g. +(2) Estimates of local inverse: The induced bijection Ψ−1 : BF β(0, ε) → Ψ−1(BF β(0, ε)) ∩ +BEβ(0, κ1ε) obeys the estimate +∥Ψ−1(g)∥Eβ ⩽ κ1∥g∥F β for all g ∈ BF β(0, ε). +(2.2.4) +Moreover, if N ∋ ν ⩽ R + r − 2β, then we have that +Ψ−1 : BF β(0, ε) ∩ F β+ν → Eβ+ν +(2.2.5) +with the estimate +∥Ψ−1(g)∥Eβ+ν ≲ ∥g∥F β+ν, +(2.2.6) +where the implied constant is independent of g. +(3) Basic continuous dependence: The map Ψ−1 obeys the Lipschitz bound +∥Ψ−1(g0) − Ψ−1(g1)∥Eβ−µ ⩽ κ2∥g0 − g1∥F β−µ +(2.2.7) +for all g0, g1 ∈ BF β(0, ε). +Remark 2.22. The inequality 3r + 4µ + 2 < R is equivalent to β = 2(r + µ) + 1 < (r + R)/2. +Furthermore, when �N� is finite, the mapping hypotheses require that R + µ ⩽ N, and so we obtain +the necessary relation 3r + 5µ + 2 < N. +We present the proof of Theorem 2.21 in Section 2.4 after first establishing two other theorems +that prove separate components of the theorem under different hypotheses. We now turn to the +statement of the second part of our inverse function theorem, beginning with some notation. Due to +the derivative loss in the nonlinear operators under consideration, the higher regularity of the local +inverse map is most conveniently phrased in terms of some variation on Gateaux derivatives rather +than the usual Fr´echet notion of differentiability. This is analogous to what is done in Section I.3 +and elsewhere in Hamilton [40]. +Definition 2.23 (Continuous Gateaux differentiability). Let X and Y be Banach spaces over a +common field, U ⊆ X an open set, and f : U → Y . + +32 +NOAH STEVENSON AND IAN TICE +(1) We say that f is continuously Gateaux differentiable on U if there exists a continuous map +Df : U × X → Y such that for all x ∈ U we have that Df(x) ∈ L(X; Y ) and for all z ∈ X +lim +t→0 t−1(f(x + tz) − f(x)) = Df(x)z. +(2.2.8) +(2) For N ∋ ℓ ⩾ 2, we say that f is ℓ-times continuously Gateaux differentiable if f is continu- +ously Gateaux differentiable and Df : U × X → Y is (ℓ − 1)-times continuously Gateaux +differentiable. +We can now state the second part of our Nash-Moser inverse function theorem, the proof of which +is in Section 2.5. +Theorem 2.24 (Further conclusions of the inverse function theorem). Assume the hypotheses of +Theorem 2.21 and additionally that the Banach scale E consists of reflexive spaces. The following +additional conclusions hold for the local inverse map Ψ−1 : BF β(0, ε) → Eβ. +(1) Continuity: For s ∈ [β, R + r − β − µ) ∩ N the map +Ψ−1 : BF β(0, ε) ∩ F s → Es +(2.2.9) +is continuous. +(2) Continuous differentiability: For s ∈ [β, R + r − β − µ) ∩ N, the map +Ψ−1 : BF β(0, ε) ∩ F s → Es−µ +(2.2.10) +is differentiable in the Fr´echet sense with DΨ−1 = L ◦ Ψ−1. Moreover, when viewing DΨ−1 +as map +DΨ−1 : (BF β(0, ε) ∩ F s) × F s−µ → Es−µ +(2.2.11) +we have that it is continuous. +(3) Higher regularity: Let N ∋ ℓ ⩾ 2 and assume that the map Ψ is ℓ-times continuously Gateaux +differentiable. For s ∈ [β + (ℓ(ℓ + 1)/2 − 1)µ, R + r − β − µ) ∩ N we have that the map +Ψ−1 : BF β(0, ε) ∩ F s → Es− ℓ(ℓ+1) +2 +µ +(2.2.12) +is also ℓ-times continuously Gateaux differentiable. +2.3. Local surjectivity and injectivity. We begin this section by proving an existence result +modeled on the main theorem of Baldi and Haus [5], but adapted to our specific setting. We +emphasize that the following theorem only requires the existence of right inverses in the mapping +hypotheses. +Theorem 2.25 (Local surjectivity). Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over +the same field, and suppose that (E, F, Ψ) satisfies the RI mapping hypotheses of Definition 2.20 +with parameters (µ, r, R) such that 2(r + µ) + 1 < (r + R)/2. Set β = 2(r + µ) + 1 ∈ �N� and let +Aβ > 0 be such that ∥∆jf∥F β ⩽ Aβ ∥f∥F β for all j ∈ N and f ∈ F β. Let Ci = Ci(R) denote the +constants from the mapping hypotheses for i ∈ {1, 2}. +Then there exist K1, K2, K3, K4 ∈ R+ such that for every g ∈ F β satisfying +∥g∥F β < +δR +2[(1 + Aβ)K1 + K2 + K3 + K4 + (1 + Aβ)2K1(K1 + K3)] +(2.3.1) +there exists a sequence {(uj, vj, hj, yj, fj, ej)}∞ +j=0 ⊆ (BEr(0, δR)∩ER)×EN ×ER ×F N ×F N ×F R−µ +satisfying the following. +(1) When j = 0 we have the identities +u0 = 0, +v0 = S0u0 = 0, +y0 = 0, +f0 = ∆0g = S1g, +(2.3.2) +h0 = L(u0)f0 = L(0)S1g, +e0 = Ψ(u0 + h0) − Ψ(u0) − f0, + +COMPRESSIBLE TRAVELING WAVES +33 +while for j ⩾ 1 we have the recursive relations +uj = uj−1 + hj−1, +vj = Sjuj, +yj = −Sj +j−1 +� +n=0 +en − +j−1 +� +n=0 +yn, +fj = ∆jg + yj, +(2.3.3) +hj = L(vj)fj, +ej = Ψ(uj + hj) − Ψ(uj) − fj. +(2) For j ∈ N we have the estimates +∥hj∥Es ⩽ K1(∥g∥F β 2−j + ∥∆jg∥F β)2j(s−β) for all s ∈ [r, R] ∩ N, +(2.3.4) +∥vj∥Es ⩽ K2∥g∥F β2j(s−β) for all β ⩽ s ∈ �N�, +(2.3.5) +∥uj − vj∥Es ⩽ K3∥g∥F β2j(s−β) for all s ∈ [0, R] ∩ N, +(2.3.6) +∥uj∥Eβ ⩽ K4∥g∥F β. +(2.3.7) +and +∥ej∥F s ≲ C1∥g∥F β2j(s+µ+r−2β) for all s ∈ [r − µ, R − µ] ∩ N. +(2.3.8) +(3) There exists u ∈ Eβ such that uj → u in Eβ as j → ∞, ∥u∥Eβ ⩽ K4 ∥g∥F β, and Ψ(u) = g. +(4) If we know additionally that ν ∈ N is such that ν ⩽ R + r − 2β and g ∈ F β+ν, then actually +uj → u in Eβ+ν as j → ∞ and we have the estimate ∥u∥Eβ+ν ≲ ∥g∥F β+ν. Here the implied +constant depends on K1, . . . , K4, Aβ, Aβ+ν, µ, r, δR, and R. +Proof. We divide the proof into several steps. The first three steps establish a trio of crucial claims +that will be used in the fourth step to inductively construct the sequence. The convergence result is +then proved in the fifth step. The higher regularity assertions of the fourth item are then proved in +the sixth step. Throughout the proof we will utilize the sequence {ξn}∞ +n=0 ⊆ [0, ∞) defined by +ξn = ∥g∥F β 2−n + ∥∆ng∥F β . +(2.3.9) +Step 1: An estimate in the style of Littlewood-Paley theory. Let ℓ, k ∈ N satisfy 0 ⩽ ℓ ⩽ k. We +claim that if {hj}k +j=0 ⊆ ER are given and satisfy (2.3.4) for 0 ⩽ j ⩽ k, then +��� +k +� +n=ℓ +hn +��� +Eβ ≲ K1 +� +k +� +n=ℓ +ξ2 +n +�1/2 +, +(2.3.10) +where ξn is defined by (2.3.9) +To prove the claim, we begin by noting that for any j ∈ N, 0 ⩽ n ⩽ k, and s = β + sgn(j − n) ∈ +[r, R] ∩ N we may bound, via (2.1.20) and (2.3.4), +∥∆jhn∥Eβ ≲ 2j(β−s)∥hn∥Es ≲ K1ξn2j(β−s)2n(s−β) = K1ξn2(j−n)(β−s) = K1ξn2−|j−n|. +(2.3.11) +We estimate the sum on the left hand side of (2.3.10) via the ‘Littlewood-Paley’ characteriza- +tion (2.1.22), the bound (2.3.11), and Young’s convolution inequality (setting ξn = 0 for n ∈ Z such +that 0 < n): +��� +k +� +n=ℓ +hn +��� +2 +Eβ ≲ +∞ +� +j=0 +���∆j +k +� +n=ℓ +hn +��� +2 +Eβ ⩽ +∞ +� +j=0 +� +k +� +n=ℓ +∥∆jhn∥Eβ +�2 +≲ K2 +1 +∞ +� +j=0 +� +k +� +n=ℓ +ξn2−|j−n|�2 +⩽ K2 +1 +� +j∈Z +� � +n∈Z +ξn1[ℓ,k](n)2−|j−n|�2 +⩽ K2 +1 +� � +n∈Z +ξ2 +n1[ℓ,k](n) +�� � +n∈Z +2−|n|�2 +≲ K2 +1 +k +� +n=ℓ +ξ2 +n. +(2.3.12) +This is (2.3.10). + +34 +NOAH STEVENSON AND IAN TICE +Step 2: An estimate from Taylor’s theorem. Let k ∈ N. We claim that if g ∈ F β satisfies (2.3.1) +and {(uj, vj, hj)}k +j=0 ⊆ (BEr(0, δR) ∩ ER) × EN × ER are given and satisfy (2.3.4)–(2.3.7) for +0 ⩽ j ⩽ k, as well as the conditions u0 = v0 = 0 and uj = uj−1 + hj−1 if 1 ⩽ j ⩽ k, then +{uj}k +j=0, {uj + hj}k +j=0, {vj}k +j=0 ⊆ BEr(0, δR) ⊆ BEr(0, δr) +(2.3.13) +and +∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1∥g∥F β2j(s+µ+r−2β) +(2.3.14) +for all s ∈ [r − µ, R − µ] ∩ N. +To prove this claim we begin by employing (2.3.1). +Indeed, (2.3.1) and (2.3.7) imply that +∥uj∥Er ⩽ ∥uj∥Eβ ⩽ K4 ∥g∥F β < +δR +2 +⩽ +δr +2 , (2.3.5) and (2.3.1) imply that ∥vj∥Er ⩽ ∥vj∥Eβ ⩽ +K2 ∥g∥F β < +δR +2 +⩽ +δr +2 for 0 ⩽ j ⩽ k, and (2.3.4) and (2.3.1) imply that ∥hj∥Er ⩽ ∥hj∥Eβ ⩽ +K1 (∥g∥F β + Aβ ∥g∥F β) = K1(1 + Aβ) ∥g∥F β < δR +2 ⩽ δr +2 for the same range of j. Together, these +three bounds imply the inclusions (2.3.13). +Now that (2.3.13) is established, we know that Ψ(uj + hj), Ψ(uj), and DΨ(vj) are well-defined. +We may then employ the fundamental theorem of calculus, Taylor’s theorem, and the convexity of +BEr(0, δr) to write +Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj = (DΨ(uj) − DΨ(vj))hj + +� 1 +0 +(1 − t)D2Ψ(uj + thj)[hj, hj]dt += +� 1 +0 +D2Ψ((1 − t)vj + tuj)[hj, uj − vj]dt + +� 1 +0 +(1 − t)D2Ψ(uj + thj)[hj, hj]dt = Ij + IIj. +(2.3.15) +By using the tame C2 estimate from the mapping hypotheses of Definition 2.20, we readily deduce +that for r − µ ⩽ s ∈ �N − µ�, +∥Ij∥F s ≲ C1⟨∥uj − vj∥Er⟩ ∥uj − vj∥Es+µ ∥hj∥Er ++ C1⟨∥vj∥Es+µ⟩ ∥uj − vj∥Er ∥hj∥Er + C1 ∥uj − vj∥Er ∥hj∥Es+µ +(2.3.16) +and +∥IIj∥F s ≲ C1⟨∥hj∥Er⟩ ∥hj∥Es+µ ∥hj∥Er + C1⟨∥uj∥Es+µ⟩ ∥hj∥2 +Er . +(2.3.17) +Synthesizing these bounds, we find that +∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1⟨∥uj − vj∥Er⟩ ∥uj − vj∥Es+µ ∥hj∥Er ++ C1 ∥uj − vj∥Er ∥hj∥Es+µ + C1⟨∥hj∥Er⟩ ∥hj∥Es+µ ∥hj∥Er ++ C1⟨∥uj, vj∥Es+µ⟩ ∥hj∥Er (∥hj∥Er + ∥uj − vj∥Er) +(2.3.18) +for r − µ ⩽ s ∈ �N − µ�. +Next we turn our attention to estimates for uj and vj. Note that if 1 ⩽ j ⩽ k, then the identity +u = �j−1 +n=0 hn and (2.3.4) imply that +∥uj∥ER ⩽ +j−1 +� +n=0 +∥hn∥ER ⩽ K1(1 + Aβ) ∥g∥F β +j−1 +� +n=0 +2n(R−β) ≲ K1(1 + Aβ) ∥g∥F β 2j(R−β). +(2.3.19) +In turn, when we couple this with the smoothing estimates from Definition 2.11, the interpolation +result from Lemma 2.19, and (2.3.7), this implies that +∥vj∥Es ≲ ∥uj∥Es ≲ ∥uj∥ +R−s +R−β +Eβ +∥uj∥ +s−β +R−β +ER +≲ K +R−s +R−β +4 +(K1(1 + Aβ)) +s−β +R−β ∥g∥F β 2j(s−β) +(2.3.20) +for all s ∈ N ∩ [β, R]. However, (2.3.1) implies that K +R−s +R−β +4 +(K1(1 + Aβ)) +s−β +R−β ∥g∥F β ⩽ 1, so we may +further bound ∥vj∥Es ≲ ∥uj∥Es ≲ 2j(s−β) for all s ∈ N ∩ [β, R]. + +COMPRESSIBLE TRAVELING WAVES +35 +On the other hand, (2.3.7) and (2.3.1) give us that ∥vj∥Es ≲ ∥uj∥Es ≲ ∥uj∥Eβ ≲ K4 ∥g∥F β ≲ 1 +for all s ∈ N ∩ [r, β]. Upon combining these, we find that +∥uj∥Es + ∥vj∥Es ≲ 2j max{0,s−β} for all s ∈ N ∩ [r, R] and 0 ⩽ j ⩽ k, +(2.3.21) +where we note that these bounds are trivial for j = 0 since u0 = v0 = 0. +Now we combine (2.3.18) and (2.3.21) with (2.3.4)–(2.3.7) to bound, for 0 ⩽ j ⩽ k and s ∈ +N ∩ [r − µ, R − µ], +∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s +≲ C1⟨K3 ∥g∥F β 2j(r−β)⟩K3 ∥g∥F β 2j(s+µ−β) · K1(1 + Aβ) ∥g∥F β 2j(r−β) ++ C1K3 ∥g∥F β 2j(r−β) · K1(1 + Aβ) ∥g∥F β 2j(s+µ−β) ++ C1⟨K1(1 + Aβ) ∥g∥F β 2j(r−β)⟩K1(1 + Aβ)2j(s+µ−β) · K1(1 + Aβ)2j(r−β) ++ C1⟨2j max{0,s+µ−β}⟩K1(1 + Aβ) ∥g∥F β 2j(r−β) · (K1(1 + Aβ) ∥g∥F β 2j(r−β) + K3 ∥g∥F β 2j(r−β)). +(2.3.22) +For the last term we note that +� +r − µ ⩽ s ⩽ β − µ ⇒ 2r − 2β ⩽ s + µ + r − 2β +β − µ ⩽ s ⩽ R − µ ⇒ s + µ − β + 2r − 2β ⩽ s + µ + r − 2β. +(2.3.23) +Thus, upon regrouping and using (2.3.1), we then see that +∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1(1 + Aβ)2K1(K1 + K3) ∥g∥2 +F β 2j(s+µ+r−2β) +≲ C1 ∥g∥F β 2j(s+µ+r−2β) +(2.3.24) +for all s ∈ N ∩ [r − µ, R − µ] and 0 ⩽ j ⩽ k. This is (2.3.14), and so the proof of the claim is +complete. +Step 3: A recursive identity. Suppose that {(yj, ej)}k +j=0 ⊆ F N × F R−µ is given for 2 ⩽ k ∈ N +and satisfies y0 = 0 and the recursive condition +yj = −Sj +j−1 +� +n=0 +en − +j−1 +� +n=0 +yn for 1 ⩽ j ⩽ k. +(2.3.25) +We claim that +yj = −Sjej−1 − ∆j−1 +j−2 +� +n=0 +en for 2 ⩽ j ⩽ k. +(2.3.26) +To see this, note that from (2.3.25) we deduce that for any 1 ⩽ j ⩽ k the identity �j +n=0 yn = +−Sj +�j−1 +n=0 en. +Hence, for 2 ⩽ j ⩽ k we have yj = −Sj +�j−1 +n=0 en − Sj−1 +�j−2 +n=0 en, but by the +definitions of the ∆n and en, this is the same as yj = −Sjej−1 −∆j−1 +�j−2 +n=0 en, and therefore (2.3.26) +holds. +Step 4: Inductive construction of the sequence. We now aim to inductively construct the desired +sequence under some assumptions on the constants K1, K2, K3, K4, which we will work out as we +proceed. +We begin by seeding the sequence, i.e. constructing its elements with j = 0. Define u0 = 0 ∈ +BEr(0, δR) ∩ ER, v0 = S0u0 = 0 ∈ EN, y0 = 0 ∈ F N, f0 = ∆0g = S1g ∈ F N, h0 = L(u0)f0 = +L(0)S1g ∈ ER, and e0 = Ψ(h0) − Ψ(0) − f0 ∈ F R−µ. By construction, the bounds (2.3.5), (2.3.6), +and (2.3.7) hold trivially when j = 0 for all possible choices of K2, K3, K4 > 0. On the other hand, + +36 +NOAH STEVENSON AND IAN TICE +we can apply the tame estimate for L(0) from the mapping hypotheses to see that for s ∈ N ∩ [r, R], +∥h0∥Es ⩽ C2 ∥S1g∥F s + C2(1 + ∥0∥Es+µ) ∥S1g∥F r ≲ C2 ∥∆0g∥F β + C2 ∥g∥F β +⩽ K1 +� +∥g∥F β 2−0 + ∥∆jg∥F β +� +20(s−β), +(2.3.27) +provided the constant K1 satisfies the bound +C2 ≲ K1, +(2.3.28) +which we henceforth assume holds. We then have that (2.3.4) is satisfied for j = 0. +We now proceed to the inductive step. Let k ∈ N and suppose that +{(uj, vj, hj, yj, fj, ej)}k +j=0 ⊆ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ +(2.3.29) +are given and satisfy (2.3.2), (2.3.4)–(2.3.7) for 0 ⩽ j ⩽ k, and (2.3.3) if 1 ⩽ j ⩽ k. We will +construct (uk+1, vk+1, hk+1, yk+1, fk+1, ek+1) ∈ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ +and show that (2.3.4)–(2.3.7) continue to hold for j = k + 1. +First, we define uk+1 = uk + hk ∈ ER. +The claim established in Step 2 guarantees that +uk+1 ∈ Br(0, δR) ⊆ Br(0, δr). Moreover, we may use a telescoping argument to see that uk+1 = +uk+1 − u0 = �k +n=0 hn. Thus, the claim established in Step 1 shows that +∥uk+1∥Eβ ≲ K1 +� ∞ +� +n=0 +ξ2 +n +�1/2 +≲ K1 ∥g∥F β , +(2.3.30) +where in the last inequality we have used the Littlewood-Paley bound (2.1.22). Thus, uk+1 ∈ +BEr(0, δR) ∩ ER satisfies (2.3.7) with j = k + 1, provided that K1 and K4 satisfy +K1 ≲ K4, +(2.3.31) +which we henceforth assume. +Second, we define vk+1 = Sk+1uk+1 ∈ EN. We then use (2.3.4) and the smoothing bounds (2.1.17)– +(2.1.20) to estimate +∥vk+1 − uk+1∥ER = ∥(I − Sk+1)uk+1∥ER ≲ ∥uk+1∥ER ⩽ +k +� +n=0 +∥hn∥ER ⩽ K1 +k +� +n=0 +ξn2n(R−β) +⩽ K1 +� ∞ +� +n=0 +ξ2 +n +�1/2� +k +� +n=0 +22n(R−β)�1/2 +≲ K1 ∥g∥F β 2(k+1)(R−β). +(2.3.32) +On the other hand, (2.3.30) and the smoothing bounds (2.1.17)–(2.1.20) show that +∥vk+1 − uk+1∥E0 = ∥(I − Sk+1)uk+1∥E0 ≲ 2−(k+1)β ∥(I − Sk+1)uk+1∥Eβ +≲ 2−(k+1)β ∥uk+1∥Eβ ≲ K1 ∥g∥F β 2−(k+1)β. +(2.3.33) +Upon combining these two estimates and interpolating with the help of Lemma 2.19, we find that +for s ∈ N ∩ [0, R], +∥vk+1 − uk+1∥Es ≲ ∥vk+1 − uk+1∥1−s/R +E0 +∥vk+1 − uk+1∥s/R +ER ≲ K1 ∥g∥F β 2(k+1)(s−β), +(2.3.34) +and so vk+1 ∈ EN satisfies (2.3.6) with j = k + 1 provided that K1 and K3 satisfy +K1 ≲ K3, +(2.3.35) +which we henceforth assume. + +COMPRESSIBLE TRAVELING WAVES +37 +Continuing with vk+1, we observe that for β ⩽ s ∈ �N� we can use the smoothing bounds of +Definition 2.11 to estimate +∥vk+1∥Es = ∥Sk+1uk+1∥Es ≲ 2(k+1)(s−β) ∥Sk+1uk+1∥Eβ ≲ 2(k+1)(s−β) ∥uk+1∥Eβ +≲ K1 ∥g∥F β 2(k+1)(s−β), +(2.3.36) +where we have again used (2.3.30). Thus, vk+1 obeys the bound (2.3.5) for j = k + 1 provided that +K1 and K2 satisfy +K1 ≲ K2, +(2.3.37) +which we henceforth assume, and in this case (2.3.1) in turn implies that +∥vk+1∥Er ⩽ ∥vk+1∥Eβ ⩽ K2 ∥g∥F β < δR. +(2.3.38) +Third, we introduce some useful estimates for the terms {ej}k +j=0. For 0 ⩽ j ⩽ k we know that +hj = L(vj)fj, and so fj = DΨ(vj)hj by the RI mapping hypotheses. We may thus invoke the claim +established in Step 2 in order to see that for 0 ⩽ j ⩽ k (2.3.14) implies that +∥ej∥F s = ∥Ψ(uj + hj) − Ψ(uj) − fj∥F s = ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s +≲ C1 ∥g∥F β 2j(s+µ+r−2β) +(2.3.39) +for all s ∈ N ∩ [r − µ, R − µ]. +Fourth, we turn our attention to defining yk+1. If k = 0, then we simply set y1 = −S1e0 ∈ F N, +while if k ⩾ 1 then we set yk+1 = −Sk+1 +�k +n=0 en − �k +n=0 yn ∈ F N. We may then invoke the claim +of Step 3 to see that the formula +yj = −Sjej−1 − ∆j−1 +j−2 +� +n=0 +en +(2.3.40) +holds whenever 1 ⩽ j ⩽ k + 1, provided that we understand sums over empty ranges to mean zero. +Next we use (2.3.39) to estimate yk+1. Initially we use the smoothing operator properties together +with (2.3.39) to estimate +∥Sk+1ek∥F s ≲ +� +∥ek∥F s +if r − µ ⩽ s ⩽ R − µ +2(k+1)(s−R+µ) ∥ek∥F R−µ +if R − µ ⩽ s ∈ �N� +≲ C1 ∥g∥F β 2k(s+µ+r−2β) (2.3.41) +for every r − µ ⩽ s ∈ �N�. Similarly, if k ⩾ 1 then we can use the fact that R + r − 2β > 0 to bound +��� +k−1 +� +j=0 +∆kej +��� +F s ⩽ +k−1 +� +j=0 +∥∆kej∥F s ≲ 2k(s−R+µ) +k−1 +� +j=0 +∥∆kej∥F R−µ ≲ 2k(s−R+µ) +k−1 +� +j=0 +∥ej∥F R−µ +≲ C1 ∥g∥F β 2k(s−R+µ) +k−1 +� +j=0 +2j(R+r−2β) ≲ C1 ∥g∥F β 2k(s+µ+r−2β) +(2.3.42) +for every r − µ ⩽ s ∈ �N�. Synthesizing (2.3.40), (2.3.41), and (2.3.42), we deduce that +∥yk+1∥F s ≲ C1 ∥g∥F β 2k(s+µ+r−2β) +(2.3.43) +for every r − µ ⩽ s ∈ �N�. +As the penultimate update we define fk+1 = ∆k+1g + yk+1 ∈ F N. We know that vk+1 satisfies +(2.3.38), so the operator L(vk+1) exists, and we may make the final update by setting hk+1 = +L(vk+1)fk+1 ∈ ER. The tame estimates for L(vk+1) then provide for the bound +∥hk+1∥Es ⩽ C2(∥∆k+1g∥F s + ∥yk+1∥F s) + C2⟨∥vk+1∥Es+µ⟩(∥∆k+1g∥F r + ∥yk+1∥F r) +(2.3.44) + +38 +NOAH STEVENSON AND IAN TICE +for s ∈ N ∩ [r, R]. From (2.3.5), which we established above holds for j = k + 1, and (2.3.1) we may +estimate ∥vk+1∥Es+µ ≲ K4 ∥g∥F β 2(k+1)(s+µ−β) ≲ 2(k+1)(s+µ−β). By plugging this and (2.3.43) into +(2.3.44), we then find that +∥hk+1∥Es ≲ C2 +� +∥∆k+1g∥F β 2(k+1)(s−β) + C1 ∥g∥F β 2k(s+µ+r−2β)� ++ C2⟨2(k+1)(s+µ−β)⟩ +� +∥∆k+1g∥F β 2(k+1)(r−β) + C1 ∥g∥F β 2k(2r+µ−2β)� +≲ C2(1 + C1)2(k+1)(s−β)(∥∆k+1g∥F β (2(k+1)(r−s) + 2(k+1)(µ+r−β)) ++ ∥g∥F β (2(k+1)(µ+r−β) + 2(k+1)(µ+2r−s−β) + 2(k+1)(2r+2µ−β)). +(2.3.45) +To consolidate all of the exponents we recall that that r − s ⩽ 0 and 1 = β − 2r − 2µ > 0, which +show that µ + r − β ⩽ 2µ + 2r − β = −1 ⩽ 0 and µ + 2r − s − β ⩽ µ + r − β ⩽ −1. Hence, the +previous estimate implies that ∥hk+1∥Es ≲ C2(1 + C1)2(k+1)(s−β)(∥∆k+1g∥F β + ∥g∥F β 2−(k+1)), and +we deduce that hk+1 obeys (2.3.4) for j = k + 1, provided that K1 satisfies +C2(1 + C1) ≲ K1, +(2.3.46) +which we assume holds. +We have now established conditions on K1, K2, K3, and K4 that are sufficient for the construction +of (uk+1, vk+1, hk+1, yk+1, fk+1, ek+1) ∈ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ, namely +(2.3.28), (2.3.31), (2.3.35), (2.3.37), and (2.3.46). +It is a simple matter to choose parameters +satisfying these conditions, and so the inductive step is complete provided these parameters are +chosen. We thus have the desired sequence. Note that (2.3.8) follows as above from the claim +established in Step 2. +Step 5: Convergence of the sequence. With the sequence in hand from Step 4, we use the claim +from Step 1 with 0 ⩽ ℓ ⩽ k and the fact that uk+1 − uℓ = �k +n=ℓ hℓ to see that +∥uk+1 − uℓ∥Eβ ⩽ +k +� +n=ℓ +∥hn∥Eβ ≲ K1 +� ∞ +� +n=ℓ +∥g∥2 +F β 2−2n + ∥∆ng∥2 +F β +�1/2 +→ 0 +(2.3.47) +as ℓ → ∞. Thus, {uj}∞ +j=0 is a Cauchy sequence in Eβ, and hence convergent to some u ∈ Eβ. +Sending j → ∞ in (2.3.7) shows that ∥u∥Eβ ⩽ K4 ∥g∥F β. +It remains only to show that Ψ(u) = g. To this end, we again telescope and use that Ψ(0) = +Ψ(u0) = 0 and (2.3.3) to write, for k ⩾ 2, +Ψ(uk+1) = +k +� +j=0 +(Ψ(uk+1) − Ψ(uk)) = +k +� +j=0 +(Ψ(uk + hk) − Ψ(uk)) = +k +� +j=0 +(ej + fj) += +k +� +j=0 +∆jg + +k +� +j=0 +ej + +k +� +j=0 +yj = Sk+1g + +k +� +j=0 +ej − Sk +k−1 +� +j=0 +ej = Sk+1g + ek + (I − Sk) +k−1 +� +j=0 +ej. +(2.3.48) +Now, we know from the properties of the smoothing operators, which are given in Definition 2.11, +that ∥(I − Sk+1)g∥F β → 0 as k → ∞, while (2.3.8) implies that +∥ek∥F β−µ ≲ C1 ∥g∥F β 2k(r−β) → 0 as k → ∞ +(2.3.49) + +COMPRESSIBLE TRAVELING WAVES +39 +and (observing that β − µ − 1 ⩾ 2r + µ ⩾ 1) +���(I − Sk) +k−1 +� +j=0 +ej +��� +F β−µ−1 ⩽ +k−1 +� +j=0 +∥(I − Sk)ej∥F β−µ−1 ≲ 2−k +k−1 +� +j=0 +∥(I − Sk)ej∥F β−µ +≲ C1∥g∥F β2−k +k−1 +� +j=0 +2j(r−β) ≲ C1∥g∥F β2−k → 0 as k → ∞. +(2.3.50) +Upon combining these, we find that ∥Ψ(uk+1) − g∥F β−µ−1 → 0 as k → ∞ but since uk → u in Eβ +as k → ∞, the continuity of Ψ guarantees that Ψ(uk+1) → Ψ(u) in F β−µ. Thus, Ψ(u) = g. +Step 6: Higher regularity. Let N ∋ ν ⩽ R + r − 2β and now suppose additionally that g ∈ F β+ν. +We wish to prove that the solution u ∈ Eβ constructed in Step 5 actually belongs to Eβ+ν and +satisfies ∥u∥Eβ+ν ≲ ∥g∥F β+ν. We will establish this via finite induction. The proposition to be proved +inductively is the following statement, depending on ν ∈ {0, 1, . . . , ν}: if j ∈ N and s ∈ N ∩ [r, R], +then +∥hj∥Es ≲ 2j(s−(β+ν))(∥∆jg∥F β+ν + 2−j∥g∥F β+ν) +(2.3.51) +for an implicit constant depending on ν, as well as K1, . . . , K4, Aβ, Aβ+ν, µ, r, δR, and R. The case +ν = 0 was established in the fourth step, in particular in the verification of (2.3.4). Assume now +that for some ν ∈ {0, 1, . . . , ν − 1} we have that the induction hypothesis (2.3.51) holds at level ν. +We wish to prove it at level ν + 1. +We begin by mimicking Step 1 and deriving improved bounds on the sequence {uj}∞ +j=0. For +n ∈ N we set ξν +n = ∥∆ng∥F β+ν + 2−n∥g∥F β+ν. By arguing as in the first step, we can show that for +any 0 ⩽ j ⩽ k we have the estimate +��� +k +� +n=ℓ +hn +��� +Eβ+ν ≲ +� +k +� +n=ℓ +(ξν +n)2�1/2 +. +(2.3.52) +Since g ∈ F β+ν, we deduce from (2.1.22) that {ξν +n}∞ +n=0 ∈ ℓ2(N); hence, the sequence of partial sums +{�j +n=0 hn}∞ +j=0 ⊂ Eβ+ν is Cauchy. We have already established convergence of the scheme, giving +that u = �∞ +n=0 hj in Eβ. Therefore, u ∈ Eβ+ν and, from (2.3.52), we deduce that for all j ∈ N +∥uj∥Eβ+ν ≲ +� j−1 +� +n=0 +(ξν +n)2�1/2 +≲ ∥g∥F β+ν. +(2.3.53) +Now we derive improved bounds on the sequence {vj}j∈N. From properties (2.1.17) and (2.1.18) +of the smoothing operators and the improved bounds of (2.3.53), we deduce for β + ν ⩽ s ∈ �N� +that +∥vj∥Es = ∥Sjuj∥Es ≲ 2j(s−(β+ν))∥uj∥Eβ+ν ≲ 2j(s−(β+ν))∥g∥F β+ν. +(2.3.54) +Next, we bound the sequence {uj − vj}j∈N. Thanks to Lemma 2.19, for s ∈ N ∩ [0, R] we have +the bound +∥uj − vj∥Es ≲ ∥uj − vj∥1−s/R +E0 +∥uj − vj∥s/R +ER . +(2.3.55) +The E0-norm term we bound using properties (2.1.17) and (2.1.19) of the smoothing operators and +estimate (2.3.53): +∥uj − vj∥E0 = ∥(I − Sj)uj∥E0 ≲ 2−j(β+ν)∥uj∥Eβ+ν ≲ 2−j(β+ν)∥g∥Eβ+ν. +(2.3.56) + +40 +NOAH STEVENSON AND IAN TICE +On the other hand, the ER term is handled via (2.1.17), the telescoping identity uj = �j−1 +n=0 hn, +induction hypothesis (2.3.51), and the bound ξν +n ≲ ∥g∥F β+ν: +∥uj − vj∥ER ⩽ +j−1 +� +n=0 +∥(I − Sj)hn∥ER ≲ +j−1 +� +n=0 +∥hn∥ER ≲ ∥g∥F β+ν +j−1 +� +n=0 +2n(R−(β+ν)) ≲ 2j(R−(β+ν))∥g∥F β+ν. +(2.3.57) +We synthesize (2.3.55), (2.3.56), and (2.3.57) to get +∥uj − vj∥Es ≲ 2j(s−(β+ν))∥g∥F β+ν. +(2.3.58) +Our next endeavor is to derive improved bounds on the sequence {ej}j∈N. We turn to esti- +mate (2.3.18) for s ∈ [r − µ, R − µ] ∩ N. To handle the right hand side, we invoke the following +bounds: +max{∥uj − vj∥Er, ∥hj∥Er} ≲ 2j(r−β) min{1, 2−jν∥g∥F β+ν}, +max{∥uj − vj∥Es+µ, ∥hj∥Es+µ} ≲ 2j(s+µ−β) min{1, 2−jν∥g∥F β+ν}, +∥uj, vj∥Es+µ ≲ 2j max{0,s+µ−β}, +(2.3.59) +which are consequences of (2.3.4), (2.3.6), (2.3.1), (2.3.51), (2.3.58), and finally (2.3.21). In this +way we acquire the estimate +∥ej∥F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν. +(2.3.60) +Now we estimate the sequence {yj}j∈N. For this we recall identity (2.3.26). By arguing as +in (2.3.41), but using the bounds established in (2.3.60) for {ej}j∈N, we learn that for j ∈ N and +r − µ ⩽ s ∈ �N� +∥Sj+1ej∥F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν. +(2.3.61) +Similarly, by arguing as in (2.3.42), but instead using (2.3.60) with s = R − µ and the fact that +R + r − 2β − ν > 0 (since ν < ν), we gain the bound +��� +j−1 +� +n=0 +∆jen +��� +F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν. +(2.3.62) +We combine (2.3.61) and (2.3.62) to see that for j ∈ N and r − µ ⩽ s ∈ �N�, +∥yj∥F s ≲ 2(j−1)(s+µ+r−2β−ν)∥g∥F β+ν. +(2.3.63) +At last, we are ready to obtain an improved estimate on the sequence {hj}j∈N and close the induc- +tion. By using the identity hj = L(vj)(∆jg + yj) with the right inverse estimates of equation (2.2.2), +we find that for j ∈ N, +∥hj∥Es ≲ ∥∆jg∥F s + ∥yj∥F s + ⟨∥vj∥Es+µ⟩(∥∆jg∥F r + ∥yj∥Er) for s ∈ [r, R] ∩ N. +(2.3.64) +We estimate ∥yj∥F s and ∥yj∥F r according to the improved estimates of (2.3.63), the ∆jg norms are +handled via smoothing estimate (2.1.20), i.e. ∥∆jg∥F k = 2j(k−(β+ν+1))∥∆jg∥F β+ν+1 for k ∈ {r, s}, +and the vj-term is estimated according to (2.1.17) and (2.3.7) in the case s ⩽ β and via (2.3.5). +Hence estimate (2.3.64) yields +∥hj∥Es ≲ 2j(s−(β+ν+1))∥∆jg∥F β+ν+1 + 2j(s+µ+r−2β−ν)∥g∥F β+ν ++ ⟨2j(s−β)⟩(2j(r−(β+ν+1))∥∆jg∥F β+ν+1 + 2j(µ+2r−2β−ν)∥g∥F β+ν). +(2.3.65) +Upon regrouping, we acquire the bound +∥hj∥Es ≲ 2j(s−(β+ν+1))(1 + ⟨2j(s−β)⟩2j(r−s))∥∆jg∥F β+ν+1 ++ 2j(s−(β+ν+1))2−j(2j(2+µ+r−β) + ⟨2j(s−β)⟩2j(2+µ+2r−β−s)))∥g∥F β+ν. +(2.3.66) + +COMPRESSIBLE TRAVELING WAVES +41 +Since r ⩽ min{s, β}, we have that +1 + ⟨2j(s−β)⟩2j(r−s) ≲ 1. +(2.3.67) +The inequalities 1 ⩽ µ + r and 2(µ + r) < β imply that +2j(2+µ+r−β) ≲ 1. +(2.3.68) +The inequality µ ⩾ 1 implies that min{2β, s + β} ⩾ β ⩾ 1 + 2(µ + r) ⩾ 2 + µ + 2r and hence +⟨2j(s−β)⟩2j(2+µ+2r−β−s) ≲ 1. +(2.3.69) +We combine inequalities (2.3.66), (2.3.67), (2.3.68), and (2.3.69) to see that +∥hj∥Es ≲ 2j(s−(β+ν+1))(∥∆jg∥F β+ν+1 + 2−j∥g∥F β+ν). +(2.3.70) +This means that the inductive proposition (2.3.51) has been verified for ν + 1, and hence the +induction is complete. +We therefore know that (2.3.51) holds for ν = ν. +We then argue as +in (2.3.52) and (2.3.53) with the sequence {ξν +n}n∈N to deduce that u ∈ Eβ+ν with the estimate +∥u∥Eβ+ν ≲ ∥g∥F β+ν. +□ +We now complement the previous local surjectivity result with the following local injectivity +result, which has no analog in Baldi and Haus [5] but is analogous to a result of Hamilton [40]. We +emphasize that in the following we only need left inverses and weaker forms of the tame estimates +than in the LRI mapping hypotheses. +Theorem 2.26 (Local injectivity). Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over +the same field, and suppose that E is terminally dense in the sense of Definition 2.2. Let σ, µ ∈ N +be such that σ + µ ∈ �N�. Assume that there exists γσ ∈ R+ and a C2 map Ψ : BEσ+µ(0, γσ) → F σ +such that the following hold. +(1) For every u0 ∈ BEσ+µ(0, γσ) we have the bound +��D2Ψ(u0)[v, w] +�� +F σ ≲ ⟨∥u0∥Eσ+µ⟩ (∥v∥Eσ+µ ∥w∥Eσ + ∥v∥Eσ ∥w∥Eσ+µ). +(2.3.71) +(2) For every u0 ∈ BEσ+µ(0, γσ) ∩ EN there exists a bounded linear operator L(u0) : F σ → Eσ +satisfying the following two conditions. +(a) L(u0)DΨ(u0)v = v for every v ∈ Eσ+µ. +(b) We have the estimate ∥L(u0)f∥Eσ ≲ ⟨∥u0∥Eσ+µ⟩ ∥f∥F σ for every f ∈ F σ. +Then there exists 0 < δinj,σ ⩽ γσ/4 such that if u0 − u1 ∈ BEσ+µ(0, 2δinj,σ), with ui ∈ BEσ+µ(0, γσ) +for i ∈ {0, 1}, then +∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ . +(2.3.72) +In particular, the restriction Ψ : BEσ+µ(0, δinj,σ) → F σ is injective. +Proof. Suppose initially that u0 − u1 ∈ BEσ+µ(0, 2δ) ∩ EN and ui ∈ BEσ+µ(0, γσ) for some 0 < δ ⩽ +γσ/4. We may use Taylor’s theorem to write +Ψ(u1) − Ψ(u0) = DΨ(u0)(u1 − u0) + +� 1 +0 +(1 − t)D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] dt, (2.3.73) +with the understanding that this equality holds in the space F σ. We then apply the bounded linear +map L(u0) : F σ → Eσ and rearrange to see that +u1 − u0 = L(u0) +� +Ψ(u1) − Ψ(u0) − +� 1 +0 +(1 − t)D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] dt +� +. (2.3.74) + +42 +NOAH STEVENSON AND IAN TICE +Next we couple the identity (2.3.74) to the estimate for L(u0) and the trivial bound ⟨∥u0∥Eσ+µ⟩ ≲ 1 +to see that +∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ + +� 1 +0 +(1 − t) +��D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] +�� +F σ dt. +(2.3.75) +For the latter term we use the C2 estimate to bound +��D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] +�� +F σ ≲ +⟨∥(1 − t)u0 + tu1∥Eσ+µ⟩ ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ ≲ ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ +(2.3.76) +for every t ∈ [0, 1]. We then plug this bound into (2.3.75) to see that +∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ + ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ +≲ ∥Ψ(u1) − Ψ(u0)∥F σ + δ ∥u1 − u0∥Eσ . +(2.3.77) +From this we readily deduce the existence of 0 < δinj ⩽ γσ/4 such that if δ ⩽ δinj then we can absorb +the right-most term onto the left to conclude that +∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ for all u0 − u1 ∈ Bσ+µ(0, 2δinj) ∩ EN. +(2.3.78) +Estimate (2.3.78) is not quite the desired result since it requires u0, u1 ∈ EN, but we can use +the fact that E is terminally dense to promote this result. Indeed, given u0, u1 ∈ BEσ+µ(0, γσ) +such that u0 − u1 ∈ BEσ+µ(0, 2δinj) we can pick {un +i }n∈N ⊆ EN such that un +i → ui in Eσ+µ +for i ∈ {0, 1}. Since BEσ+µ(0, 2δinj) and BEσ+µ(0, γσ) are open, we may assume without loss of +generality, that {un +0 − un +1}n∈N ⊆ BEσ+µ(0, 2δinj) and for i ∈ {0, 1}, {un +i }n∈N ⊂ BEσ+µ(0, γσ) ∩ EN. +We then apply (2.3.78) to the sequence to see that ∥un +1 − un +0∥Eσ ≲ ∥Ψ(un +1) − Ψ(un +0)∥F σ for all +n ∈ N. By sending n → ∞ and using the continuity of Ψ : BEσ+µ(0, γσ) → F σ, we deduce that +∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ for all u0 − u1 ∈ BEσ+µ(0, 2δinj), with u0, u1 ∈ BEσ+µ(0, γσ) +which is (2.3.72). +□ +2.4. Proof of the inverse function theorem. We are now ready for the proofs of the inverse +function theorem. The proofs are given under conditions I and II separately. +Proof of Theorem 2.21, assuming I. Assume that condition I is satisfied, i.e. E and F are LP- +smoothable. +First note that Lemma 2.18 implies that E is terminally dense. This and the LRI mapping +hypotheses from Definition 2.20 imply the hypotheses of Theorem 2.26 with σ = β − µ and +γσ = δR ⩽ δr. Let δinj,σ > 0 be the constant from Theorem 2.26. Then the theorem tells us that +Ψ : BEβ(0, δinj,σ) → F β−µ is injective and obeys the estimate (2.3.72). +On the other hand, by hypothesis, we know that the inequality 2(r + µ) < β < (r + R)/2 is +satisfied. Let K1, K2, K3, and K4 be the constants from Theorem 2.25 and let εsurj > 0 denote +the constant appearing on the right side of (2.3.1). Then Theorem 2.25 guarantees that for every +g ∈ BF β(0, εsurj) there exists u ∈ Eβ satisfying Ψ(u) = g as well as the bound +∥u∥Eβ ⩽ K4 ∥g∥F β . +(2.4.1) +Set ε = min{εsurj, δinj,σ/K4}, κ1 = K4, and κ2 > 0 to be the constant on the right side of +(2.3.72). Since κ1ε ⩽ δinj,σ, the above analysis shows that for every g ∈ BF β(0, ε) there exists +a unique u ∈ BEβ(0, κ1ε) such that Ψ(u) = g. Moreover, the induced map Ψ−1 : BF β(0, ε) → +Ψ−1(BF β(0, ε)) ∩ BEβ(0, κ1ε) satisfies (2.2.4) and (2.2.7) in light of (2.4.1) and (2.3.72). +Now suppose that N ∋ ν ⩽ R + r − 2β. From the fourth item of Theorem 2.25, we deduce (2.2.5) +and (2.2.6), i.e. that the inverse Ψ−1 sends BF β(0, ε) ∩ F β+ν to Eβ+ν with the tame estimate +∥Ψ−1(g)∥Eβ+ν ≲ ∥g∥F β+ν. +□ + +COMPRESSIBLE TRAVELING WAVES +43 +A similar, but slightly more involved argument is needed to prove the inverse function theorem +under assumption II. +Proof of Theorem 2.21, assuming II. Assume that condition II is satisfied, i.e. E is tame and F is +a direct summand of E. +Since E is tame, there exists an LP-smoothable Banach scale G = {Gs}s∈�N� such that E is a +tame direct summand of G with lifting and restriction operators λE : E0 → G0 and ρE : G0 → E0. +Similarly, since F is a tame direct summand of E we can pick associated lifting and restriction +operators λF : F 0 → E0 and ρF : E0 → F 0. +Before proceeding, we need to introduce three bits of notation. First, we define H = G3, endowed +with the 2−norm for the sake of definiteness, which makes H into an LP-smoothable Banach scale +thanks to Lemma 2.17. Next, for s ∈ �N� write Qs = ∥ρE∥L(Gs;Es). Then by construction we have +that +ρE(BGs(0, δ/Qs)) ⊆ BEs(0, δ) for all s ∈ �N�, 0 < δ. +(2.4.2) +Third, we define the map Φ : BHr(0, δr/Qr) → Gr−µ via +Φ(u, v, w) = λE(λFΨ(ρEu) + (I − λFρF)ρEv) + (I − λEρE)w, +(2.4.3) +which is well-defined in light of (2.4.2) and the fact that Ψ : BEr(0, δr) → F r−µ. +We now claim that (H, G, Φ) satisfy the RI mapping hypotheses with parameters (µ, r, R). Clearly, +Φ(0) = 0. Now let r − µ ⩽ s ∈ �N − µ�. By the mapping properties of Ψ and the lifting and +restriction operators, we readily deduce that Φ : BHr(0, δr/Qr) ∩ Hs+1 → Gs is C2 and satisfies +DΦ(u0, v0, w0)(a, b, c) = λE(λFDΨ(ρEu0)ρEa + (I − λFρF)ρEb) + (I − λEρE)c +(2.4.4) +and +D2Φ(u0, v0, w0)[(a1, b1, c1), (a2, b2, c2)] = λEλFD2Ψ(ρEu0)[ρEa1, ρEa2]. +(2.4.5) +In turn, the tame C2 estimate for D2Ψ and (2.4.5) imply that +��D2Φ(u0, v0, w0)[(a1, b1, c1), (a2, b2, c2)] +�� +Hs +⩽ C′ +1(s)[∥(a1, b1, c1)∥Hs+µ ∥(a2, b2, c2)∥Hr + ∥(a1, b1, c1)∥Hr ∥(a2, b2, c2)∥Hs+µ] ++ C′ +1(s) ⟨∥(u0, v0, w0)∥Hs+µ⟩ ∥(a1, b1, c1)∥Hr ∥(a2, b2, c2)∥Hr +(2.4.6) +for +C′ +1(s) = C1(s) ∥λEλF∥L(F s;Gs) (1 + Qs+µ)(Qr + Q2 +r). +(2.4.7) +This proves the first and second items of the RI mapping hypotheses. +To complete the proof of the claim, it remains to show that the third item of the RI mapping +hypotheses is satisfied. Let 0 < δR ⩽ δr be given by the RI mapping hypotheses for (E, F, Ψ). For +(u0, v0, w0) ∈ BHr(0, δR/QR) ∩ HN we then define the bounded linear operator Λ(u0, v0, w0) : Gs → +Hs, for r ⩽ s ⩽ R, via +Λ(u0, v0, w0)ξ = (λEL(ρEu0)ρFρEξ, λEρEξ, ξ), +(2.4.8) +which is well-defined since ρEu0 ∈ BEr(0, δR) ∩ EN whenever (u0, v0, w0) ∈ BHr(0, δR/QR) ∩ HN. +We now use (2.4.4) to verify that Λ(u0, v0, w0) is a right inverse of DΦ(u0, v0, w0), using the identities +ρFλF = 1 and ρEλE = 1: +DΦ(u0, v0, w0)L(u0, v0, w0)ξ = λE(λFDΨ(ρEu0)ρEλEL(ρEu0)ρFρEξ + (I − λFρF)ρEλEρEξ) ++ (I − λEρE)ξ = λE(λFρFρEξ + (I − λFρF)ρEξ) + (I − λEρE)ξ = λEρEξ + (I − λEρE)ξ = ξ +(2.4.9) + +44 +NOAH STEVENSON AND IAN TICE +for all ξ ∈ Gr. Moreover, for ξ ∈ Gs, the tame estimate for L(u0) allows us to bound +∥Λ(u0, v0, w0)ξ∥Hs = ∥λEL(ρEu0)ρFρEξ∥Gs + ∥λEρEξ∥Gs + ∥ξ∥Gs +⩽ C2(s) ∥λE∥L(Es;Gs) (∥ρFρEξ∥F s + ⟨∥ρEu0∥Es+µ⟩ ∥ρFρEξ∥F r) ++ ⟨∥λEρE∥L(Gs)⟩ ∥ξ∥Gs ⩽ C′ +2(s) ∥ξ∥Gs + ⟨∥(u0, v0, w0)∥Hs+µ⟩ ∥ξ∥F r , +(2.4.10) +where the constant C′ +2(s) depends on C2(s) as well as on the quantities ∥λEρE∥L(Gs), ∥λE∥L(Es;Gs), +∥ρFρE∥L(Gs;F s), and ∥ρE∥L(Gs+µ;Es+µ). Thus, the third item of the RI hypotheses is satisfied by the +triple (H, G, Φ) with parameters (µ, r, R), and the claim is proved. We emphasize, though, that we +are not asserting that the LRI hypotheses are satisfied, as the left inverse condition fails in general +for DΦ and Λ. +With the claim in hand, we now consider β = 2(r + µ) + 1 ∈ �N� and invoke Theorem 2.25 for +the triple (H, G, Φ). Let K1, K2, K3, and K4 be the constants from Theorem 2.25 and let ε′ +surj > 0 +denote the constant appearing on the right side of (2.3.1). Then Theorem 2.25 guarantees that for +every ξ ∈ BGβ(0, ε′ +surj) there exists (u′, v′, w′) ∈ Hβ = (Gβ)3 satisfying Φ(u′, v′, w′) = ξ as well as +the bound +∥u′∥Gβ + ∥v′∥Gβ + ∥w′∥Gβ = ∥(u′, v′, w′)∥Hβ ⩽ K4∥ξ∥Gβ. +(2.4.11) +Set εsurj = ε′ +surj/∥λEλF∥L(F β;Gβ) and note that +λEλF(BF β(0, εsurj)) ⊆ BGβ(0, ε′ +surj) +(2.4.12) +by construction. Consequently, for any g ∈ BF β(0, εsurj) there exists (u′, v′, w′) ∈ Hβ such that +Φ(u′, v′, w′) = λEλFg, which unravels to +λE(λFΨ(ρEu′) + (I − λFρF)ρEv′) + (I − λEρE)w′ = λEλFg. +(2.4.13) +Applying ρE and using the identity ρEλE = 1, this implies the identity +λFΨ(ρEu′) + (I − λFρF)ρEv′ = λFg, +(2.4.14) +to which we apply ρF and using the identity ρFλF = 1 to see that +Ψ(ρEu′) = g. +(2.4.15) +Thus, if we set u = ρEu′ ∈ Eβ, then Ψ(u) = g, and (2.4.11) implies that +∥u∥Eβ ⩽ κ1 ∥g∥F β +(2.4.16) +for κ1 = K4 ∥ρE∥L(Gβ;Eβ) ∥λEλF∥L(F β;Gβ), which in particular means that u ∈ BEβ(0, κ1εsurj). +On the other hand, Lemma 2.18 implies that E is terminally dense. This and the LRI mapping +hypotheses imply that the hypotheses of Theorem 2.26 are satisfied by the triple (E, F, Ψ) with +σ = β − µ and γσ = δR ⩽ δr. Let δinj,σ > 0 be the constant from Theorem 2.26. Then the theorem +tells us that Ψ : BEβ(0, δinj,σ) → F β−µ is injective and obeys the estimate (2.3.72). +Set ε = min{εsurj, δinj,σ/κ1} and κ2 > 0 to be the constant on the right side of (2.3.72). Since κ1ε ⩽ +δinj,σ, the above analysis shows that for every g ∈ BF β(0, ε) there exists a unique u ∈ BEβ(0, κ1ε) +such that Ψ(u) = g. Moreover, the induced map Ψ−1 : BF β(0, ε) → Ψ−1(BF β(0, ε)) ∩ BEβ(0, κ1ε) +satisfies (2.2.4) and (2.2.7) in light of (2.4.16) and (2.3.72). +Finally, if we assume that N ∋ ν ⩽ R + r − 2β and g ∈ BF β(0, ε) ∩ F β+ν, then we are assured by +the fourth item of Theorem 2.25 that there exists (u′, v′, w′) ∈ Hβ+ν such that Φ(u′, v′, w′) = λEλFg. +By unraveling as in (2.4.13)–(2.4.15), we find that for u = ρEu′ ∈ Eβ+ν we have that Ψ(u) = g and +∥u∥Eβ+ν ≲ ∥g∥F β+ν. +□ + +COMPRESSIBLE TRAVELING WAVES +45 +2.5. Refinements. In this subsection we aim to strengthen the conclusions of Theorem 2.21. In +particular, we will study the continuity and higher order smoothness of the inverse map Ψ−1 +provided by the theorem. First, we analyze the right and left linear inverse map L by making an +extension to backgrounds outside of the terminal space EN, and then proving various continuity and +differentiability assertions. Second, we return to the map Ψ−1 and show a more refined continuity +estimate than the basic assertion of (2.2.7). Then we prove differentiability of Ψ−1 and relate +the derivative to the operator L. Once this is done, we conclude by reading off higher regularity +assertions. +We now enumerate further properties of the family of inverses L. +Theorem 2.27 (Extension and regularity of the right and left inverse). Under the LRI mapping +hypotheses set forth in Definition 2.20 and the additional assumptions that r + µ ⩽ R and that the +Banach scale {Es}s∈�N� consists of reflexive spaces, we have that the following properties of L hold. +(1) Existence of extension: There exists a family of bounded linear maps L : BEr(0, δR)∩Er+µ → +L(F r; Er) such that the following extension properties are satisfied. +(a) L = L on the subset BEr(0, δR) ∩ EN. +(b) If g ∈ F r and u0 ∈ BEr(0, δR) ∩ Er+µ, then we have that DΨ(u0)L(u0)g = g. Addi- +tionally, if u ∈ Er+µ then L(u0)DΨ(u0)u = u. +(c) For s ∈ [r, R] ∩ N we have that L : BEr(0, δR) ∩ Es+µ → L(F s; Es) with the tame +estimate ∥L(u0)g∥Es ≲ ∥g∥F s + ⟨∥u0∥Es+µ⟩∥g∥F r. +(2) Continuity: For s ∈ [r, R−µ]∩N, if we view L as mapping L : (BEr(0, δR)∩Es+µ)×F s → Es, +then this map is continuous. +(3) Higher regularity: Assume that, for some N ∋ ℓ ⩾ 2, the map Ψ is ℓ-times continuously +Gateaux differentiable in the sense of Definition 2.23. If for s ∈ [r, R − ℓµ] ∩ N we view L as +a mapping +L : (BEr(0, δR) ∩ Es+ℓµ) × F s+(ℓ−1)µ → Es, +(2.5.1) +then L is (ℓ − 1)-times continuously Gateaux-differentiable. +Proof. We divide the proof into several steps. +Step 1: Constructing L. First, we provide a continuity estimate on the right and left inverse +map L (a priori only defined for EN-backgrounds). +Suppose that u0, w0 ∈ BEr(0, δR) ∩ EN, +s ∈ [r, R − µ] ∩ N, and g ∈ F s+µ. We will estimate the difference +L(u0)g − L(w0)g = −L(u0)(DΨ(u0) − DΨ(w0))L(w0)g += − +� 1 +0 +L(u0)D2Ψ((1 − t)w0 + tu0)(u0 − w0, L(w0)g) dt +(2.5.2) +in Es. +By applying the tame estimates of (2.2.1) and (2.2.2) repeatedly and the embedding +inequalities of the first item of Definition 2.2, we get the somewhat crude estimate +∥L(u0)g − L(w0)g∥Es ≲ ⟨∥u0, w0∥Es+µ⟩3∥u0 − w0∥Es+µ∥g∥F r ++ ⟨∥u0, w0∥Es+µ⟩∥u0 − w0∥Er(∥g∥F s+µ + ⟨∥w0∥Es+2µ⟩∥g∥F r). +(2.5.3) +While forgoing tameness, this estimate is strong enough to allow us to define our extension. +Indeed, let u0 ∈ BEr(0, δR) ∩ Er+µ and fix some g ∈ F r. +We claim that the sequence +{L(Tju0)Tjg}∞ +j=ℓ ⊂ Er is Cauchy, where the operators {Tj}∞ +j=0 are from Lemma 2.18 and ℓ ∈ N is +the first index for which Tju0 ∈ BEr(0, δR) for all N ∋ j ⩾ ℓ. We verify this by first estimating +∥L(Tj+ku0)Tj+kg − L(Tju0)Tjg∥Er ⩽ ∥L(Tj+ku0)(Tj+kg − Tjg)∥F r ++ ∥(L(Tj+ku0) − L(Tju0))Tjg∥Er = Ij,k + IIj,k. +(2.5.4) + +46 +NOAH STEVENSON AND IAN TICE +For Ij,k we simply apply estimate (2.2.2) and Lemma 2.18: +Ij,k ≲ ⟨∥Tj+ku0∥Er+µ⟩∥Tj+kg − Tjg∥F r ≲ ⟨∥u0∥Er+µ⟩(nr +j(g) + nr +j+k(g)). +(2.5.5) +Hence limj,k→∞ Ij,k = 0. On the other hand, for IIj,k, we first employ estimate (2.5.3): +IIj,k ≲ ⟨∥Tj+ku0, Tju0∥Er+µ⟩3∥Tj+ku0 − Tju0∥Er+µ∥Tjg∥F r ++ ⟨∥Tj+ku0, Tju0∥Er+µ⟩∥Tj+ku0 − Tju0∥Er(∥Tjg∥F r+µ + ⟨∥Tju0∥Er+2µ⟩∥Tjg∥F r). +(2.5.6) +Thanks again to Lemma 2.18, we are free to make the following bounds: +∥Tj+ku0∥Er+µ, ∥Tju0∥Er+µ ≲ ∥u0∥F r+µ, +∥Tjg∥F r ≲ ∥g∥F r, +∥Tj+ku0 − Tju0∥Er ≲ 2−jµ(nr+µ +j +(u0) + nr+µ +j+k(u0)), +∥Tjg∥F r+µ ≲ 2jµ∥g∥F r, +∥Tj+ku0 − Tju0∥Er+µ ≲ nr+µ +j +(u0) + nr+µ +j+k(u0) +∥Tju0∥Er+2µ ≲ 2jµ∥u0∥Er+µ. +(2.5.7) +Upon combining (2.5.6) and (2.5.7), we acquire the estimate +IIj,k ≲ ⟨∥u0∥Er+µ⟩3∥g∥F r(nr+µ +j +(u0) + nr+µ +j+k(u0)), +(2.5.8) +and hence limj,k→∞ IIj,k = 0. We deduce that the sequence {L(Tju0)Tjg}∞ +j=ℓ ⊂ Er is Cauchy. +Hence there exists +L(u0)g = lim +j→∞ L(Tju0)Tjg, +(2.5.9) +and this defines L as a family of linear maps. +Step 2: Properties of L. We now examine the restriction of L to higher regularity (larger s) +spaces in the scale. Suppose that u0 ∈ BEr(0, δR) ∩ Es+µ and that g ∈ F s for some s ∈ [r, R] ∩ N. +For N ∋ j sufficiently large we can apply estimate (2.2.2) and again Lemma 2.18 to see that +∥L(Tju0)Tjg∥Es ≲ ∥Tjg∥F s + ⟨∥Tju0∥Es+µ⟩∥Tjg∥F r ≲ ∥g∥F s + ⟨∥u0∥Es+µ⟩∥g∥F r. +(2.5.10) +Hence, the sequence {L(Tju0)Tjg}∞ +j=ℓ ⊂ Es is bounded by the right hand expression above. The +space Es is reflexive, and this sequence already converges in Er, so the limit L(u0)g belongs to +Es and has norm bounded above by the right side of (2.5.10) thanks to the weak sequential lower +semicontinuity of the norm. +We next prove that L is a family of right and left inverses for DΨ. First suppose that g ∈ F r +and u0 ∈ BEr(0, δR) ∩ Er+µ. For j ∈ N sufficiently large, we have that +DΨ(Tju0)L(Tju0)Tjg = Tjg. +(2.5.11) +Since Ψ : BEr(0, δr) ∩ Er → F r−µ is C2, {L(Tju0)Tjg}∞ +j=0 ⊂ Er converges to L(u0)g, and we have +the convergences Tju0 → u0 in Er+µ �→ Er and Tjg → g in F r as j → ∞, we may send j → ∞ in +(2.5.11) to see that DΨ(u0)L(u0)g = g. +On the other hand, if we assume that u ∈ Er+µ and u0 ∈ BEr(0, δR) ∩ Er+µ, then we have +L(Tju0)TjDΨ(u0)u = u + L(Tju0)(TjDΨ(u0) − DΨ(Tju0))u. +(2.5.12) +The left hand side converges in Er to L(u0)DΨ(u0)u as j → ∞. For the right hand side we may +estimate +∥L(Tju0)(TjDΨ(u0) − DΨ(Tju0))u∥Er ≲ ⟨∥u0∥Er+µ⟩∥(TjDΨ(u0) − DΨ(Tju0))u∥F r +≲ ⟨∥u0∥Er+µ⟩(∥(I − Tj)DΨ(u0)u∥F r + ∥(DΨ(u0) − DΨ(Tju0))u∥F r), +(2.5.13) +and the right hand side of this evidently converges to zero as j → ∞ thanks again to the properties +of I − Tj from Lemma 2.18, the continuity properties of DΨ, and the fact that u, u0 ∈ Er+µ. Hence +by sending j → ∞ in (2.5.12) we learn that L(u0)DΨ(u0)u = u. + +COMPRESSIBLE TRAVELING WAVES +47 +Now we are ready to prove that L is actually an extension of L. Suppose that u0 ∈ BEr(0, δR)∩EN +and that g ∈ F r. Then by the right and left inverse properties, for every j ∈ N we have the identity +L(u0)Tjg = L(u0)DΨ(u0)L(u0)Tjg = L(u0)Tjg. +(2.5.14) +Upon sending j → ∞ and using that L(u0) and L(u0) are bounded, we deduce that L(u0)g = L(u0)g. +This completes the proof of all of the assertions of the first item. +Step 3: Continuity. We now study continuity. We have established that for s ∈ [r, R − µ] ∩ N, +u0, w0 ∈ BEr(0, δR) ∩ Es+2µ, and g ∈ F s+µ the sequence {L(Tju0)Tjg − L(Tjw0)Tjg}∞ +j=0 ⊂ Es+µ +converges weakly up to a subsequence in Es+µ and strongly in Er to the limit L(u0)g − L(w0)g. +Therefore, upon invoking the log-convexity of Lemma 2.19, we obtain strong convergence in Es up +to a subsequence. By passing along this subsequence in (2.5.3) and then taking the supremum over +∥g∥F s+µ ⩽ 1, we obtain the Lipschitz estimate +∥L(u0) − L(w0)∥L(F s+µ,Es) ≲ ⟨∥u0, w0∥Es+2µ⟩3∥u0 − w0∥Es+µ. +(2.5.15) +We use (2.5.15) as an intermediate step in deriving the stated continuity in the second item of +the Theorem statement. Let u0, w0 ∈ BEr(0, δR) ∩ Es+µ and g, h ∈ F s. For j ∈ N sufficiently large, +we consider the difference +L(u0)g − L(w0)h = (L(u0)g − L(Tju0)Tjg) ++ (L(Tju0)Tjg − L(Tjw0)Tjh) + (L(Tjw0)Tjh − L(w0)h) = Ij + IIj + IIIj +(2.5.16) +and estimate the three terms individually. For Ij, we note that (2.5.4), (2.5.5), (2.5.6), and (2.5.7) +hold with r replaced by s (by the same proof), and hence we may send k → ∞ to acquire the bound +∥Ij∥Es ≲ ⟨∥u0∥Es+µ⟩3� +∥g∥F sns+µ +j +(u0) + ns +j(g) +� +. +(2.5.17) +For IIj we simply apply the local Lipschitz estimate (2.5.15) and use properties of the operators Tj: +∥IIj∥Es ≲ ⟨2jµ∥g∥F s⟩⟨2jµ∥u0, w0∥Es+µ⟩3∥u0 − w0, g − h∥Es+µ×F s. +(2.5.18) +For IIIj, we begin by bounding as in (2.5.17): +∥IIIj∥Es ≲ ⟨∥w0∥Es+µ⟩3� +∥h∥F sns+µ +j +(w0) + ns +j(h) +� +. +(2.5.19) +Now, by Lemma 2.18, we may estimate +ns+µ +j +(w0) ≲ ns+µ +j +(u0) + ∥u0 − w0∥Es+µ and ns +j(h) ≲ ns +j(g) + ∥g − h∥F s , +(2.5.20) +and hence deduce that +∥L(u0)g − L(w0)h∥Es ≲ 23µj⟨∥u0, w0∥Es+µ, ∥g, h∥F s⟩3∥u0 − w0, g − h∥Es+µ×F s ++ ⟨∥u0, w0∥Es+µ, ∥g, h∥F s⟩3(ns+µ +j +(u0) + ns +j(g)). +(2.5.21) +By taking j large relative to u0 and g and then taking w0 and h sufficiently close u0 and g to we +see that L is continuous as a map from Es+µ × F s to Es, but not necessarily uniformly so. This +completes the proof of the second item. +Step 4: Higher regularity. Finally, we prove the third assertion. By arguing as in the derivation +of (2.5.10), we learn that for s ∈ [r, R − 2µ] ∩ N, w0, h0, w0 + h0 ∈ BEr(0, δR) ∩ Es+2µ, g ∈ F s+µ, +and τ ∈ (0, 1) we have the decomposition +τ −1(L(w0 + τh0)g − L(w0)g) + L(w0)D2Ψ(w0)(h0, L(w0)g) = Iτ + IIτ +(2.5.22) +where +Iτ = −(L(w0 + τh0) − L(w0)) +� 1 +0 +D2Ψ(w0 + τh0)(h0, L(w0)g) dt +(2.5.23) + +48 +NOAH STEVENSON AND IAN TICE +and +IIτ = −L(w0) +� 1 +0 +(D2Ψ(w0 + tτh0) − D2Ψ(w0))(h0, L(w0)g) dt. +(2.5.24) +Since Ψ is µ-tamely C2, L satisfies the continuity assertions of the second item, and L obeys the +same tame estimates as L, it holds that ∥Iτ, IIτ∥Es → 0 as τ → 0. This proves that the map +L : (BEr(0, δR) ∩ Es+2µ) × F s+µ → Es +(2.5.25) +is Gateaux differentiable with derivative given by +DL(u0, g0)[w, h] = −L(u0)D2Ψ(u0)(w, L(u0)g0) + L(u0)h +(2.5.26) +for u0 ∈ BEr(0, δR) ∩ Es+2µ, w ∈ Es+2µ, and g0, h ∈ F s+µ. Now suppose that N ∋ ℓ ⩾ 3 and +that Ψ is ℓ-times continuously Gateaux differentiable. By a simple induction argument using the +formula (2.5.26), we find the remaining conclusions of the third item. +□ +Now that we have a refined understanding of the mapping properties of the family of right and +left inverses L, we return to studying the local inverse map Ψ−1, which we recall is granted by the +conclusions of Theorem 2.21. Indeed, we now prove Theorem 2.24. +Proof of Theorem 2.24. Throughout the proof we will use the operator L from Theorem 2.27, which +is an extension of the operator L from the LRI mapping hypotheses. By a very mild abuse of +notation, we write L in place of L in what follows. +Under either hypothesis I or II, the Banach scales E and F are tame, and so there exist smoothing +operators {Tj}∞ +j=0 as in Lemma 2.18. We aim to establish that if s ∈ [β, R + r − β − µ) ∩ N and +g ∈ BF β(0, ε) ∩ F s, then Ψ−1(Tjg) → Ψ−1(g) in the space Es as j → ∞. By Taylor expanding Ψ at +Tjg to second order as in the proof of Lemma 2.26 and using the left invertibility of DΨ(Tjg) by +L ◦ Ψ−1(Tjg), we arrive at the identity +Ψ−1(Tj+1g) − Ψ−1(Tjg) = L ◦ Ψ−1(Tjg)(Tj+1 − Tj)g +− L ◦ Ψ−1(Tjg) +� 1 +0 +(1 − t)D2Ψ((1 − t)Ψ−1(Tjg) + tΨ−1(Tj+1g))(Ψ−1(Tj+1g) − Ψ−1(Tjg))⊗2 dt += Ij + IIj. +(2.5.27) +We will estimate the right hand side of the above in the spaces Es+σ for σ ∈ {−1, 0, 1}. From the +assumed tame structure, we have the estimate +∥L ◦ Ψ−1(Tjg)h∥Es+σ ≲ ⟨∥Ψ−1(Tjg)∥Es+µ+σ⟩∥h∥F r + ∥h∥F s+σ ≲ ⟨2j(µ+σ)∥g∥F s⟩∥h∥F r + ∥h∥F s+σ. +(2.5.28) +For h = (Tj+1 − Tj)g we estimate +� +∥(Tj+1 − Tj)g∥F r ≲ 2j(r−s)ms +j(g), +∥(Tj+1 − Tj)g∥F s+σ ≲ 2jσms +j(g), +(2.5.29) +and hence (since µ + r ⩽ s) ∥Ij∥Es+σ ≲ 2jσ⟨∥g∥F s⟩ms +j(g), where we recall that the seminorms ms +j +are from Lemma 2.18. On the other hand, for some t ∈ [0, 1], we take +ht = D2Ψ((1 − t)Ψ−1(Tjg) + tΨ−1(Tj+1g))(Ψ−1(Tj+1g) − Ψ−1(Tjg))⊗2 +(2.5.30) +and estimate for ℓ ∈ {r, s + σ} +∥ht∥Eℓ ≲ ⟨∥Tjg, Tj+1g∥F ℓ+µ⟩∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥2 +Er ++ ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Eℓ+µ. +(2.5.31) + +COMPRESSIBLE TRAVELING WAVES +49 +If j is sufficiently large, say j ⩾ J(g), then we have that Tjg, Tj+1g ∈ BF β(0, ε), and hence we can +apply the estimate from the third conclusion of Theorem 2.21, to bound +� +∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er +∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er+µ +≲ ∥(Tj+1 − Tj)g∥Eβ−µ ≲ 2j(β−µ−s)ms +j(g). +(2.5.32) +On the other hand, we trivially bound +∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Es+σ+µ ≲ ∥Tj+1g, Tjg∥F s+σ+µ ≲ 2j(µ+σ)∥g∥F s. +(2.5.33) +Therefore, we get that +∥ht∥Es+σ ≲ 22j(β−µ−s)+j(µ+σ)⟨∥g∥F s⟩ms +j(g)2 + 2j(β+σ−s)∥g∥F sms +j(g) ≲ 2jσ⟨∥g∥F s⟩2ms +j(g) (2.5.34) +and ⟨2j(µ+σ)∥g∥F s⟩∥ht∥Er ≲ 2jσ⟨∥g∥F s⟩3ms +j(g). Upon synthesizing these bounds we arrive at the +estimate ∥IIj∥Es+σ ≲ 2jσ⟨∥g∥F s⟩3ms +j(g), and hence (for σ ∈ {−1, 0, 1} and N ∋ j ⩾ J(g)) we have +∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Es+σ ≲ 2jσ⟨∥g∥F s⟩3ms +j(g). +(2.5.35) +We now use (2.5.35) to prove that the sequence {�n +j=J(g)(Ψ−1(Tj+1g) − Ψ−1(Tjg))}∞ +n=J(g) is +Cauchy in Es. Again, the Banach scales E and F are tame in the sense of Definition 2.14, and +so there exist LP-smoothable Banach scales E and F with lifting and restriction pairs λE, ρE and +λF, ρF witnessing the definition of tameness for E and F, respectively. Let ξj(g) = λE(Ψ−1(Tj+1g)− +Ψ−1(Tjg)) ∈ E +R+r−β and ηj(g) = ⟨∥g∥F s⟩3(Sj+1 − Sj) ◦ λFg ∈ F +R+r−β. By the continuity of λE, +(2.5.35), and properties of the LP-smoothing operators from Definition 2.11, we deduce that for +j, k ∈ N with j ⩾ J(g) we have the bound +∥∆kξj(g)∥Es ≲ 2−|j−k|∥ηj(g)∥F s. +(2.5.36) +Hence, we obtain the following estimates for ℓ, n ∈ N with ℓ ⩾ J(g): +��� +ℓ+n +� +j=ℓ +ξj(g) +��� +2 +Es ≲ +∞ +� +k=0 +� ℓ+n +� +j=ℓ +∥∆kξj(g)∥Es +�2 +≲ +∞ +� +k=0 +� ℓ+n +� +j=ℓ +2−|j−k|∥ηj(g)∥F s +�2 +≲ +ℓ+n +� +j=ℓ +∥ηj(g)∥2 +F s, (2.5.37) +where in the last inequality above we have employed Young’s convolution inequality, as was done +in (2.3.12). Since g ∈ F s, we have the inclusion {∥ηj(g)∥F s}∞ +j=0 ∈ ℓ2(N) and hence from (2.5.37), +we deduce that {�n +j=J(g) ξj(g)}∞ +n=J(g) ⊂ E +s is Cauchy. The map ρE is continuous and linear and +thus by taking the image of this sequence we learn that +� +n +� +j=J(g) +(Ψ−1(Tj+1g) − Ψ−1(Tjg)) +�∞ +n=J(g) = {Ψ−1(Tn+1g) − Ψ−1(TJ(g)g)}∞ +n=0 ⊂ Es +(2.5.38) +is Cauchy, as desired. On the other hand, the third conclusion of Theorem 2.21 yields: +∥Ψ−1(Tn+1g) − Ψ−1(g)∥Eβ−µ ≲ ∥(I − Tn+1)g∥F β−µ → 0 +as +n → ∞. +(2.5.39) +Therefore, the limit in Es of the above sequence, which exists by the satisfaction of the Cauchy +condition, necessarily is Ψ−1(g). By tracing back through the estimates, we find the following +quantitative rate of convergence for n ⩾ J(g): +∥Ψ−1(Tn+1g) − Ψ−1(g)∥Es ≲ ⟨∥g∥F s⟩3� ∞ +� +j=n +ms +j(g)2�1/2 +. +(2.5.40) +We now have all the tools we need to establish that the map Ψ−1 : BF β(0, ε) ∩ F s → Es is +continuous for every s ∈ [β, R + r − β − µ) ∩ N. Let g, h, g + h ∈ BF β(0, ε) ∩ F s, and assume that +∥h∥F s is sufficiently small so that J(g + h) ⩽ J(g) + 1, where once more we let J(g + h) denote the + +50 +NOAH STEVENSON AND IAN TICE +first index for which j ⩾ J(g + h) implies that Tj(g + h) ∈ BF β(0, ε). For any N ∋ n ⩾ J(g) + 1, we +may then estimate +∥Ψ−1(g) − Ψ−1(g + h)∥Es ⩽ ∥Ψ−1(Tn+1g) − Ψ−1(g)∥Es ++ ∥Ψ−1(Tn+1g) − Ψ−1(Tn+1(g + h))∥Es + ∥Ψ−1(Tn+1(g + h)) − Ψ−1(g + h)∥Es += In + IIn + IIIn. +(2.5.41) +For In and IIIn we employ the quantitative rate of convergence (2.5.40) along with the inequality +(which is true by Lemma 2.18) +� ∞ +� +j=n +ms +j(g + h)2�1/2 +≲ ∥h∥F s + +� ∞ +� +j=n +ms +j(g)2�1/2 +(2.5.42) +to see that +In + IIIn ≲ ⟨∥g, h∥F s⟩3� +∥h∥F s + +� ∞ +� +j=n +ms +j(g)2�1/2� +. +(2.5.43) +It remains to handle IIn. As before, we have +Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g) = L ◦ Ψ−1(Tn+1g) +� +Tn+1h− +� 1 +0 +(1 − t)D2Ψ((1 − t)Ψ−1(Tn+1g) + tΨ−1(Tn+1(g + h)))(Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g))⊗2 dt +� +(2.5.44) +as a consequence of Taylor’s theorem. Hence we have the estimate +IIn ≲ ⟨2µn∥g∥F s⟩ +� +∥h∥F s+ +⟨2µn∥g, h∥F s⟩∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Es+µ∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Er� +. +(2.5.45) +We then crudely estimate +∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Es+µ ≲ 2nµ∥g, h∥F s +(2.5.46) +and also apply the third conclusion of Theorem 2.21 to see that +∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Er ⩽ ∥Tn+1h∥F β−µ ≲ ∥h∥F s. +(2.5.47) +Synthesizing these bounds shows that IIn ≲ ⟨2µn∥g, h∥F s⟩3∥h∥F s, and hence +∥Ψ−1(g)−Ψ−1(g+h)∥Es ≲ ⟨∥g, h∥F s⟩3� +∥h∥F s+ +� ∞ +� +j=n +ms +j(g)2�1/2� ++⟨2µn∥g, h∥F s⟩3∥h∥F s. (2.5.48) +By taking n large relative to g and then taking h small, we see that the above estimate proves that +Ψ−1 is continuous at g. This completes the proof of the first item. +We continue by studying the differentiability of the inverse map Ψ−1. Suppose that s ∈ [β, R + +r − β − µ) ∩ N, g, h, g + h ∈ BF β(0, ε) ∩ F s. By expanding Ψ to second order via Taylor’s theorem +with integral remainder and utilizing the left invertibility of L (see e.g. (2.5.44)), we derive the +equality +Ψ−1(g + h) − Ψ−1(g) − L ◦ Ψ−1(g)h = +− L ◦ Ψ−1(g) +� 1 +0 +(1 − t)D2Ψ((1 − t)Ψ−1(g) + tΨ−1(g + h))(Ψ−1(g + h) − Ψ−1(g))⊗2 dt. +(2.5.49) + +COMPRESSIBLE TRAVELING WAVES +51 +By taking the norm of both sides in the space Es−µ, employing tame estimates, and utilizing the +third item of Theorem 2.21, we find that +∥Ψ−1(g + h) − Ψ−1(g) − L ◦ Ψ−1(g)h∥Es−µ +≲ ⟨∥g, h∥F s⟩2∥Ψ−1(g + h) − Ψ−1(g)∥Es∥Ψ−1(g + h) − Ψ−1(g)∥Er +≲ ⟨∥g, h∥F s⟩2∥Ψ−1(g + h) − Ψ−1(g)∥Es∥h∥F β−µ. +(2.5.50) +Since F s �→ F β−µ and we have already established that Ψ−1 is continuous with respect to the +F s and Es topologies, estimate (2.5.50) shows that Ψ−1 is differentiable at g when viewed as a +map from F s to Es−µ, and we have the derivative formula DΨ−1(g)h = L ◦ Ψ−1(g)h. Moreover, +since Ψ−1 is continuous relative to the F s and Es topologies, and L is continuous relative to the +Es × F s−µ and Es−µ topologies, we find, by composition of continuity, that DΨ−1 is continuous +with respect to the F s × F s−µ and Es−µ topologies. This proves the second item. +The third item now follows by pairing the final item of Theorem 2.27 and the derivative formula +DΨ−1 = L ◦ Ψ−1 with a simple induction argument. +□ +3. Nonlinear analysis of traveling free boundary compressible Navier-Stokes +With our tools from nonlinear analysis now established in abstract form, we return to our main +goal: the analysis of the traveling wave free boundary compressible Navier-Stokes equations, (1.4.9). +In this section we verify most of the ‘nonlinear’ hypotheses for our Nash-Moser inverse function +theorem and make preparations for the linear analysis that follows in subsequent sections. +We select a nonlinear mapping with tame Banach scales for the domain and codomain. In +Section 3.1 we verify condition II of Theorem 2.21. To check that our rather complicated operator +is 1-tamely C2, we analyze each of the atomic nonlinearities individually in Section 3.2 and then +synthesize in Section 3.3 via the calculus of tame maps from Section 2.1. Once this is done, in +Section 3.4 we then use our newly developed understanding of the nonlinear map to decompose its +derivative into a principal part and a remainder term. We then conclude in Section 3.5 with some +preliminary results for our linear analysis. +3.1. Banach scales for the traveling wave problem. We begin by defining some function +spaces that will comprise our Banach scales. First, we introduce spaces that will play a role in the +domain of our nonlinear map. For s ∈ {−1, 0} ∪ R+ we define +Xs = H1+s(Ω) × H2+s(Ω; Rn) × H5/2+s(Σ), +(3.1.1) +where H5/2+s(Σ) denotes the anisotropic Sobolev space defined in (B.1.1), with Σ identified with +Rn−1, and we endow Xs with the Hilbert norm +∥q, u, η∥Xs = +� +∥q∥2 +H1+s + ∥u∥2 +H2+s + ∥η∥2 +H5/2+s. +(3.1.2) +We single out an important closed subspace of Xs: +Xs = {(q, u, η) ∈ Xs : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = 0}. +(3.1.3) +Second, we define some spaces that will play a role in the codomain of our nonlinear map. For +s ∈ {−1, 0} ∪ R+ we define the space +Ys = +� +L2(Ω) × (0H1(Ω; Rn))∗ +if s = −1, +H1+s(Ω) × Hs(Ω; Rn) × H1/2+s(Σ; Rn) +if s ⩾ 0 +(3.1.4) +and endow it with the norm +� +� +� +∥g, F∥Ys = +� +∥g∥2 +L2 + ∥F∥2 +(0H1)∗ +if s = −1, +∥g, f, k∥Ys = +� +∥g∥2 +H1+s + ∥f∥2 +Hs + ∥k∥2 +H1/2+s +if s ⩾ 0. +(3.1.5) + +52 +NOAH STEVENSON AND IAN TICE +We also single out an important subspace of Ys: +Ys = +� +� +� +� +(g, F) ∈ Ys : +� b +0 g(·, y) dy ∈ ˙H−1(Σ) +� +if s = −1, +� +(g, f, k) ∈ Ys : +� b +0 g(·, y) dy ∈ ˙H−1(Σ) +� +if s ⩾ 0, +(3.1.6) +which we endow with the norm +� +� +� +∥g, F∥Ys = +� +∥g, F∥2 +Y−1 + [ +� b +0 g(·, y) dy]2 +˙H−1 +if s = −1, +∥g, f, k∥Ys = +� +∥g, f, k∥2 +Ys + [ +� b +0 g(·, y) dy]2 +˙H−1 +if s ⩾ 0. +(3.1.7) +The spaces Ys and Ys are clearly complete and are Hilbert spaces in the case s ⩾ 0. +Remark 3.1. In the notation of (C.1.1), we have that Ys = ˆH1+s(Ω) × Hs(Ω; Rn) × H1/2+s(Σ; R) +for R ∋ s ⩾ 0, while Y−1 = ˆH0(Ω) × (0H1(Ω; Rn))∗. +Third, we introduce spaces that will contain the stress and forcing data tuple (T, G, F). For +R ∋ s ⩾ 0 we define the space +Ws = H1+s(Rn; Rn×n) × Hs(Rn; Rn) × Hs(Rn; Rn) +(3.1.8) +and endow it with the norm +∥T , G, F∥Ws = +� +∥T ∥2 +H1+s + ∥G∥2 +Hs + ∥F∥2 +Hs. +(3.1.9) +Finally, for s ∈ N we set +Es = Xs × W1+s × R +and +Fs = Ys × W1+s × R +(3.1.10) +and endow these with the norm from Remark 2.3. +From these spaces we now define the following Banach scales: +XXX = {Xs}s∈N, +YYY = {Ys}s∈N, +W +W +W = {W1+s}s∈N, +EEE = {Es}s∈N, +FFF = {Fs}s∈N. +(3.1.11) +It is a simple matter to check that these all satisfy the conditions required by Definition 2.2 to be +Banach scales. +In the next result we check the second condition in the statement of Theorem 2.21 for these +scales. +Lemma 3.2 (Tameness of domain and codomain). The Banach scale EEE is tame, and the Banach +scale FFF is a tame direct summand of EEE in the sense of Definition 2.14. +Proof. We begin by proving that EEE is a tame Banach scale. The scale W +W +W is tame in light of Lemma +2.17 and Example 2.13, as it is the product of scales of L2-based Sobolev spaces on all of Rn. Since +W +W +W is tame, this lemma also shows that it suffices to prove that R and XXX = {Xs}s∈N are tame. R is +handled by Example 2.12, so it remains only to handle XXX. +For s ∈ N write Xs = H1+s(Rn) × H2+s(Rn; Rn) × H5/2+s(Σ). Thanks to Lemma B.6, Exam- +ple 2.13, and Lemma 2.17 the Banach scale {Xs}s∈N is LP smoothable. We now show that {Xs}s∈N +is tame by showing it is a tame direct summand of {Xs}s∈N. To this end, we define an auxiliary +mapping E1 : H1/2+s(Σ0; Rn) × H1/2+s(Σ) → H1+s(Ω; Rn) via E1(φ, ϕ) = w, where w is the unique +solution to the PDE +� +� +� +� +� +� +� +� +� +−∆w = 0 +in Ω, +w = φ +on Σ0, +(I − en ⊗ en)w = 0 +on Σ, +w · en = ϕ +on Σ. +(3.1.12) + +COMPRESSIBLE TRAVELING WAVES +53 +Also, let EΩ denote a Stein extension operator, in the sense of Definition A.2, and write RΩ for +the linear operator corresponding to restriction from Rn to Ω. We then define the lifting operators +λX : Xs → Xs and the restriction operators ρX : Xs → Xs via +λX(q, u, η) = (EΩq, EΩu, η) and ρX(q, u, η) = (RΩq, RΩu−E1(TrΣ0u, TrΣ(u·en)+∂1η), η). (3.1.13) +In light of Proposition B.4, it is straightforward to check that λX and ρX indeed define bounded +linear maps between their stated domains and codomains. Additionally, we have that ρXλX = idX0, +which completes the proof that {Xs}s∈N is a tame direct summand of {Xs}s∈N, and hence is tame. +In turn, this completes the proof that EEE is tame. +We continue by showing that FFF is a tame direct summand of EEE. For this, it is clearly sufficient to +prove that YYY is a tame direct summand of XXX. In fact, we have the stronger result that these spaces +are isomorphic. To see this, consider the map Γ : Xs → Ys defined by Γ(q, u, η) = (g, f, k), where +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · u = g +in Ω, +−∂1u + ∇(q + η) − ∆u = f +in Ω, +−(q − Du)en − ∆∥ηen = k +on Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0. +(3.1.14) +This is well-defined by virtue of Proposition B.4. The proof of Theorem 6.6 in Leoni and Tice [60] +establishes that this map is a Banach isomorphism, although the theorem itself states the isomorphism +between slightly different spaces, which is needed in [60] due to a change of unknown made by +taking a particular linear combination of q and η. +□ +3.2. Analysis of atomic nonlinearities. In this subsection we put to use the abstract analysis of +smooth tame structures from Section 2.1 and the tools for verifying smoothness and tame estimates +from Appendices D.1 and D.2 in the study of the nonlinear expressions appearing in the equations +under consideration, namely system (1.4.9). The majority of the results in this subsection are a +recasting of the tame calculus estimates from Appendix D.2 in the language of Section 2.1. As such, +some of the proofs are not much more than mere referrals; it is the statements that are important +as we move forward. Once these are established, we conclude this subsection by considering two +nonlinearities that play a distinguished role in our analysis of (1.4.9). These aid in understanding +the mapping properties of the continuity equation, which is rather subtle. +The Banach scales we work with in this subsection are built from combinations of the following +atomic scales: +H(U; V ) = {Hs(U; V )}s∈N, +H(Rd) = {Hs(Rd)}s∈N, +W(U; V ) = {W s,∞(U; V )}s∈N, +WH(κ) = {W s,∞((0, b); Hs +(κ)(Rd))}s∈N, +H(κ)(Rd) = {Hs +(κ)(Rd)}s∈N, +(3.2.1) +where κ ∈ R+, d ∈ N+, U ⊆ Rd is a Stein extension domain (see Definition A.2), V is a finite +dimensional vector space over R, and the spaces Hs, Hs +(κ) are defined in Appendix B.1. We will +often suppress the domain and codomain in this notation when they are clear from context. It is a +simple matter to check that these are indeed Banach scales. +We begin by looking at products. Recall the spaces of (strongly)-tame maps introduced in +Definition 2.4. +Lemma 3.3 (Smooth tameness of products, 1). Let V1, V2 and W be real finite dimensional vector +spaces, B : V1 × V2 → W be a bilinear map, and U ⊆ Rd be a Stein-extension domain (see +Definition A.2). The following inclusions hold when viewing B as a product for functions taking +values in V1 and V2. + +54 +NOAH STEVENSON AND IAN TICE +(1) B ∈ sT ∞ +0,1+⌊d/2⌋(H(U; V1) × H(U; V2); H(U; W)). +(2) B ∈ sT ∞ +0,0(H(U; V1) × W(U; V2); H(U; W)). +Proof. By using the universality of tensor products to factor B = L ◦ ⊗ for a linear map L : +V1 ⊗ V2 → W and then working component-wise, we see that this is a direct application of the +high-low product estimates of Corollary D.7. +□ +We now handle more complicated products also involving sums. +Lemma 3.4 (Smooth tameness of products, 2). Let m ∈ N+, V , W, and V1, . . . , Vm be real finite +dimensional vector spaces, B : V1 × · · · × Vm × V → W be (m + 1)-linear, and U ⊆ Rd be a Stein +extension domain (see Definition A.2). The (m + 1)-linear map PB defined via PB((gj, ψj)m +j=1, ϕ) = +B(g1 + ψ1, . . . , gm + ψm, ϕ) satisfies the inclusion +PB ∈ sT ∞ +0,1+⌊d/2⌋ +� m +� +j=1 +(W(U; Vj) × H(U; Vj)) × H(U; V ); H(U; W) +� +. +(3.2.2) +Proof. We proceed via induction on m ∈ N+. The base case, m = 1, is a simple application of +Lemma 3.3. Now suppose that m ∈ N+ and the result holds for m. We wish to prove it for m + 1, +so let B : V1 × · · · Vm × Vm+1 × V → W be an (m + 2)-linear map on some finite dimensional real +vector spaces. We claim that there exists a bilinear map B1 : V1 × (V2 ⊗ · · · ⊗ Vm+1 ⊗ V ) → W and +an (m + 1)-linear map B2 : V2 × · · · × Vm+1 × V → V2 ⊗ · · · ⊗ Vm+1 ⊗ V such that +B(v1, . . . , vm+1, v) = B1(v1, B2(v2, . . . , vm+1, v)). +(3.2.3) +Indeed, by the universality property of tensor products there exists a linear map L : V1 ⊗ · · · ⊗ +Vm+1 ⊗ V → W defined via +L(v1 ⊗ · · · ⊗ vm+1 ⊗ v) = B(v1, . . . , vm+1, v), +(3.2.4) +so we set B1(v1, τ) = L(v1 ⊗ τ) and B2(v2, . . . , vm+1, v) = v2 ⊗ · · · ⊗ vm+1 ⊗ v. We then apply the +induction hypothesis to the operator PB2 to see that +PB2 ∈ sT ∞ +0,1+⌊d/2⌋ +� m+1 +� +j=2 +(W(U; Vj) × H(U; Vj)) × H(U; V ); H(U; V2 ⊗ · · · ⊗ Vm+1 ⊗ V ) +� +(3.2.5) +Similarly, we apply the base case to the operator PB1 and acquire the inclusion +PB1 ∈ sT ∞ +0,1+⌊d/2⌋(W(U; V1) × H(U; V1) × H(U; V2 ⊗ · · · ⊗ Vm+1 ⊗ V ); H(U; W)). +(3.2.6) +Equation (3.2.3) implies that PB((gj, ψj)m+1 +j=1 , ϕ) = PB1((g1, ϕ1), PB2((gj, ψj)m+1 +j=2 , ϕ)). Hence, the +inclusion PB ∈ sT ∞ +0,1+⌊d/2⌋ is a consequence of the composition of smooth tame maps, Lemma 2.8. +□ +We also consider products with members of the anisotropic Sobolev spaces, which are defined in +Appendix B.1. +Lemma 3.5 (Smooth tameness of products, 3). The following hold. +(1) Let rd ∈ N+ be as defined in the second item of Proposition B.2. We have that the map +�rd +j=1 H0 +(1)(Rd) ∋ (η1, . . . , ηrd) �→ �rd +j=1 ηj ∈ H0(Rd) belongs to sT ∞ +0,0(�rd +j=1 H(1); H). +(2) For any χ ∈ W ∞,∞(0, b) and ν ∈ N+ we have that the map �ν +j=1 H0 +(1)(Rd) ∋ (η1, . . . , ην) �→ +χ �ν +j=1 ηj ∈ W 0,∞((0, b); H0 +(ν)(Rd)) belongs to sT ∞ +0,0(�ν +j=1 H(1); WH(ν)). +Proof. The fact that the maps in the statement are well-defined and smooth follows from mul- +tilinearity and Proposition B.2. Note that for s ∈ N, the product map of the first item actu- +ally smoothly maps �rd +j=1 H0 +(1)(Rd) → Hs(Rd) and the map of the second item smoothly maps +�ν +j=1 H0 +(1)(Rd) → W s,∞((0, b); Hs +(ν)(Rd)). Then the strong tame estimates on all of the derivatives +follow from Remark 2.6 and these improved mapping properties. +□ + +COMPRESSIBLE TRAVELING WAVES +55 +The next key nonlinear structure we handle is superposition as a Sobolev multiplier. +Lemma 3.6 (Smooth tameness of superposition multipliers). Let V1, V2, W, and Z be real finite +dimensional vector spaces, and B : V1 × V2 → W be a bilinear map. Let U ⊆ Rn and O ⊆ Z +be Stein extension domains (see Definition A.2), and f ∈ C∞ +b (O; V1). +Finally, suppose that +Y ⊆ W 2+⌊n/2⌋,∞(U; Z) × H2+⌊n/2⌋(U; Z) is an open set with the property that if (g, ψ) ∈ Y , then +(g + ψ)(U) ⊆ O. Then the nonlinear operator P defined via P(g, ψ, ϕ) = B(f(g + ψ), ϕ) belongs to +sT ∞ +0,2+⌊n/2⌋(Y × H2+⌊n/2⌋(U; V2), W(U; Z) × H(U; Z) × H(U; V2); H(U; W)). +(3.2.7) +Proof. By working component-wise and using the universality of tensor products, we see that as a +consequence of Theorem D.3 and the smoothness of the Sobolev multiplier times Sobolev to Sobolev +pairing, for any s ⩾ 2 + ⌊n/2⌋ the map +P : (Y × H2+⌊n/2⌋(U, V2)) ∩ (W s,∞(U; Z) × Hs(U; Z) × Hs(U; V2)) → Hs(U; W) +(3.2.8) +is well-defined and smooth. Thus, we need only check that the derivatives of P obey the required +tame estimates. Consider first the case of zero derivatives, which we refer to as the base case. +Employing the third item of Corollary D.9, working component-wise, and using the universality of +tensor products, we readily deduce that the map in (3.2.8) belongs to the space sT 0 +0,2+⌊n/2⌋. +Now for k ∈ N+ we consider the case of k-derivatives of P. A short computation reveals the +derivative formula +DkP(g, ψ, ϕ)(gj, ψj, ϕj)k +j=1 = B(Dkf(g + ψ)(gj + ψj)k +j=1, ϕ) + +k +� +i=1 +B(Dk−1f(g + ψ)(gj + ψj)j̸=i, ϕi). +(3.2.9) +Again by the universality of tensor products, there are linear maps +Lk : Lk(Z; V1) ⊗ +� +k +� +j=1 +Z +� +⊗ V2 → W and Lk−1 : Lk−1(Z; V1) ⊗ +� k−1 +� +j=1 +Z +� +⊗ V2 → W +(3.2.10) +(depending only on k and B) such that +B(Dkf(g + ψ)(gj + ψj)k +j=1, ϕ) = Lk� +Dkf(g + ψ) ⊗ +k +� +j=1 +(gj + ψj) ⊗ ϕ +� +(3.2.11) +and +B(Dk−1f(g + ψ)(gj + ψj)j̸=i, ϕi) = Lk−1 +� +Dk−1f(g + ψ) ⊗ +� +j̸=i +(gj + ψj) ⊗ ϕi +� +. +(3.2.12) +We then find that the operators in (3.2.11) and (3.2.12) belong to sT 0 +0,2+⌊n/2⌋ by applying the base +case, the tameness of composition from Lemma 2.8, and the second version of tameness of products, +Lemma 3.4. This shows that, indeed, DkP belongs to sT 0 +0,2+⌊n/2⌋ for every k ∈ N. +□ +The next nonlinear structure we examine is superposition. +Lemma 3.7 (Smooth tameness of superposition). Let N ∋ k ⩾ 2 + ⌊n/2⌋. The following hold. +(1) Let V , W be real finite dimensional vector spaces, 0 ∈ O ⊆ V be a Stein extension domain +(see Definition A.2) that is star shaped with respect to the origin, U ⊆ Rn be a Stein +extension domain, and f ∈ C∞(O; W) be such that f(0) = 0 and Df ∈ C∞ +b (O; L(V ; W)). +Let Y ⊆ H2+⌊n/2⌋(U; V ) be an open set with the property that if ϕ ∈ Y , then ϕ(U) ⊆ O. +Then the operator ϕ �→ f(ϕ) belongs to sT ∞ +0,2+⌊n/2⌋(Y, H(U; V ); H(U; W)). + +56 +NOAH STEVENSON AND IAN TICE +(2) Suppose that Φ ∈ C∞(Rn; Rn) is a bi-Lipschitz homeomorphism and a C1 diffeomorphism +such that DΦ ∈ C∞ +b (Rn; Rn×n). There exists an open set 0 ∈ UΦ ⊆ W 2+⌊n/2⌋,∞(Rn; Rn) × +H2+⌊n/2⌋(Rn; Rn), depending only on Φ and the dimension, such that for any m ∈ N, the +operator P defined via P(f, g, h) = f(Φ + g + h) satisfies +P ∈ sT m +0,2+⌊n/2⌋(Hm+2+⌊n/2⌋(Rn; Rd) × UΦ, Im; H(Rn; Rd)), +(3.2.13) +where Im = {Hm+s(Rn; Rd) × W s,∞(Rn, Rn) × Hs(Rn; Rn)}s∈N. +Proof. We begin by proving the first item. Note that by using the fundamental theorem of calculus, +the map from the first item has the equivalent formula +ϕ �→ f(ϕ) = +� 1 +0 +Df(tϕ)(ϕ) dt. +(3.2.14) +For each t ∈ [0, 1], the map ϕ �→ Df(tϕ)(ϕ) is seen to be sT ∞ +0,2+⌊n/2⌋ as a consequence of our previous +result on the smooth tameness of superposition multipliers, Lemma 3.6, and the composition of +smooth tame maps, Lemma 2.8. In fact, it is a simple matter to verify that we actually have +uniformity of the defining inequalities with respect to t as well as the satisfaction of the remaining +hypotheses of Lemma 2.10. Hence, the first item follows by applying the lemma. +We next consider the second item. +Theorem D.2 establishes the existence of an open set +UΦ ⊆ W 2+⌊n/2⌋,∞(Rn; Rn) × H2+⌊n/2⌋(Rn; Rn) such that for every m ∈ N and s ⩾ 2 + ⌊n/2⌋ the +map +P : Hm+s(Rn; Rd) × (UΦ ∩ (W s,∞(Rn; Rn) × Hs(Rn; Rn))) → Hs(Rn; Rd) +(3.2.15) +is well-defined and Cm. Thus we need only verify that the derivatives of P, which are enumerated +in (D.1.7), are sT 0 +0,2+⌊n/2⌋. The case of the 0th derivative (m = 0) follows immediately from the second +item of Corollary D.10 and the bound ∥det D(g+h)−1∥L∞ ⩽ 2n∥DΦ−1∥n +L∞ for all (g, h) ∈ UΦ, which +holds as a consequence of Lemma D.1 and Theorem D.2. When m > 0, the formula (D.1.7) shows +that the mth derivative is built from simple products and the 0th derivative structure. Consequently, +the result follows in this case by supplementing this observation with Lemmas 2.8 and 3.4. +□ +The next two results, which happen to be the most subtle part of the nonlinear analysis, deal +with superposition-like nonlinearities whose argument contains a member of the anisotropic Sobolev +space. Recall that Appendix B.1 gives the notation and an enumeration of basic properties for these +specialized spaces. In the following statement rd ∈ N+, for d = n − 1, refers to the number defined +in the second item of Proposition B.2. +Lemma 3.8 (Taylor expansion trick). Let I ⊆ R be an open interval containing [−gb, 0] and +φ ∈ C∞(I). There exists a ρI ∈ R+, +N (1) +φ +∈ sT ∞ +0,2+⌊n/2⌋(BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI), H(Ω) × H(Σ); H(Ω)), +(3.2.16) +and +N (2) +φ +∈ sT ∞ +0,0(H(1)(Σ); WH(rn−1−1)) +(3.2.17) +such that N (1) +φ (0, 0) = 0, N (2) +φ (0) = 0, and +Nφ(q, η) − Nφ(0, 0) = N (1) +φ (q, η) + N (2) +φ (Π1 +Lη), +(3.2.18) +where the projectors Π are defined in (B.1.4) and we write +Nφ(q, η) = φ(−gidRn · en + q + g(I − E)η) and Nφ(0, 0) = φ(−gidRn · en). +(3.2.19) + +COMPRESSIBLE TRAVELING WAVES +57 +Proof. We begin by choosing ρI. Set RI = dist(R\I, [−gb, 0])/2 ∈ (0, ∞]. Thanks to the supercritical +Sobolev embeddings, properties of anisotropic Sobolev from Proposition B.1, and the boundedness +of the extension operator E from Lemma A.1, we see that the map +H2+⌊n/2⌋(Ω) × H2+⌊n/2⌋(Σ) ∋ (q, η) �→ q + g(I − E)η ∈ L∞(Ω) +(3.2.20) +is well-defined and continuous. Hence, the preimage of BL∞(0, RI) under this map is an open set +UI containing the origin. We take ρI = dist(0, ∂UI)/ +√ +2 so that +BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI) ⊆ UI. +(3.2.21) +Now let (q, η) ∈ BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI). Note that Nφ(q, η), as defined in (3.2.19), is +well-defined as a function from Ω to R. We write +Nφ(q, η) = φ(g0 + g + ϕ), +(3.2.22) +where g0 = −gidRn · en, g = g(I − idRn · en/b)Π1 +Lη, and ϕ = q + g(I − E)Π1 +Hη. Hence, +Nφ(q, η) − Nφ(0, 0) = (φ(g0 + g + ϕ) − φ(g0 + g)) + (φ(g0 + g) − φ(g0)) = I + II. +(3.2.23) +For I we use the fundamental theorem of calculus to express +I = ϕ +� 1 +0 +φ′(g0 + g + τϕ) dτ, +(3.2.24) +whereas for II we use Taylor’s formula with integral remainder to write +II = +rd−1 +� +j=1 +φ(j)(g0) +j! +gj + grd +� 1 +0 +(1 − τ)rd−1 +(rd − 1)! φ(rd)(g0 + τg) dτ, +(3.2.25) +where rd = rn−1 ⩾ 1 is given by (B.1.5) and sums over empty intervals are understood as zero. This +leads us to define +N (1) +φ (q, η) = ϕ +� 1 +0 +φ′(g0 + g + τϕ) dτ + grd +� 1 +0 +(1 − τ)rd−1 +(rd − 1)! φ(rd)(g0 + τg)dτ +(3.2.26) +and +N (2) +φ (Π1 +Lη) = +rd−1 +� +j=1 +φ(j)(g0) +j! +gj. +(3.2.27) +The computation leading up to these definitions shows that decomposition (3.2.18) holds and that +N (1) +φ +and N (2) +φ +vanish at the origin. +Now we analyze the tame smoothness of N (1) +φ +and N (2) +φ . The former is sT ∞ +0,2+⌊n/2⌋ thanks to +the results on addition of smooth tame maps, Lemma 2.10, smooth tameness of superposition +multipliers, Lemma 3.6, composition of smooth tame maps, Lemma 2.8, and the third version of +smooth tameness of products, Lemma 3.5. On the other hand, the map N (2) +φ +is sT ∞ +0,0 as a consequence +of Lemma 3.5. +□ +We now use the previous result to understand the mapping properties of the vector field argument +of the divergence in the continuity equation of (1.4.9). +Lemma 3.9 (A vector field decomposition). Let I ⊆ R be an open interval containing [−gb, 0], +φ ∈ C∞(I), and ρI be as in Lemma 3.8. +Let the open set OI be given by BH2+⌊n/2⌋(0, ρI) × +H2+⌊n/2⌋(Ω; Rn) × BH3+⌊n/2⌋(0, ρI), and consider the Banach scale +J = {Hs(Ω) × Hs(Ω; Rn) × H1+s(Σ)}s∈N. +(3.2.28) +There exist +M(1) +φ +∈ sT ∞ +0,2+⌊n/2⌋(OI, J; H(Ω; Rn)) +(3.2.29) + +58 +NOAH STEVENSON AND IAN TICE +and +e1 · M(2) +φ +∈ sT ∞ +0,0(H(1)(Σ); WH(rn−1)) +(3.2.30) +such that M(1) +φ (0, 0, 0) = 0, e1 · M(2) +φ (0) = 0, +Tr∂Ω(M(1) +φ (q, u, η) · en) = Tr∂Ω(Nφ(q, η)) +� +(TrΣ(u · en) + ∂1η)1Σ + TrΣ0(u · en)1Σ0 +� +, +(3.2.31) +and +Mφ(q, u, η) − Mφ(0, 0, 0) = M(1) +φ (q, u, η) + e1 · M(2) +φ (Π1 +Lη)e1, +(3.2.32) +where we write +Mφ(q, u, η) = Nφ(q, η)(u − Mηe1) and Mφ(0, 0, 0) = −Nφ(0, 0)e1 +(3.2.33) +for Nφ(q, η), Mη, and Π given by (3.2.19), (1.4.10), and (B.1.4), respectively. +Proof. The precise form of the decomposition (3.2.32), which uses Lemma 3.8 as well as the identities +Mηe1 = (1 + ∂nEη)e1 − ∂1Eηen and ∂nEΠ1 +Lη = Π1 +Lη/b, is given by +M(1) +φ (q, u, η) = Nφ(q, η)(u − ∂nEΠ1 +Hηe1 + ∂1Eηen) − N (1) +φ (q, η)(1 + Π1 +Lη/b)e1 +(3.2.34) +and +e1 · M(2) +φ (Π1 +Lη) = −(Nφ(0, 0)Π1 +Lη/b + N (2) +φ (Π1 +Lη)(1 + Π1 +Lη/b)). +(3.2.35) +That M(1) +φ +is sT ∞ +0,2+⌊n/2⌋ is a consequence of the composition of tame maps, Lemma 2.8, the second +product result, Lemma 3.4, the mapping properties of Nφ, Lemma 3.8, and the properties of +anisotropic Sobolev spaces enumerated in (B.1.25) and Proposition B.1. That e1 · M(2) +φ +is sT ∞ +0,0 is a +consequence of Lemmas 3.5 and 3.8. +□ +3.3. Smooth tameness of the nonlinear operator. We begin by defining a nonlinear operator +associated with the PDE (1.4.9). We will employ the Banach scales defined in Section 3.1. For some +r ⩾ 2 + ⌊n/2⌋, 0 < ρ ⩽ ρWD, with ρWD determined by Theorem 3.17, and s ⩾ r, we set +Ψ : (BXr(0, ρ) ∩ X1+s) × Ws × R+ → Ys × Ws × R +(3.3.1) +via +Ψ(q, u, η, T , G, F, γ) = (Ψ(q, u, η, γ) + Φ(q, u, η, T , G, F), T , G, F, γ), +(3.3.2) +where +Ψ : (BXr(0, ρ) ∩ X1+s) × R+ → Ys and Φ : (BXr(0, ρ) ∩ Xs) × Ws → Ys +(3.3.3) +are the principal and auxiliary parts of Ψ, which are defined via +Ψ(q, u, η, γ) = (Ψ1(q, u, η), Ψ2(q, u, η, γ), Ψ3(q, u, η, γ)) +(3.3.4) +and +Φ(q, u, η, T , G, F) = (0, Φ2(q, η, G, F), Φ3(η, T )), +(3.3.5) +where +Ψ1(q, u, η) = ∇ · (σq,η(u − Mηe1)), +(3.3.6) +Ψ2(q, u, η, γ) = γ2σq,ηM−t +η ((u − γMηe1) · ∇(M−1 +η u)) + σq,η∇(q + gη) − γM−t +η ∇ · (Sσq,η +Aη (M−1 +η u)Mt +η), +(3.3.7) +Ψ3(q, u, η, γ) = −((P − Pext) ◦ σq,η − γSσq,η +Aη (M−1 +η u))Mt +ηen − ςH (η)Mt +ηen, +(3.3.8) +Φ2(q, η, G, F) = −JηM−t +η (σq,ηG ◦ Fη + F ◦ Fη), +(3.3.9) +and +Φ3(η, T ) = −T ◦ FηMt +ηen. +(3.3.10) +Here we recall that Fη is defined by (1.4.4), Aη and Jη by (1.4.5), Sτ +A by (1.4.8), Mη by (1.4.10), +and σq,η by (1.4.11). We also recall the assumptions on ς, µ, and λ stated in (1.1.7). + +COMPRESSIBLE TRAVELING WAVES +59 +Armed with the results from the previous two subsections, we now endeavor to study the smooth +tameness of the map Ψ from (3.3.1). Our strategy is to handle the Ψ and Φ pieces separately +and component-wise. Our first result studies the continuity equation piece, Ψ1, which we recall +is defined in (3.3.6). This Ψ1 term is the source of the derivative loss and also requires the most +careful analysis among the smooth tameness verification results. +Proposition 3.10 (Smooth tameness of the continuity equation). There exists a ρcont ∈ (0, ∞), +depending only on the domain of the inverse enthalpy (see (1.1.11)) such that the following hold +(1) There exists constants c, C ∈ R+ such that for all (q, u, η) ∈ BX1+⌊n/2⌋(0, ρcont) we have the +estimate c ⩽ σq,η ⩽ C where σq,η is defined in (1.4.11). +(2) Ψ1 ∈ sT ∞ +1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρcont),XXX; { ˆH1+s(Ω)}s∈N), where we recall that the spaces ˆHs(Ω) +are defined in (C.1.1) (see also Remark 3.1). +Proof. We begin by proving the second item. First, we use Lemma 3.9 to set ρcont = ρ(Hmin,Hmax), +where Hmin and Hmax are defined by (1.1.11), to see that ∇ · MH−1(0, 0, 0) = 0, and to equate +Ψ1(q, u, η) = ∇ · (M(1) +H−1(q, u, η)) + ∂1(e1 · M(2) +H−1(Π1 +Lη)) = ΨI +1(q, u, η) + ΨII +1 (Π1 +Lη). +(3.3.11) +In the above H−1 refers to the inverse enthalpy function (see 1.1.10). As a direct consequence of +Lemma 3.9, we obtain the inclusion +ΨI +1 ∈ sT ∞ +1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρcont),XXX; {H1+s(Ω)}s∈N). +(3.3.12) +On the other hand, the boundary conditions built into the definition of the space X1+⌊n/2⌋ in (3.1.3), +together with condition (3.2.31), imply that +� b +0 +ΨI +1(q, u, η)(·, y) dy = (∇∥, 0) · +� b +0 +M(1) +H−1(q, u, η)(·, y) dy, +(3.3.13) +and hence we deduce that ΨI +1 also maps smoothly into the space ˆH0(Ω). This fact combined +with (3.3.12) proves that ΨI +1 is sT ∞ +1,1+⌊n/2⌋ with respect to the Banach scales stated in the hypotheses. +We next handle the ΨII +1 piece. By the norm in the anisotropic Sobolev spaces given in (B.1.25), +we readily see that the embedding W 0,∞((0, b); ∂1H0 +(rn−1)(Σ)) �→ H0(Ω) holds and hence we deduce +from conclusion (3.2.30) of Lemma 3.9 that ΨII +1 ∈ sT ∞ +0,0(H(1)(Σ); H(Ω)). +By inspection of the anisotropic norm from (B.1.25) again, we also deduce the embedding +W 0,∞((0, b); ∂1H0 +(rn−1)(Σ)) �→ W 0,∞((0, b); ˙H−1(Σ)), +(3.3.14) +and hence the map +H0 +(1)(Σ) ∋ ζ �→ +� b +0 +ΨII +1 (ζ)(·, y) dy ∈ ˙H−1(Σ) +(3.3.15) +is well-defined and smooth. These facts merge to show that ΨII +1 is sT ∞ +0,0 with respect to the Banach +scales H(1)(Σ) and { ˆHs(Ω)}s∈N. We synthesize our results for ΨI +1 and ΨII +1 to complete the proof of +the second item. +We now deduce the bounds stated in the first item. By unpacking the meaning of ρcont = ρI +from Lemma 3.8, for I = (Hmin, Hmax), we find that for (q, u, η) ∈ BX1+⌊n/2⌋(0, ρcont) we have the +inclusion +(−gidRn · en + q + g(I − E)η)(Ω) ⊆ (h⋆, h⋆) ⋐ (Hmin, Hmax), +(3.3.16) +for some universal h⋆, h⋆ ∈ (Hmin, Hmax). Hence, by applying the inverse enthalpy, we find that +H−1(h⋆) ⩽ σq,η ⩽ H−1(h⋆), which is the first item with c = H−1(h⋆), C = H−1(h⋆) ∈ R+. +□ +Next we examine the Ψ2 piece of the momentum equation, which we recall is defined in (3.3.7). +Proposition 3.11 (Smooth tameness of the momentum equation, 1). There exits a ρmome ∈ (0, ρcont] +such that the following hold. + +60 +NOAH STEVENSON AND IAN TICE +(1) For (q, u, η) ∈ BX1+⌊n/2⌋(0, ρmome), we have the bounds 1/2 ⩽ Jη ⩽ 3/2, where Jη is defined +in (1.4.5). +(2) Ψ2 ∈ sT ∞ +0,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρmome) × R+,XXX × R; H(Ω; Rn)). +Proof. We begin by selecting ρmome. The map +H3/2+⌊n/2⌋(Σ) ∋ η �→ Jη − 1 = ∂nEη ∈ L∞(Ω) +(3.3.17) +is bounded thanks to the supercritical Sobolev embedding and the continuity properties of E from +Proposition A.1. Denote the preimage of the set BL∞(0, 1/2) under this map by O ⊆ H3/2+⌊n/2⌋(Σ). +O is open and we set �ρmome = dist(0, ∂O) ∈ R+ and ρmome = min{�ρmome, ρcont}, where the latter is +defined in Proposition 3.10. +With this choice of ρmome we have that 1/2 ⩽ Jη ⩽ 3/2, and hence the inverse matrix M−1 +η +exists +whenever ∥η∥H3/2+⌊n/2⌋ < ρmome. Consequently, all of the expressions appearing in Ψ2 are pointwise +well-defined. We next decompose +Ψ2(q, u, η, γ) = γ2ΨI +2(q, u, η) + ΨII +2 (q, η) + γΨIII +2 (q, u, η), +(3.3.18) +where +� +� +� +� +� +ΨI +2(q, u, η) = σq,ηM−t +η [(u − Mηe1) · ∇(M−1 +η u)], +ΨII +2 (q, η) = σq,η∇(q + gη), +ΨIII +2 (q, u, η) = −M−t +η ∇ · (Sσq,η +Aη (M−1 +η u)Mt +η). +(3.3.19) +The γ dependence in (3.3.18) is simple, and it is sufficient to study the pieces of (3.3.19) individually. +For the first piece, we use Lemma 3.8 to write +σq,η = ϱ + N (1) +H−1(q, η) + N (2) +H−1(Π1 +Lη), +(3.3.20) +where H−1 is the inverse of the enthalpy (see (1.1.10)). The embeddings of the second item of +Proposition B.1 allow us to view the latter map as N (2) +H−1 ∈ sT ∞ +0,0(H(1)(Σ); W(Ω)). Thanks to the +smooth tameness of superposition multipliers proved in Proposition 3.6, the map (u, η) �→ ∇(M−1 +η u) +is sT ∞ +0,1+⌊n/2⌋ with respect to the Banach scales {H2+s(Ω; Rn) × H5/2+s(Σ)}s∈N and {Hs(Ω; Rn)}s∈N. +By applying the product of tame maps result from Lemma 3.4 and the composition of smooth +tame maps result from Lemma 2.8, we then see that the map (u, η) �→ (u − Mηe1) · ∇(M−1 +η u) +is also sT ∞ +0,1+⌊n/2⌋ with respect to the aforementioned Banach scales. By invoking Lemmas 3.6 +and 2.8 again, we get that the same conclusion is true for the map (u, η) �→ M−t +η [(u − Mηe1) · +∇(M−1 +η u)]. Finally, by using Lemmas 3.8, 3.4, and 2.8, we deduce that (q, u, η) �→ (ϱ + N (1) +H−1(q, η) + +N (2) +H−1(Π1 +Lη))M−t +η [(u − Mηe1) · ∇(M−1 +η u)], i.e. ΨI +2, is sT ∞ +0,1+⌊n/2⌋ with respect to the Banach scales +{Xs}s∈N and {Hs(Ω; Rn)}s∈N. +Now we examine the ΨII +2 piece. We again employ the density decomposition in (3.3.20). Then we +see that ΨII +2 is sT ∞ +0,1+⌊n/2⌋ for the Banach scales {H1+s(Ω) × H5/2+s(Σ)}s∈N and {Hs(Ω; Rn)}s∈N +as a consequence of Lemmas 3.8 and 3.4. +Finally, we examine the ΨIII +2 +piece, decomposing further: +ΨIII +2 (q, u, η) = ΨIII1 +2 +(q, u, η) + ΨIII2 +2 +(q, u, η) + ΨIII3 +2 +(q, u, η) + ΨIII4 +2 +(q, u, η), +(3.3.21) +where +� +� +� +� +� +� +� +� +� +ΨIII1 +2 +(q, u, η) = −M−t +η ∇ · +� +µ(σq,η)J−1 +η ∇(M−1 +η u)MηMt +η +� +, +ΨIII2 +2 +(q, u, η) = −M−t +η ∇ · (µ(σq,η)J−1 +η Mt +η∇(M−1 +η u)tMt +η), +ΨIII3 +2 +(q, u, η) = 2M−t +η ∇ · (µ(σq,η)J−1 +η (∇ · u)Mt +η), +ΨIII4 +2 +(q, u, η) = −M−t +η ∇ · (λ(σq,η)J−1 +η (∇ · u)Mt +η). +(3.3.22) + +COMPRESSIBLE TRAVELING WAVES +61 +We handle these four terms in more or less the same way. The µ and λ viscosity coefficients are +decomposed as +� +µ(σq,η) = µ(ϱ) + N (1) +µ◦H−1(q, η) + N (2) +µ◦H−1(Π1 +Lη), +λ(σq,η) = λ(ϱ) + N (1) +λ◦H−1(q, η) + N (2) +λ◦H−1(Π1 +Lη), +(3.3.23) +via Lemma 3.8, while the Mη, M−1 +η , Mt +η, M−t +η , and J−1 +η +terms are viewed as superposition mul- +tipliers and thus are handled via Lemma 3.6. Hence, the fact that each ΨIIIj +2 +, j ∈ {1, . . . , 4}, is +sT ∞ +0,1+⌊n/2⌋ with respect to the Banach scales {Xs}s∈N and {Hs(Ω; Rn)}s∈N is a consequence of the +aforementioned lemmas, the second result on products, Lemma 3.4, and the result on composition +of smooth tame maps, Lemma 2.8. +□ +Now we examine the Ψ3 piece of the dynamic boundary condition, which we recall is defined +in (3.3.8). +Proposition 3.12 (Smooth tameness of the dynamic boundary condition, 1). With ρdyna = ρmome ∈ +R+ as in Proposition 3.11, we have +Ψ3 ∈ sT ∞ +0,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρdyna) × R+,XXX × R; {H1/2+s(Σ; Rn)}s∈N). +(3.3.24) +Proof. As before, we begin the proof by decomposing +Ψ3(q, u, η, γ) = ΨI +3(q, η) + γΨII +3 (q, u, η) + ΨIII +3 (η) +(3.3.25) +where +� +� +� +� +� +ΨI +3(q, η) = −(P ◦ σq,η − Pext)Mt +ηen, +ΨII +3 (q, u, η) = Sσq,η +Aη (M−1 +η u)Mt +ηen, +ΨIII +3 (η) = −ςH (η)Mt +ηen. +(3.3.26) +For the first piece we recall that H−1 is the inverse of the enthalpy, as in (1.1.10), and use the fact +that σq,η = H−1(−gb + q) and Pext = P ◦ ϱ(b) = P ◦ H−1(−gb) on Σ to rewrite +ΨI +3(q, η) = (P ◦ H−1(· − gb) − P ◦ H−1(−gb))(q)Mt +ηen. +(3.3.27) +From this, the operator ΨI +3 is readily seen to belong to sT ∞ +0,1+⌊n/2⌋ with respect to the Banach scales +{H1+s(Ω) × H5/2+s(Σ)}s∈N and {H1/2+s(Σ; Rn)}s∈N, as a consequence of the first item of Lemma +3.7 regarding the tame smoothness of superposition, the result on tame smoothness of superposition +multipliers, Lemma 3.6, and the result on the composition of smooth tame maps, Lemma 2.8. +The fact that ΨII +3 is sT ∞ +0,1+⌊n/2⌋ for the scales {Xs}s∈N and {H1/2+s(Σ; Rn)}s∈N follows from an +argument similar to the analysis of the ΨIII +2 -term from the proof of Lemma 3.11. +Finally, we handle ΨIII +3 . The mean curvature operator has the expression H (η) = ∇∥ ·( � +H (∇∥η)), +where the map +� +H ∈ C∞(Rn−1; Rn−1) is given by +� +H (v) = ⟨v⟩−1v. From this we deduce that ΨIII +3 +is sT ∞ +0,1+⌊n/2⌋ by combining the conclusions of Lemmas 3.7 (the first item), 3.6, and 2.8. +□ +We now synthesize our previous results to deduce the smooth tameness of the principal part +nonlinear operator from (3.3.4). +Theorem 3.13 (Smooth tameness of the principal part nonlinear operator). There exists a ρprin ∈ +R+ such that +Ψ ∈ sT ∞ +1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρprin) × R+,XXX × R;YYY). +(3.3.28) +Proof. We set ρprin = min{ρcont, ρmome, ρdyna} and apply Propositions 3.10, 3.11, and 3.12. +□ +The remainder of this subsection is devoted to the study of the auxiliary piece Φ of the nonlinear +operator Ψ. The next result handles the Φ2 piece, which we recall is defined in (3.3.9). + +62 +NOAH STEVENSON AND IAN TICE +Proposition 3.14 (Smooth tameness of the momentum equation, 2). There exists a ρbulk ∈ R+ +such that for every m ∈ N, +Φ2 ∈ sT m +0,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρbulk) × Wm+2+⌊n/2⌋, {Xs × Wm+s}s∈N; H(Ω; Rn)). +(3.3.29) +Proof. Thanks to the smooth tameness of superposition multipliers, Lemma 3.6, the decomposition +of σq,η = N (1) +H−1(q, η) + N (2) +H−1(Π1 +Lη) + ϱ from Lemma 3.8, and the result on composition of smooth +tame maps, Lemma 2.8, we see that it is sufficient to prove that the nonlinear operator +Λ : Hm+2+⌊n/2⌋(Rn; Rn) × BH9/2+⌊n/2⌋(Σ)(0, ρ) → H0(Ω; Rn) given by Λ(I, η) = I ◦ Fη +(3.3.30) +is, for some ρ > 0, well-defined and sT m +0,2+⌊n/2⌋ for the scales {Hm+s(Rn; Rn) × H5/2+s(Σ)}s∈N and +{Hs(Ω; Rn)}. For this we let EΩ and RΩ denote the Stein extension (see Definition A.2) and the +restriction operators for Ω, respectively, and note that we have the equivalent formula +Λ(I, η) = RΩI(idRn + EΩE(Π1 +Lη + Π1 +Hη)). +(3.3.31) +Hence, according to Lemma D.2 and properties of the maps E and EΩ (see Lemma A.1 and +Example 2.16), there exists a ρcomp ∈ R+ (depending only on the dimension and Ω) such that +whenever ρ ⩽ ρcomp the map (3.3.30) is sT m +0,2+⌊n/2⌋ with respect to the aforementioned Banach +scales. +□ +The penultimate result of this subsection considers the Φ3 piece of Ψ, which is defined in (3.3.10). +Proposition 3.15 (Smooth tameness of the dynamic boundary condition, 2). For ρsurf = ρbulk ∈ +R+, where the latter is from Proposition 3.14, we have that for every m ∈ N, +Φ3 ∈ sT m +0,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρsurf) × Wm+2+⌊n/2⌋, {Xs × Wm+s}s∈N; {H1/2+s(Σ; Rn)}s∈N). +(3.3.32) +Proof. We argue is in the proof of Proposition 3.14 and reduce to studying the map +�Λ : Hm+2+⌊n/2⌋(Rn; Rn) × BH9/2+⌊n/2⌋(Σ)(0, ρ) → H0(Σ; Rn) +(3.3.33) +given by �Λ(T , η) = TrΣ(T ◦ Fη). We then use the equivalent formula +�Λ(T , η) = TrΣRΩT (idRn + EΩE(Π1 +Lη + Π1 +Hη)), +(3.3.34) +and conclude as before. +□ +Remark 3.16. The parameter m ∈ N appearing in Propositions 3.14 and 3.15 (and subsequently +in Theorem 3.17) gives the data terms (T , G, F) m-extra derivatives to ensure that the composition +type nonlinearities of Φ2 and Φ3 are Cm into the correct codomain space. One sees that this extra +regularity is necessary upon inspection of the derivative formulae (D.1.6) and (D.1.7). +At last, we are ready to deduce the smooth tameness of the nonlinear operator Ψ, which we recall +is defined in (3.3.1). In the following statement WD is an acronym for ‘well-defined’. +Theorem 3.17 (Smooth tameness of the nonlinear operator). There exists a ρWD ∈ R+ such that +the following hold. +(1) For N ∋ s ⩾ 2 + ⌊n/2⌋ and (q, u, η) ∈ Xs ∩ BX2+⌊n/2⌋(0, ρWD) we have that the flattening +map Fη defined in (1.4.4) is a smooth diffeomorphism from Ω to Ω[η] that extends to a +Cs+2−⌊n/2⌋ diffeomorphism from Ω to Ω[η]. +(2) For every m ∈ N, Ψ ∈ sT m +1,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρWD) × Wm+1+⌊n/2⌋ × R+, Om; Pm), where +we have denoted Om = {Xs × Wm−1+s × R}s∈N and Pm = {Ys × Wm+s−1 × R}s∈N. + +COMPRESSIBLE TRAVELING WAVES +63 +Proof. We set ρWD = min{ρprin, ρbulk, ρsurf} and apply Theorem 3.13 and Propositions 3.14 and 3.15. +This immediately gives the second item. For the first item, we note that Proposition 3.11 guarantees +that Jη > 0 and Jη, 1/Jη ∈ L∞(Ω), which means that Fη is a continuous bijection from Ω to Ω[η]. +On the other hand, the identity ∂n(Fη · en) = Jη = det(∇Fη) and the inverse function theorem +guarantee that Fη : Ω → Ω[η] is a smooth diffeomorphism. The regularity of Fη : Ω → Ω[η] and its +inverse now follow from Sobolev embeddings and Proposition B.1. +□ +Remark 3.18. In the case m = 2, the second item of Theorem 3.17 is stating that Ψ ∈ sT 2 +1,2+⌊n/2⌋ +for the Banach scales EEE and FFF, which are introduced in (3.1.10). +3.4. Derivative splitting. We now turn our attention to the study of the derivative of the map +Ψ from (3.3.1). When written in full, it is rather complicated, so our focus now is to identify a +principal part and handle the remainder terms. Let +(q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρWD), +(T0, G0, F0) ∈ W3+⌊n/2⌋, +(3.4.1) +(q, u, η) ∈ X2+⌊n/2⌋, +(T , G, F) ∈ W2+⌊n/2⌋, +γ0 ∈ R+, γ ∈ R, +(3.4.2) +and write the 6−tuple +θ0 = (q0, u0, η0, T0, G0, F0). +(3.4.3) +The derivative of the map Ψ has the formula +DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = +� +DΨ(q0, u0, η0, γ0)(q, u, η, γ) + DΦ(θ0)(q, u, η, T , G, F), T , G, F, γ +� +. +(3.4.4) +We will decompose the above into a principal part, A, and remainder terms, P, Q, and R (these +symbols will come equipped with various adornments, but for brevity we will often refer to them +without these in the main text). The derivative DΨ is split as follows: +DΨ(w0, γ0)(q, u, η, γ) = +w0,γ0 +A (q, u, η) + +w0,γ0 +P (q, u, η, γ), +(3.4.5) +where for brevity we define the triple +w0 = (q0, u0, η0). +(3.4.6) +The A piece is meant to be as close as possible to DΨ(0, 0, 0, γ0), but to retain the entirety of the +structure responsible for derivative loss in the continuity equation. To that end, we set +w0,γ0 +A (q, u, η) = +� w0 +A1(q, u, η), +γ0 +A2(q, u, η), +γ0 +A3(q, u, η) +� +, +(3.4.7) +where +w0 +A1(q, u, η) = ∇ · (ϱu) + ∇ · (vw0(q + gη)), +(3.4.8) +γ0 +A2(q, u, η) = DΨ2(0, 0, 0, γ0)(q, u, η, 0) = −γ2 +0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu, +(3.4.9) +and +γ0 +A3(q, u, η) = DΨ3(0, 0, 0, γ0)(q, u, η, 0) = −(ϱq − γ0Sϱu)en − ς∆∥ηen. +(3.4.10) +Here vw0 : Ω → Rn is the vector field +vw0 = M(H−1)′(q0, u0, η0) = (H−1)′(−gidRn · en + q0 + g(I − E)η0)(u0 − Mη0e1), +(3.4.11) +where the M notation is from Lemma 3.9 and (H−1)′ refers to the derivative of the inverse enthalpy +(see equation (1.1.10)). Note that since ϱ(y) = H−1(−gy) for y ∈ [0, b] we have that +v0 = −(H−1)′(−gidRn · en)e1 = g−1ϱ′e1. +(3.4.12) +The P piece is, of course, the remainder of the derivative of Ψ and has the formula +w0,γ0 +P (q, u, η, γ) = +� w0 +P 1(q, u, η), +w0,γ0 +P 2 (q, u, η, γ), +w0,γ0 +P 3 (q, u, η, γ) +� +(3.4.13) + +64 +NOAH STEVENSON AND IAN TICE +where +w0 +P 1(q, u, η) = ∇ · ((σq0,η0 − ϱ)(u − ˙M[η]e1)) − g∇ · ((vw0 − ϱ′e1/g)Eη) − ∇ · (ϱ ˙M[η]e1) − ∇ · (ϱ′Eηe1), +(3.4.14) +˙M[η] : Ω → Rn×n is the matrix field given by +˙M[η] = +�∂nEηI(n−1)×(n−1) +0(n−1)×1 +−E(∇∥η) +0 +� +, +(3.4.15) +and +w0,γ0 +P j (q, u, η, γ) = (DΨj(w0, γ0) − DΨj(0, 0, 0, γ0))(q, u, η, 0) + DΨj(w0, γ0)(0, 0, 0, γ) +(3.4.16) +for j ∈ {2, 3}. +Before we define a similar decomposition of the DΦ piece of DΨ, we will prove some basic +properties about the A + P decomposition. First, we have the following lemma which, aside from +the final item, is a reprise of Lemma 3.9. +Lemma 3.19 (Properties of the derivative loss vector field). Let 0 < ρ ⩽ ρWD, where the latter +is defined in Theorem 3.17, and w0 = (q0, u0, η0) ∈ BX1+⌊n/2⌋(0, ρ) ∩ X∞. Define the vector field +vw0 : Ω → Rn as in (3.4.11). There exists a decomposition +vw0 = g−1ϱ′e1 + v(1) +q0,u0,η0 + v(2) +η0 +(3.4.17) +such that the following hold. +(1) The vector field v(1) +q0,u0,η0 has vanishing normal trace, Tr∂Ω(v(1) +q0,u0,η0 · en) = 0, satisfies the +inclusion v(1) +q0,u0,η0 ∈ H∞(Ω; Rn), and obeys the estimates +∥v(1) +q0,u0,η0∥H1+s ≲ +� +ρ +if s = 1 + ⌊n/2⌋, +∥q0, u0, η0∥Xs +if s > 1 + ⌊n/2⌋. +(3.4.18) +(2) The vector field v(2) +η0 is parallel to e1, satisfies the inclusion +v(2) +η0 · e1 ∈ W ∞,∞((0, b); H0 +(rn−1)(Σ)) �→ W ∞,∞(Ω), +(3.4.19) +and obeys the estimates +∥v(2) +η0 · e1∥W s,∞((0,b);H0 +(rn−1)(Σ)) ≲ ρ for s ⩾ 1 + ⌊n/2⌋, +(3.4.20) +where rn−1 ∈ N is from the second item of Proposition B.2 with d = n − 1. +(3) We have the inclusion ∇ · vw0 ∈ H∞(Ω; Rn) along with the estimates +∥∇ · vw0∥Hs ≲ +� +ρ +if s = 1 + ⌊n/2⌋, +∥q0, u0, η0∥Xs +if s > 1 + ⌊n/2⌋. +(3.4.21) +(4) If s ∈ N, q ∈ H1+s(Ω), and η ∈ H1+s(Σ), then we have that ∇ · (vw0(q + gη)) ∈ ˆH1+s(Ω), +where the latter space is defined in (C.1.1) (see also Remark 3.1), as well as the estimates +∥∇ · (vw0(q + gη))∥ ˆHs ≲ ∥q, η∥H1+s×H1+s + +� +0 +if s ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ +if ⌊n/2⌋ < s. +(3.4.22) +In the above, the implicit constants depend on the dimension, the physical parameters, s, and ρWD. + +COMPRESSIBLE TRAVELING WAVES +65 +Proof. We apply Lemma 3.9 to obtain identity (3.4.17) with ϱ′e1/g = M(H−1)′(0, 0, 0), v(1) +q0,u0,η0 = +M(1) +(H−1)′(q0, u0, η0), and v(2) +η0 = e1 · M(2) +(H−1)′(Π1 +Lη0)e1. The qualitative smoothness assertions in the +first, second, and third items now follow immediately the lemma. We will prove the quantitative +bounds via the fundamental theorem of calculus. Since M(1) +(H−1)′(0, 0, 0) = 0, we have that +v(1) +q0,u0,η0 = +� 1 +0 +DM(1) +(H−1)′(tq0, tu0, tη0)(q0, u0, η0) dt, +(3.4.23) +and hence, by using the strong smooth tameness assertions in Lemma 3.9 and the derivative estimates +on smooth tame maps from Lemma 2.7, we find that for s ⩾ 1 + ⌊n/2⌋, +∥v(1) +q0,u0,η0∥H1+s ⩽ +� 1 +0 +∥DM(1) +(H−1)′(tq0, tu0, tη0)[q0, u0, η0]∥H1+s dt ≲ ∥q0, u0, η0∥Xs. +(3.4.24) +The same technique proves the quantitative bounds asserted in the second item. +Next, we justify the divergence estimates of the third item. By using (3.4.17), we compute that +∇ · vw0 = ∇ · v(1) +q0,u0,η0 + ∂1(v(2) +η0 · e1). +(3.4.25) +From this identity, the quantitative estimates of the first and second items, and the properties of +band limited anisotropic Sobolev spaces from Proposition B.1, we deduce that for s ⩾ 1 + ⌊n/2⌋, +∥∇ · v(1) +q0,u0,η0∥Hs ≲ ∥v(1) +q0,u0,η0∥H1+s ≲ ∥q0, u0, η0∥Xs +(3.4.26) +and +∥∂1(v(2) +η0 · e1)∥H1+s ≲ ∥v(2) +η0 · e1∥W 1+s,∞((0,b);H0 +(rn−1)(Σ)) ≲ ρ. +(3.4.27) +These complete the proof of the third item. +Finally, to prove the fourth item we first note that (3.4.17) yields the formula +∇ · (vw0(q + gη)) = ∇ · (vw0(q + gΠ1 +Hη)) + g∇ · (v(1) +q0,u0,η0Π1 +Lη) ++ g∂1(e1 · v(2) +η0 Π1 +Lη) + ϱ′∂1Π1 +Lη = I + II + III + IV. +(3.4.28) +For I we use Proposition B.1, the first and second items above, and Corollary D.7 to estimate +∥I∥Hs ≲ ∥vw0(q + gΠ1 +Hη)∥H1+s ⩽ ∥v(1) +q0,u0,η0(q + gΠ1 +Hη)∥H1+s + ∥(v(2) +η0 + ϱ′e1/g)(q + gηH)∥H1+s +≲ ∥q, η∥H1+s×H1+s + +� +0 +if s ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ +if ⌊n/2⌋ < s. +(3.4.29) +On the other hand, since Tr∂Ω(vq0,u0,η0 · en) = 0, we use estimate (B.2.12) from Proposition B.10 to +bound +�� b +0 +I(·, y) dy +� +˙H−1 ≲ ∥vw0(q + gΠ1 +Hη)∥L2 ≲ ∥q, η∥L2×H0. +(3.4.30) +From (3.4.29) and (3.4.30) we deduce that +∥I∥ ˆHs ≲ ∥q, η∥H1+s×H1+s + +� +0 +if s ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ +if ⌊n/2⌋ < s. +(3.4.31) +For II, we use the first item along with Proposition B.1 and Corollary D.7 again to estimate +∥II∥Hs ≲ ∥v(1) +q0,u0,η0∥L2∥Π1 +Lη∥W 1+s,∞ + ∥v(1) +q0,u0,η0∥H1+s∥Π1 +Lη∥L∞ ≲ ∥q0, u0, η0∥Xs∥Π1 +Lη∥H0. (3.4.32) +On the other hand, since Tr∂Ω(v(1) +q0,u0,η0 · en) = 0, we have +�� b +0 +II(·, y) dy +� +˙ +H−1 ≲ ∥v(1) +q0,u0,η0Π1 +Lη∥L2 ≲ ∥Π1 +Lη∥H0. +(3.4.33) + +66 +NOAH STEVENSON AND IAN TICE +From (3.4.32) and (3.4.33) we deduce that ∥II∥ ˆHs is controlled by the right hand side of (3.4.22). +We now estimate III, using the second item and the algebraic properties of the anisotropic Sobolev +spaces from Proposition B.2 to get +∥III∥Hs ≲ ∥e1 · v(2) +η0 Π1 +Lη∥W s,∞((0,b);H0 +(rn−1+1)(Σ)) ≲ ∥e1 · v(2) +η0 ∥W s,∞H0 +(rn−1)∥Π1 +Lη∥H0 ≲ ∥Π1 +Lη∥H0 +(3.4.34) +and +�� b +0 +III(·, y) dy +� +˙H−1 ⩽ +���Π1 +Lη +� b +0 +e1·v(2) +η0 (·, y) dy +��� +H0 ≲ +� b +0 +∥e1·v(2) +η0 (·, y)∥H0 dy·∥Π1 +Lη∥H0 ≲ ∥Π1 +Lη∥H0. +(3.4.35) +The bounds (3.4.34) and (3.4.35) imply that ∥III∥ ˆHs is controlled by the right hand side of (3.4.22). +The norm ∥IV∥ ˆHs is trivially controlled by the same quantity thanks to (B.1.25). The proof of the +fourth item is then complete upon synthesizing these bounds on I, II, III, and IV. +□ +We now introduce a new scale of adapted spaces pertinent to the analysis of the A piece of DΨ. +As we will see, these are the natural domains on which A is an isomorphism of Banach spaces. To +define the spaces, we simply build an extra condition into the spaces {Xs}s∈N, which we recall were +defined in (3.1.3), to create a new family in which the effect of the derivative loss is mitigated. What +is notable is that these domains depend on the background point w0 = (q0, u0, η0) in a non-trivial +way. For s ∈ {−1, 0} ∪ R+ and (q0, u0, η0) as in the hypotheses of Lemma 3.19 we define the space +q0,u0,η0 +Xs += {(q, u, η) ∈ Xs : ∇ · (vw0q) ∈ H1+s(Ω)}. +(3.4.36) +and equip it with the norm +∥q, u, η∥q0,u0,η0 +Xs += +� +∥q, u, η∥2 +Xs + ∥∇ · (vw0q)∥2 +H1+s. +(3.4.37) +Remark 3.20. A simple modification of the proof of the first item of Proposition B.7 reveals that +the spaces +q0,u0,η0 +Xs +are Hilbert. +The following result captures that the A + P decomposition of DΨ from (3.4.5) is such that A +contains all of the derivative loss and P is a small correction term without derivative loss. Note that +the conclusions in what follows are stronger than what can be deduced from a direct application of +the smooth-tameness verification result, namely Theorem 3.13. +Proposition 3.21 (Properties of the A+P decomposition of DΨ). Let ρ ∈ R+ and w0 = (q0, u0, η0) +be as in the hypotheses of Lemma 3.19 and let I ⋐ R+ be an interval with γ0 ∈ I. The following +hold. +(1) For every s ∈ N, the map +w0,γ0 +A +: +q0,u0,η0 +Xs +→ Ys is well-defined and continuous. +(2) For every N ∋ s ⩾ 1 + ⌊n/2⌋, the map +w0,γ0 +P +: Xs × R → Ys is well-defined, continuous, and +obeys the estimates +�� +w0,γ0 +P (q, u, η, γ) +�� +Ys ≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥X1+s∥q, u, η, γ∥X⌊n/2⌋×R. +(3.4.38) +The implicit constants depend on the dimension, the physical parameters, s, ρWD, and I. +Proof. Upon inspecting the components of +w0,γ0 +A +in (3.4.7), we see that the second and third are +trivially well-defined and continuous, so to prove the first item we heed to Remark 3.1 and reduce +to showing that the map +q0,u0,η0 +Xs +∋ (q, u, η) �→ +w0 +A1(q, u, η) ∈ ˆH1+s(Ω) +(3.4.39) + +COMPRESSIBLE TRAVELING WAVES +67 +is well-defined and continuous. As a consequence of the fourth item of Lemma 3.19, the identity +� b +0 +∇ · (ϱu)(·, y) dy = (∇∥, 0) · +� b +0 +(ϱu)(·, y) dy − ϱ(b)∂1η +(3.4.40) +(which is true by virtue of the boundary condition build into the Xs spaces, as defined in (3.1.3)), +and the norm equivalence (B.1.25) from Proposition B.4, we see that the operator in (3.4.39) +continuously maps into ˆH0(Ω). On the other hand, by employing the first, second, and third items +of Lemma 3.19, the equality +∇ · (vw0(q + gη)) = g∇ · (vw0η) + ∇ · (vw0q), +(3.4.41) +and the definition of the norm in (3.4.37), we readily deduce that +w0 +A1 maps boundedly into H1+s(Ω). +This completes the proof of the first item. +We now turn to the proof of the second item. Recall from (3.4.13) that +w0,γ0 +P +has components +w0,γ0 +P i +for i ∈ {1, 2, 3}. We divide the remainder of the proof into three steps, each of which handles +one of these components. +Step 1: Estimates on +w0 +P 1. We first aim to exploit some hidden cancellation by observing that +the sum of the final two terms in (3.4.14) vanishes. Indeed, since +˙M[η]e1 = ∂nEηe1 − ∂1Eηen we +have that ∇ · ( ˙M[η]e1) = 0, and therefore +− ∇ · (ϱ′Eηe1) − ∇ · (ϱ ˙M[η]e1) = −ϱ′∂1Eη − ϱ′(∂nEηe1 − ∂1Eηen) · en = 0. +(3.4.42) +Thus, we have the more accessible formula +w0 +P 1(u, η) = ∇ · ((σq0,η0 − ϱ)(u − ˙M[η]e1)) − g∇ · ((vw0 − ϱ′e1/g)Eη) = I + II, +(3.4.43) +and we will deal with I and II separately. +The term I is handled with the Taylor expansion trick of Lemma 3.8, i.e. +we decompose +σq0,η0 − ϱ = σ(1) +q0,η0 + σ(2) +η0 , where +σ(1) +q0,η0 = N (1) +H−1(q0, η0) and σ(2) +η0 = N (2) +H−1(Π1 +Lη). +(3.4.44) +Employing the above decomposition and recalling (3.4.15), we may further rewrite +I = ∇ · ((σq0,η0 − ϱ)(u + ∂1Eηen) − σ(1) +q0,η0∂nEηe1 − σ(2) +η0 ∂nE0Π1 +Hηe1) − ∂1(σ(2) +η0 Π1 +Lη)/b = ∇ · I1 + ∂1I2. +(3.4.45) +To handle ∇·I1 we will first derive a Sobolev estimate for I1. Indeed, thanks to multiple applications +of Corollary D.7 and Lemma A.1 , we are free to bound, for any s ∈ N, +∥I1∥Hs ≲ ∥σ(1) +q0,η0, σ(2) +η0 ∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥u, η∥Hs×H1/2+s ++ +� +0 +if s ⩽ 1 + ⌊n/2⌋, +∥σ(1) +q0,η0, σ(2) +η0 ∥Hs×W s,∞∥u, η∥H1+⌊n/2⌋×H3/2+⌊n/2⌋ +if 1 + ⌊n/2⌋ < s. +(3.4.46) +Then we use Lemma 3.8 combined with the fundamental theorem of calculus as in the first part of +the proof of Lemma 3.19 to acquire the bounds +∥σ(1) +q0,η0, σ(2) +η0 ∥Hs×W s,∞ ≲ +� +∥q0, u0, η0∥X1+⌊n/2⌋ +if s ⩽ 2 + ⌊n/2⌋, +∥q0, u0, η0∥Xs−1 +if 2 + ⌊n/2⌋ < s. +(3.4.47) +Upon synthesizing the bounds (3.4.46) and (3.4.47), we find that +∥I1∥Hs ≲ ρ∥q, u, η∥Xs−2 + +� +0 +if s ⩽ 2 + ⌊n/2⌋, +∥q0, u0, η0∥Xs−1∥q, u, η∥X⌊n/2⌋−1 +if 2 + ⌊n/2⌋ < s, +(3.4.48) + +68 +NOAH STEVENSON AND IAN TICE +which is the aforementioned Sobolev estimate. Then, in light of (3.4.48), the fact that Tr∂Ω(I1·en) = +0, and divergence - normal trace compatibility estimates from Proposition C.1, we arrive at the +bound +∥∇ · I1∥ ˆH1+s ⩽ ∥I1∥H2+s ≲ ρ∥q, u, η∥Xs + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋−1 +if ⌊n/2⌋ < s. (3.4.49) +Having dispatched I1, we turn our attention to the I2 term in (3.4.45). The product σ(2) +η0 Π1 +Lη is +handled via Lemma 3.8 and the algebra properties of band limited members of H0(Σ) enumerated +in Proposition B.2, applied to σ(2) +η0 (·, y)Π1 +Lη for y ∈ [0, b]: +∥∂1I2∥ ˆH1+s ≲ +� +∂1 +� +Π1 +Lη +� b +0 +σ(2) +η0 (·, y) dy +�� +˙H−1 + +max +0⩽j⩽1+ν sup +y∈[0,b] +∥∂1∂j +nI2(·, y)∥H1+s(Rn−1) +≲ +��� +� b +0 +σ(2) +η0 (·, y) dy +��� +H0∥Π1 +Lη∥H0 + +max +0⩽j⩽1+s sup +y∈[0,b] +[∂1∂j +n(σ(2) +η0 (·, y)Π1 +Lη)] ˙H−1 +≲ +max +0⩽j⩽1+s sup +y∈[0,b] +∥∂j +nσ(2) +η0 (·, y)∥H0∥Π1 +Lη∥H0 ≲ ρ∥Π1 +Lη∥H0, +(3.4.50) +where in deducing the final inequality we again use the tame mapping properties of σ(2) +η0 from +Lemma 3.8, combined with the fundamental theorem of calculus as before. This completes the +handling of I2, and hence of I. +Next, we consider II from equation (3.4.43). Thanks to the decomposition of vw0 from Lemma 3.19, +II has the further decomposition +II = −g∇ · ((v(1) +q0,u0,η0 + v(2) +η0 )E0Π1 +Hη + v(1) +q0,u0,η0EΠ1 +Lη) − g∂1(idRn · env(2) +η0 · e1Π1 +Lη)/b = ∇ · II1 + ∂1II2. +(3.4.51) +As in the proof of (3.4.48), we use Corollary D.7 and the mapping properties of v(1) +q0,u0,η0 and v(2) +η0 +from Lemma 3.19 to see that for any s ∈ N, +∥II1∥Hs ≲ ∥v(1) +q0,u0,η0, v(2) +η0 ∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥η∥Hs−1/2 ++ +� +0 +if s ⩽ 1 + ⌊n/2⌋, +∥v(1) +q0,u0,η0, v(2) +η0 ∥Hs×W s,∞∥η∥H1/2+⌊n/2⌋ +if 1 + ⌊n/2⌋ < s, +≲ ρ∥q, u, η∥Xs−3 + +� +0 +if s ⩽ 2 + ⌊n/2⌋, +∥q0, u0, η0∥Xs−1∥q, u, η∥X⌊n/2⌋−2 +if 2 + ⌊n/2⌋ < s. +(3.4.52) +In turn, (3.4.52), the fact that Tr∂Ω(II1 · en) = 0, and Proposition C.1 provide the bound +∥∇ · II1∥H1+s ≲ ∥II1∥H2+s ≲ ρ∥q, u, η∥Xs−1 + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋−1 +if ⌊n/2⌋ < ν. +(3.4.53) +For II2, we argue as in (3.4.50), estimating the product (idRn ·en)(v(2) +η0 ·e1)ηL by invoking the second +item from Lemma 3.19 as well as the algebra properties of band-limited members of H0(Rn−1) from +Proposition B.2; this results in the bound +∥∂1II2∥ ˆH1+s ≲ ρ∥Π1 +Lη∥H0. +(3.4.54) +Finally, we synthesize (3.4.43), (3.4.45), (3.4.49), (3.4.50), (3.4.51), (3.4.52), and (3.4.54) to get +the +w0 +P 1 estimate +�� +q0,u0,η0 +P 1 +(u, η) +�� ˆH1+s ≲ ρ∥q, u, η∥Xs + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋−1 +if ⌊n/2⌋ < s. +(3.4.55) + +COMPRESSIBLE TRAVELING WAVES +69 +Step 2: Estimates on +w0,γ0 +P 2 . This is simpler than the previous step, as we can use the tame +calculus conclusions of Proposition 3.11. Recall that +w0,γ0 +P 2 (q, u, η, γ) = (DΨ2(w0, γ0) − DΨ2(0, 0, 0, γ0))(q, u, η, 0) + DΨ2(w0, γ0)(0, 0, 0, γ) = J1 + J2, +(3.4.56) +where Ψ2 is the nonlinear map defined in (3.3.7). We may use that Ψ2 is C2 paired with the +fundamental theorem of calculus to write +J1 = +� 1 +0 +D2Ψ2(tw0, γ0)((q0, u0, η0, 0), (q, u, η, 0)) dt. +(3.4.57) +Hence, by the strong tame estimates on the second derivative from Proposition 3.11, the log-convexity +of the norm of the Xs-spaces (see Lemma 3.25), and Young’s inequality we get that for ν ⩾ 1+⌊n/2⌋: +∥J1∥Hs ≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥Xs∥q, u, η∥X1+⌊n/2⌋. +(3.4.58) +On the other hand, by inspection we see that DΨ2(0, γ0)(0, 0, 0, γ) = 0. +Thus by a similar +fundamental theorem of calculus argument, we have that +J2 = +� 1 +0 +D2Ψ2(tw0, γ0)((w0, 0), (0, 0, 0, γ)) dt. +(3.4.59) +This lets us use the strong tame estimates on the second derivative again to bound +∥J2∥Hs ≲ |γ|∥q0, u0, η0∥Xs. +(3.4.60) +By combining equations (3.4.58) and (3.4.60), we obtain the stated bounds for +w0,γ0 +P 2 . +Step 3: Estimates on +w0,γ0 +P 3 . We perform the same analysis, utilizing the fundamental theorem of +calculus and the tame C2-estimates as in the second step, but this time we employ Proposition 3.12 +in place of Proposition 3.11. +□ +The remainder of this subsection is devoted to the decomposition of the DΦ piece of DΨ, which +is derived from equations (3.3.9) and (3.3.10), into subcomponents Q and R. The key observation +here is that Φ : BX2+⌊n/2⌋(0, ρWD) × W3+⌊n/2⌋ → Y0 is linear in the second factor, and hence for θ0 +as in (3.4.3) we can write, +DΦ(θ0)(q, u, η, T , G, F) = +θ0Q(q, u, η) + +q0,u0,η0 +R +(T , G, F) +(3.4.61) +where +θ0Q(q, u, η) = DΦ(θ0)(q, u, η, 0, 0, 0) +(3.4.62) +and +q0,u0,η0 +R +(T , G, F) = DΦ(q0, u0, η0, 0, 0, 0)(0, 0, 0, T , G, F). +(3.4.63) +The following result reveals the point of the Q + R decomposition of DΦ: Q is small when the +background is small, and R is independent of (q, u, η) and thus enjoys a useful decoupling. In +contrast with the A + P decomposition (Proposition 3.21), the next result is less delicate and only +relies on the smooth-tameness results from Propositions 3.14 and 3.15. +Proposition 3.22 (Properties of the Q + R decomposition of DΦ). Let 0 < ρ ⩽ ρWD, where ρWD +is defined in Theorem 3.17, and +θ0 = (q0, u0, η0, T0, G0, F0) ∈ (BX2+⌊n/2⌋(0, ρ) × BW3+⌊n/2⌋(0, ρ)) ∩ (X∞ × W∞). +(3.4.64) +The following hold for N ∋ s ⩾ 2 + ⌊n/2⌋. + +70 +NOAH STEVENSON AND IAN TICE +(1) The map +θ0Q : Xs → Ys is well-defined, continuous, and obeys the estimate +�� +θ0Q(q, u, η) +�� +Ys ≲ ρ∥q, u, η∥Xs + ∥θ0∥Xs×W1+s∥q, u, η∥X2+⌊n/2⌋. +(3.4.65) +(2) The map +q0,u0,η0 +R +: W1+s → Ys is well-defined, continuous, and obeys the estimate +�� +q0,u0,η0 +R +(T , G, F) +�� +Ys ≲ ∥T , G, F∥W1+s + ⟨∥q0, u0, η0∥Xs⟩∥T , G, F∥W3+⌊n/2⌋. +(3.4.66) +In the above the implicit constants depend on the dimension, the physical parameters, s, and ρWD. +Proof. That +θ0Q is well-defined, continuous, and, when ρ = ρWD and s ⩾ 2 + ⌊n/2⌋, obeys a tame +estimate of the form +�� +θ0Q(q, u, η) +�� +Ys ≲ ∥q, u, η∥Xs + ⟨∥θ0∥Xs×W1+s⟩∥q, u, η∥X2+⌊n/2⌋ +(3.4.67) +follows from the formula (3.4.62), the tame smoothness assertions of Propositions 3.14 and 3.15, +and the tame estimates on derivatives from Lemma 2.7. We can improve the above estimate by +leveraging the fact that for fixed (q0, u0, η0) the map (T0, G0, F0) �→ +θ0Q is linear; indeed, by arguing +as in the proof of Lemma 2.5, we deduce from estimate (3.4.67) that actually +�� +θ0Q(q, u, η) +�� +Ys ≲ ∥T0, G0, F0∥W3+⌊n/2⌋∥q, u, η∥Xs + ∥θ0∥Xs×W1+s∥q, u, η∥X2+⌊n/2⌋. +(3.4.68) +The first item now follows. The assertions and estimate of the second item are a direct consequence +of formula (3.4.63), the tame smoothness assertions of Propositions 3.14 and 3.15,and the tame +estimates on derivatives from Lemma 2.7. +□ +3.5. Prelude to linear analysis. Recall that Ψ is the nonlinear operator associated with the +PDE (1.4.9) and is defined in (3.3.2). Our goal now is to prove that this map satisfies the hypotheses +of the inverse function theorem, which are enumerated in Definition 2.20, Theorem 2.21, and +Theorem 2.24. The previous analysis of this section, in particular Lemma 3.2 and Theorem 3.17, +shows that the ‘nonlinear hypotheses,’ other than a trivial issue with the domain and the first item +of Definition 2.20, of this inverse function theorem are satisfied. +It only remains, then, to show that the ‘linear hypotheses’ given in the third item of Definition 2.20 +are satisfied. In other words, we aim to prove the following assertion about the derivative, DΨ. +Given N ∋ s ⩾ 2 + ⌊n/2⌋ and an interval I ⋐ R+, there exists an existence and estimates parameter +ρEE(s) ∈ (0, ρWD] (also depending on I), where ρWD ∈ R+ is from Theorem 3.17, with the property +that whenever (recall that θ0 is defined in (3.4.3)) +(θ0, γ0) ∈ (BX2+⌊n/2⌋(0, ρEE(s)) × BW3+⌊n/2⌋(0, ρEE(s)) × I) ∩ (X1+s × W2+s × R+) +(3.5.1) +and +(g, f, k, T , G, F, γ) ∈ Ys × W1+s × R, +(3.5.2) +there exists a unique (q, u, η) ∈ Xs solving DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ); +moreover, if we set Ξ = (g, f, k, T , G, F, γ) and Θ = (q, u, η, T , G, F, γ) then the solution tuple obeys +the tame estimate +∥Θ∥Xs×W1+s×R ≲ ∥Ξ∥Ys×W1+s×R + ⟨∥θ0∥X1+s×W2+s⟩∥Ξ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R +(3.5.3) +for an implicit constant depending only on the dimension, the various physical parameters, s, I, +and ρEE(s). +Thanks to the analysis of the previous subsection, namely Propositions 3.21 and 3.22, we have +that the behavior of DΨ is governed by the principal part operator A defined in (3.4.7). Thus +our above goal is essentially achieved, modulo minor supplementary analysis, as soon as we prove +the following assertion. For any s ∈ N and γ0 ∈ I ⋐ R+an interval, there exists an existence and + +COMPRESSIBLE TRAVELING WAVES +71 +estimates principal part parameter ρEEP(s) ∈ (0, ρWD] (also depending on I) with the property that +whenever +w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρEEP(s)) ∩ X∞ and (g, f, k) ∈ Ys, +(3.5.4) +there exists a unique (q, u, η) ∈ Xs solving +w0,γ0 +A (q, u, η) = (g, f, k); moreover, the solution obeys the +tame estimates +∥q, u, η∥Xs ≲ ∥g, f, k∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+s⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(3.5.5) +Again we allow for implied constants to depend on the dimension, the physical parameters, s, I, and +ρEEP(s). We will see that a necessary condition for the estimate (3.5.5) to hold is that the operator +w0,γ0 +A +: +q0,u0,η0 +Xs +→ Ys is a Banach isomorphism, where we recall the adapted spaces are defined +in (3.4.36). This expresses one of the core difficulties of the linear analysis: we either have that the +family of operators { +w0,γ0 +A } ⊂ L(X1+s; Ys) are defined on the common Banach space X1+s but not +isomorphisms, or else each individual operator +w0,γ0 +A +is defined on a larger adapted Banach space +q0,u0,η0 +Xs +making it an isomorphism onto Ys, but the spaces { +q0,u0,η0 +Xs +} are inequivalent. Consequently, +we cannot port invertibility from one operator to the next via the method of continuity, even if we +were to establish the estimates of (3.5.5) a priori. +We overcome this difficulty via an elliptic regularization procedure which we now describe. For +m ∈ N+ we define the 2mth-order linear elliptic differential operator +Lm = (−1)m +n +� +j=1 +∂2m +j +. +(3.5.6) +We will consider a sequence of operators obtained by adding a vanishing contribution of this +operator to the continuity equation component of A and also adding a vanishing contribution of +(−∆∥)m−1/4 to the kinematic boundary condition. These new operators have the following domains: +for m, N ∈ N+ and s ∈ {−1} ∪ N we define the space +Xs +m,N = {(q, u, η) ∈ Xs : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = N−1(−∆∥)m−1/4η +(q, η) ∈ H1+s+2m(Ω) × H1+s+2m(Σ), Tr∂Ω(∂m +n q) = · · · = Tr∂Ω(∂2m−1 +n +q) = 0} +(3.5.7) +and endow it with the norms +∥q, u, η∥Xs +m,N = +� +∥q, u, η∥2 +Xs + N−2∥q, η∥2 +H1+s+2m×H1+s+2m, +(3.5.8) +and +∥q, u, η∥q0,u0,η0 +Xs +m,N += +� +∥q, u, η∥2 +Xs +m,N + ∥∇ · (vq0,u0,η0q)∥2 +H1+s. +(3.5.9) +We note that, in light of the fourth item of Lemma 3.19, ∥·∥q0,u0,η0 +Xs +m,N +is a norm equivalent to ∥·∥Xs +m,N , +with equivalence constants depending on s, m, N, and ρWD. +For s ∈ N we then define the regularized principal part operator +w0,γ0 +Am,N : Xs +m,N → Ys +(3.5.10) +via +w0,γ0 +Am,N(q, u, η) = (N−1Lm(q + gη) + +w0,γ0 +A1 (q, u, η), +γ0 +A2(q, u, η), +γ0 +A3(q, u, η)), +(3.5.11) +where the Ai terms are the same as in (3.4.7). + +72 +NOAH STEVENSON AND IAN TICE +The upshot is that the operators { +w0,γ0 +Am,N} are all contained in the space L(Xs +m,N, Ys) and, as we +will see, obey a family of nice a priori estimates. Thus the method of continuity is available for the +regularized operators’ existence theory. +Once we have existence for the regularized operators, we would like to show that we can pass to +the limit as N → ∞ and obtain existence for A. This is achieved by carefully proving N-independent +a priori estimates for Am,N by inductively working up from the base case of a priori estimates for +weak solutions. This inductive estimate procedure will also be done in tandem with the operator A +to obtain the sought-after a priori estimates of (3.5.5). +As a result, we will need to set notation for weak formulations of the operators A and Am,N. +First we recall the definition of the space 0H1 from (1.7.3). The operator associated with the weak +formulation of the momentum equation is +γ0 +I : X−1 → (0H1(Ω; Rn))∗ +(3.5.12) +defined by +⟨ +γ0 +I (q, u, η), w⟩(0H1)∗,0H1 = +� +Ω +−γ2 +0ϱ∂1u · w − q∇ · (ϱw) + gϱ∇η · w + γ0Sϱu : ∇w +− ς⟨∆∥η, TrΣ(w · en)⟩H−1/2,H1/2 +(3.5.13) +for (q, u, η) ∈ X−1 and w ∈ 0H1(Ω; Rn). We next define a family of operators associated with the +weak formulation of the full problem by setting +w0,γ0 +J +: +q0,u0,η0 +X−1 +→ Y−1 +(3.5.14) +via +w0,γ0 +J (q, u, η) = +� +∇ · (ϱu) + ∇ · (vw0(q + gη)), +γ0 +I (q, u, η) +� +, +(3.5.15) +where we recall that the vector field vw0 is defined in (3.4.11). Furthermore, given τ ∈ [0, 1] and +m, N ∈ N+ with m ⩾ 2 we define the map +w0,γ0 +J τ +m,N : X−1 +m,N → Y−1 +(3.5.16) +via +w0,γ0 +J τ +m,N(q, u, η) = (∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη), +γ0 +I (q, u, η)). +(3.5.17) +We now give a basic result on the well-definedness and continuity of these operators. +Lemma 3.23 (Well-definedness check for linear analysis). Let ρ ∈ R+ and w0 = (q0, u0, η0) be as +in the hypotheses of Lemma 3.19. Let I ⋐ R+ be an interval with γ0 ∈ I. The following hold. +(1) The weak formulation operator of (3.5.14) is well-defined and bounded. +(2) For N ∋ m ⩾ 2, N ∈ N+, and τ ∈ [0, 1], the regularized weak formulation operator of (3.5.16) +is well-defined and bounded. +(3) For N ∋ m ⩾ 2 and N ∈ N+ the regularization of the principal part operator defined by +(3.5.10) is well-defined and bounded. +Proof. For the first item we see that it suffices to check that the first component of +γ0,w0 +J +is well- +defined and maps boundedly into ˆH0(Ω). For this we can argue in a manner similar to the first +part of the proof of Proposition 3.21, where we studied (3.4.39). The only difference is in handling +the ∇ · (vw0q) term. That this term maps boundedly into L2(Ω) is immediate from the definition of +the norm on +q0,u0,η0 +X−1 . We then use Proposition B.10, paired with the fact that vw0 ∈ L∞(Ω; Rn), to +obtain that q �→ +� b +0 ∇ · (vw0q)(·, y) dy maps boundedly into ˙H−1(Ω). + +COMPRESSIBLE TRAVELING WAVES +73 +The second and third items follow as soon as we check that +X−1 +m,N ∋ (q, u, η) �→ ∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) ∈ ˆH0(Ω) +(3.5.18) +is a well-defined and bounded linear map. We handle the τ∇ · (vw0(q + gη)) term in the same +way that we handled (3.4.41) from Proposition 3.21. For the N−1Lm(q + gη) term we see that it +obviously maps into L2(Ω). To acquire the divergence compatibility condition, we use the Neumann +boundary conditions built into the domain X−1 +m,N to compute +� b +0 +1 +N Lm(q + gη)(·, y) dy = 1 +N +n−1 +� +j=1 +∂2m +j +� +gbη + +� b +0 +q(·, y) dy +� +, +(3.5.19) +but since m ⩾ 1, this provides the bound +�� b +0 +1 +N Lm(q + gη)(·, y) dy +� +˙H−1 ≲ 1 +N (∥∇∥η∥H2m−2 + ∥q∥H2m−1) ≲ ∥q, u, η∥X−1 +m,N . +(3.5.20) +Finally we handle the ∇ · (ϱu) term of (3.5.18). Evidently, this maps into L2(Ω), leaving us to check +the divergence compatibility condition. By integrating, we learn that +� b +0 +∇ · (ϱu)(·, y) dy = (∇∥, 0) · +� b +0 +(ϱu)(·, y) dy − ϱ(b)∂1η + ϱ(b) +N (−∆∥)m−1/4η, +(3.5.21) +and since m ⩾ 2 this yields the estimate +�� b +0 +∇ · (ϱu)(·, y) dy +� +˙H−1 ≲ ∥u∥L2 + ∥η∥H0 + 1 +N ∥∇∥η∥H2m−5/2 ≲ ∥q, u, η∥X−1 +m,N . +(3.5.22) +□ +We now consider the relationship between the weak and strong operators, J , J τ +m,N and A, +Am,N, by introducing a functional that maps the strong form of the data to the weak formulation +of the data. For s ∈ N we define the strong-to-weak data map +K : Hs(Ω; Rn) × H1/2+s(Σ; Rn) → (0H1(Ω; Rn))∗ +(3.5.23) +given by +⟨K (f, k), w⟩(0H1)∗,0H1 = +� +Ω +f · w + +� +Σ +k · w, +(3.5.24) +where w ∈ 0H1(Ω; Rn), (f, k) ∈ Hs(Ω; Rn) × H1/2+s(Σ; Rn), and s ∈ N. +Lemma 3.24 (Strong and weak solutions). Under the hypotheses of Lemma 3.23, the following +hold for (g, f, k) ∈ Y0. +(1) If (q, u, η) ∈ +q0,u0,η0 +X0 +, then +w0,γ0 +A (q, u, η) = (g, f, k) if and only if +w0,γ0 +J (q, u, η) = (g, K (f, k)). +(2) If m, N ∈ N+, m ⩾ 2, and (q, u, η) ∈ X0 +m,N, then +w0,γ0 +Am,N(q, u, η) = (g, f, k) if and only if +w0,γ0 +J 1 +m,N(q, u, η) = (g, K (f, k)). +Proof. These follow directly from integration by parts. +□ +This section is concluded with the following simple lemma on log-convexity, which we recall is +defined in Section 1.7. + +74 +NOAH STEVENSON AND IAN TICE +Lemma 3.25 (Log-convexity of the norms). The following Banach scales are log-convex for the +stated norms: +� +Xs, ∥·∥Xs� +s∈N, +�q0,u0,η0 +Xs +, ∥·∥q0,u0,η0 +Xs +� +s∈N, +� +Xs +m,N, ∥·∥Xs +m,N +� +s∈N, +� +Xs +m,N, ∥·∥q0,u0,η0 +Xs +m,N +� +s∈N, +� +Ys, ∥·∥Ys� +s∈N, +(3.5.25) +where we take m, N ∈ N and (q0, u0, η0) be as in the hypotheses of Lemma 3.19. +Proof. Lemma B.5 shows that the anisotropic Sobolev spaces {Hs(Rd)}s∈N are log-convex. The +result then follows from the log-convexity of standard Sobolev spaces (see Theorem D.4 and Corollary +D.5) and the evident preservation of log-convexity under products. +□ +4. Analysis of steady transport equations and their regularizations +In the previous section we introduced the regularized principal part operator +w0,γ0 +Am,N (defined +in equation (3.5.10)), which perturbs the operator +w0,γ0 +A +(defined in equation (3.4.7)) by adding a +multiple of the elliptic operator Lm (defined in (3.5.6)) to the part of +w0,γ0 +A +corresponding to the +linearized continuity equation. In this section we focus our attention on solutions to such regularized +steady transport equations, with the aim of deriving precise estimates that are independent of the +regularization parameter. These will play an essential role in our subsequent linear analysis. Along +the way, we also develop some estimates of solutions to the standard steady transport equation. +To be precise, we now study equations of the type +αf + ∇ · (vf) + εLmf = g in Ω, +(4.0.1) +where α, g : Ω → R are given functions, v : Ω → Rn is a given vector field, ε ⩾ 0 is small, Lm is +the linear elliptic operator given by (3.5.6), and f : Ω → R is the unknown. The key results of this +section are Proposition 4.4 and Theorem 4.8. +In what follows, a key player is a vector field X ∈ W ∞,∞(Ω; Rn) such that Tr∂Ω(X · en) = 0. +Assume X0 ∈ H∞(Ω; Rn), X1 ∈ W ∞,∞(Ω; Rn), ρmax ∈ R+ is arbitrary but fixed, 0 < ρ ⩽ ρmax, +r ∈ N, +X = X0 + X1, and (DX0, DX1) ∈ BHr(0, ρ) × BW r,∞(0, ρ). +(4.0.2) +We will frequently reference this equation when we need to quantify a vector field of this type. +4.1. Preliminary tame estimates. Given m ∈ N+, we define a bilinear form associated to Lm +via +Bm : Hm(Ω) × Hm(Ω) → R via Bm(ϕ0, ϕ1) = +n +� +j=1 +� +Ω +∂m +j ϕ0∂m +j ϕ1. +(4.1.1) +We have recorded a number of basic properties of Bm in Appendix C.2. Our first result here +considers an estimate for Bm in which we are able to a save a derivative thanks to integration by +parts. +Lemma 4.1 (A bilinear estimate). Let m ∈ N+. Suppose that ϕ ∈ Hm(Ω), X ∈ W ∞,∞(Ω; Rn) +satisfies Tr∂Ω(X · en) = 0 as well as (4.0.2) with r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, and that +∇ · (Xϕ) ∈ Hm(Ω). Then we have the estimates +|Bm(ϕ, ∇ · (Xϕ))| ≲ ρ∥ϕ∥2 +Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2 +(4.1.2) +and +|Bm(ϕ, ∇ · (Xϕ))| ≲ ρ∥ϕ∥2 +Hm + ∥ϕ∥Hm +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.1.3) + +COMPRESSIBLE TRAVELING WAVES +75 +Here the implicit constants depend on m, ρmax, and the dimension. +Proof. Since ϕ, ∇ · (Xϕ) ∈ Hm(Ω) and X ∈ W ∞,∞(Ω; Rn), we can apply the Leibniz rule to see +that +∇ · (X∂m +j ϕ) = ∂m +j ∇ · (Xϕ) − +m +� +k=1 +�m +k +� +∇ · (∂k +j X∂m−k +j +ϕ) ∈ L2(Ω), +(4.1.4) +and so ∂m +j ϕ ∈ H0 +X(Ω), the space defined by (B.2.1) in Appendix B.2. By introducing commutators +and employing Proposition B.9, we obtain the identity +Bm(∇ · (Xϕ), ϕ) = +n +� +j=1 +� +Ω +�∇ · X +2 +∂m +j ϕ + ∇ · ([∂m +j , X]ϕ) +� +∂m +j ϕ. +(4.1.5) +Hence, we have the estimate +|Bm(∇ · (Xϕ), ϕ)| ≲ +� +∥DX∥L∞∥ϕ∥Hm + +n +� +j=1 +∥[∂m +j , X]ϕ∥H1 +� +∥ϕ∥Hm. +(4.1.6) +According to Corollary D.8, we have the estimate +∥[∂m +j , X0]ϕ∥H1 ≲ ∥∂jX0∥H1+⌊n/2⌋∥ϕ∥Hm + +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥∂jX0∥Hm∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m, +(4.1.7) +and +∥[∂m +j , X1]ϕ∥H1 ≲ ∥∂jX1∥L∞∥ϕ∥Hm + ∥∂jX1∥W m,∞∥ϕ∥L2. +(4.1.8) +Estimates (4.1.7) and (4.1.8) combine to show that +n +� +j=1 +∥[∂m +j , X]ϕ∥H1 ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.1.9) +This establishes (4.1.3). +We now continue to prove (4.1.2). In the latter case of (4.1.9), we use interpolation and Young’s +inequality to bound +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ ≲ ρ− +1+⌊n/2⌋ +m−1−⌊n/2⌋ ∥DX0, DX1∥ +m +m−1−⌊n/2⌋ +Hm×W m,∞∥ϕ∥L2 + ρ∥ϕ∥Hm +≲ρmax ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2 + ρ∥ϕ∥Hm. +(4.1.10) +Together, (4.1.9) and (4.1.10) provide the bound +n +� +j=1 +∥[∂m +j , X]ϕ∥H1 ≲ ρ∥ϕ∥Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.1.11) +Now we return to (4.1.6) and plug in (4.1.11) to derive the estimate +|Bm(∇ · (Xϕ), ϕ)| ≲ ρ∥ϕ∥2 +Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2, +(4.1.12) +which is (4.1.2). +□ +For the next two results we use the notation ∇Xϕ = X · ∇ϕ. The following rather technical +lemma considers traces of the normal derivatives of ∇Xϕ when X has vanishing normal trace and +ϕ has some vanishing normal derivative traces. The point is that the higher norms on normal +derivative traces depend only on X and a lower norm of ϕ. + +76 +NOAH STEVENSON AND IAN TICE +Lemma 4.2 (A trace estimate). Suppose that X ∈ W ∞,∞(Ω; Rn) satisfies Tr∂Ω(X · en) = 0 and +decomposes as X = X0 + X1 with DX0 ∈ H∞(Ω; Rn×n) and DX1 ∈ W ∞,∞(Ω; Rn×n). Assume +additionally that ϕ ∈ H2m+1(Ω) and satisfies the Neumann conditions +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω. +(4.1.13) +The we have the following normal derivative estimates for ℓ ∈ {0, 1, . . . , m − 1}: +∥Tr∂Ω(∂m+ℓ +n +∇Xϕ)∥H−1/2 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m, +(4.1.14) +and +∥Tr∂Ω(∂m+ℓ +n +∇Xϕ)∥H−1/2 ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞⟩∥ϕ∥Hm ++ ⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.1.15) +Here the implicit constants depend on the domain, the dimension, m and ℓ. +Proof. We begin by computing the argument of the trace in the stated estimate. +Fix ℓ ∈ +{0, 1, . . . , m − 1}. According to the Leibniz rule, we have +∂m+ℓ +n +∇Xϕ = +m+ℓ +� +k=0 +�m + ℓ +k +� +∂m+ℓ−k +n +X · ∇∂k +nϕ += +m+ℓ +� +k=0 +�m + ℓ +k +� +∂m+ℓ−k +n +X · (∇∥, 0)∂k +nϕ + +m+ℓ +� +k=0 +�m + ℓ +k +� +∂m+k−ℓ +n +(X · en)∂k+1 +n +ϕ. +(4.1.16) +Applying Tr∂Ω and using that ∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 and X · en = 0 on ∂Ω, we compute +Tr∂Ω(∂m+ℓ +n +∇Xϕ) = +m−1 +� +k=0 +�m + ℓ +k +� +Tr∂Ω(∂m+ℓ−k +n +X · (∇∥, 0)∂k +nϕ) ++ +m−2 +� +k=0 +�m + ℓ +k +� +Tr∂Ω(∂m+ℓ−k +n +(X · en)∂k+1 +n +ϕ) + +� +0 +if ℓ < m − 1, +Tr∂Ω(X · en∂2m +n ϕ) +if ℓ = m − 1. += +m−2 +� +k=0 +�m + ℓ +k +� +Tr∂Ω(∂m+ℓ−k +n +X · ∇∂k +nϕ) + +�m + ℓ +m − 1 +� +Tr∂Ω(∂ℓ+1 +n +X · (∇∥, 0)∂m−1 +n +ϕ) = I + II. +(4.1.17) +We then take H−1/2-norms and handle I and II separately. +We bound I via the embedding H1/2 �→ H−1/2 and the H1 → H1/2 continuity of the trace map: +∥I∥H−1/2 ≲ +m−2 +� +k=0 +∥∂m+ℓ−k +n +X · ∇∂k +nϕ∥H1 ≲ +m−2 +� +k=0 +∥Dm+ℓ−kX ⊗ Dk+1ϕ∥L2 ++ +m−1 +� +k=0 +∥Dm+1+ℓ−kX ⊗ Dk+1ϕ∥L2 = III + IV. +(4.1.18) + +COMPRESSIBLE TRAVELING WAVES +77 +For III and IV we employ the splitting X = X0 + X1 and Corollary D.7: +III ⩽ +m−2 +� +k=0 +� +∥Dm−1−k(D1+ℓX0) ⊗ Dk+1ϕ∥L2 + ∥Dm−1−k(D1+ℓX1) ⊗ Dk+1ϕ∥L2 +� +≲ ∥D1+ℓX0, D1+ℓX1∥H1+⌊n/2⌋,W 1+⌊n/2⌋,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥D1+ℓX0, D1+ℓX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m, +(4.1.19) +and similarly, +IV ⩽ +m−1 +� +k=0 +� +∥Dm−1−k(D2+ℓX0) ⊗ Dk+1ϕ∥L2 + ∥Dm−1−k(D2+ℓX1) ⊗ Dk+1ϕ∥L2 +� +≲ ∥D2+ℓX0, D2+ℓX1∥H1+⌊n/2⌋,W 1+⌊n/2⌋,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥D2+ℓX0, D2+ℓX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.1.20) +Hence, the estimate on I we obtain is +∥I∥H−1/2 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.1.21) +For II we make a perfect divergence to exploit the negative norm in the H−1/2 space, i.e. +∂ℓ+1 +n +X · (∇∥, 0)∂m−1 +n +ϕ = (∇∥, 0) · (∂ℓ+1 +n +X∂m−1 +n +ϕ) − (∇∥, 0) · (∂ℓ+1 +n +X)∂m−1 +n +ϕ. +(4.1.22) +Then again by the continuity of the trace map and Corollary D.7, we find that +∥II∥H−1/2 ≲ ∥∂ℓ+1 +n +X∂m−1 +n +ϕ∥H1 + ∥(∇∥, 0) · (∂ℓ+1 +n +X)∂m−1 +n +ϕ∥H1 ≲ ∥Dℓ+1X ⊗ Dm−1ϕ∥H1 ++ ∥Dℓ+2X ⊗ Dm−1ϕ∥H1 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm ++ +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.1.23) +Combining the estimates for I and II completes the proof of estimate (4.1.14). +For the remaining estimate, we first take 1 + ⌊n/2⌋ < m and use interpolation and Young’s +inequality to bound +∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ ≲ ∥ϕ∥Hm + ∥DX0, DX1∥ +m +m−(1+⌊n/2⌋) +H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥L2 +≲ ∥ϕ∥Hm + ⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.1.24) +Hence, for any m we see that (4.1.15) follows. +□ +As an application of the previous result, we develop the following estimate for the map Bm. As +was the case for Lemma 4.1, the point of this estimate is that the right hand side depends on fewer +derivatives that one might expect from a na¨ıve inspection of the left hand side. +Lemma 4.3 (Another bilinear estimate). Suppose that m ∈ N+ and X ∈ W ∞,∞(Ω; Rn) satisfies +Tr∂Ω(X · en) = 0 and decomposes as X = X0 + X1, where DX0 ∈ H∞(Ω; Rn×n) and DX1 ∈ + +78 +NOAH STEVENSON AND IAN TICE +W ∞,∞(Ω; Rn). Assume that ϕ ∈ H3m(Ω) satisfies the Neumann conditions ∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω. Then +|Bm(∇Xϕ, Lmϕ)| ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩∥ϕ∥H3m∥ϕ∥Hm ++ ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 ++ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩6∥ϕ∥Hm∥ϕ∥L2. +(4.1.25) +Here the implicit constant depends on the domain, the dimension, and m. +Proof. We begin by integrating by parts. Thanks to Lemma C.6, we have the identity +Bm(∇Xϕ, Lmϕ) = +� +Ω +Lm(∇Xϕ)Lmϕ + +m−1 +� +ℓ=0 +� � +Σ +− +� +Σ0 +� +(−1)ℓ∂m+ℓ +n +(∇Xϕ)∂m−1−ℓ +n +Lmϕ. +(4.1.26) +Hence, by trace theory, we have the estimate +|Bm(∇Xϕ, Lmϕ)| ≲ +��� +� +Ω +Lm(∇Xϕ)Lmϕ +��� + +m−1 +� +ℓ=0 +∥Tr∂Ω(∂m+ℓ +n +(∇Xϕ))∥H−1/2∥Lmϕ∥Hm−ℓ = I + II. +(4.1.27) +We will handle I and II separately. +For I we first use ∇Xϕ = ∇ · (ϕX) − ϕ∇ · X to expand +� +Ω +Lm(∇Xϕ)Lmϕ = +� +Ω +∇ · Lm(ϕX)Lmϕ − Lm(ϕ(∇ · X))Lmϕ += +� +Ω +∇ · (XLmϕ)Lmϕ + ∇ · ([Lm, X]ϕ)Lmϕ − Lm(ϕ(∇ · X))Lmϕ, +(4.1.28) +and then employ Proposition B.9 on the first term on the right. This yields the bound +I ⩽ +��� +� +Ω +∇ · X +2 +(Lmϕ)2��� + +��� +� +Ω +∇ · ([Lm, X]ϕ)Lmϕ +��� + +��� +� +Ω +Lm(ϕ(∇ · X))Lmϕ +��� +≲ +� +∥DX∥L∞∥ϕ∥H2m + +n +� +j=1 +∥[∂2m +j +, X]ϕ∥H1 + ∥(∇ · X)ϕ∥H2m +� +∥ϕ∥H2m. +(4.1.29) +We next use Corollary D.7, followed by interpolation and Young’s inequality, to estimate +∥(∇ · X)ϕ∥H2m ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥H2m ++ +� +0 +if 2m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥H2m×W 2m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < 2m, +≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.1.30) +On the other hand, for j ∈ {1, . . . , n}, we use Corollary D.8 and another interpolation and Young +inequality argument to estimate +∥[∂2m +j +, X]ϕ∥H1 ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥H2m ++ +� +0 +if 2m ⩽ 1 + ⌊n/2, +∥DX0, DX1∥H2m×W 2m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < 2m, +≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.1.31) +Upon synthesizing (4.1.29),(4.1.30), and (4.1.31), we learn that +I ≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥2 +H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H2m∥ϕ∥L2. +(4.1.32) + +COMPRESSIBLE TRAVELING WAVES +79 +Now we turn our attention to II. First we input the estimate (4.1.15) from Lemma 4.2: +II ⩽ ∥ϕ∥Hm +m−1 +� +ℓ=0 +⟨∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞⟩∥Lmϕ∥Hm−ℓ ++ ∥ϕ∥L2 +m−1 +� +ℓ=0 +⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥Lmϕ∥Hm−ℓ. +(4.1.33) +Next we utilize the interpolation inequalities +∥Lmϕ∥Hm−ℓ ≲ ∥Lmϕ∥ +ℓ +m−1 +H1 ∥Lmϕ∥ +m−1−ℓ +m−1 +Hm +, +(4.1.34) +∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞ +≲ ∥DX0, DX1∥ +m−1−ℓ +m−1 +H2+⌊n/2⌋×W 2+⌊n/2⌋,∞∥DX0, DX1∥ +ℓ +m−1 +H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞, +(4.1.35) +and +∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞ ≲ ∥DX0, DX1∥ +m−1−ℓ +m−1 +H1+m×W 1+m,∞∥DX0, DX1∥ +ℓ +m−1 +H2m×W 2m,∞, +(4.1.36) +together with Young’s inequality to bound +II ⩽ ∥ϕ∥Hm� +⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩∥Lmϕ∥Hm ++ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩∥Lmϕ∥H1 +� ++ ∥ϕ∥L2 +� +⟨∥DX0, DX1∥H1+m×W 1+m,∞⟩2+⌊n/2⌋∥Lmϕ∥Hm ++ ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥Lmϕ∥H1 +� +. +(4.1.37) +Upon combining (4.1.32) and (4.1.37), we see that +I + II ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩(∥ϕ∥2 +H2m + ∥ϕ∥Hm∥ϕ∥H3m) ++ ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 ++ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩∥ϕ∥Hm∥ϕ∥H2m+1. +(4.1.38) +We then interpolate again: ∥ϕ∥2 +H2m ≲ ∥ϕ∥Hm∥ϕ∥H3m, and if m ⩾ 2 then we use Young’s inequality +to bound +⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩∥ϕ∥H2m+1 +≲ ∥ϕ∥H3m + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩ +3m +m−1 ∥ϕ∥L2. +(4.1.39) +By combining these with (4.1.38), we prove (4.1.25). +□ +4.2. Some results on steady transport equations and elliptic regularizations. With the +lemmas from Section 4.1 in hand, we may now begin to derive the required precise estimates for +steady transport equations and their regularizations. The former is much simpler, and our first +result gives all of the necessary a priori estimates. +Proposition 4.4 (A priori estimates for steady transport). Let m ∈ N. Suppose that ϕ ∈ Hm(Ω), +X ∈ W ∞,∞(Ω; Rn) satisfies (4.0.2) with r = 1+⌊n/2⌋ and 0 < ρ ⩽ ρmax, and that ∇·(Xϕ) ∈ Hm(Ω). +Suppose additionally that Λ ∈ W ∞,∞(Ω) is such that Λ > 0, 1/Λ ∈ L∞(Ω), and +Λϕ + ∇ · (Xϕ) = ψ in Ω. +(4.2.1) + +80 +NOAH STEVENSON AND IAN TICE +There exists a ρ(m) ∈ R+, depending only on m, Λ, and the dimension, such that if 0 < ρ ⩽ ρ(m), +then +∥ϕ∥Hm ≲ ∥ψ∥Hm + +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.2.2) +Here the implicit constant depends only on m, ρ(m), Λ, and the dimension. +Proof. We begin by multiplying (4.2.1) by ϕ and applying Proposition B.9 to obtain the identity +� +Ω +� +Λ + ∇ · X +2 +� +ϕ2 = +� +Ω +ϕψ. +(4.2.3) +Selecting ρ(0) sufficiently small, depending on ∥1/Λ∥L∞, and using Cauchy-Schwarz on the right +hand side, we obtain the a priori estimate +∥ϕ∥L2 ≲ ∥ψ∥L2. +(4.2.4) +Now suppose that m ⩾ 1 and 0 < ρ ⩽ ρ(0). Plugging (4.2.1) into the bilinear form Bm from (4.1.1), +we have the identity Bm(Λϕ + ∇ · (Xϕ), ϕ) = Bm(ψ, ϕ), and hence +Bm(Λϕ, ϕ) ⩽ ∥ψ∥Hm∥ϕ∥Hm + |Bm(∇ · (Xϕ), ϕ)|. +(4.2.5) +By applying G˚arding’s inequality, Lemma C.5, we thus obtain the bound +∥ϕ∥2 +Hm ≲ ∥ψ∥Hm∥ϕ∥Hm + ∥ϕ∥2 +L2 + |Bm(∇ · (Xϕ), ϕ)|. +(4.2.6) +Now we invoke Cauchy’s inequality for the ∥ψ∥Hm∥ϕ∥Hm term, estimate (4.2.4) for the ∥ϕ∥2 +L2 term, +and estimate (4.1.3) from Lemma 4.1 for the final term above; these then yield the bound +∥ϕ∥2 +Hm ≲ ∥ψ∥2 +Hm + ρ∥ϕ∥2 +Hm ++ ∥ϕ∥Hm +� +0 +if m ⩽ 1 + ⌊n/2, +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m. +(4.2.7) +We define ρ(m) ∈ (0, ρ(0)] to be sufficiently small so that when taking ρ ⩽ ρ(m) we can absorb the +right hand side’s ∥ϕ∥2 +Hm-contribution with the left. Once this is done, estimate (4.2.2) follows from +one last application of Cauchy’s inequality to the final term in (4.2.7). +□ +The remainder of this subsection develops corresponding a priori estimates for regularized steady +transport equations, with the right uniformities with respect to the approximation parameters. This +is more complicated than Proposition 4.4 since the theory now has to account for a rather unhappy +marriage of elliptic and hyperbolic structures. Our first result in this direction handles the case of +low regularity, namely L2, data. +Proposition 4.5 (A priori estimate for regularized steady transport with data in L2). Let m ∈ N+ +and Lm be the operator given in (3.5.6). Suppose that ψ ∈ L2(Ω), ϕ ∈ H2m(Ω), X ∈ W ∞,∞(Ω; Rn) +satisfies (4.0.2) with r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, Λ ∈ W ∞,∞(Ω) satisfies Λ > 0 and +1/Λ ∈ L∞(Ω), and N ∈ N+. Assume additionally that the equations +� +N−1ΛLmϕ + ∇ · (Xϕ) = ψ +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω +(4.2.8) +are satisfied. If +N ≳ ⟨∥DX0, DX1∥Hm×W m,∞⟩4+2⌊n/2⌋, +(4.2.9) +then we have the a priori estimate +∥ϕ∥H2m ≲ N∥ϕ, ψ∥L2×L2, +(4.2.10) +where the implied constants depend on m, the dimension, Λ, and ρmax. + +COMPRESSIBLE TRAVELING WAVES +81 +Proof. We multiply the first equation in (4.2.8) by N−1Lmϕ and integrate over Ω. By applying +Lemma C.6 on the ∇ · (Xϕ)-term, we acquire the identity +1 +N2 +� +Ω +Λ(Lmϕ)2 + 1 +N Bm(∇ · (Xϕ), ϕ) = +� +Ω +ψ 1 +N Lmϕ. +(4.2.11) +Now we use the hypotheses on Λ paired with Cauchy’s inequality and absorption: +1 +N2 ∥Lmϕ∥2 +L2 ≲ ∥ψ∥2 +L2 + 1 +N |Bm(∇ · (Xϕ), ϕ)|. +(4.2.12) +We now handle the Bm(∇·(Xϕ), ϕ) term by using estimate (4.1.2) from Lemma 4.1 (with ρ = ρmax) +and then interpolating: +1 +N |Bm(∇ · (Xϕ), ϕ)| ≲ 1 +N ∥ϕ∥2 +Hm + 1 +N ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2 +≲ 1 +N ∥ϕ∥H2m∥ϕ∥L2 + +1 +N1/2 ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋ ∥ϕ∥1/2 +H2m +N1/2 ∥ϕ∥3/2 +L2 . +(4.2.13) +Therefore, if (4.2.9) is satisfied, then +1 +N |Bm(∇ · (Xϕ), ϕ)| ≲ 1 +N ∥ϕ∥H2m∥ϕ∥L2 + ∥ϕ∥1/2 +H2m +N1/2 ∥ϕ∥3/2 +L2 . +(4.2.14) +Upon returning to (4.2.12), we see that +∥Lmϕ∥L2 ≲ N∥ψ∥L2 + N1/2∥ϕ∥1/2 +H2m∥ϕ∥1/2 +L2 + N3/4∥ϕ∥1/4 +H2m∥ϕ∥3/4 +L2 . +(4.2.15) +As the boundary conditions in (4.2.8) are satisfied, we may then invoke the a priori estimate for Lm +from Lemma C.7 to acquire the bound +∥ϕ∥H2m ≲ ∥ϕ, Lmϕ∥L2×L2 ≲ N∥ϕ, ψ∥L2×L2 + N1/2∥ϕ∥1/2 +H2m∥ϕ∥1/2 +L2 + N3/4∥ϕ∥1/4 +H2m∥ϕ∥3/4 +L2 . (4.2.16) +The proof is complete upon applying Young’s inequality in order to absorb the ∥ϕ∥H2m terms from +the right onto the left. +□ +Our next two results consider what happens when the data for a regularized steady transport +equation is of higher regularity than L2. The first of these analyzes the bonus regularity of the +solution gained from the vanishing elliptic term. +Proposition 4.6 (A priori estimate for regularized steady transport with high regularity data, +1). Let m ∈ N+. Suppose that ψ ∈ Hm(Ω), ϕ ∈ H3m(Ω), X ∈ W ∞,∞(Ω; Rn) satisfies (4.0.2) with +r = 2 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, Λ ∈ W ∞,∞(Ω) satisfies Λ > 0 and 1/Λ ∈ L∞(Ω), and N ∈ N+. +Assume additionally that (4.2.8) is satisfied. If +N ≳ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩4+2⌊n/2⌋, +(4.2.17) +then we have the a priori estimate +∥ϕ∥H3m ≲ N∥ψ, ϕ∥Hm×Hm + N⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.2.18) +Here the implied constants depend only on m, the dimension, Λ, and ρmax. +Proof. We rearrange the first equation in (4.2.8) as +N−1ΛLmϕ = ψ − (∇ · X)ϕ − ∇Xϕ +(4.2.19) +and then take the Bm-product of the above equation with N−1Lmϕ. After applying G˚arding’s +inequality, Lemma C.5, followed by the L2-a priori estimate of Proposition 4.5, we find that +N−2∥Lmϕ∥2 +Hm ≲ ∥ϕ, ψ∥2 +L2×L2 + N−1∥ψ, (∇ · X)ϕ∥Hm×Hm∥Lmϕ∥Hm + N−1|Bm(∇Xϕ, Lmϕ)|. +(4.2.20) + +82 +NOAH STEVENSON AND IAN TICE +Hence, by Cauchy’s inequality we get the bound +N−2∥Lmϕ∥2 +Hm ≲ ∥ϕ, ψ∥2 +L2×L2 + ∥ψ, (∇ · X)ϕ∥2 +Hm×Hm + N−1|Bm(∇Xϕ, Lmϕ)|. +(4.2.21) +Now we apply the a priori estimates for Lm of Lemma C.7 to learn that +N−2∥ϕ∥2 +H3m ≲ ∥ϕ, ψ∥2 +L2×L2 + ∥ψ, (∇ · X)ϕ∥2 +Hm×Hm + N−1|Bm(∇Xϕ, Lmϕ)|. +(4.2.22) +By Corollary D.7 and a familiar argument using interpolation and Young’s inequality, we deduce +that +∥(∇ · X)ϕ∥Hm ≲ ∥ϕ∥Hm + +� +0 +if m ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < m, +≲ ∥ϕ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.2.23) +On the other hand, Lemma 4.3 shows that +|Bm(∇Xϕ, Lmϕ)| ≲ ∥ϕ∥H3m∥ϕ∥Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 ++ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩6∥ϕ∥Hm∥ϕ∥L2. +(4.2.24) +Inserting (4.2.23) and (4.2.24) into (4.2.22) and using Cauchy’s inequality then shows that +N−2∥ϕ∥2 +H3m ≲ ∥ϕ, ψ∥2 +Hm×Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩4+2⌊n/2⌋∥ϕ∥2 +L2 ++ N−1⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩6∥ϕ∥2 +Hm, +(4.2.25) +and upon combining this with hypothesis (4.2.17) we deduce the bound +N−2∥ϕ∥2 +H3m ≲ ∥ϕ, ψ∥2 +Hm×Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩4+2⌊n/2⌋∥ϕ∥2 +L2. +(4.2.26) +This is the stated estimate. +□ +Next, we aim to estimate the norm of the solution to a regularized steady transport equation in +the same space as the regular data, but independently of the approximation parameter. +Proposition 4.7 (A priori estimate for regularized steady transport with high regularity data, 2). +Let m, N ∈ N+. Suppose that ψ ∈ Hm(Ω), ϕ ∈ H3m(Ω), X ∈ W ∞,∞(Ω; Rn) satisfies (4.0.2) with +r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax. Further suppose that, for i ∈ {0, 1}, Λi ∈ W ∞,∞(Ω) satisfies +Λi > 0 and 1/Λi ∈ L∞(Ω), and that the equations +� +Λ0ϕ + N−1Λ1Lmϕ + ∇ · (Xϕ) = ψ +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω, +(4.2.27) +are satisfied. There exists a �ρ(m) ∈ R+, depending only on the dimension, m, Λ0, and Λ1 such that +if 0 < ρ ⩽ �ρ(m), then we have the a priori estimate +∥ϕ∥Hm ≲ ∥ψ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2, +(4.2.28) +where the implicit constant depends only on m, �ρ(m), Λ0, Λ1, and the dimension. +Proof. We test the first equation in (4.2.27) with Lmϕ in the L2(Ω) inner product and integrate by +parts via Lemma C.6 to obtain the identity +Bm(Λ0ϕ, ϕ) + N−1∥ +� +Λ1Lmϕ∥2 +L2 = Bm(ψ, ϕ) + Bm(∇ · (Xϕ), ϕ). +(4.2.29) +Hence, Bm(Λ0ϕ, ϕ) ⩽ ∥ψ∥Hm∥ϕ∥Hm + |Bm(∇ · (Xϕ), ϕ)|. This is exactly inequality (4.2.5) from +Proposition 4.4. +We can therefore argue in exactly the same way here to reach the desired +conclusion. +□ + +COMPRESSIBLE TRAVELING WAVES +83 +The final result of this section is the culmination of our steady transport analysis. Essentially, we +interpolate between the low regularity estimate of Proposition 4.5 and the high regularity estimates +of Propositions 4.6 and 4.7. In doing so, we aim to preserve a key structure of the previous estimates: +all norms involving the vector field are multiplied by the lowest regularity norms of the data or +solution. This requires some fine interpolation results, which are proved in Appendix A.3. +Theorem 4.8 (Estimates for regularized steady transport). Let m, N ∈ N+ and j ∈ {1, . . . , m}. +Suppose that ψ ∈ Hj(Ω), ϕ ∈ Hj+2m(Ω), and X ∈ W ∞,∞(Ω; Rn) satisfies (4.0.2) with r = 2+⌊n/2⌋ +and 0 < ρ ⩽ ρmax. Further suppose Λi ∈ W ∞,∞(Ω) satisfies Λi > 0 and 1/Λi ∈ L∞(Ω) for i ∈ {0, 1}, +and that the equations +� +Λ0ϕ + N−1Lmϕ + ∇ · (Λ1Xϕ) = ψ +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω +(4.2.30) +are satisfied. There exists a ρ(m) ∈ R+, depending only on m, Λ0, and Λ1, such that if +max{ρ, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} ⩽ ρ(m) +(4.2.31) +and +N ≳ ⟨∥X0, X1∥H2+⌊n/2⌋+m×W 2+⌊n/2⌋+m,∞⟩4+2⌊n/2⌋, +(4.2.32) +then we have the a priori estimate +∥ϕ, ∇ · (Λ1Xϕ), N−1ϕ∥Hj×Hj×Hj+2m ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2. (4.2.33) +Here the implied constants depend only on Λ0, Λ1, m, the dimension, and ρ(m). +Proof. We let ρ ∈ R+ be such that +max{ρ, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} ⩽ ρ. +(4.2.34) +Throughout the proof we will take ρ to be ever smaller to meet various criteria. Ultimately, we then +define ρ(m) to be the value for ρ we have at the end of the proof. +For ℓ ∈ N, we will also make use of the Banach space +Z(m, ℓ) = {φ ∈ Hℓ+2m(Ω) : Tr∂Ω(∂m +n φ) = · · · = Tr∂Ω(∂2m−1 +n +φ) = 0} +(4.2.35) +and the bounded linear map L : Z(m, ℓ) → Hℓ(Ω) defined via Lϕ = Λ0ϕ + N−1Lmϕ + ∇ · (Λ1Xϕ). +We divide the remainder of the proof into several steps. +Step 1: Establishing invertibility. We claim that L is a Banach isomorphism for every ℓ ∈ N. +This follows from standard elliptic theory arguments once we establish the fact that the bilinear +form B defined by +Hm(Ω) × Hm(Ω) ∋ (ϕ0, ϕ0) B +�→ +� +Ω +Λ0ϕ0ϕ1 + ∇ · (Λ1Xϕ0)ϕ1 + N−1Bm(ϕ0, ϕ1) ∈ R +(4.2.36) +is coercive when ρ is sufficiently small. To prove coercivity, we first use Proposition B.9 to compute +B(ϕ, ϕ) = +� +Ω +� +Λ0 + 2−1∇ · (Λ1X) +� +ϕ2 + N−1Bm(ϕ, ϕ). +(4.2.37) +In light of (4.0.2), (4.2.34), and the Sobolev embeddings, we may bound +∥∇ · (Λ1X)∥L∞(Ω) ⩽ ∥Λ1∇ · X∥L∞(Ω) + ∥∇Λ1 · X∥L∞(Ω) ≲ ρ, +(4.2.38) +where the implicit constant depends on Λ0 and Λ1. Therefore, if we take 0 < ρ ⩽ ρ(0) sufficiently +small, we get that +B(ϕ, ϕ) ≳ ∥ϕ∥2 +L2 + N−1Bm(ϕ, ϕ), +(4.2.39) +and hence Lemma C.5 shows that B is coercive. + +84 +NOAH STEVENSON AND IAN TICE +Step 2: Low-norm estimates on the inverse. Assume that ψ ∈ L2(Ω), ϕ ∈ Z(m, 0), and that +Lϕ = ψ. According to (4.2.39), we have the estimate ∥ϕ∥2 +L2 ≲ B(ϕ, ϕ) = +� +Ω ψϕ, and hence +∥L−1ψ∥L2 = ∥ϕ∥L2 ≲ ∥ψ∥L2. +(4.2.40) +Now we may invoke Proposition 4.5 (the hypotheses of which are satisfied thanks to (4.2.32)) +followed by (4.2.40) to see that ∥L−1ψ∥H2m = ∥ϕ∥H2m ≲ N∥ψ − Λ0ϕ, ϕ∥L2×L2 ≲ N∥ψ∥L2. Thus, +we have shown that L−1 maps L2(Ω) into Z(m, 0) ⊂ H2m(Ω) with the operator bounds +∥L−1ψ, N−1L−1ψ∥L2×H2m ≲ ∥ψ∥L2. +(4.2.41) +Step 3: High-norm estimates on the inverse. Now assume that ψ ∈ Hm(Ω) and ϕ ∈ Z(m, m) ⊂ +H3m(Ω) satisfy Lϕ = ψ. Note that this equation is equivalent to +�Λ0ϕ + ∇ · (Xϕ) + N−1�Λ1Lmϕ = �ψ, +(4.2.42) +where �Λ0 = Λ0/Λ1, �Λ1 = 1/Λ1, and �ψ = (ψ − ϕX · ∇Λ1)/Λ1. We take ρ ⩽ �ρ(m), where the latter is +given by Proposition 4.7, and then apply the proposition to gain the bound +∥L−1ψ∥Hm ≲ ∥ �ψ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ �ψ∥L2. +(4.2.43) +Thanks to Corollary D.7, interpolation, and Young’s inequality, we have that +∥ �ψ∥Hm ≲ ∥ψ∥Hm + ρ∥ϕ∥Hm + ⟨∥X0 · ∇Λ1, X1 · ∇Λ1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2. +(4.2.44) +On the other hand, (4.2.41) provides the bound +∥ �ψ∥L2 ≲ ∥ψ∥L2. +(4.2.45) +We combine (4.2.40), (4.2.43), (4.2.44), and (4.2.45) and then take ρ ⩽ ρ(m) ⩽ ρ(0) to be sufficiently +small so that the right hand side’s ∥ϕ∥Hm-contribution can be absorbed by the left; this results in +the bound +∥L−1ψ∥Hm ≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+m×W 1+m,∞⟩2+⌊n/2⌋∥ψ∥L2. +(4.2.46) +We next apply Proposition 4.6 followed by estimates (4.2.46) and (4.2.41): +N−1∥ϕ∥H3m ≲ ∥ψ − Λ0ϕ, ϕ∥Hm + ⟨∥D(Λ1X0), D(Λ1X1)∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2 +≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2. +(4.2.47) +The culmination of this analysis is that we have shown that L−1 maps Hm(Ω) into Z(m, m) ⊂ H3m(Ω) +with the operator bounds +∥L−1ψ, N−1L−1ψ∥Hm×H3m ≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2. +(4.2.48) +Step 4: Interpolation and conclusion. Let EΩ denote a Stein extension operator for Ω (see Defini- +tion A.2) and let RΩ denote the operator given by restriction to Ω of functions defined on Rn (see Ex- +ample 2.16). We define a map T ∈ L(L2(Rn); H2m(Rn))∩L(Hm(Rn); H3m(Rn)) via the formula T = +EΩL−1RΩ. Thanks to the previous steps, we know that for A = ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋, +T satisfies the operator bounds +� +∥Tψ∥L2 ≲ ∥ψ∥L2 +for all ψ ∈ L2(Rn), +∥Tψ∥Hm ≲ ∥ψ∥Hm + A∥ψ∥L2 +for all ψ ∈ Hm(Rn), +(4.2.49) +and +� +∥Tψ∥H2m ≲ N∥ψ∥L2 +for all ψ ∈ L2(Rn), +∥Tψ∥H3m ≲ N∥ψ∥Hm + NA∥ψ∥L2 +for all ψ ∈ Hm(Rn). +(4.2.50) +We are therefore in a position to apply Proposition A.8 to deduce that T ∈ L(Hj(Rn); Hj+2m(Rn)) +for j ∈ {0, 1, . . . , m}, with the estimates +∥Tψ, N −1Tψ∥Hj×Hj+2m ≲ ∥ψ∥Hj + A∥ψ∥Hj−m for all ψ ∈ Hj(Rn). +(4.2.51) + +COMPRESSIBLE TRAVELING WAVES +85 +By utilizing that L−1 = RΩTEΩ, we can port (4.2.51) to an estimate on L−1, namely: +∥L−1ψ, N −1L−1ψ∥Hj×Hj+2m ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2 +(4.2.52) +for all ψ ∈ Hj(Ω). +It remains to obtain an estimate on ∇ · (Λ1XL−1ψ). For this we rearrange the equation as +∇ · (Λ1Xϕ) = ψ − Λ0ϕ − N−1Lmϕ, take the norm in Hj(Ω) of both sides, and apply the established +estimates of (4.2.52). This yields the bound +∥∇ · (Λ1Xϕ)∥Hj ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2. +(4.2.53) +The proof is complete upon combining (4.2.52) and (4.2.53). +□ +5. Analysis of weak solutions to the principal part linear equations +In this section we study weak solutions to the PDE +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (ϱu) + ∇ · (vw0(q + gη)) = g +in Ω, +−γ2 +0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu = f +in Ω, +−(ϱq − γ0Sϱu)en − ς∆∥ηen = k +on Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0. +(5.0.1) +Here the given data are g : Ω → R, f : Ω → Rn, and k : Σ → Rn, as well as γ0 ∈ R+ and a vector +field vw0 : Ω → Rn defined via a fixed triple w0 = (q0, u0, η0) as in (3.4.11) (see also Lemma 3.19). +The unknowns are q : Ω → R, u : Ω → Rn, and η : Σ → R. In other words, we are interested in the +weak formulation principal part linear operator +w0,γ0 +J , which we recall is defined in (3.5.14). +The above system is not elliptic in the sense of Agmon, Douglis, and Nirenberg [3]. Because +of this and various other linear effects of the derivative loss, we are led to consider the following +regularized version of (5.0.1) with parameters τ ∈ [0, 1], m, N ∈ N+, and m ⩾ 2: +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) = g +in Ω, +−γ2 +0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu = f +in Ω, +−(ϱq − γ0Sϱu)en − ς∆∥ηen = k +on Σ, +u · en + ∂1η = N−1(−∆∥)m−1/4η +on Σ, +u = 0 +on Σ0, +∂m +n q = · · · = ∂2m−1 +n +q = 0 +on ∂Ω, +(5.0.2) +where the linear elliptic operator Lm is defined in (3.5.6). In other words, we are also considering +in this section the weak formulation regularized principal part linear operators +w0,γ0 +J τ +m,N, which are +defined in (3.5.16). +The strategy is as follows. We begin in Section 5.1 by proving a priori estimates for weak solutions +to the systems (5.0.1) and (5.0.2) that are appropriately uniform with respect to the background +solution, N, and τ. In Section 5.2 we develop the existence theory for (5.0.2) from our a priori +estimates and the method of continuity, which is the reason for including the homotopy parameter τ. +As it turns out, the problem with τ = 0 can be solved by taking a two parameter limit of solutions +to similar equations, which we solve with the help of the Lax-Milgram lemma. +5.1. Estimates. In this subsection we prove a priori estimates for weak solutions to (5.0.1) +and (5.0.2). We first require the following technical lemma. + +86 +NOAH STEVENSON AND IAN TICE +Lemma 5.1. Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.17, and let w0 = +(q0, u0, η0) be as in Lemma 3.19. Suppose that η ∈ H0(Σ) has Fourier support in the punctured ball +BRn−1(0, 1) \ {0}. Then we have the estimate +��� +� +Ω +(∇ · vw0)η2��� ≲ ρ∥η∥2 +H0, +(5.1.1) +where the implied constant depends only on the dimension, the various physical parameters, and +ρWD. +Proof. First note that the support hypotheses on η imply that η ∈ H∞(Σ) (see, for instance, +Proposition B.1). Second, we compute ∇ · vw0 = ∇ · (vw0 − ϱ′/ge1), and use this, integration by +parts, Fubini-Tonelli, the fundamental theorem of calculus, and the fact that Tr∂Ω(vw0 · en) = 0 to +rewrite +1 +2 +� +Ω +∇·(vw0 −ϱ′/ge1)η2 = +� +Ω +∇·((vw0 −ϱ′/ge1)η)η = +� +Rn−1 +� +(∇∥, 0)· +� b +0 +(vw0 −ϱ′/ge1)η +� +η. (5.1.2) +In turn, this readily implies that +���1 +2 +� +Ω +(∇ · vw0)η2��� ⩽ +� +(∇∥, 0) · +� b +0 +(vw0 − ϱ′/ge1)η +� +˙H−1[η] ˙H1. +(5.1.3) +By Proposition B.4, specifically (B.1.25), [η] ˙H1 ≲ ∥η∥H0, so it remains to estimate the ˙H−1 term on +the right. For this we use the decomposition (3.4.17) of Lemma 3.19, which allows us to rewrite +(∇∥, 0) · +� b +0 +(vw0 − ϱ′/ge1)η = (∇∥, 0) · +� b +0 +v(1) +q0,u0,η0η + ∂1 +� +η +� b +0 +v(2) +η0 (·, y) · e1 dy +� += I1 + I2. (5.1.4) +We bound I1 by using the fact that the integrand belongs to L2(Ω) (and tacitly using Proposition B.1, +Remark B.3, and the first item of Lemma 3.19): +[I1] ˙H−1 ≲ ∥v(1) +q0,u0,η0η∥L2 ≲ ∥v(1) +q0,u0,η0∥L2∥η∥L∞ ≲ ρ∥η∥H0. +(5.1.5) +We bound I2 using the algebra properties of the specialized Sobolev spaces (see Proposition B.2) +and the second item of Lemma 3.19: +[I2] ˙H−1 ⩽ +���η +� � b +0 +v(2) +η0 (·, y) dy +���� +H0 ≲ +��� +� b +0 +v(2) +η0 (·, y) dy +��� +H0∥η∥H0 ≲ ρ∥η∥H0. +(5.1.6) +Combining these bounds yields (5.1.1). +□ +With the lemma in hand, we are ready to study estimates of weak solutions to the principal part +equations (5.0.1). Recall that the spaces +q0,u0,η0 +X−1 +and Y−1 are defined in equations (3.1.3) and (3.1.6), +while the weak formulation operator +w0,γ0 +J +is defined in (3.5.14). Also recall the surface tension ς +and viscosity µ, λ hypotheses set forth in (1.1.7). +Proposition 5.2 (A priori estimates for weak solutions). Let 0 < ρ ⩽ ρWD, where the latter is +defined in Theorem 3.17, let w0 = (q0, u0, η0) be as in Lemma 3.19, and let γ0 ∈ I, where I ⋐ R+ is +some interval. Suppose that (q, u, η) ∈ +q0,u0,η0 +X−1 +and (g, F) ∈ Y−1 satisfy the equation +w0,γ0 +J (q, u, η) = (g, F). +(5.1.7) +There exists a ρweak ∈ R+ such that if 0 < ρ ⩽ ρweak, then we have the a priori estimate +∥q, u, η∥q0,u0,η0 +X−1 +≲ ∥g, F∥Y−1. +(5.1.8) +The implicit constants and ρweak depend on the physical parameters, the dimension, ρWD, and I. + +COMPRESSIBLE TRAVELING WAVES +87 +Proof. We divide the proof into several steps. +Step 1: Reduction to g = 0. We claim first that it suffices to prove the result under the +specialized assumption that g = 0. Indeed, suppose this special case has been proved and let (q, u, η), +w0 = (q0, u0, η0), γ0, and (g, F) be related as in (5.1.7). With the help of the operator B0 from +Proposition C.2, we define w ∈ 0H1(Ω; Rn) via w = u − ϱ−1B0g. Since B0g ∈ H1 +0(Ω; Rn), we have +that TrΣ(w · en) = TrΣ(u · en) = −∂1η, and hence (q, w, η) ∈ +q0,u0,η0 +X−1 . Recalling that +γ0 +I is defined +in (3.5.12), we calculate that +γ0 +I (q, w, η) = F − +γ0 +I (0, B0g/ϱ, 0) and ∇ · (ϱw) + ∇ · (vw0(q + gη)) = 0. +(5.1.9) +Thus, we can apply the special case to obtain the estimate +∥q, w, η∥q0,u0,η0 +X−1 +≲ ∥F − +γ0 +I (0, B0g/ϱ, 0)∥(0H1)∗ ≲ ∥g, F∥Y−1. +(5.1.10) +To switch from w to u in this bound we use the estimate provided by Proposition C.2, namely +∥u∥H1 ≲ ∥w∥H1 + ∥g∥ ˆH0. By chaining this together with the previous estimate we prove the result +in general, which completes the proof of the claim. In the remaining steps we will prove the result +in the special case that g = 0. +Step 2: A priori bound on u. We claim that the bound +∥u∥2 +H1 ≲ ∥F∥2 +(0H1)∗ + ρ(∥q∥2 +L2 + ∥η∥2 +H3/2) +(5.1.11) +holds. We make the following notational simplification for the Fourier space decompositions of η +from (B.1.4): +ηκ,L = Πκ +Lη and ηκ,H = Πκ +Hη. +(5.1.12) +According to Proposition B.1, we have that ηκ,H ∈ L2(Σ) and ηκ,L ∈ H∞(Σ). We test the equation +γ0 +I (q, u, ηκ,H) = F − +γ0 +I (0, 0, ηκ,L) +(5.1.13) +with u; the left hand side of the resulting identity reads +⟨ +γ0 +I (q, u, ηκ,H), u⟩ = +� +Ω +γ0 +�µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2� +− (q + gηκ,H)∇ · (ϱu) ++ ⟨(gϱ − ς∆∥)ηκ,H, TrΣ(u · en)⟩H−1/2,H1/2, +(5.1.14) +and we next aim to estimate the latter two terms on the right side this expression. To this end, we +recall that +TrΣ(u · en) = −∂1η and − ∇ · (ϱu) = ∇ · (vw0(q + gη)). +(5.1.15) +These allow us to compute +⟨(gϱ − ς∆∥)ηκ,H, TrΣ(u · en)⟩H−1/2,H1/2 = −⟨(gϱ − ς∆∥)ηκ,H, ∂1ηκ,H⟩H−1/2,H1/2 = 0 +(5.1.16) +and (employing the integration by parts trick of Proposition B.9) +� +Ω +−(q + gηκ,H)∇ · (ϱu) = +� +Ω +∇ · vw0 +2 +(q + gηκ,H)2 + ∇ · (vw0ηκ,L)(q + gηκ,H) = I1 + I2. +(5.1.17) +We handle I1 first by expanding +I1 = +� +Ω +∇ · vw0 +2 +(q + gη1,H)2 + g +� +Ω +∇ · vw0(q + gη1,H)(ηκ,H − η1,H) ++ +� +Ω +g2∇ · vw0 +2 +(ηκ,H − η1,H)2 = I1,1 + I1,2 + I1,3. +(5.1.18) +According to the third item of Lemma 3.19 and Proposition B.1, we may estimate +|I1,1| ⩽ ∥∇ · vw0∥L∞∥q + gη1,H∥2 +L2 ≲ ρ(∥q∥2 +L2 + ∥η∥2 +H0) +(5.1.19) + +88 +NOAH STEVENSON AND IAN TICE +and +|I1,2| ≲ ∥∇ · vw0∥L2∥q + gη1,H∥L2∥ηκ,H − η1,H∥L∞ ≲ ρ(∥q∥2 +L2 + ∥η∥2 +H0). +(5.1.20) +For I1,3, we instead use Lemma 5.1: +|I1,3| ≲ ρ∥η∥2 +H0. +(5.1.21) +Upon piecing together the previous three estimates, we deduce that +|I1| ≲ ρ(∥q∥2 +L2 + ∥η∥2 +H0). +(5.1.22) +Now we turn our attention to the term I2. Again, we first decompose +I2 = +� +Ω +∇ · (vw0ηκ,L)(q + gη1,H) + g +� +Ω +∇ · (vw0ηκ,L)(ηκ,H − η1,H) = I2,1 + I2,2. +(5.1.23) +I2,1 is handled via fourth item of Lemma 3.19: +|I2,1| ⩽ ∥∇ · (vw0ηκ,L)∥L2∥q + gη1,H∥L2 ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0). +(5.1.24) +For I2,2 we instead integrate by parts and use Fubini-Tonelli: +|I2,2| = g +��� +� +Rn−1 +� +(∇∥, 0) · +� b +0 +vw0ηκ,L +� +(ηκ,H − η1,H) +��� ≲ +� +(∇∥, 0) · +� b +0 +vw0ηκ,L +� +˙H−1[ηκ,H − η1,H] ˙ +H1. +(5.1.25) +By decomposing vw0 = ϱ′g−1e1 + v(1) +q0,u0,η0 + v(2) +η0 +(see (3.4.17)) and arguing as in the proof of +Lemma 5.1, we acquire the bound +� +(∇∥, 0) · +� b +0 +vw0ηκ,L +� +˙H−1 ≲ ∥ηκ,L∥H0. +(5.1.26) +Hence, +|I2| ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0), +(5.1.27) +and upon combining this with the I1 estimate we deduce that +��� +� +Ω +−(q + gηκ,H)∇ · (ϱu) +��� ≲ ρ(∥q∥2 +L2 + ∥η∥2 +H0) + ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0). +(5.1.28) +With (5.1.28) and (5.1.16) in hand, we return to (5.1.14) to obtain the inequality +� +Ω +γ0 +�µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2� +≲ ∥F − I (0, 0, ηκ,L)∥(0H1)∗∥u∥H1 + ρ(∥q∥2 +L2 + ∥η∥2 +H0) ++ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0). +(5.1.29) +This holds for all κ ∈ (0, 1) and the implicit constant is independent of κ. Thus we may send κ → 0 +and use the fact that, as a consequence of the dominated convergence theorem and the definition of +the norm on the anisotropic Sobolev spaces (B.1.2), ∥ηκ,L∥H0 → 0 to arrive at the bound +γ0 +� +Ω +µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2 ≲ ∥F∥(0H1)∗∥u∥H1 + ρ(∥q∥2 +L2 + ∥η∥2 +H0). +(5.1.30) +Recall that the assumptions on µ, λ ∈ C∞(R+) are that µ > 0 and λ > 0 if n = 2, while µ > 0 +and λ ⩾ 0 if n ⩾ 3. Thus, the bound (5.1.11) follows from (5.1.30), the inclusion γ0 ∈ I ⋐ R+, and +either Proposition A.4 in the case n ⩾ 3 or else Proposition A.3 in the case n = 2. +Step 3: A priori bound on ∇∥η. Next we claim that we have the a priori bound +∥∇∥η∥H1/2 ≲ ∥u∥H1 + ∥F∥(0H1)∗. +(5.1.31) +We first consider the case that surface tension is positive: ς > 0. To prove this we will utilize the +operator B2 from Corollary C.4. Recall from the previous step that for κ ∈ (0, 1) we have the +decomposition η = ηκ,L + ηκ,H defined in (5.1.12). We test identity (5.1.13) with wκ ∈ 0H1(Ω; Rn), +defined via +wκ = −ϱ−1B2(ϱ(b)⟨∇∥⟩−1∆∥ηκ,H), +(5.1.32) + +COMPRESSIBLE TRAVELING WAVES +89 +noting that ∇ · (ϱwκ) = 0, TrΣ(wκ · en) = −⟨∇∥⟩−1∆∥ηκ,H, and +∥wκ∥H1 ≲ ∥⟨∇∥⟩−1∆∥ηκ,H∥ ˙H−1∩H1/2 ≍ ∥⟨∇∥⟩1/2|∇∥|ηκ,H∥L2 ≲ ∥∇∥ηκ,H∥H1/2. +(5.1.33) +The result is the identity +⟨ +γ0 +I (q, u, ηκ,H), wκ⟩ = +� +Ω +γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇wκ + gϱ∇ηκ,L · wκ ++ ⟨(gϱ − ς∆∥)ηκ,H, −⟨∇∥⟩−1∆∥ηκ,H⟩H−1/2,H1/2. +(5.1.34) +As +∥∇∥ηκ,H∥2 +H1/2 ≲ ⟨(gϱ − ς∆∥)ηκ,H, −⟨∇∥⟩−1∆∥ηκ,H⟩H−1/2,H1/2, +(5.1.35) +we obtain the estimate +∥∇∥ηκ,H∥2 +H1/2 ≲ (∥u∥H1 + ∥F∥(0H1)∗ + ∥ηκ,L∥H3/2)∥wκ∥H1. +(5.1.36) +Then (5.1.31) follows from this and (5.1.33) upon sending κ → 0, which is valid since the implicit +constants are independent of κ. This proves the claim in the case of positive surface tension. +Now we consider the case of vanishing surface tension, ς = 0, in dimension n = 2. For this we +simply look to the boundary condition satisfied by u, namely ∂1η = −TrΣ(u · en). Since n = 2 and η +is defined on Σ ≃ R, we have ∂1η = ∇∥η. Therefore we have ∥∇∥η∥H1/2 = ∥TrΣ(u·en)∥H1/2 ≲ ∥u∥H1, +so (5.1.31) holds. +Step 4: A priori bound on q. We claim that we have the a priori bound +∥q∥L2 ≲ ∥u∥H1 + ∥∇∥η∥H1/2 + ∥F∥(0H1)∗. +(5.1.37) +To see this, we first let w = ϱ−1Bq ∈ 0H1(Ω; Rn), where B is constructed in Corollary C.3. By +construction, we have that ∇ · (ϱw) = q and +∥w∥H1 ≲ ∥q∥L2. +(5.1.38) +Then we we test the identity +γ0 +I (q, u, η) = F with w to see that +� +Ω +γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w + gϱ∇η · w − q2 − ς⟨∆∥η, TrΣ(w · en)⟩H−1/2,H1/2 = ⟨F, w⟩(0H1)∗,0H1. +(5.1.39) +Estimate (5.1.37) readily follows from this, (5.1.38), and the estimates established in the previous +steps. This proves the claim. +Step 5: A priori bound on ∂1η. Next we claim that +[∂1η] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0. +(5.1.40) +First, we note that by using the decomposition of vw0 from Lemma 3.19, the continuity equation is +equivalently written as +0 = ∇ · (ϱu + vw0q) + g∇ · (v(1) +q0,u0,η0η) + g∂1(v(2) +η0 · e1η) + ϱ′∂1η. +(5.1.41) +We integrate this in the nth coordinate over (0, b); after recalling the identities TrΣ(u · en) + ∂1η = 0 +and (B.2.11) from Proposition B.10, this results in the equality +ϱ(0)∂1η = (∇∥, 0) · +� b +0 +(ϱu + vw0q + gv(1) +q0,u0,η0η) + g∂1 +� +η +� � b +0 +v(2) +η0 (·, y) · e1 dy +�� +. +(5.1.42) +Hence, we may use the estimates from Lemma 3.19, the fact that +� b +0 v(2) +η0 (·, y) dy has rn−1 as +a band limit, the algebra properties of the anisotropic Sobolev spaces in Proposition B.2, and + +90 +NOAH STEVENSON AND IAN TICE +estimate (B.2.12) from Proposition B.10 to bound +ϱ(0)[∂1η] ˙H−1 ≲ ∥ϱ + vw0q + gv(1) +q0,u0,η0η∥L2 + g +��� +� � b +0 +v(2) +η0 (·, y) dy +� +η +��� +H0 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0. +(5.1.43) +This proves the claim since ϱ(0) > 0. +Step 6: Conclusion. We now synthesize the claims of the previous steps to conclude. First, +we take the bound from (5.1.11) and plug it into the right hand side of (5.1.31); this yields the +inequality +∥∇∥η∥H1/2 ≲ ∥F∥(0H1)∗ + √ρ(∥q∥L2 + ∥η∥H3/2). +(5.1.44) +Second, we take (5.1.44) and (5.1.11) and insert them into the right hand side of (5.1.37) to get +∥q∥L2 ≲ ∥F∥(0H1)∗ + √ρ(∥q∥L2 + ∥η∥H3/2). +(5.1.45) +Now, while heeding to (B.1.25), we sum the estimates (5.1.11), (5.1.40), (5.1.44), and (5.1.45), and +then use (5.1.11) and (5.1.45) on the right hand side; the resulting estimate is +∥q, u, η∥X−1 ≲ ∥F∥(0H1)∗ + √ρ∥q, u, η∥X−1. +(5.1.46) +We choose ρweak ∈ R+ sufficiently small so that when taking ρ ⩽ ρweak we can absorb the X−1 +contribution onto the left side and obtain the clean a priori bound +∥q, u, η∥X−1 ≲ ∥F∥(0H1)∗. +(5.1.47) +It remains to only estimate the L2(Ω)-norm of ∇ · (vw0q) in terms of the data. This is now a simple +matter since we can isolate it in the continuity equation via −∇ · (vw0q) = ∇ · (ϱu) + g∇ · (vw0η) +and then note that the right hand side is controlled by ∥q, u, η∥X−1 (see in particular the fourth +item of Lemma 3.19). +□ +Before our next a priori estimates result, we need a lemma from the theory of regularized steady +transport equations. +Lemma 5.3 (Regularized steady transport lemma). Let 0 < ρ ⩽ ρWD, where the latter is defined +in Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, m, N ∈ N+, τ ∈ [0, 1], ϕ ∈ H2m(Ω), and +ψ ∈ L2(Ω). Suppose that +� +N−1Lmϕ + τ∇ · (vw0ϕ) = ψ +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω, +(5.1.48) +where Lm is defined in (3.5.6). If N ≳ ⟨τ∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, then we have the a priori estimate +∥ϕ∥H2m ≲ N∥ϕ, ψ∥L2×L2, +(5.1.49) +where the implicit constant depends only on the physical parameters, the dimension, m, and ρWD. +Proof. Most of the work in verifying this result has already been executed in Section 4. We invoke +Proposition 4.5 with Λ = 1 and the decomposed vector field X = X0 + X1, where X = τvw0, +X0 = τv(1) +q0,u0,η0, and X1 = τ(v(2) +η0 + g−1ϱ′e1), with v(1) +q0,u0,η0 and v(2) +η0 as in Lemma 3.19. The estimate +(5.1.49) then follows from properties of the vector field vq0,u0,η0 stated in Lemma 3.19, namely: for +s ∈ {1 + ⌊n/2⌋, m} we have ∥DX0, DX1∥Hs×W s,∞ ⩽ ∥X0, X1∥H1+s×W 1+s,∞ ≲ τ⟨∥q0, u0, η0∥Xs⟩. +□ +Our next result studies estimates on weak solutions to the regularized equations (5.0.2). Recall +that the spaces X−1 +m,N, the norms ∥·∥q0,u0,η0 +X−1 +m,N +, and the mappings +w0,γ0 +J τ +m,N are defined in (3.5.7), (3.5.9), +and (3.5.16), respectively. + +COMPRESSIBLE TRAVELING WAVES +91 +Proposition 5.4 (A priori estimates for regularized weak solutions). Let 0 < ρ ⩽ ρWD, where the +latter is from Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, and γ0 ∈ I, where I ⋐ R+ is +some interval. Suppose that m, N ∈ N+, τ ∈ [0, 1], (q, u, η) ∈ X−1 +m,N, and that (g, F) ∈ Y−1 satisfy +the equation +w0,γ0 +J τ +m,N(q, u, η) = (g, F). There exists a ρweak,reg ∈ R+ such that if +0 < ρ ⩽ ρweak,reg and N ≳ ⟨∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, +(5.1.50) +then we have the a priori estimate +∥q, u, η∥q0,u0,η0 +X−1 +m,N +≲ ∥g, F∥Y−1. +(5.1.51) +The implicit constants and ρweak,reg depend on the dimension, physical parameters, m, ρWD, and I. +Proof. We proceed in much the same way as in the proof of Proposition 5.2, breaking to steps that +mirror the structure of the argument used there. +Step 1: Reduction to the case g = 0. We claim that it suffices to prove the result in the special +case that g = 0. Indeed, the exact same argument used in the first step of Proposition 5.2 proves +the claim here. In the remaining steps we will prove the result in the special case that g = 0. +Step 2: A priori bound on u. We claim that we have the a priori bound +∥u∥2 +H1 ≲ ∥F∥2 +(0H1)∗ + ρ(∥q∥2 +L2 + ∥η∥2 +H3/2). +(5.1.52) +To prove the claim we again use the Fourier space decomposition of (5.1.12), which again yields +(5.1.13). We then test (5.1.13) with u to arrive at (5.1.14) as in the proof of Proposition 5.2. Note, +though, that in the present context we have the identities +TrΣ(u · en) = −∂1η + N−1(−∆∥)m−1/4η +(5.1.53) +and +− ∇ · (ϱu) = τ∇ · (vw0(q + gη)) + N−1Lm(q + gη), +(5.1.54) +which are somewhat different from (5.1.15). We insert (5.1.53) and (5.1.54) into (5.1.14) and +integrate by parts to see that +� +Ω +γ0 +�µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2� ++ τ∇ · vw0 +2 +(q + gηκ,H)2 + 1 +N +n +� +j=1 +(∂m +j (q + gηκ,H))2 ++ g +� +Ω +(q + gηκ,H) +� +τ∇ · (vw0ηκ,L) + 1 +N Lm,∥ηκ,L +� ++ 1 +2∥(gϱ − ς∆∥)1/2(∆∥)m/2−1/8ηκ,H∥2 +L2 += ⟨F − I (0, 0, ηκ,L), u⟩(0H1)∗,0H1. +(5.1.55) +By combining (5.1.55), the Korn inequalities from Propositions A.4 and A.3 as well as the end of +the second step of Proposition 5.2, and the fact that τ ∈ [0, 1], we obtain the estimate +∥u∥2 +H1 ≲ +��� +� +Ω +∇ · vw0 +2 +(q + gηκ,H)2��� + g +��� +� +Ω +(q + gηκ,H) +� +τ∇ · (vw0ηκ,L) + 1 +N Lm,∥ηκ,L +���� ++ (∥F∥(0H1)∗ + ∥ηκ,L∥H3/2)∥u∥H1 = I1 + I2 + I3. +(5.1.56) +The term I1 is identical to the term I1 that appeared in (5.1.17), so we may use the same argument +used in the second step of the proof of Proposition 5.2 to arrive at the estimate +I1 ≲ ρ(∥q∥2 +L2 + ∥η∥2 +H0). +(5.1.57) +For the term I2 we bound +I2 ⩽ g +��� +� +Ω +(q + gηκ,H)τ∇ · (vw0ηκ,L) +��� + g +N +��� +� +Ω +(q + gηκ,H)Lm,∥ηκ,L +��� = I2,1 + I2,2. +(5.1.58) + +92 +NOAH STEVENSON AND IAN TICE +Arguing as in (5.1.23)–(5.1.27), we find that +I2,1 ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0). +(5.1.59) +For the remaining piece we exploit the fact that ηκ,L and ηκ,H have disjoint Fourier supports, together +with (B.1.25) to see that +I2,2 = g +N +��� +� +Ω +qLm,∥ηκ,L +��� ≲ 1 +N ∥q∥L2∥Lm,∥ηκ,L∥L2. ≲ 1 +N ∥q∥L2∥ηκ,L∥H0. +(5.1.60) +Finally, we plug (5.1.57), (5.1.59), and (5.1.60) into (5.1.56) and then use Cauchy’s inequality on I3 +to deduce that +∥u∥2 +H1 ≲ ∥F∥2 +(0H1)∗ + ρ(∥q∥2 +L2 + ∥η∥2 +H0) + ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0). +(5.1.61) +The implicit constant is independent of κ, so we may send κ → 0 to obtain (5.1.52) from this. The +claim is proved. +Step 3: A priori bound on ∇∥η. We claim that we have the a priori bound +∥∇∥η∥H1/2 ≲ ∥u∥H1 + ∥F∥(0H1)∗. +(5.1.62) +Indeed, the claim is proved in exactly the same way as the claim from third step of the proof of +Proposition 5.2. +Step 4: A priori bound on q. We claim that we have the a priori bound +∥q∥L2 ≲ ∥u∥H1 + ∥∇∥η∥H1/2 + ∥F∥(0H1)∗. +(5.1.63) +Again, this is proved in exactly the same way as the fourth step in the proof of Proposition 5.2. +Step 5: High norm a priori bounds on η and q. We next claim that we have the bound +∥η1,H∥H2m + ∥q∥H2m ≲ N(∥u∥H1 + ∥q∥L2 + ∥∇∥η∥H1/2 + ρ∥η∥H0). +(5.1.64) +To see this, we begin by rewriting the equation +∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) = 0 +(5.1.65) +with the help of the splitting (5.1.12) with κ = 1: +N−1Lm(q + gη1,H) + τ∇ · (vw0(q + gη1,H) = −∇ · (ϱu) − gτ∇ · (vw0η1,L) − gN−1Lm,∥η1,L. (5.1.66) +As ∂m +n (q + gη1,H) = · · · = ∂2m−1 +n +(q + gη1,H) = 0, we are in a position to apply Lemma 5.3, provided +that N ≳ ⟨∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, which we are free to assume; this yields the bound +∥q + gη1,H∥H2m ≲ N(∥q∥L2 + ∥u∥H1 + ρ∥η∥H0 + ∥∇∥η∥H1/2). +(5.1.67) +Next we obtain a bound on η1,H in H2m(Σ) through the normal trace boundary condition. Indeed, +we test equation (5.1.53) with N−1(−∆∥)m+1/4η1,H to see that +N−2∥(−∆∥)mη1,H∥2 +L2 ⩽ N−1∥TrΣ(u · en)∥H1/2∥(−∆∥)m+1/4η1,H∥H−1/2, +(5.1.68) +and hence +∥η1,H∥H2m ≲ N∥u∥H1. +(5.1.69) +We then obtain (5.1.64) by combining (5.1.67) and (5.1.69). The claim is proved. +Step 6: A priori bounds on ∂1η. We claim that we have the a priori bound +[∂1η] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + N−1∥q∥H2m + ρ∥η∥H0. +(5.1.70) + +COMPRESSIBLE TRAVELING WAVES +93 +As in the fifth step of the proof of Proposition 5.2, we integrate equation (5.1.54) in the nth +coordinate from 0 to b and isolate the non-small ∂1η contributions: +((1 − τ)ϱ(b) + τϱ(0))∂1η = (∇∥, 0) · +� b +0 +(ϱu + τvw0q + τgv(1) +q0,u0,η0η) + τg∂1 +� +η +� b +0 +v(2) +η0 (·, y) · e1 dy +� ++ 1 +N Lm,∥ +� b +0 +(q + gη) + ϱ(b) +N (−∆∥)m−1/4η = I1 + I2 + I3 + I4. +(5.1.71) +Fix κ ∈ (0, 1) and set ηκ = (Π1/κ +L +− Πκ +L)η. We take the L2-inner product of (5.1.71) with |∇∥|−2∂1ηκ. +The terms involving I1 and I2 are estimated via Cauchy-Schwarz +⟨I1 + I2, |∇∥|−2∂1ηκ⟩L2,L2 ≲ [I1 + I2] ˙H−1[∂1ηκ] ˙H−1, +(5.1.72) +but then the argument used in the fifth step of the proof of Proposition 5.2 shows that +[I1] ˙H−1 + [I2] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0, +(5.1.73) +so +⟨I1 + I2, |∇∥|−2∂1ηκ⟩L2,L2 ≲ (∥u∥L2 + ∥q∥L2 + ρ∥η∥H0) [∂1ηκ] ˙H−1. +(5.1.74) +On the other hand, for the term involving I3 we have +⟨I3, |∇∥|−2∂1ηκ⟩L2,L2 = 1 +N +� +|∇∥|−1Lm,∥ +� b +0 +q, |∇∥|∂1ηκ +� +L2,L2 ≲ 1 +N ∥q∥H2m−1[ηκ] +˙ +H−1. +(5.1.75) +while for the I4 term we compute ⟨I4, |∇∥|−2∂1ηκ⟩L2,L2 = 0. All together, these combine to show +that +[∂1ηκ] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + N−1∥q∥H2m + ρ∥η∥H0, +(5.1.76) +and since the implicit constant is independent of κ, we can send κ → 0 to arrive at (5.1.70). The +claim is proved. +Step 7: Conclusion. Finally, we derive the desired estimate (5.1.51) by combining the bounds +from the previous steps and taking ρ to be sufficiently small to absorb, exactly as in the proof of +the sixth step of Proposition 5.2. +□ +5.2. Existence of solutions to the regularization. In this subsection we prove the existence of +weak solutions to the regularized problem (5.0.2) and then quickly deduce qualitative regularity. +First we have our main existence result for weak solutions. +Theorem 5.5 (Existence of regularized weak solutions). Let 0 < ρ ⩽ ρWD, where the latter is +from Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, m, N ∈ N+ with m ⩾ 2, τ ∈ [0, 1], +(g, F) ∈ Y−1, and γ0 ∈ I for some interval I ⋐ R+. If (5.1.50) is satisfied, then there exists a +unique (q, u, η) ∈ X−1 +m,N satisfying +w0,γ0 +J τ +m,N(q, u, η) = (g, F). +(5.2.1) +Proof. In functional analytic terms, we aim to prove that for every τ ∈ [0, 1] the operator +w0,γ0 +J τ +m,N ∈ +L(X−1 +m,N; Y−1), which is well-defined thanks to Lemma 3.23, is an isomorphism. We divide the proof +of this into several steps. +Step 1: Reduction to proving existence with g = 0 and τ = 0. We will achieve the reduction to +τ = 0 through the method of continuity (see, for instance, Theorem 5.2 in Gilbarg and Trudinger [35]). +Indeed, the convex homotopy of operators [0, 1] ∋ τ �→ +w0,γ0 +J τ +m,N ∈ L(X−1 +m,N; Y−1) satisfies τ-uniform a +priori estimates thanks to Proposition 5.4. Thus, the method of continuity guarantees that +w0,γ0 +J τ +m,N is + +94 +NOAH STEVENSON AND IAN TICE +an isomorphism for every τ ∈ [0, 1] if and only if +w0,γ0 +J 0 +m,N is an isomorphism, so we reduce to proving +that +w0,γ0 +J 0 +m,N is an isomorphism. +Next, we note that the a priori estimate of Proposition 5.4 guarantees that +w0,γ0 +J 0 +m,N is injective, so +we further reduce, by way of the bounded inverse theorem, to proving that this map is surjective. +In turn, we reduce to proving the existence of solutions to (5.2.1) with g = 0 and τ = 0 with the +help of the operator B0, exactly as in the first step in the proof of Proposition 5.2. +We have now shown that it suffices to establish the existence of solutions to (5.2.1) with g = 0 +and τ = 0. Thus, in the remainder of the proof we set g = 0 and τ = 0. +Step 2: Existence of a two parameter family of approximate solutions. +We introduce the +approximation parameters M, K ∈ N+ and fix data F ∈ (0H1(Ω; Rn))∗. We then claim that there +exists a collection +{(qM,K, uM,K, ηM,K)}M,K∈N+ ⊂ H2m(Ω) × 0H1(Ω; Rn) × H2m(Σ) +(5.2.2) +satisfying +γ0 +I (qM,K, uM,K, ηM,K) = F, +(5.2.3) +� +N−1� +K−1 + Lm +� +(qM,K + gηM,K) = −∇ · (ϱuM,K) +in Ω, +∂m +n (qM,K + gηM,K) = · · · = ∂2m−1 +n +(qM,K + gηM,K) = 0 +on ∂Ω, +(5.2.4) +and +TrΣ(uM,K · en) + ∂1ηM,K = N−1(M−1 + (−∆∥)m−1/4)ηM,K. +(5.2.5) +The high-level idea for producing these approximate solutions is as follows. First, we note that +equation (5.2.5) determines ηM,K as a function of uM,K. Then equation (5.2.4) determines qM,K as +a function of uM,K and ηM,K. These allow us to rewrite (5.2.3) as an equation relating F and uM,K +alone, and it turns out that this can be solved by utilizing the Lax-Milgram lemma. +To prove the claim we begin by recalling that the space 0H1(Ω; Rn) is defined by (1.7.3) and +defining the bounded linear maps pK : 0H1(Ω; Rn) → H2m(Ω) and ηM : 0H1(Ω; Rn) → H2m(Σ) via +pK(u) = −N(K−1 + Lm)−1(∇ · (ϱu)) and ηM(u) = N(M−1 + (−∆∥)m−1/4 − N∂1)−1TrΣ(u · en). +(5.2.6) +The map ηM is well-defined and bounded in light of the symbol inversion result in Lemma C.9, +while pK is well-defined and bounded by virtue of Lemma C.8. These maps allow us to define the +bilinear form BM,K : 0H1(Ω; Rn) × 0H1(Ω; Rn) → R via +BM,K(u, w) = ⟨ +γ0 +I (pK(u) − gηM(u), u, ηM(u)), w⟩(0H1)∗,0H1. +(5.2.7) +Thanks to integration by parts and the definitions of pK and ηM, we have the equivalent formulation +BM,K(u, w) = +� +Ω +γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w − pK(u)∇ · (ϱw) ++ ⟨(gϱ − ς∆∥)ηM(u), TrΣ(w · en)⟩H−1/2,H1/2 += +� +Ω +γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w + 1 +N +� +Ω +1 +K pK(u)pK(w) + +n +� +j=1 +∂m +j pK(u)∂m +j pK(w) ++ ⟨(gϱ − ς∆∥)ηM(u), −∂1ηM(w) + N−1(M−1 + (−∆∥)m−1/4)ηM(w)⟩H−1/2,H1/2. +(5.2.8) +The boundedness of BM,K is straightforward to check, but it is also coercive since by anti-symmetry +we have +� +Ω +γ2 +0ϱu ⊗ e1 : ∇u = 0, +(5.2.9) + +COMPRESSIBLE TRAVELING WAVES +95 +by orthogonality we have +� +Ω +Sϱu : ∇u = +� +Ω +µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2, +(5.2.10) +by squaring we have +1 +N +� +Ω +1 +K pK(u)2 + +n +� +j=1 +(∂m +j pK(u))2 ⩾ 0, +(5.2.11) +and by anti-symmetry and symmetry considerations again we have +⟨(gϱ − ς∆∥)ηM(u), −γ∂1ηM(w) + N−1(M−1 + (−∆∥)m−1/4)ηM(w)⟩H−1/2,H1/2 += ⟨(gϱ − ς∆∥)ηM(u), N−1(M−1 + (−∆∥)m−1/4)ηM(u)⟩ ⩾ 0, +(5.2.12) +which together with the Korn inequalities of Propositions A.4 and A.3 imply the coercivity estimate +BM,K(u, u) ⩾ γ0 +� +Ω +µ(ϱ) +2 +|D0u|2 + λ(ϱ)(∇ · u)2 ≳ ∥u∥2 +H1 . +(5.2.13) +Recall that our conditions on µ, λ ∈ C∞(R+) are that µ, λ > 0 if n = 2, and µ > 0, λ ⩾ 0 if n ⩾ 3. +The hypotheses of the Lax-Milgram lemma (see, for instance Theorem 6 in Chapter 6 of Lax [58]) +are satisfied, so we are therefore granted uM,K ∈ 0H1(Ω; Rn) with the property that +BM,K(uM,K, w) = ⟨F, w⟩(0H1)∗,0H1 for all w ∈ 0H1(Ω; Rn). +(5.2.14) +We then set ηM,K = ηK(uM,K) ∈ H2m(Σ) and qM,K = pK(uM,K) − gηM(uM,K) ∈ H2m(Ω). By +construction, the collection {(qM,K, uM,K, ηM,K)}M,K∈N+ satisfies the claimed inclusions and satisfies +(5.2.3), (5.2.4), and (5.2.5). +Step 3: Estimates on the two parameter family of approximate solutions. We claim that the +approximate solutions (5.2.2) obey the K-independent bounds +∥qM,K, uM,K, ηM,K∥H2m×H1×H2m ≲m,M,N ∥F∥(0H1)∗. +(5.2.15) +To prove the claim, we first take w = uM,K in (5.2.14) and then use the coercive inequality (5.2.13) +to get the control +sup +M,K∈N+∥uM,K∥H1 ≲ ∥F∥(0H1)∗, +(5.2.16) +where the implied constant only depends on the physical parameters. Next, we employ the continuity +of the map ηM to see that +∥ηM,K∥H2m ≲M N∥uM,N∥H1 ≲M N∥F∥(0H1)∗. +(5.2.17) +Obtaining a K-independent bound on the qM,K is slightly more involved. We test the weak +formulation (5.2.14) with w = ϱ−1B(qM,K + gηM,K) ∈ 0H1(Ω; Rn), where B is the right inverse to +the divergence constructed in Corollary C.3. Since ∇ · (ϱw) = qM,K + gηM,K = pK(uM,K), the +identity BM,K(uM,K, w) = ⟨F, w⟩ and the bounds (5.2.16) and (5.2.17) imply that +∥qM,K + gηM,K∥L2 ≲ ∥uM,K∥H1 + ∥ηM,K∥H3/2 + ∥F∥(0H1)∗ ≲N,M ∥F∥(0H1)∗. +(5.2.18) +To get a higher-regularity estimate, we rewrite the first equation in (5.2.4) as (1 + Lm)(qM,K + +gηM,K) = K−1 +K (qM,K + gηM,K) − N∇ · (ϱuM,K) and then apply Lemma C.8 with κ = 1 (and the +already established K-independent bounds) to arrive at the estimate +∥qM,K + gηM,K∥H2m ≲m (1 − K−1)∥qM,K + gηM,K∥L2 + N∥uM,K∥H1 ≲m,M,N ∥F∥(0H1)∗. (5.2.19) +Then (5.2.17) and (5.2.19) imply that ∥qM,K∥H2m ≲m,M,N ∥F∥(0H1)∗, and we then combine this +with (5.2.16) and (5.2.17) to complete the verification of (5.2.15), and hence the proof of the claim. + +96 +NOAH STEVENSON AND IAN TICE +Step 4: Existence of a one parameter family of approximate solutions. With the fixed data +F ∈ (0H1(Ω; Rn))∗ as in the previous steps, we claim that there exists a sequence +{(qM, uM, ηM)}M∈N+ ⊂ H2m(Ω) × 0H1(Ω; Rn) × H2m(Σ) +(5.2.20) +such that +γ0 +I (qM, uM, ηM) = F, +(5.2.21) +� +N−1Lm(qM + gηM) = −∇ · (ϱuM) +in Ω, +∂m +n (qM + gηM) = · · · = ∂2m−1 +n +(qM + gηM) = 0 +on ∂Ω, +(5.2.22) +and +TrΣ(uM · en) + ∂1ηM = N−1(M−1 + (−∆∥)m−1/4)ηM. +(5.2.23) +The existence of this sequence follows by taking a weak subsequential limit in the K parameter in +our previously constructed collection. More precisely, for each fixed M ∈ N+ we have established +in the previous step that the K-independent bounds (5.2.15) hold. Thus, by weak compactness, +there exist (qM, uM, ηM) ∈ H2m(Ω) × 0H1(Ω; Rn) × H2m(Σ) that are a weak subsequential limit +of the sequence {(qM,K, uM,K, ηM,K)}K∈N+. Routine weak convergence arguments applied to the +identities (5.2.3), (5.2.4), and (5.2.5) then show that (5.2.21), (5.2.22), and (5.2.23) hold. This +completes the construction and the proof of the claim. +Step 5: Estimates on the one parameter family of approximate solutions. We claim that the one +parameter family of approximate solutions from (5.2.20) obeys the M-independent bounds +∥qM, uM, ηM∥H2m×H1×H2m ≲m,N ∥F∥(0H1)∗. +(5.2.24) +To see this, we first employ the weak sequential lower semicontinuity of the norm and (5.2.16) to +obtain the estimate +sup +M∈N+∥uM∥H1 ≲ ∥F∥(0H1)∗. +(5.2.25) +Next, as in the third step of the proof of Proposition 5.2, in the case that ς > 0, we test (5.2.21) +against the function wM = −ϱ−1B2(ϱ(b)⟨∇∥⟩−1∆∥ηM) ∈ 0H1(Ω; Rn), where B2 is defined in Corol- +lary C.4. This yields the identity +⟨F, wM⟩(0H1)∗,0H1 = +� +Ω +γ0(γ0ϱuM ⊗e1 +SϱuM) : ∇wM +⟨(gϱ−ς∆∥)ηM, −⟨∇∥⟩−1∆∥ηM⟩H−1/2,H1/2, +(5.2.26) +which in turn yields the estimate +∥∇∥ηM∥2 +H1/2 ≲ (∥F∥(0H1)∗ + ∥uM∥H1)∥wM∥H1 ≲ ∥F∥(0H1)∗∥wM∥H1. +(5.2.27) +The continuity of B2 shows that ∥wM∥H1 ≲ ∥∇∥ηM∥H1/2, and hence +∥∇∥ηM∥H1/2 ≲ ∥F∥(0H1)∗. +(5.2.28) +On the other hand, if ς = 0 and n = 2, estimate (5.2.28) follows by testing (5.2.23) with +∂1⟨∇∥⟩ηM = ∇∥⟨∇∥⟩ηM ∈ H−1/2(Σ) and using orthogonality, Cauchy-Schwarz, boundedness of +traces, and (5.2.25). +Now, as in the fourth step of Proposition 5.2, we employ wM = ϱ−1Bq ∈ 0H1(Ω; Rn), where B is +constructed in Corollary C.3, as a test function in (5.2.21) in order to see that +⟨F, wM⟩(0H1)∗,0H1 = +� +Ω +γ0(γ0ϱuM ⊗ e1 + SϱuM) : ∇wM + gϱ∇ηM · wM − +� +Ω +q2 +M +− ς⟨∆∥ηM, TrΣ(wM · en)⟩. +(5.2.29) +From this we readily deduce the bound ∥qM∥2 +L2 ≲ (∥F∥(0H1)∗ +∥uM∥H1 +∥∇∥ηM∥H1/2)∥wM∥H1, but +this combines with our already established bounds and the continuity estimate ∥wM∥H1 ≲ ∥qM∥L2 +to show the low regularity bound ∥qM∥L2 ≲ ∥F∥(0H1)∗. + +COMPRESSIBLE TRAVELING WAVES +97 +We now have low regularity estimates on qM and ηM. To promote these to high regularity +bounds, we proceed as in the fifth step of the proof of Proposition 5.4. The first identity in (5.2.22) +is equivalent to N−1Lm(qM + gηH +M) = −∇ · (ϱuM) − gN−1Lm,∥ηL +M, where we have decomposed +ηL +M = Π1 +LηM and ηH +M = Π1 +HηM. Since ∂m +n (qM + gηH +M) = · · · = ∂2m−1 +n +(qM + gηH +M) = 0, we can apply +Lemma 5.3 with τ = 0 and exploit the Fourier supports of ηL +M and ηH +M to obtain the estimate +∥qM + gηH +M∥H2m ≲ N(∥uM∥H1 + ∥Lm,∥ηL +M∥L2 + ∥qM + gηH +M∥L2) +≲ N(∥uM∥H1 + ∥∇∥ηM∥H1/2 + ∥qM∥L2). +(5.2.30) +Now we test the identity (5.2.23) against N−1(−∆∥)m+1/4ηH +M ∈ H−1/2(Σ) and employ the Fourier +support of ηH +M as well as the bound (5.2.24) to arrive at the estimate +∥ηH +M∥H2m ≲ N∥uM∥H1 ≲ N∥F∥(0H1)∗. +(5.2.31) +Together, (5.2.30) and (5.2.31) imply that +∥qM, ηH +M∥H2m×H2m ≲ N∥F∥(0H1)∗. +(5.2.32) +In light of Proposition B.1 and equation (B.1.25), it remains only to obtain M-uniform bounds +on [∂1ηM] ˙H−1. First, we integrate (5.2.22) over (0, b) in the nth-coordinate and recall (5.2.23) to +acquire the equality +ϱ(b)∂1ηM = 1 +N Lm,∥ +� b +0 +(qM + gηM) + ϱ(b) +N +� 1 +M + (−∆∥)m−1/4� +ηM + (∇∥, 0) · +� b +0 +ϱuM. +(5.2.33) +For κ ∈ (0, 1) we write ηκ +M = (Π1/κ +L +− Πκ +L)ηM ∈ H∞(Σ) and then take the L2 inner product of +(5.2.33) with |∇∥|−2∂1ηκ +M ∈ H∞(Σ). The right hand side ηM terms all vanish due to the ∂1 operator, +and we arrive at the equality +ϱ(b)[∂1ηκ +M]2 +˙H−1 = +�|∇∥|−1 +N +Lm,∥ +� b +0 +qM + |∇∥|−1(∇∥, 0) · +� b +0 +ϱuM, |∇∥|−1∂1ηκ +M +� +L2,L2, +(5.2.34) +from which we readily deduce that +[∂1ηM] ˙H−1 = lim +κ→0[∂1ηκ +M] ˙H−1 ≲m N−1∥qM∥H2m−1 + ∥uM∥L2 ≲m ∥F∥(0H1)∗. +(5.2.35) +Then (5.2.24) follows by combining (5.2.25), (5.2.32), and (5.2.35) and recalling the equivalent norm +on H2m(Σ) given in (B.1.25). The claim is proved. +Step 6: Conclusion. The M-uniform bounds (5.2.24) guarantee the existence of a weak subse- +quential limit for the sequence (5.2.20), say (q, u, η) ∈ H2m(Ω) × 0H1(Ω; Rn) × H2m(Σ). Routine +weak convergence arguments applied to the identities (5.2.21), (5.2.22), and (5.2.23) then reveal +that (q, u, η) satisfy +γ0 +I (q, u, η) = F, +� +N−1Lm(q + gη) = −∇ · (ϱu) +in Ω, +∂m +n (q + gη) = · · · = ∂2m−1 +n +(q + gη) = 0 +on ∂Ω, +(5.2.36) +and TrΣ(u·en)+∂1η = N−1(−∆∥)m−1/4η. Therefore (q, u, η) ∈ X−1 +m,N satisfy +w0,γ0 +J 0 +m,N(q, u, η) = (0, F), +and so the proof is complete in light of the first step. +□ +As a consequence of the existence of weak solutions to the regularization, we now show that the +operators associated to the strong formulation of the regularization, namely +w0,γ0 +Am,N defined in (3.5.10), +are automatically isomorphisms. The catch is that at this point we cannot guarantee the inverses +come with estimates independent of N, so will will have to work harder in subsequent sections to +verify this. + +98 +NOAH STEVENSON AND IAN TICE +Corollary 5.6 (Isomorphisms induced by the regularization). Let 0 < ρ ⩽ ρWD, where the latter is +from Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, m, N ∈ N+ with m ⩾ 2, and γ0 ∈ I for +an interval I ⋐ R+. Suppose that (5.1.50) holds. Then for ν ∈ N+, the map +w0,γ0 +Am,N : Xν +m,N → Yν +(5.2.37) +defined by (3.5.11) is a Banach isomorphism. +Proof. That the map (5.2.37) is well-defined is a consequence of the third item of Lemma 3.23. +We begin by proving injectivity. Suppose that (q, u, η) ∈ ker +w0,γ0 +Am,N ⊆ Xν +m,N. Lemma 3.24 shows +that strong solutions are weak solutions, and hence +w0,γ0 +J 1 +m,N(q, u, η) = (0, 0). We then deduce that +(q, u, η) = 0 by invoking the a priori estimates of Proposition 5.4. +We now turn to the proof of surjectivity. Suppose that +(g, f, k) ∈ Yν, +(5.2.38) +and, by utilizing Theorem 5.5 and the map from (3.5.23), define (q, u, η) via +(q, u, η) = +� w0,γ0 +J 1 +m,N +�−1(g, K (f, k)) ∈ X−1 +m,N. +(5.2.39) +We claim that we have the higher-regularity inclusion (q, u, η) ∈ Xν +m,N. Once this is shown, another +application of Lemma 3.24 reveals that +w0,γ0 +Am,N(q, u, η) = (g, f, k), which establishes surjectivity. +To prove the claim we will employ a finite induction argument to promote the regularity of the +triple (q, u, η) one step at a time. To this end, it is useful to unpack the definition of weak solution +in a more helpful way. Equation (5.2.39) is equivalent to: (q, u, η) ∈ X−1, +� +N−1Lmq = g − ∇ · (vw0(q + gη)) − ∇ · (ϱu) − gN−1Lm,∥η +in Ω, +∂m +n q = · · · = ∂2m−1 +n +q = 0 +on ∂Ω, +(5.2.40) +N−1(−∆∥)m−1/4Π1 +Hη = −N−1(−∆∥)m−1/4Π1 +Lη + ∂1η + TrΣ(u · en), +TrΣ0(u) = 0, +(5.2.41) +and for all w ∈ 0H1(Ω; Rn) it holds that +� +Ω +γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w = ⟨ �F, w⟩(0H1)∗,0H1, +(5.2.42) +where �F = K ( �f, �k) for �f = f − ϱ∇(q + gη) and �k = k + (ϱq + ς∆∥η)en. +The identity (5.2.42) means, in other words, that u is a weak solution to the elliptic boundary +value problem +� +� +� +� +� +−γ2 +0ϱ∂1u − γ0∇ · Sϱu = �f +in Ω, +γ0Sϱuen = �k +on Σ, +u = 0 +on Σ0. +(5.2.43) +From the inclusions (5.2.38) and (q, η) ∈ H2m(Ω) × H2m(Σ) and the norm (B.1.25) we deduce that +�f ∈ Hmin{ν,2m−1}(Ω; Rn) and �k ∈ Hmin{1/2+ν,2m−2}(Σ; Rn), +(5.2.44) +and so the standard elliptic regularity gain for the problem (5.2.43) (see, for instance, Agmon, +Douglis, and Nirenberg [3]) guarantees the inclusion +u ∈ Hmin{2+ν,2m−1/2}(Ω; Rn) �→ H2(Ω; Rn), +(5.2.45) +where the embedding holds since m ⩾ 2. With the improved u regularity from (5.2.45) in hand, +we return to (5.2.41) to see that Π1 +Hη ∈ H1+2m(Σ), and hence, by Proposition B.1, η ∈ H1+2m(Σ). +In turn, we use use the improved u and η regularity together with the fourth item of Lemma 3.19 +in (5.2.40), appealing to Lemma C.7 to deduce the improvement q ∈ H1+2m(Ω). + +COMPRESSIBLE TRAVELING WAVES +99 +Proceeding by finite induction, we assume now that for some 0 ⩽ ν ⩽ ν − 1 we have the inclusion +(q, u, η) ∈ Hν+1+2m(Ω) × Hν+2(Ω; Rn) × Hν+1+2m(Σ). +(5.2.46) +This implies that �f ∈ Hmin{ν,ν+2m}(Ω; Rn) and �k ∈ Hmin{1/2+ν,ν−1+2m}(Σ; Rn), and so elliptic +regularity for (5.2.43) implies that u ∈ Hmin{2+ν,ν+1/2+2m}(Ω; Rn) �→ Hν+3(Ω; Rn). +We then +argue exactly as above to use the improved u regularity to promote to η ∈ Hν+2+2m(Σ) and +q ∈ Hν+2+2m(Ω). Thus, (5.2.46) holds with ν replaced by ν + 1. By finite induction, (5.2.46) then +also holds for ν = ν, and so (q, u, η) ∈ Xν +m,N, which completes the proof of the claim. +□ +6. Analysis of strong solutions to the linearization +Previously, we have established estimates for weak solutions to (5.0.1) and (5.0.2), and for the +latter we proved existence and qualitative regularity in Corollary 5.6. The majority of this section is +devoted to building a series of tools to aid in the estimation of the higher regularity norms of these +weak solutions when given sufficiently regular data. In other words, we require a specific quantitative +understanding of the regularity for systems (5.0.1) and (5.0.2); for the former we will develop tame +estimates of the solution norms with respect to the background w0 = (q0, u0, η0) and γ0, while for the +latter we will prove high regularity bounds that are independent of the approximation parameter. +We proceed as follows. Section 6.1 analyzes the equations satisfied by the tangential derivatives of +the solutions to systems (5.0.1) and (5.0.2). Section 6.2 reduces the regularity promotion of solutions +to estimates on tangential derivatives via analysis of the so-called normal system. Section 6.3 +combines results from the tangential derivative and normal system analysis to derive the sought-after +precise a priori estimates for the principal part equations. We then conclude in Section 6.4 by +deducing the existence of strong solutions to (5.0.1) and then to the full linearization. +6.1. Analysis of tangential derivatives. In order to study the tangential derivatives of solutions +to (5.0.1) and (5.0.2), we must first understand the commutators of the operators +w0,γ0 +A , +w0,γ0 +Am,N, +w0,γ0 +J , +and +w0,γ0 +J 1 +m,N with the tangential derivatives ∂j for j ∈ {1, . . . , n − 1}. These operators are nearly +tangentially translation invariant, and so nearly commute with the tangential derivatives; the failure +of each to commute is precisely due to the appearance of ∇ · (vw0(q + gη)) in their continuity +equations. The following definition and subsequent lemma capture this almost tangential translation +invariance. We recall that the ˆHs(Ω) spaces are defined in (C.1.1). +Definition 6.1 (The principal part commutator). Let 0 < ρ ⩽ ρWD, where the latter is defined in +Theorem 3.17, and let w0 = (q0, u0, η0) be as in Lemma 3.19. For j ∈ {1, . . . , n − 1} we define the +map +w0 +C j : H1+s(Ω) × H1+s(Σ) → ˆHs(Ω) via +w0 +C j(q, η) = ∇ · (∂jvw0(q + gη)). +We now check that the C maps are well-defined and then explore their mapping properties. +Lemma 6.2 (Mapping properties of C ). Under the hypotheses of Definition 6.1, we have the +following estimates for s ∈ N: +�� +w0 +C j(q, η) +�� ˆHs ≲ ρ∥q, η∥H1+s×H1+s + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ +if ⌊n/2⌋ < s. (6.1.1) +Here the implicit constant depends only on s, the physical parameters, the dimension, and ρWD. +Proof. We first use Lemma 3.19 (in particular the splitting (3.4.17)) to compute +∂jvw0 = ∂jv(1) +q0,u0,η0 + ∂jv(2) +η0 . +(6.1.2) +Then, given N ∋ s ⩾ 1 + ⌊n/2⌋, we use the first item of Lemma 3.19 to estimate +∥∂jv(1) +q0,u0,η0∥Hs ⩽ ∥v(1) +q0,u0,η0∥H1+s ≲ ∥q0, u0, η0∥Xs. +(6.1.3) + +100 +NOAH STEVENSON AND IAN TICE +For the other piece, we may use the second item of Lemma 3.19 to deduce that v(2) +η0 = (v(2) +η0 · e1)e1 +and that v(2) +η0 has rn−1 as a band-limit. This, in addition to the second item of the aforementioned +lemma, (B.1.25), and Proposition B.1, allow us to estimate +∥∂jv(2) +η0 ∥Hs ≲ sup +0⩽p⩽s +sup +y∈[0,b] +∥∂j∂p +nv(2) +η0 (·, y)∥Hs−p(Rn−1) ⩽ sup +0⩽p⩽s +sup +y∈[0,b] +∥∂p +nvη0(·, y) · e1∥H0 ≲ ∥Π1 +Lη0∥H0. +(6.1.4) +Together, (6.1.2), (6.1.3), and (6.1.4) imply the bound ∥∂jvw0∥Hs ≲ ∥q0, u0, η0∥Xs for s ⩾ 1 + ⌊n/2⌋. +With this established, proving the stated estimates is a simple application of Corollary D.7. Indeed, +it shows that for any s ∈ N and (q, η) ∈ H1+s(Ω) × H1+s(Σ), we have +�� +w0 +C j(q, η) +�� +Hs ≲ ∥∂jvw0(q + gη)∥H1+s ≲ ρ∥q, Π1 +Hη, Π1 +Lη∥H1+s×H1+s×W 1+s,∞ ++ +� +0 +if s < ⌊n/2⌋, +∥∂jvq0,u0,η0∥H1+s∥q, Π1 +Hη, Π1 +Lη∥H1+⌊n/2⌋×H1+⌊n/2⌋×W 1+⌊n/2⌋,∞ +if ⌊n/2⌋ ⩽ s, +≲ ρ∥q, η∥H1+s×H1+s + +� +0 +if s < ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ +if ⌊n/2⌋ ⩽ s. +(6.1.5) +Additionally, since Tr∂Ω(vw0 · en) = 0 and ∂jvw0 ∈ (L∞ ∩ L2)(Ω), we have the estimate +�� b +0 +w0 +C j(q, η)(·, y) dy +� +˙H−1 ≲ ∥∂jvw0(q + gη)∥L2 ≲ ρ∥q, Π1 +Hη, Π1 +Lη∥L2×L2×L∞ ≲ ρ∥q, η∥L2×H0. +(6.1.6) +Then (6.1.1) follows by combining (6.1.5) and (6.1.6). +□ +We now apply the mapping properties of C to obtain low regularity estimates for first order +tangential derivatives. +Proposition 6.3 (Low norm estimates on tangential derivatives). Under the hypotheses of Defini- +tion 6.1, the following hold for γ0 ∈ I with I ⋐ R+ an interval, (g, f, k) ∈ Y0, and j ∈ {1, . . . , n−1}. +(1) If (q, u, η) ∈ +q0,u0,η0 +X0 +satisfies +w0,γ0 +A (q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ +q0,u0,η0 +X−1 +and obeys +the estimate +�� +w0,γ0 +J (∂jq, ∂ju, ∂jη) +�� +Y−1 ≲ ρ∥q, u, η∥X0 + ∥g, f, k∥Y0. +(6.1.7) +(2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ X0 +m,N satisfy +w0,γ0 +Am,N(q, u, η) = (g, f, k), then +(∂jq, ∂ju, ∂jη) ∈ X−1 +m,N and obeys the estimate +�� +w0,γ0 +J 1 +m,N(∂jq, ∂ju, ∂jη) +�� +Y−1 ≲ ρ∥q, u, η∥X0 + ∥g, f, k∥Y0. +(6.1.8) +Here the implicit constants depend on the dimension, the physical parameters, ρWD, and I. +Proof. We will prove only the first item; the proof of the second follows from a nearly identical +argument. We begin with three observations. +First, recall the bounded linear map K : L2(Ω; Rn) × H1/2(Σ; Rn) → (0H1(Ω; Rn))∗ defined by +(3.5.23). For each j ∈ {1, . . . , n − 1} we construct a related bounded linear map Kj : L2(Ω; Rn) × +H1/2(Σ; Rn) → (0H1(Ω; Rn))∗ as follows. For (f, k) ∈ L2(Ω; Rn)×H1/2(Σ; Rn) we initially define the +linear map Kj(f, k) : 0H1(Ω; Rn)∩H2(Ω; Rn) → R via ⟨Kj(f, k), w⟩ = ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 = +− +� +Ω f · ∂jw − +� +Σ k · ∂jw. For such f, k, and w, we may bound +|⟨Kj(f, k), w⟩| ⩽ ∥f∥L2∥∂jw∥L2 + ∥k∥H1/2∥∂jTrΣw∥H−1/2 ≲ ∥f, k∥L2×H1/2∥w∥H1, +(6.1.9) + +COMPRESSIBLE TRAVELING WAVES +101 +and from this and the density of 0H1(Ω; Rn) ∩ H2(Ω; Rn) in 0H1(Ω; Rn) we deduce that Kj(f, k) +uniquely extends to an element Kj(f, k) ∈ (0H1(Ω; Rn))∗ such that +⟨Kj(f, k), w⟩(0H1)∗,0H1 = ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 +(6.1.10) +for all w ∈ H2(Ω; Rn) ∩ 0H1(Ω; Rn) and ∥Kj(f, k)∥(0H1)∗ ≲ ∥f, k∥L2×H1/2. In particular, the latter +estimate shows that the induced map Kj is bounded and linear with the domain and codomain +stated above. +Second, we recall the map +γ0 +I : X−1 → (0H1(Ω; Rn))∗ defined by (3.5.12). Suppose that (q, u, η) ∈ +X0. For j ∈ {1, . . . , n − 1} we have that (∂jq, ∂ju, ∂jη) ∈ X−1, which means that +γ0 +I (∂jq, ∂ju, ∂jη) +defines an element of (0H1(Ω; Rn))∗. +We may compute the action of this functional on any +w ∈ H2(Ω; Rn) ∩ 0H1(Ω; Rn) by integrating by parts: +⟨ +γ0 +I (q, u, η), −∂jw⟩(0H1)∗,0H1 = − +� +Ω +−γ2 +0ϱ∂1u · ∂jw − q∇ · (ϱ∂jw) + gϱ∇η · ∂jw + γ0Sϱu : ∇∂jw ++ ς⟨∆∥η, TrΣ(∂jw · en)⟩H−1/2,H1/2 = +� +Ω +−γ2 +0ϱ∂1∂ju · w − ∂jq∇ · (ϱw) + gϱ∇∂jη · w ++ +� +Ω +γ0Sϱ∂ju : ∇w − ς⟨∆∥∂jη, TrΣ(w · en)⟩H−1/2,H1/2 = ⟨I (∂jq, ∂ju, ∂jη), w⟩(0H1)∗,0H1. +(6.1.11) +For the third observation, suppose that q ∈ H1(Ω) satisfies ∇ · (vw0q) ∈ H1(Ω). Testing against +members of C∞ +c (Ω) and appealing to Lemma 6.2, we see that for each j ∈ {1, . . . , n − 1} we have +the distributional identity +∇ · (vw0∂jq) = ∂j∇ · (vw0q) − +w0 +C j(q, 0) ∈ L2(Ω). +(6.1.12) +In particular, this identity establishes that ∂jq ∈ H0 +vw0(Ω), where this space is defined in Appen- +dix B.2. +Having established these three observations, we are now ready to prove the first item. Suppose +that (q, u, η) ∈ +q0,u0,η0 +X0 +and (g, f, k) ∈ Y0 satisfy the strong from equation +w0,γ0 +A (q, u, η) = (g, f, k), +which (due to the assumed level of regularity) is equivalent to the weak form equation +w0,γ0 +J (q, u, η) = +(g, K (f, k)) (see Lemma 3.24). In turn, the weak formulation unpacks into the pair of equations +∇ · (ϱu) + ∇ · (vw0(q + gη)) = g in L2(Ω) and I (q, u, η) = K (f, k) in (0H1(Ω; Rn))∗. +(6.1.13) +Let j ∈ {1, . . . , n − 1}. For the first equation in (6.1.13) we apply ∂j and use (6.1.12) from the third +observation to see that +∇ · (ϱ∂ju) + ∇ · (vw0∂j(q + gη)) = ∂jg − +w0 +C j(q, η). +(6.1.14) +For the second equation in (6.1.13) we let w ∈ H2(Ω; Rn) ∩ 0H1(Ω; Rn), test against −∂jw, and use +(6.1.10) and (6.1.11) from the first and second observations to see that +⟨ +γ0 +I (∂jq, ∂ju, ∂jη), w⟩(0H1)∗,0H1 = ⟨ +γ0 +I (q, u, η), −∂jw⟩(0H1)∗,0H1 += ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 = ⟨Kj(f, k), w⟩(0H1)∗,0H1. +(6.1.15) +Since H2(Ω; Rn) ∩ 0H1(Ω; Rn) is dense in 0H1(Ω; Rn), we deduce that +γ0 +I (∂jq, ∂ju, ∂jη) = Kj(f, k). +(6.1.16) +By combining (6.1.14) and (6.1.16), we deduce that +w0,γ0 +J (∂jq, ∂ju, ∂jη) = (∂jg − +w0 +C j(q, η), Kj(f, k)). +(6.1.17) + +102 +NOAH STEVENSON AND IAN TICE +Therefore, (6.1.7) follows by taking the norm in Y−1 of (6.1.17) and applying both Lemma 6.2 and +the boundedness of Kj established in the first observation. +□ +The next result is a higher regularity version of the previous one. +Proposition 6.4 (High norm estimates on tangential derivatives). Under the hypotheses of Defi- +nition 6.1, the following hold for s ∈ N+, j ∈ {1, . . . , n − 1}, γ0 ∈ I ⋐ R+ for I an interval, and +(g, f, k) ∈ Ys. +(1) If (q, u, η) ∈ +q0,u0,η0 +Xs +satisfies +w0,γ0 +A (q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ +q0,u0,η0 +Xs−1 and obeys +the estimate +�� +w0,γ0 +A (∂jq, ∂ju, ∂jη) +�� +Ys−1 ≲ ρ∥q, u, η∥Xs ++ ∥g, f, k∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.1.18) +(2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ Xs +m,N satisfy +w0,γ0 +Am,N(q, u, η) = (g, f, k), then +(∂jq, ∂ju, ∂jη) ∈ Xs−1 +m,N and obeys the estimate +�� +w0,γ0 +Am,N(∂jq, ∂ju, ∂jη) +�� +Ys−1 ≲ ρ∥q, u, η∥Xs ++ ∥g, f, k∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.1.19) +Here the implicit constants depend on s, the dimension, the physical parameters, ρWD, and I. +Proof. We will only prove the first item, as the second follows from a nearly identical argument. +Employing the identity ∇ · (vw0∂jq) = ∂j∇ · (vw0q) − +w0 +C j(q, 0) and Lemma 6.2, it is evident +that the differentiated triple (∂jq, ∂ju, ∂jη) belongs to the space +q0,u0,η0 +Xs−1 . +Applying ∂j to the +equations in +w0,γ0 +A (q, u, η) = (g, f, k) and rearranging provides the identity +w0,γ0 +A (∂jq, ∂ju, ∂jη) = +� +∂jg − +w0 +C j(q, η), ∂jf, ∂jk +� +. Then (6.1.18) follows by taking the Ys−1 norm of both sides of this +identity and applying the estimates from Lemma 6.2. +□ +To conclude this subsection, we iterate Proposition 6.4 and combine with Proposition 6.3 to +obtain low norm estimates on high-order tangential derivatives. +Theorem 6.5 (Synthesis of tangential derivative analysis). Let 0 < ρ ⩽ ρWD, where the latter is +as in Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, s ∈ N, γ0 ∈ I ⋐ R+ for an interval I, +(g, f, k) ∈ Ys, and α ∈ Nn \ {0} be a multiindex such that α · en = 0 and |α| ⩽ 1 + s. The following +hold. +(1) If (q, u, η) ∈ +q0,u0,η0 +Xs +satisfies +w0,γ0 +A (q, u, η) = (g, f, k), then (∂αq, ∂αu, ∂αq) ∈ +w0,γ0 +X−1 and we +have the estimate +�� +w0,γ0 +J (∂αq, ∂αu, ∂αη) +�� +Y−1 ≲ ρ∥q, u, η∥Xs ++ ∥g, f, k∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.1.20) + +COMPRESSIBLE TRAVELING WAVES +103 +(2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ Xs +m,N satisfy +w0,γ0 +Am,N(q, u, η) = (g, f, k), then +(∂αq, ∂αu, ∂αη) ∈ X−1 +m,N and we have the estimate +�� +w0,γ0 +J 1 +m,N(∂αq, ∂αu, ∂αη) +�� +Y−1 ≲ ρ∥q, u, η∥Xs ++ ∥g, f, k∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.1.21) +The implicit constants depend on s, the dimension, the physical parameters, ρWD, and I. +Proof. Again we only prove the first item, as the second follows from a nearly identical argument. +First we claim that if (q, u, η) ∈ +q0,u0,η0 +Xs +and α ∈ Nn satisfies 1 ⩽ |α| ⩽ s and α · en = 0, then +(∂αq, ∂αu, ∂αη) ∈ +q0,u0,η0 +Xs−|α| and +�� +w0,γ0 +A (∂αq, ∂αu, ∂αη) +�� +Ys−|α| ≲ ρ∥q, u, η∥Xs ++ +�� +w0,γ0 +A (q, u, η) +�� +Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.1.22) +We establish this via strong induction on |α|. +The base case of the claim, |α| = 1, was already established in Proposition 6.4. Suppose now that +for a fixed 1 ⩽ ν ⩽ s − 1 the claim holds for all α ∈ Nn such that α · en = 0 and 1 ⩽ |α| ⩽ ν. Let +α ∈ Nn with α·en = 0 be such that |α| = ν+1. Let β, γ ∈ Nn\{0} be such that α = β+γ and |γ| = 1. +By repeated applications of the induction hypothesis, it follows that (∂βq, ∂βu, ∂βη) ∈ +q0,u0,η0 +Xs−|β| and, in +turn, that (∂αq, ∂αu, ∂αη) ∈ +q0,u0,η0 +Xs−|α|. Moreover, the induction hypothesis also provides the estimates +�� +w0,γ0 +A (∂αq, ∂αu, ∂αη) +�� +Ys−|α| ≲ ρ∥∂βq, ∂βu, ∂βη∥Xs−|β| + +�� +w0,γ0 +A (∂βq, ∂βu, ∂βη) +�� +Ys−|β| ++ +� +0 +if s − |β| ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s−|β|∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ +if ⌊n/2⌋ < s − |β|, +≲ ρ +� +∥q, u, η∥Xs + ∥∂βq, ∂βu, ∂βη∥Xs−|β| +� ++ +�� +w0,γ0 +A (q, u, η) +�� +Ys ++ +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s, ++ +� +0 +if s − |β| ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s−|β|∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ +if ⌊n/2⌋ < s − |β|. +(6.1.23) +To morph this estimate into the correct form, we note the following two facts. First, the ∂β-continuity +estimates: ∥∂βq, ∂βu, ∂βη∥Xs−|β| ≲ ∥q, u, η∥Xs and ∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ ≲ ∥q, u, η∥X|β|+⌊n/2⌋. Sec- +ond, by the log-convexity of the norm in the X-spaces (see Lemma 3.25) and Young’s inequality, we +have that if ⌊n/2⌋ < s − |β|, then +∥q0, u0, η0∥X1+s−|β|∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ ≲ ∥q0, u0, η0∥X1+s−|β|∥q, u, η∥X|β|+⌊n/2⌋ +≲ +� +∥q0, u0, η0∥X1+⌊n/2⌋∥q, u, η∥Xs +� +|β| +s−⌊n/2⌋ � +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +�1− +|β| +s−⌊n/2⌋ +≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋. +(6.1.24) + +104 +NOAH STEVENSON AND IAN TICE +Upon combining these facts with (6.1.23), we verify that (6.1.22) holds, which completes the proof +of the inductive step and hence the claim. +Now, if α ∈ Nn is such that α · en and 1 ⩽ |α| ⩽ s, then the sought-after estimate (6.1.20) is true +thanks to the estimate (6.1.22) established in the above claim and the trivial bounds +�� +w0,γ0 +J (q, u, η) +�� +Y−1 ≲ +�� +w0,γ0 +A (q, u, η) +�� +Y0 ⩽ +�� +w0,γ0 +A (q, u, η) +�� +Ys−|α|. +(6.1.25) +It then only remains to prove (6.1.20) in the case that |α| = s + 1. In this case we may write +∂α = ∂j∂β for some j ∈ {1, . . . , n − 1} and |β| = s. Applying Proposition 6.3, followed by (6.1.22), +we arrive at the estimate +�� +w0,γ0 +J (∂αq, ∂αu, ∂αη) +�� +Y−1 ≲ ρ∥∂βq, ∂βu, ∂βη∥X0 + ∥ +w0,γ0 +A (∂βq, ∂βu, ∂βη)∥Y0 +≲ ρ∥q, u, η∥Xs + ∥ +w0,γ0 +A (q, u, η)∥Ys + +� +0 +if s ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s, +(6.1.26) +which completes the proof in the case |α| = s + 1. +□ +6.2. Analysis of normal systems. A useful technique in the study of the dynamic compressible +Navier-Stokes system, originally developed by Matsumura and Nishida [75], is to take linear +combinations of a normal, or vertical, derivative of the continuity equation with certain components +of the momentum equation in order to reveal a subtle dissipative structure for the normal derivative +of the density. Our goal now is to implement a version of this technique for our traveling wave +problem. The result will essentially be a bound on various high norms of a solution in terms of the +data and norms of tangential derivatives alone. +We begin with a computation that motivates the definition of the normal system. Suppose that +w0,γ0 +A1 (q, u, η) = g and +γ0 +A2(q, u, η) = f, where the operators Ai are as defined in (3.4.8) and (3.4.9), +and consider the linear combination +∂n +w0 +A1(q, u, η) + +ϱ +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) +γ0 +A2(q, u, η) · en = ∂ng + +ϱ +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))f · en. +(6.2.1) +On the left hand side the ∂2 +n(u · en)-terms cancel each other out. We may then rearrange the above +equation to create a (differentiated) steady transport equation in q, i.e. (6.2.1) is equivalent to +∂n +� +ϱ2 +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))q + ∇ · (vw0q) +� += +w0,γ0 +N 0 (q, u, η, g, f), +(6.2.2) +where +w0,γ0 +N 0 (q, u, η, g, f) = +� +ϱ2 +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) +�′ +q + ∂ng +− ∂n(ϱ′u · en) − ϱ′∂nu · en − (∇∥, 0) · ∂n(ϱu) − g∂n∇ · (vw0η) ++ +ϱ +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) +� +f · en + γ2 +0∂1u · en + γ0µ(ϱ)∆∥u · en ++ γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))(∇∥, 0) · ∂nu + γ0ϱ′(µ′(ϱ)D0uen · en + γ0λ′(ϱ)∇ · u) +� +. +(6.2.3) +The key point is that (6.2.3) depends only on tangential derivatives of q, u, ∂nu, η along with ∂ng +and f. +If we perform the same manipulations under the regularized hypotheses N−1Lm(q + gη) + +w0 +A1(q, u, η) = g and +γ0 +A2(q, u, η) = f, where we recall that Lm is defined in (3.5.6), then we obtain + +COMPRESSIBLE TRAVELING WAVES +105 +the equation +∂n +� +ϱ2 +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))q + ∇ · (vw0q) + 1 +N Lmq +� += +w0,γ0 +N 0 (q, u, η, g, f). +(6.2.4) +The identities (6.2.2) and (6.2.4) are only half of the normal system in that they only allow us to +gain control of the normal derivative of q in terms of lower order and tangential derivatives. Next, +we see how to obtain similar control of the normal derivatives u. For this we only need to examine +the equation +γ0 +A2(q, u, η) = f. If j ∈ {1, . . . , n − 1}, then we take the jth-component of this equation +and isolate the ∂2 +n(u · ej) term: ∂2 +n(u · ej) = +γ0 +N j(q, u, η, f), where +γ0 +N j(q, u, η, f) = +1 +γ0µ(ϱ) +� +− γ2 +0ϱ∂1u · ej + ϱ∂j(q + gη) − γ0µ(ϱ)∆∥u · ej +− γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))∂j∇ · u − γ0D0uen · ej(µ(ϱ))′ − f · ej +� +. +(6.2.5) +On the other hand, if we take the nth-component of +γ0 +A2(q, u, η) = f and isolate the ∂2 +n(u · en) +contribution, then we get the equation ∂2 +n(u · en) = +γ0 +N n(q, u, η, f), where +γ0 +N n(q, u, η, f) = +1 +γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) +� +− γ2 +0ϱ∂1u · en + ϱ∂nq − γ0µ(ϱ)∆∥u · en +− γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))(∇∥, 0) · ∂nu − γ0ϱ′(µ′(ϱ)D0uen · en + λ′(ϱ)∇ · u) − f · en +� +. +(6.2.6) +We record the output of these calculations in the following lemma. +Lemma 6.6 (Existence of the normal systems). Let 0 < ρ ⩽ ρWD, where the latter is defined +in Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, γ0 ∈ I for I ⋐ R+ an interval, and +(g, f, k) ∈ Y0. The following hold upon setting Λγ0(ϱ) = γ−1 +0 ϱ2(2(1 − 1/n)µ(ϱ) + λ(ϱ))−1. +(1) If (q, u, η) ∈ +q0,u0,η0 +X0 +satisfies +w0,γ0 +A (q, u, η) = (g, f, k), then +∂n(Λγ0(ϱ)q + ∇ · (vw0q)) = +w0,γ0 +N 0 (q, u, η, f) +(6.2.7) +and +∂2 +nu = ( +γ0 +N 1, . . . , +γ0 +N n)(q, u, η, f). +(6.2.8) +(2) If m, N ∈ N+, with m ⩾ 2, and (q, u, η) ∈ X0 +m,N satisfy the equations +w0,γ0 +Am,N(q, u, η) = +(g, f, k), then +∂n(Λγ0(ϱ)q + ∇ · (vw0q) + N−1Lmq) = +w0,γ0 +N 0 (q, u, η, f), +(6.2.9) +and (6.2.8) holds. +We next study the mapping properties of the N mappings, starting with N 0. +Lemma 6.7 (Boundedness of N 0). Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.17, +w0 = (q0, u0, η0) be as in Lemma 3.19, γ0 ∈ I for I ⋐ R+ an interval, and s ∈ N. The linear map +w0,γ0 +N 0 : Xs × H1+s(Ω) × Hs(Ω; Rn) → Hs(Ω) from (6.2.3) is well-defined, bounded, and obeys the + +106 +NOAH STEVENSON AND IAN TICE +following estimate. +�� +w0,γ0 +N 0 (q, u, η, g, f) +�� +Hs ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xs−1 ++ ∥g, f∥H1+s×Hs + +� +0 +if s < ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+s⟩∥η∥H1+⌊n/2⌋ +if ⌊n/2⌋ ⩽ s. +(6.2.10) +Here the implicit constant depends on the dimension, the physical parameters, ρWD, and I. +Proof. We decompose +w0,γ0 +N 0 (q, u, η, g, f) = +0,γ0 +N 0(q, u, η, g, f) − g∂n∇ · ((vw0 − ϱ′e1/g)η). +(6.2.11) +By inspection, it is clear that for any s ⩾ 0 we have the estimate +�� +0,γ0 +N 0(q, u, η, g, f) +�� +Hs ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xs−1 + ∥g, f∥H1+s×Hs. +(6.2.12) +For the remaining piece we appeal to the fourth item of Lemma 3.19 to estimate +∥∂n∇ · ((vw0 − ϱ′e1/g)η)∥Hs ⩽ ∥∇ · (vw0η)∥H1+s + ∥∂1η∥Hs +≲ ∥η∥H2+s + +� +0 +if s < ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+s⟩∥η∥H1+⌊n/2⌋ +if ⌊n/2⌋ ⩽ s. +(6.2.13) +We then conclude by noting that +∥η∥H5/2+s ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j η∥H3/2+s. +(6.2.14) +□ +Next, we study the N j maps for 1 ⩽ j ⩽ n, as defined by (6.2.5) and (6.2.6). +Lemma 6.8 (Boundedness of N j, for 1 ⩽ j ⩽ n). Suppose that γ0 ∈ I for I ⋐ R+ an interval, +s ∈ N, and i ∈ {1, . . . , n}. Then the linear map +γ0 +N i : Xs × Hs(Ω; Rn) → Hs(Ω) from (6.2.5) when +j < n or (6.2.6) when j = n is well-defined, bounded, and satisfies the following estimates. If +1 ⩽ i ⩽ n − 1, then +∥ +γ0 +N i(q, u, η, f)∥Hs ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xs−1 + ∥f∥Hs, +(6.2.15) +and if j = n, then +∥ +γ0 +N n(q, u, η, f)∥Hs ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xs−1 + ∥∂nq∥Hs + ∥f∥Hs. +(6.2.16) +Proof. These estimates are clear by inspection. +□ +The next result is an application of some of the analysis from Section 4 to the specific steady +transport structure appearing here in the normal system. + +COMPRESSIBLE TRAVELING WAVES +107 +Lemma 6.9 (Steady transport estimate). Let 0 < ρ ⩽ ρWD, where the latter is defined in +Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, γ0 ∈ I for I ⋐ R+ an interval, and ν ∈ N. +Suppose that ϕ, ψ ∈ Hν(Ω) satisfy ∇ · (vw0ϕ) ∈ Hν(Ω) and +Λγ0(ϱ)ϕ + ∇ · (vw0ϕ) = ψ +in Ω, +(6.2.17) +where Λγ0(ϱ) is defined as in Lemma 6.6. There exists a ρST,ν ∈ R+, depending only on the physical +parameters, ν, and the dimension, and I, such that if ρ ⩽ ρST,ν then we have the estimate +∥ϕ, ∇ · (vw0ϕ)∥Hν×Hν ≲ ∥ψ∥Hν + +� +0 +if ν ⩽ 1 + ⌊n/2⌋, +∥q0, u0, η0∥Xν∥ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < ν. +(6.2.18) +The implicit constant depends on ν, the dimension, the physical parameters, ρST,ν, and I. +Proof. Most of the work was already carried out in Section 4 in the sense that we endeavor to apply +Proposition 4.4 with the decomposed vector field X = g +ϱ′ vw0 = X0 + X1, where X0 = g +ϱ′ v(1) +q0,u0,η0 and +X1 = e1 + g +ϱ′ v(2) +η0 , which is split according to the decomposition of vw0 from Lemma 3.19. Write +Λ = gΛγ0(ϱ)/γϱ′. By hypothesis, �ϕ = ϱ′ϕ/g ∈ Hν(Ω) satisfies +Λ�ϕ + ∇ · (X �ϕ) = ψ in Ω and ∇ · (X �ϕ) = ∇ · (vw0ϕ) ∈ Hν(Ω). +(6.2.19) +Thus, we may employ Proposition 4.4 to see that there exists ρ(ν) ∈ R+ (depending only ν, the +physical parameters, and I) such that if ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞ ⩽ ρ(ν), then we have the +estimate +∥�ϕ∥Hν ≲ ∥ψ∥Hν + +� +0 +if ν ⩽ 1 + ⌊n/2⌋, +∥DX0, DX1∥Hν×W ν,∞∥�ϕ∥H1+⌊n/2⌋ +if 1 + ⌊n/2⌋ < ν. +(6.2.20) +Next we note that, with implicit constants depending only on the physical parameters and s ∈ +{ν, 1 + ⌊n/2⌋}, we have the estimates +∥ϕ∥Hs ≍ ∥�ϕ∥Hs and ∥DX0, DX1∥Hs×W s,∞ ≲ ∥v(1) +q0,u0,η0, v(2) +η0 ∥H1+s×W 1+s,∞. +(6.2.21) +By invoking the first and second items of Lemma 3.19, we also see that +∥v(1) +q0,u0,η0, v(2) +η0 ∥H1+s×W 1+s,∞ ≲ ∥q0, u0, η0∥Xs. +(6.2.22) +The claimed bound on ϕ in the Hν(Ω) norm now follows by combining these bounds and taking +ρST,ν small enough. +It remains to establish a bound on ∇ · (vw0ϕ) in the same space. The equation (6.2.17) is +equivalent to ∇ · (vw0ϕ) = ψ − Λ(ρ)ϕ, so by taking the norm in Hν(Ω) and utilizing the established +bounds on ∥ϕ∥Hν, we complete the proof of (6.2.18). +□ +We are now ready to identify a recursive estimate for the norm of a solution to the principal part +equations. The previous normal system identification and boundedness results merge in this next +proposition and allow us to control the solution’s norm in terms of the data and a lower norm of +the tangentially differentiated solution. +Proposition 6.10 (Synthesis of normal system results, 1). Let 0 < ρ ⩽ ρWD, where the latter is +from Theorem 3.17, w0 = (q0, u0, η0) be as in Lemma 3.19, γ0 ∈ I for I ⋐ R+ an interval, and +ν ∈ N. Suppose that (q, u, η) ∈ +q0,u0,η0 +Xν +and (g, f, k) ∈ Yν satisfy +w0,γ0 +A (q, u, η) = (g, f, k). There +exists a ρnormal,ν ∈ R+ , depending only on the physical parameters, the dimension, ν, and I, such + +108 +NOAH STEVENSON AND IAN TICE +that that if ρ ⩽ ρnormal,ν then we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 ++ ∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.23) +The implicit constant depends on ν, ρnormal,ν, the dimension, the physical parameters, and I. +Proof. Suppose first that ρ ⩽ ρST,1+ν, where this parameter is from Lemma 6.9. We begin by +proving the stated bounds on q. Denote ψ = Λγ0(ϱ)q + ∇ · (vw0q), where Λγ0(ϱ) is from Lemma 6.6. +According to the steady transport estimate, Lemma 6.9, we have that +∥q, ∇ · (vw0q)∥H1+ν×H1+ν ≲ +� +∥ψ∥H1+ν +if ν ⩽ ⌊n/2⌋, +∥ψ∥H1+ν + ⟨∥q0, u0, η0∥X1+ν⟩∥q∥H1+⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.24) +To go from this to the desired bounds on q we will carefully estimate ∥ψ∥H1+ν by splitting into +three pieces: +∥ψ∥H1+ν ≲ ∥ψ∥L2 + ∥∇ψ∥Hν ≲ ∥ψ∥L2 + +n−1 +� +j=1 +∥∂jψ∥Hν + ∥∂nψ∥Hν. +(6.2.25) +For the L2-norm in (6.2.25) we bound via the definition of the norm on +q0,u0,η0 +X−1 : +∥ψ∥L2 ≲ ∥q∥L2 + ∥∇ · (vw0q)∥L2 ≲ ∥q, u, η∥q0,u0,η0 +X−1 . +(6.2.26) +Next, we consider the tangential (1 ⩽ j ⩽ n − 1) derivative terms in (6.2.25) by employing the +commutator operator from Definition 6.1 to write +∂jψ = Λγ0(ϱ)∂jq + ∇ · (vw0∂jq) + +w0 +C j(q, 0). +(6.2.27) +The first and second terms are readily dealt with using the definition of the space +q0,u0,η0 +Xν−1 : +∥Λγ0(ϱ)∂jq + ∇ · (vw0∂jq)∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 +Xν−1 . +(6.2.28) +We next use Lemma 6.2 on the +w0 +C j(q, 0)-term in (6.2.27) and then combine with estimate (6.2.28) +to see that +∥∂jψ∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 +Xν−1 + ρ∥q∥H1+ν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q∥H1+⌊n/2⌋ +if ⌊n/2⌋ < ν, +(6.2.29) +for all 1 ⩽ j ⩽ n − 1. +In order to handle the normal (j = n) derivative in (6.2.25), we use the existence of the normal +system, Lemma 6.6, which shows that ∂nψ = +w0,γ0 +N 0 (q, u, η, g, f). Then we use the boundedness of +the linear map N 0, Proposition 6.7, to bound +∥∂nψ∥Hν ≲ +1 +� +σ=0 +n +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 ++ ∥g, f∥H1+ν×Hν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.30) + +COMPRESSIBLE TRAVELING WAVES +109 +We have now handled all of the terms on the right side of (6.2.25); by combining (6.2.24), (6.2.25), +(6.2.26), (6.2.28), (6.2.29), and (6.2.30), we deduce the estimate +∥q, ∇ · (vw0q)∥H1+ν×H1+ν ≲ ρ∥q∥H1+ν ++ +1 +� +σ=0 +n−1 +� +j=1 +∥∂jq, ∂ju, ∂jη∥q0,u0,η0 +Xν−1 + ∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.31) +Next, we turn our attention to the estimate of u in (6.2.23). Note that +∥u∥H2+ν ≲ ∥u∥L2 + +n−1 +� +j=1 +∥∂ju∥H1+ν + ∥∂nu∥H1+ν. +(6.2.32) +The L2-norm and the sum of tangential (1 ⩽ j ⩽ n − 1) derivatives are trivially controlled by the +right hand side of (6.2.23). For the normal (j = n) derivative, we split as before: +∥∂nu∥H1+ν ⩽ ∥∂nu∥L2 + +n−1 +� +j=1 +∥∂j∂nu∥Hν + ∥∂2 +nu∥Hν. +(6.2.33) +The first two terms in (6.2.33) are again trivially bounded by the right hand side of (6.2.23). +According to Lemma 6.6, ∂2 +nu satisfies (6.2.8), so we may employ the boundedness of the linear +maps N 1, . . . , N n, proved in Lemma 6.8, to estimate +∥∂2 +nu∥Hν ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 + ∥∂nq∥Hν + ∥f∥Hν. +(6.2.34) +For the ∥∂nq∥Hν term on the right we insert the already established bounds for ∥q∥H1+ν from +(6.2.31). We then synthesize these bounds to deduce the sought-after estimate of u in (6.2.23). In +remains only to handle η. However, the estimate for η in (6.2.23) follows directly from (6.2.14). +□ +Remark 6.11. We are free to choose the function N ∋ ν �→ ρnormal,ν ∈ R+ to be nonincreasing +without altering the statement or conclusion of Proposition 6.10. +Next, we consider the normal system corresponding to the regularized principal part operator +w0,γ0 +Am,N. We require the following preliminary result, which is analogous to Lemma 6.9 from the +unregularized case. +Lemma 6.12 (Regularized steady transport estimate). Let 0 < ρ ⩽ ρWD, where the latter is +defined in Theorem 3.17, w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, +m, N ∈ N+, and ν ∈ {1, . . . , m}. Suppose that ϕ ∈ Hν+2m(Ω) and ψ ∈ Hν(Ω) satisfy +� +Λγ0(ϱ)ϕ + ∇ · (vw0ϕ) + N−1Lmϕ = ψ, +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +in ∂Ω, +(6.2.35) +where Λγ0(ϱ) and Lm are defined in Lemma 6.6 and (3.5.6), respectively. There exists a ρRST,m ∈ R+, +depending only on the physical parameters, the dimension, m, and I, such that if ρ ⩽ ρRST,m and +N ≳ ⟨∥q0, u0, η0∥X1+⌊n/2⌋+m⟩4+2⌊n/2⌋, then we have the estimate +∥ϕ, ∇ · (vw0ϕ), N−1ϕ∥Hν×Hν×Hν+2m ≲ ∥ψ∥Hν + ⟨∥q0, u0, η0∥X2m⟩2+⌊n/2⌋∥ψ∥L2, +(6.2.36) +where the implied constants depend on m, the dimension, the physical parameters, ρRST,m, and I. + +110 +NOAH STEVENSON AND IAN TICE +Proof. Once more, the goal is to invoke the work from Section 4 by applying Theorem 4.8. To this +end, we set Λ0 = Λγ0(ϱ) and Λ1 = g−1ϱ′, and we use Lemma 3.19 to define the decomposed vector +field X = X0 + X1, where X0 = g +ϱ′ v(1) +q0,u0,η0 and X1 = e1 + g +ϱ′ v(2) +η0 . Thanks to the first and second +items of the lemma, we have that +∥X0, X1∥H1+s×W 1+s,∞ ≲ ⟨∥q0, u0, η0∥Xs⟩ for N ∋ s ⩾ 1 + ⌊n/2⌋. +(6.2.37) +By the additional fact that ∇Λ1 = γg−1ϱ′′en, we see that +max{∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} +≲ ∥q0, u0, η0∥X2+⌊n/2⌋ ⩽ ρ. +(6.2.38) +Thus, we may take 0 < ρRST,m ≲ ρ(m), where the latter is defined in Theorem 4.8, in order to reach +the asserted conclusion. +□ +The next result is the analog of Proposition 6.10 for the m, N-regularized principal part operator. +There are a few key differences, the most glaring of which is the finite range, depending on m, in +which the estimate holds. Another more minor distinction is that while the following estimates are +no longer tame (due to the appearance of the ⟨·⟩2+⌊n/2⌋ term), they still place all of the high norms +of the background onto the low norms of the solution. +Proposition 6.13 (Synthesis of normal system results, 2). Let 0 < ρ ⩽ ρWD, where the latter is +defined in Theorem 3.17, w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, +m, N ∈ N+ with m ⩾ 2, and ν ∈ {0, . . . , m − 1}. Suppose that (q, u, η) ∈ Xν +m,N and (g, f, k) ∈ Yν +satisfy +w0,γ0 +Am,N(q, u, η) = (g, f, k). There exists a ρreg,normal,ν,m ∈ R+, depending only on the physical +parameters, the dimension, ν, m, and I such that if ρ ⩽ ρreg,normal,ν,m and +N ≳ ⟨∥q0, u0, η0∥X1+⌊n/2⌋+m⟩4+2⌊n/2⌋, +(6.2.39) +then we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N ++ ∥g, f, k∥Yν ++ ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.40) +The implicit constants depend on the dimension, the physical parameters, m, ρreg,normal,ν,m, and I. +Proof. The proof largely mirrors that of Proposition 6.10, but since we have to appeal to estimates +for solutions to the regularized steady transport equation, the estimates we get are somewhat +different in a few key places. As such, we work through most of the argument in detail, appealing +to the proof of Proposition 6.10 when possible. +Suppose first that ρ ⩽ ρRST,m. We begin with the bounds on q by setting ψ = Λγ0(ϱ)q + ∇ · +(vw0q) + N−1Lmq. According to the regularized steady transport estimate, Lemma 6.12, we have +the bound +∥q, ∇ · (vw0q), N−1q∥H1+ν×H1+ν×H1+ν+2m ≲ ∥ψ∥H1+ν + ⟨∥q0, u0, η0∥X2m⟩2+⌊n/2⌋∥ψ∥L2, +(6.2.41) +and to parlay this into our desired estimate we split +∥ψ∥H1+ν ⩽ ∥ψ∥L2 + +n−1 +� +j=1 +∥∂jψ∥Hν + ∥∂nψ∥Hν. +(6.2.42) +For the L2-norm we bound as in (6.2.26): +∥ψ∥L2 ≲ ∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.43) + +COMPRESSIBLE TRAVELING WAVES +111 +We study the tangential (1 ⩽ j ⩽ n − 1) derivatives in (6.2.42) via the identity +∂jψ = Λγ0(ϱ)∂jq + ∇ · (vw0∂jq) + N−1Lm∂jq + +w0 +C j(q, 0), +(6.2.44) +where +w0 +C j is as in Definition 6.1, which shows that +∥∂jψ∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 +Xν−1 +m,N ++ +�� +w0 +C j(q, 0) +�� +Hν. +(6.2.45) +By the boundedness of C j established in Lemma 6.2, we obtain the estimate +∥ +w0 +C j(q, 0)∥Hν ≲ ρ∥q∥H1+ν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q∥H1+⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.46) +In the case that ⌊n/2⌋ < ν, we use interpolation (see Lemma 3.25) and Young’s inequality to further +estimate ∥q0, u0, η0∥X1+ν∥q∥H1+⌊n/2⌋ ≲ ρ∥q∥H1+ν + ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q∥L2. Hence, for any ν we +have the bound +�� +w0 +C j(q, 0) +�� +Hν ≲ ρ∥q∥H1+ν + ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q∥L2 +(6.2.47) +Upon combining (6.2.45) and (6.2.47) and then summing over 1 ⩽ j ⩽ n − 1, we acquire the +tangential bound +n−1 +� +j=1 +∥∂jψ∥Hν ≲ ρ∥q∥H1+ν + +n−1 +� +j=1 +∥∂jq, ∂ju, ∂jη∥q0,u0,η0 +Xν−1 +m,N ++ ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q, u, η∥X−1. +(6.2.48) +It remains to estimate the ∥∂nψ∥Hν-term in (6.2.42). For this, we recall the regularized normal +system, Lemma 6.6, which says that ∂nψ = +w0,γ0 +N 0 (q, u, η, g, f). Therefore, we may apply the continuity +properties of N 0, Lemma 6.7, to see that +∥∂nψ∥Hν ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 + ∥g, f∥H1+ν×Hν ++ +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.49) +Again, in the latter case of ⌊n/2⌋ < ν we use interpolation and Young’s inequality to bound +⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ ≲ ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2⟩∥η∥H3/2+ν + ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2+ν⟩∥η∥H3/2, +(6.2.50) +and hence in all cases +∥∂nψ∥Hν ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 + ∥g, f∥H1+ν×Hν + ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2+ν⟩∥q, u, η∥X−1. +(6.2.51) +Upon piecing together (6.2.42), (6.2.43), (6.2.48), and (6.2.51), we arrive at the bound +∥ψ∥H1+ν ≲ ρ∥q∥H1+ν + +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N ++ ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥X2+⌊n/2⌋+ν⟩∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.52) + +112 +NOAH STEVENSON AND IAN TICE +Now we insert (6.2.43) and (6.2.52) into (6.2.41) and choose ρreg,normal,ν,m ∈ R+ such that +ρreg,normal,ν,m ⩽ ρRST,m and sufficiently small to allow absorption of the ∥q∥H1+ν term on the +right by the left hand side. This yields +∥q, ∇ · (vw0q), N−1q∥H1+ν×H1+ν×H1+ν+2m ≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N ++ ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 +X−1 +m,N +, +(6.2.53) +and this implies the asserted bound on q in (6.2.40). +Finally, the proof of the estimates for u and η is mostly a reprise of the latter part of the proof of +Proposition 6.10. We get equations (6.2.32), (6.2.33), and (6.2.34) exactly as before. Then we insert +our new bound for q from (6.2.53) into the ∥∂nq∥Hν-term. This yields the u bound of (6.2.40). The +η bound is again trivial. +□ +Remark 6.14. For fixed m, we are free to choose the function {0, . . . , m − 1} ∋ ν �→ ρreg,normal,ν,m ∈ +R+ to be nonincreasing without altering the statement or conclusion of Proposition 6.13. +The final result of this subsection iterates the conclusions of Propositions 6.10 and 6.13 to alter +the form of the estimate’s right hand side to one in which higher order tangential derivatives appear +in a low norm. +Theorem 6.15 (Synthesis of normal system results, 3). Let 0 < ρ ⩽ ρWD, where the latter is +defined in Theorem 3.17, r ∈ {1 + ⌊n/2⌋, 2 + ⌊n/2⌋}, w0 = (q0, u0, η0) ∈ BXr(0, ρ) ∩ X∞, γ0 ∈ I for +I ⋐ R+ an interval, ν ∈ N, and (g, f, k) ∈ Yν. The following hold. +(1) If r = 1 + ⌊n/2⌋, ρ ⩽ ρnormal,ν, and (q, u, η) ∈ +q0,u0,η0 +Xν +satisfy +w0,γ0 +A (q, u, η) = (g, f, k), then +we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ ρ∥q, u, η∥Xν + +� +|α|⩽1+ν +α·en=0 +∥∂αq, ∂αu, ∂αη∥q0,u0,η0 +X−1 ++ ∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.54) +The implied constant depends on ν, the dimension, the physical parameters, ρnormal,ν, and I. +(2) If r = 2 + ⌊n/2⌋, N ∋ m ⩾ max{2, 1 + ν}, N ∈ N satisfies (6.2.39), ρ ⩽ ρreg,normal,ν,m, and +(q, u, η) ∈ Xν +m,N satisfy +w0,γ0 +Am,N(q, u, η) = (g, f, k), then there exists χν,m ∈ R+, depending +only on ν, m, and the dimension, and C1/ρ ∈ R+, depending only on 1/ρ, ν, m, such that +we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ ρ∥q, u, η∥q0,u0,η0 +Xν +m,N ++ +� +|α|⩽1+ν +α·en +∥∂αq, ∂αu, ∂αη∥q0,u0,η0 +X−1 +m,N ++ ∥g, f, k∥Yν + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.55) +Here the implied constant depends on m, the dimension, the various physical parameters, +ρreg,normal,ν,m, and I. +Proof. We begin by proving the first item via an induction argument. The proposition to be +proved inductively is as follows: if s ∈ N+ satisfies s ⩽ ν + 1 for some ν ∈ N, ρ ⩽ ρnormal,ν, and + +COMPRESSIBLE TRAVELING WAVES +113 +(q, u, η) ∈ +q0,u0,η0 +Xν +, then we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ ρ∥q, u, η∥Xν + +� +|α|⩽s +α·en=0 +∥∂αq, ∂αu, ∂αη∥q0,u0,η0 +Xν−s ++ +�� +w0,γ0 +A (q, u, η) +�� +Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.56) +The base case, s = 1, was established in Proposition 6.10. Suppose now that N ∋ s ⩾ 2 is such that +the proposition holds at the level s − 1. We will now verify it at level s. Suppose that s ⩽ ν + 1 +for some ν ∈ N, ρ ⩽ ρnormal,ν, and (q, u, η) ∈ +q0,u0,η0 +Xν +. We first invoke Proposition 6.10 to obtain the +estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 ++ +�� +w0,γ0 +A (q, u, η) +�� +Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.2.57) +Fix σ ∈ {0, 1} and j ∈ {1, . . . , n − 1}. +To handle the ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +term appearing +in (6.2.57), we invoke the induction hypothesis at level s − 1 (while heeding to Remark 6.11) in +order to learn that +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 ≲ ρ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 + +� +|α|⩽s−1 +α·en=0 +∥∂α∂σ +j q, ∂α∂σ +j u, ∂α∂σ +j η∥q0,u0,η0 +Xν−s ++ +�� +w0,γ0 +A (∂σ +j q, ∂σ +j u, ∂σ +j η) +�� +Yν−1 + +� +0 +if ν ⩽ 1 + ⌊n/2⌋, +⟨∥q0, u0, η0∥Xν⟩∥∂σ +j q, ∂σ +j u, ∂σ +j η∥X⌊n/2⌋ +if 1 + ⌊n/2⌋ < ν. +(6.2.58) +We estimate the first term on the right hand side trivially via ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥Xν−1 ≲ ∥q, u, η∥Xν. +For the term involving the operator +w0,γ0 +A , we invoke the first item of Proposition 6.4 (noting that +ν > 0) to bound +�� +w0,γ0 +A (∂σ +j q, ∂σ +j u, ∂σ +j η) +�� +Yν−1 ≲ ρ∥q, u, η∥Xν ++ +�� +w0,γ0 +A (q, u, η) +�� +Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s. +(6.2.59) +Upon combining inequalities (6.2.57), (6.2.58), and (6.2.59) with the fact that +� +0 +if ν ⩽ 1 + ⌊n/2⌋, +⟨∥q0, u0, η0∥Xν⟩∥∂σ +j q, ∂σ +j u, ∂σ +j η∥X⌊n/2⌋ +if 1 + ⌊n/2⌋ < ν, +≲ ρ∥q, u, η∥Xν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < s, +(6.2.60) +we establish (6.2.56) for the level s, and thus prove the proposition at the s level. The proposition +then holds for all s ∈ N+ by induction. By taking s = 1 + ν in (6.2.56) we obtain (6.2.54), which +completes the proof of the first item. + +114 +NOAH STEVENSON AND IAN TICE +Next, we turn our attention to the second item. The strategy here is the same as in the first +item, but due to the dissimilarities between (6.2.54) and (6.2.55), we cannot apply precisely the +same argument. Our new proposition to be proved inductively is as follows: if s ∈ N+, then for all +N ∋ ν ⩾ s − 1 and N ∋ m ⩾ max{2, 1 + ν}, there exists χ ∈ R+, depending on s, m, and ν, and +C1/ρ, depending on s, 1/ρ, ν, and m, such that for all N ∈ N satisfying (6.2.39), ρ ⩽ ρreg,normal,ν,m, +and (q, u, η) ∈ Xν +m,N we have the estimate +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ ρ∥q, u, η∥q0,u0,η0 +Xν +m,N ++ +� +|α|⩽s +α·en=0 +∥∂αq, ∂αu, ∂αη∥q0,u0,η0 +Xν−s +m,N ++ +�� +w0,γ0 +Am,N(q, u, η) +�� +Yν + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.61) +As before, the base case, s = 1, was established already, this time in Proposition 6.13. +Suppose now that N ∋ s ⩾ 2 is such that the induction proposition holds at level s−1. We will now +verify it at level s. Suppose that N ∋ ν ⩾ s − 1, N ∋ m ⩾ max{2, 1 + ν}, N ∈ N satisfies (6.2.39), +ρ ⩽ ρreg,normal,ν,m, and (q, u, η) ∈ Xν +m,N. As before, we first invoke Proposition 6.13 and obtain the +estimate +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ +1 +� +σ=0 +n−1 +� +j=1 +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N ++ +�� +w0,γ0 +Am,N(q, u, η) +�� +Yν ++ ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.62) +For σ ∈ {0, 1} and j ∈ {1, . . . , n − 1}, we invoke the (s − 1)-induction hypotheses while again +heeding Remark 6.14 to bound +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N +≲ ρ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N ++ +� +|α|⩽s−1 +α·en=0 +∥∂α∂σ +j q, ∂α∂σ +j u, ∂α∂σ +j η∥q0,u0,η0 +Xν−s +m,N ++ +�� +w0,γ0 +Am,N(∂σ +j q, ∂σ +j u, ∂σ +j η) +�� +Yν−1 + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +X−1 +m,N +. +(6.2.63) +Now we make three estimates. First, we again have the trivial estimate +∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +Xν−1 +m,N +≲ ∥q, u, η∥q0,u0,η0 +Xν +m,N +. +(6.2.64) +Second, by invoking the second item of Proposition 6.4, the log-convexity of the X-norms from +Lemma 3.25, and Young’s inequality, we get the bounds +�� +w0,γ0 +Am,N(∂σ +j q, ∂σ +j u, ∂σ +j η) +�� +Yν−1 ≲ ρ∥q, u, η∥Xν ++ +�� +w0,γ0 +Am,N(q, u, η) +�� +Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν, +≲ ρ∥q, u, η∥Xν + +�� +w0,γ0 +A (q, u, η) +�� +Yν + ρ−1−⌊n/2⌋⟨∥q0, u0, η0∥X1+ν⟩2+⌊n/2⌋∥q, u, η∥X−1. +(6.2.65) + +COMPRESSIBLE TRAVELING WAVES +115 +Third, by the log-convexity of the X-norms again and Young’s inequality, we get the bound +C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥∂σ +j q, ∂σ +j u, ∂σ +j η∥q0,u0,η0 +X−1 +m,N +≲ ρ∥q, u, η∥q0,u0,η0 +Xν +m,N ++ ρ−1/νC(1+ν)/ν +1/ρ +⟨∥q0, u0, η0∥Xmax{2m,1+m+⌊n/2⌋}⟩χ(1+ν)/ν∥q, u, η∥q0,u0,η0 +X−1 +m,N +. +(6.2.66) +Now we combine estimates (6.2.62) and (6.2.63) with the trio (6.2.64), (6.2.65), and (6.2.66); this +shows that the induction proposition holds at the level s, and hence for all s ∈ N+ by induction. +Estimate (6.2.55) from the second item now follows by taking s = 1 + ν in (6.2.61). +□ +6.3. Estimates and existence for the principal part. In this subsection we first prove a priori +estimates for systems (5.0.1) and (5.0.2). After this, we derive the existence theory for the former. +Theorem 6.16 (A priori estimates for the principal part and the regularization). Let 0 < ρ ⩽ +ρWD, where the latter is defined in Theorem 3.17, r ∈ {1 + ⌊n/2⌋, 2 + ⌊n/2⌋}, w0 = (q0, u0, η0) ∈ +BXr(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, ν ∈ N, and (g, f, k) ∈ Yν. The following hold. +(1) Let r = 1 + ⌊n/2⌋. There exists a ρest,ν ∈ R+, depending only on ν, the dimension, the +various physical parameters, and I, such that if ρ ⩽ ρest,ν, then for (q, u, η) ∈ +q0,u0,η0 +Xν +satisfying +w0,γ0 +A (q, u, η) = (g, f, k) we have the a priori estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ ∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.3.1) +The implicit constants depend on ν, the dimension, the physical parameters, ρest,ν, and I. +(2) Let r = 2 + ⌊n/2⌋ and N ∋ m ⩾ max{2, 1 + ν}. There exists a ρreg,ν,m ∈ R+, depending +only on ν, m, the various physical parameters, and I such that if ρ ⩽ ρreg,ν,m, then for +(q, u, η) ∈ Xν +m,N satisfying +w0,γ0 +Am,N(q, u, η) = (g, f, k) we have the a priori estimate +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 +X−1 +m,N +, +(6.3.2) +where χν,m ∈ R+ is from the second item of Theorem 6.15. Here the implicit constant +depends on ν, m, the dimension, the physical parameters, ρreg,ν,m, and I. +Proof. We begin by proving the first item. Assume that ρ ⩽ min{ρweak, ρnormal,ν}, where these +smallness parameters are from Propositions 5.2 and 6.10, respectively. By combining the a priori +estimates for weak solutions from Proposition 5.2 with the tangential derivative estimates from the +first item of Theorem 6.5, we learn that +� +|α|⩽1+ν +α·en=0 +∥∂αq, ∂αu, ∂αη∥q0,u0,η0 +X−1 +≲ ρ∥q, u, η∥Xν ++ ∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.3.3) +We insert the bound (6.3.3) into conclusion (6.2.54) of the first item of Theorem 6.15 to acquire the +estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ ρ∥q, u, η∥Xν +∥g, f, k∥Yν + +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. (6.3.4) +Hence, we may define ρest,ν ∈ R+ to be sufficiently small so that when 0 < ρ ⩽ ρest,ν, we may +absorb the right hand side’s ∥q, u, η∥Xν-contribution by the left and obtain (6.3.1). This completes +the proof of the first item. + +116 +NOAH STEVENSON AND IAN TICE +Next we consider the second item. The argument is basically the same, but with an extra +step. Assume 0 < ρ ⩽ min{ρweak,reg, ρreg,normal,ν,m}, where these smallness parameters are from +Propositions 5.4 and 6.13, respectively. As in the proof of the first item, we combine the conclusions +of Proposition 5.4, the second item of Theorem 6.5, and the second item of Theorem 6.15; however, +the resulting estimate is slightly different: +∥q, u, η∥q0,u0,η0 +Xν +m,N +≲ ρ∥q, u, η∥q0,u0,η0 +Xν +m,N ++ ∥g, f, k∥Yν ++ C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 +X−1 +m,N ++ +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν. +(6.3.5) +To reach the estimate (6.3.2), we note that the log-convexity of the X-norms from Lemma 3.25, +paired with Young’s inequality grants the bound +� +0 +if ν ⩽ ⌊n/2⌋, +∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ +if ⌊n/2⌋ < ν, ≲ ρ∥q, u, η∥q0,u0,η0 +Xν +m,N ++ ρ−1−⌊n/2⌋⟨∥q0, u0, η0∥X1+ν⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 +X−1 +m,N +, +(6.3.6) +and thus we may combine estimates (6.3.5) and (6.3.6), and then take 0 < ρ ⩽ ρreg,ν,m sufficiently +small to obtain (6.3.2). +□ +We now are in a position to give an existence result for the principal part of the linearization, +system (5.0.1). +Theorem 6.17 (Existence for the principal part). Let 0 < ρ ⩽ ρWD, where the latter is defined in +Theorem 3.17, and let w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval. For +each ν ∈ N, there exists a ρexi,ν ∈ R+, depending on ν, the physical parameters, and I, such that if +ρ ⩽ ρexi,ν then the map +w0,γ0 +A +: +q0,u0,η0 +Xν +→ Yν, +(6.3.7) +the action of which is given via (3.4.7), is a Banach isomorphism. +Proof. Take ρexi,ν = min{ρest,ν, ρreg,ν,max{2,1+ν}}, where these smallness parameters are from the +first and second items of Theorem 6.16. That the map (6.3.7) is well-defined is a consequence of the +first item of Proposition 3.21. This map is injective as a consequence of a priori estimate (6.3.1) +from the first item of Theorem 6.16. It remains only to verify surjectivity. +The proof of the second item of Theorem 6.16 shows that ρexi,ν ⩽ ρreg,weak, and hence we are in a +position to apply Corollary 5.6, which tells us that the maps +w0,γ0 +Amax{2,1+ν},N : Xν +max{2,1+ν},N → Yν +are Banach isomorphisms for N ∈ N sufficiently large, say N ⩾ N. Thus, given (g, f, k) ∈ Yν we can +define the sequence {(qN, uN, ηN)}∞ +N=N ⊂ Xν via (qN, uN, ηN) = +� +w0,γ0 +Amax{2,1+ν},N +�−1(g, f, k). In this +we are tacitly using that Xν +m,N �→ Xν for any m, N ∈ N+; in fact, this embedding is non-expansive. +Therefore, by applying the a priori estimate (6.3.2) from the second item of Theorem 6.16, followed +by the regularized weak solution a priori estimate from Proposition 5.4 (and by also invoking +Lemma 3.24), we obtain the uniform bounds +sup +N⩾N +� +∥qN, uN, ηN∥Xν+∥∇·(vw0qN)∥H1+ν +� +≲ ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥g, f, k∥Yν. (6.3.8) + +COMPRESSIBLE TRAVELING WAVES +117 +Therefore, we may extract a weak limit (q, u, η) ∈ Xν with ∇ · (vw0q) ∈ H1+ν(Ω) such that along +some unlabeled subsequence we have that +(qN, uN, ηN) ⇀ (q, u, η) in Xν and ∇ · (vw0qN) ⇀ ∇ · (vw0q) in H1+ν(Ω). +(6.3.9) +By routine weak convergence arguments, we readily deduce that (q, u, η) ∈ +q0,u0,η0 +Xν +and +w0,γ0 +A (q, u, η) = +(g, f, k), which completes the proof of surjectivity. +□ +6.4. Synthesis of linear analysis. In this final subsection of linear analysis, we turn to the study +of the full derivative of the nonlinear map Ψ from (3.3.1), which is associated to the PDE (1.4.9). +In other words, we consider the question of existence and tame estimates for the system +DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ). +(6.4.1) +As usual, the unknowns are q : Ω → R, u : Ω → Rn, and η : Σ → R, while the given data are +g : Ω → R, f : Ω → Rn, k : Σ → Rn, G, F : Rn → Rn, T : Rn → Rn×n, and γ ∈ R, as well as the +background tuple (θ0, γ0). We now state our main linear analysis result. +Theorem 6.18 (Analysis of the linearization). Let 0 < ρ ⩽ ρWD (recall that the latter is defined in +Theorem 3.17), I ⋐ R+ be an interval, and +(θ0, γ0) = (q0, u0, η0, T0, G0, F0, γ0) ∈ (BX3+⌊n/2⌋(0, ρ) × BW4+⌊n/2⌋(0, ρ) × I) ∩ (X∞ × W∞ × R+). +(6.4.2) +For every N ∋ ν ⩾ 3 + ⌊n/2⌋ there exists a ρν ∈ R+, depending on ν, the physical parameters, and +I, such that if 0 < ρ ⩽ ρν, then the following hold. +(1) The map +DΨ(θ0, γ0) : +q0,u0,η0 +Xν +× W1+ν × R → Yν × W1+ν × R +(6.4.3) +is well-defined and a Banach isomorphism. +(2) Assume that Ξ = (g, f, k, T , G, F, γ) ∈ Yν ×W1+ν ×R and Θ = (DΨ(θ0, γ0))−1Ξ ∈ +q0,u0,η0 +Xν +× +W1+ν × R. We have the tame estimate +∥Θ∥q0,u0,η0 +Xν +×W1+ν×R ≲ ∥Ξ∥Yν×W1+ν×R + ⟨∥θ0∥X1+ν×W1+ν⟩∥Ξ∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R, +(6.4.4) +where the implicit constant depends on ν, the dimension, the physical parameters, ρν, and I. +Proof. Suppose initially that N ∋ ν ⩾ 2 + ⌊n/2⌋. The idea of the proof is to prove a priori estimates, +uniform with respect to a parameter, and utilize the method of continuity. To this end, for τ ∈ [0, 1] +we define the convex homotopy of operators +Lτ(θ0, γ0) : +q0,u0,η0 +Xν +× W1+ν × R → Yν × W1+ν × R +(6.4.5) +via +Lτ(θ0, γ0)(q, u, η, T , G, F, γ) = +�w0,γ0 +A (q, u, η) + τ +�w0,γ0 +P ++ +θ0Q +� +(q, u, η, γ) + +q0,u0,η0 +R +(T , G, F) +T , G, F, γ +� +. +(6.4.6) +Note that L1(θ0, γ0) = DΨ(θ0, γ0) from (3.4.4) and that the mapping properties of A, P, Q, and +R established in Propositions 3.21 and 3.22 ensure well-definedness and continuity of the maps +Lτ(θ0, γ0). +We now aim to prove τ-uniform a priori estimates for {Lτ(θ0, γ0)}τ∈[0,1]. Assume that +(q, u, η, T , G, F, γ) ∈ +q0,u0,η0 +Xν +× W1+ν × R and (g, f, k) ∈ Yν +(6.4.7) +are related via +Lτ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ). +(6.4.8) + +118 +NOAH STEVENSON AND IAN TICE +The first three components of the above equation are equivalent to +w0,γ0 +A (q, u, η) = (g, f, k) − τ +�w0,γ0 +P ++ +θ0Q +� +(q, u, η, γ) − +q0,u0,η0 +R +(T , G, F) = (�g, �f, �k). +(6.4.9) +Assume that 0 < ρ ⩽ ρest,ν; then we may invoke the principal part estimates from the first item of +Theorem 6.16 to see that +∥q, u, η∥q0,u0,η0 +Xν +≲ ∥g, f, k∥Yν + +�� +w0,γ0 +P (q, u, η, γ) +�� +Yν + +�� +θ0Q(q, u, η) +�� +Yν ++ +�� +q0,u0,η0 +R +(T , G, F) +�� +Yν + ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋. +(6.4.10) +According to the second item of Proposition 3.21 and the first and second items of Proposition 3.22, +we have the bounds +�� +w0,γ0 +P (q, u, η, γ) +�� +Yν ≲ ρ∥q, u, η∥Xν + ∥q0, u0, η0∥X1+ν∥q, u, η, γ∥X⌊n/2⌋×R, +(6.4.11) +�� +θ0Q(q, u, η) +�� +Yν ≲ ρ∥q, u, η∥Xν + ∥θ0∥Xν×W1+ν∥q, u, η∥X2+⌊n/2⌋, +(6.4.12) +and +�� +q0,u0,η0 +R +(T , G, F) +�� +Yν ≲ ∥T , G, F∥W1+ν + ⟨∥q0, u0, η0∥Xν⟩∥T , G, F∥W3+⌊n/2⌋. +(6.4.13) +Inserting (6.4.11), (6.4.12), and (6.4.13) into the right hand side of (6.4.10), we obtain the estimate +∥q, u, η∥q0,u0,η0 +Xν +≲ ρ∥q, u, η∥Xν + ∥g, f, k, T , G, F∥Yν×W1+ν ++ ⟨∥θ0∥X1+ν×W1+ν⟩∥q, u, η, T , G, F, γ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R. +(6.4.14) +Now we take 0 < ρ ⩽ min{ρest,ν, ρexi,ν}, where the latter smallness parameters are from The- +orems 6.16 and 6.17, to be sufficiently small so that we can absorb as usual in (6.4.14) to see +that +∥q, u, η∥q0,u0,η0 +Xν +≲ ∥g, f, k, T , G, F∥Yν×W1+ν+⟨∥θ0∥X1+ν×W1+ν⟩∥q, u, η, T , G, F, γ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R. +(6.4.15) +We need to handle the norm of (q, u, η) in X2+⌊n/2⌋ on the right hand side of (6.4.15). According +to the first item of Theorem 6.16 and (6.4.2), we know that +∥q, u, η∥X2+⌊n/2⌋ ⩽ ∥q, u, η∥ q0,u0,η0 +X2+⌊n/2⌋ ≲ ∥�g, �f, �k∥Y2+⌊n/2⌋, +(6.4.16) +where the latter is defined in (6.4.9). By arguing as in (6.4.11), (6.4.12), and (6.4.13) again but +using (6.4.2), we learn that +∥�g, �f, �k∥Y2+⌊n/2⌋ ≲ ρ∥q, u, η, γ∥X2+⌊n/2⌋×R + ∥g, f, k∥Y2+⌊n/2⌋ + ∥T , G, F∥W3+⌊n/2⌋. +(6.4.17) +By combining (6.4.16) and (6.4.17) and taking ρ smaller, if necessary, we gain the estimate +∥q, u, η∥X2+⌊n/2⌋ ≲ ∥g, f, k, T , G, F, γ∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R. +(6.4.18) +Finally, we insert (6.4.18) into (6.4.15) to acquire the estimate +∥q, u, η, T , G, F, γ∥q0,u0,η0 +Xν +×W1+ν×R ≲ ∥Lτ(θ0, γ0)(q, u, η, T , G, F, γ)∥Yν×W1+ν×R ++ ⟨∥θ0∥X1+ν×W1+ν⟩∥Lτ(θ0, γ0)(q, u, η, T , G, F, γ)∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R. +(6.4.19) +The bound (6.4.19) establishes the desired τ-uniform a prior bounds for {Lτ(θ0, γ0)}τ∈[0,1], as +long as ρ ⩽ ρν for some ρν ∈ R+. Therefore, by the method of continuity (see, for instance, Theorem +5.2 in Gilbarg and Trudinger [35]), the invertibility of L1(θ0, γ0) = DΨ(θ0, γ0) is established as soon + +COMPRESSIBLE TRAVELING WAVES +119 +as we know that L0(θ0, γ0) is invertible. The latter holds thanks to Theorem 6.17 (and the fact that +ρν ⩽ ρexi,ν) since +(L0(θ0, γ0))−1(g, f, k, T , G, F, γ) = +��w0,γ0 +A +�−1� +(g, f, k) − +q0,u0,η0 +R +(T , G, F) +� +, T , G, F, γ +� +. +(6.4.20) +This completes the proof of the first item. The second item follows from the first and estimate (6.4.19) +at τ = 1. +□ +7. Conclusion +In this section we prove our main result, Theorem 1. We begin with Theorem 7.1, which is an +abstract construction in the Nash-Moser inverse function theorem framework from Section 2. We +then employ the abstract construction for the PDE-style result in Theorem 7.3. +7.1. Abstract construction. Recall the scales of Banach spaces from (3.1.11), the nonlinear +operator Ψ from (3.3.1), and the parameter ρWD ∈ R+ from Theorem 3.17. To state the next +theorem we introduce the following notation. First, we set β = 9 + 2⌊n/2⌋. Second, for N ∋ s ⩾ β +and ρ ∈ R+ we define the open sets +Es(ρ) = BEβ(0, ρ) ∩ Es and Fs(ρ) = BFβ(0, ρ) ∩ Fs, +(7.1.1) +where Es and Fs are defined by (3.1.10). Finally, it is convenient to introduce a translation of Ψ; +namely, for Γ ∈ R+ we set +ΨΓ(q, u, η, T , G, F, γ) = Ψ(q, u, η, T , G, F, Γ + γ) − (0, 0, 0, 0, 0, 0, Γ), +(7.1.2) +defined for (q, u, η) ∈ BXβ(0, ρWD), (T , G, F) ∈ W1+β, and R+ ∋ γ > −Γ. The utility of this +definition is that ΨΓ(0) = 0. +Theorem 7.1 (Traveling waves for free boundary compressible Navier-Stokes, 1). Assume that +the parameters µ, λ, and ς satisfy (1.1.7) and that P satisfies P ′ > 0 and (1.1.9). Let Γ ∈ R+, +and let β, Es(ρ), Fs(ρ), and ΨΓ be as defined above. +There exists a nonincreasing sequence +{εν(Γ)}∞ +ν=0 ⊂ (0, min{ρWD, Γ/2}) and κ(Γ) ∈ R+ such that the following hold. +(1) Existence and uniqueness: Given f ∈ Fβ(ε0(Γ)), there exists a unique e ∈ Eβ(κ(Γ)ε0(Γ)) +such that ΨΓ(e) = f. This induces the local inverse map +Ψ +−1 +Γ +: Fβ(ε0(Γ)) → Ψ +−1 +Γ (Fβ(ε0(Γ))) ⊆ Eβ(κ(Γ)ε0(Γ)). +(7.1.3) +(2) Higher regularity, given low norm smallness: If ν ∈ N and f ∈ Fβ+ν(εν(Γ)), then Ψ +−1 +Γ (f) ∈ +Eβ+ν, and we have the tame estimate +∥Ψ +−1 +Γ (f)∥Eβ+ν ≲ ∥f∥Fβ+ν, +(7.1.4) +for an implicit constant depending only on the dimension, the physical parameters, ν, ρWD, +and Γ. +(3) Continuous dependence: For every ν ∈ N, the restricted map +Ψ +−1 +Γ +: Fβ+ν(εν(Γ)) → Eβ+ν +(7.1.5) +is continuous with respect to the norms on Fβ+ν and Eβ+ν. +(4) Continuous differentiability: For every ν ∈ N, the restricted map +Ψ +−1 +Γ +: Fβ+ν(εν+2(Γ)) → Eβ+ν−1 +(7.1.6) +is differentiable. Moreover, if we view DΨ +−1 +Γ +as a map +DΨ +−1 +Γ +: Fβ+ν(εν+2(Γ)) × Fβ+ν−1 → Eβ+ν−1, +(7.1.7) +then DΨ +−1 +Γ +is continuous. + +120 +NOAH STEVENSON AND IAN TICE +Proof. Our aim is to show that the hypotheses of Theorem 2.21 are satisfied by ΨΓ with the Banach +scales EEE and FFF defined in (3.1.11). Thanks to Lemma 3.2, we have that condition II from the +hypotheses of Theorem 2.21 is satisfied by EEE and FFF. In verifying the rest of the hypotheses we +consider the triple (EEE,FFF, ΨΓ) with parameters (µ, r, R) = (1, 3 + ⌊n/2⌋, 17 + 3⌊n/2⌋), which obey +the requisite inequality 2(r + µ) + 1 = β < (r + R)/2 with β = 9 + 2⌊n/2⌋. +We next set δr = min{ρWD, Γ/2}. Then Theorem 3.17 and Remark 3.18 prove that the second +item (C2 and µ-tameness) of Definition 2.20 is satisfied. Invoking Theorem 6.18 with ν = R and +interval I = [Γ/2, 2Γ] then shows that the third item (derivative inversion) of Definition 2.20 holds +with δR = min{ρν, δr} (where ρν is given by Theorem 6.18) and that the remainder of the LRI +mapping hypotheses from the definition are also satisfied. +Hence, the hypotheses of Theorem 2.21 are satisfied. We thus obtain an ε = ε0(Γ) such that the +first item holds, and we have the estimate +∥Ψ +−1 +Γ (f)∥Eβ ≲ ∥f∥Fβ +(7.1.8) +for all f ∈ Fβ(ε0(Γ)). In fact, by the second item of Theorem 2.21, we have that for all s ∈ +[β, R + r − β] ∩ N = [9 + ⌊n/2⌋, 11 + 3⌊n/2⌋] ∩ N it holds that if f ∈ Fβ(ε0(Γ)) ∩ Fs, then +Ψ +−1 +Γ (f) ∈ Es and estimate (7.1.8) holds with β replaced by s. Therefore, we may set εν(Γ) = ε0 for +ν ∈ {0, 1, . . . , 2 + 2⌊n/2⌋ = R + r − 2β}. +Now, given N ∋ ν > 2+2⌊n/2⌋, we define εν(Γ) ∈ (0, ε0] to be the smallness parameter ε provided +by Theorem 2.21 with parameter triple (µ, r, �R), where �R = R + ν − (2 + 2⌊n/2⌋). Note that the +LRI mapping hypotheses are satisfied in this case if we take δ �R = min{ρR+ν−(2+2⌊n/2⌋), δr}, where +the former parameter is from Theorem 6.18 (with ν = R + ν − (2 + 2⌊n/2⌋) and I = [Γ/2, 2Γ]). The +second item now follows from this definition of {εν(Γ)}∞ +ν=0, the second item of Theorem 2.21, and +uniqueness. Finally, the third and fourth items now follow directly from Theorem 2.24. +□ +Remark 7.2. As a consequence of the third conclusion of Theorem 2.24, the local inverse map Ψ +−1 +Γ +produced in Theorem 7.1 satisfies certain higher-order differentiability assertions beyond the basic +continuous differentiability result in the third item of the previous theorem. However, the precise +statements become cumbersome to enumerate due to a compounding of the derivative loss in the +formulas for higher-order derivatives of the inverse map. As such, we have chosen not to state these +precisely. +7.2. PDE construction. We utilize Theorem 7.1 to prove our main result about the free boundary +compressible Navier-Stokes equations, system (1.4.9). +Theorem 7.3 (Traveling wave for free boundary compressible Navier-Stokes, 2). Assume that the +parameters µ, λ, and ς satisfy (1.1.7) and that P satisfies P ′ > 0 and (1.1.9). Let β = 9 + 2⌊n/2⌋. +There exist a family {V(γ)}γ∈R+ of open sets of Xβ and a nonincreasing sequence {Us}∞ +s=β of open +sets of W1+β × R+ such that the following hold. +(1) Nondegeneracy: We have that {0} × R+ ⊆ �∞ +s=β Us and 0 ∈ � +γ∈R+ V(γ). +(2) Existence and uniqueness: For all (T , G, F, γ) ∈ Uβ there exists a unique (q, u, η) ∈ V(γ) +such that the traveling wave formulation for the free boundary compressible Navier-Stokes +equations, system (1.4.9), is classically satisfied with wave speed γ, data (T , G, F), and +solution (q, u, η). +(3) Higher regularity, given low norm smallness: If N ∋ s ⩾ β and (T , G, F, γ) ∈ Us∩(W1+s×R), +then the corresponding solution satisfies (q, u, η) ∈ V(γ) ∩ X1+s. +(4) Continuous dependence: For any N ∋ s ⩾ β, the solution map +Us ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (q, u, η) ∈ X1+s +(7.2.1) +is continuous with respect to the W1+s × R and X1+s norms. + +COMPRESSIBLE TRAVELING WAVES +121 +(5) No vacuum formation: There exists positive constants c, C ∈ R+ such that for all (q, u, η) ∈ +� +γ∈R+ V(γ) we have that c ⩽ σq,η ⩽ C, where σq,η is defined in (1.4.11). +(6) Flattening map diffeomorphism: For any N ∋ s ⩾ β and (q, u, η) ∈ Xs ∩ � +γ∈R+ V(γ) we +have that the flattening map Fη from (1.4.4) is a smooth diffeomorphism from Ω to Ω[η] that +extends to a Cs+2−⌊n/2⌋ diffeomorphism from Ω to Ω[η]. +Proof. For each Γ ∈ R+, we many invoke Theorem 7.1 and acquire a nonincreasing sequence +{εν(Γ)}∞ +ν=0 ⊂ (0, min{ρWD, Γ/2}) and a κ(Γ) ∈ R+ such that the various conclusions of the theorem +hold. We then define the open sets +Us = +� +Γ∈R+ +BW1+β×R((0, Γ), εs−β+1(Γ)/C) for N ∋ s ⩾ β +(7.2.2) +for some constant C ⩾ 1 (independent of s) to be determined, and +V(Γ) = BXβ(0, κ(Γ)ε0(Γ)) for Γ ∈ R+. +(7.2.3) +The first item is now clear by inspection. +For the second item, we apply the first conclusion of Theorem 7.1. Indeed, for (T , G, F, γ) ∈ Uβ we +have existence by setting (q, u, η) to be the first three components of the tuple Ψ +−1 +γ (0, 0, 0, T , G, F, 0). +Uniqueness in V(γ) follows from the fact that if Ψ(q, u, η, T , G, F, γ) = Ψ(�q, �u, �η, T , G, F, γ), then +Ψγ(q, u, η, T , G, F, 0) = Ψγ(�q, �u, �η, T , G, F, 0) and hence (q, u, η) = (�q, �u, �η) by the uniqueness +assertions of Theorem 7.1. +We now prove the third item. An immediate consequence of the second conclusion of Theorem 7.1 +is that if N ∋ s ⩾ β and (T , G, F, γ) ∈ Us ∩ (W1+s × R), then the corresponding solution satisfies +(q, u, η) ∈ V(γ) ∩ Xs. In fact, the solution is one degree more regular. To see this, we recall the map +Φ from (3.3.5) and note that the equation Ψ(q, u, η, T , G, F, γ) = (0, 0, 0, T , G, F, γ) is equivalent to +Ψ(q, u, η, 0, 0, 0, γ) = (−Φ(q, u, η, T , G, F), 0, 0, 0, γ), which in turn is equivalent to +Ψγ(q, u, η, 0, 0, 0, 0) = (−Φ(q, u, η, T , G, F), 0, 0, 0, 0). +(7.2.4) +Note that Φ(q, u, η, T , G, F) is linear in (T , G, F); consequently, Propositions 3.14 and 3.15 (with +m = 0) show that +∥Φ(q, u, η, T , G, F)∥Yβ ⩽ c∥T , G, F∥Wβ and ∥Φ(q, u, η, T , G, F)∥Y1+s ≲∥q,u,η∥Xs ∥T , G, F∥W1+s +(7.2.5) +for a constant c depending only on the dimension, the physical parameters, and ρWD. Assume the +constant C ⩾ 1 from (7.2.2) satisfies C ≳ c. We then use the inclusion (T , G, F, γ) ∈ Us∩(W1+s×R) +and (7.2.5) to see that +(−Φ(q, u, η, T , G, F), 0, 0, 0, γ) ∈ F1+s ∩ +� +Γ∈R+ +BFβ×R((0, Γ), εs−β+1(Γ)). +(7.2.6) +Hence, we can apply the first and second conclusions of Theorem 7.1 to deduce from (7.2.4) +and (7.2.6) that the third item holds. +It remains to justify the fourth item. Identity (7.1.2) and the third conclusion of Theorem 7.1 +show that for N ∋ s ⩾ β, the solution map +Us−1 ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (q, u, η) ∈ Xs +(7.2.7) +is continuous. In fact, we can do one derivative better for the target space topology by arguing as in +the proof of the third item above. Indeed, by Propositions 3.14 and 3.15 again, along with (7.2.7), +we find that the map +Us ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (−Φ(q, u, η, T , G, F), 0, 0, 0, γ) ∈ F1+s +(7.2.8) +is continuous. In light of the identity +Ψ(q, u, η, 0, 0, 0, γ) = (−Φ(q, u, η, T , G, F), 0, 0, 0, γ), +(7.2.9) + +122 +NOAH STEVENSON AND IAN TICE +the third conclusion of Theorem 7.1 (which implies that Ψ +−1 is continuous), the inclusion (7.2.6) +(which implies that the right side of (7.2.9) is in the domain of Ψ +−1), the continuity of the map (7.2.8), +and the continuity of compositions, we then complete the verification of the fourth item. +The fifth and sixth items are immediate consequences of the first conclusions of Proposition 3.10 +and Theorem 3.17. This completes the proof. +□ +We conclude this section by recording some simple consequences of Theorem 7.3. +Corollary 7.4 (Open set of data for fixed wave speed). For each γ ∈ R+ there exists a nonempty +open set (0, 0, 0) ∈ W(γ) ⊂ W1+β with the property that for all stress-force data tuples (T , G, F) ∈ +W(γ) there exists a unique (q, u, η) ∈ V(γ) such that system (1.4.9) is satisfied with solution (q, u, η), +wave speed γ, and data (T , G, F). +Proof. Given γ ∈ R+ we take W(γ) = {(T , G, F) : (T , G, F, γ) ∈ Uβ}. +□ +Corollary 7.5 (On the Eulerian formulation). Each solution to the flattened, perturbative enthalpy +formulation of the traveling wave problem for free boundary compressible Navier-Stokes, i.e sys- +tem (1.4.9), produced by Theorem 7.3, gives rise to a classical solution to the traveling Eulerian +formulation of the problem given by system (1.2.1). +Proof. The sixth item of Theorem 7.3 verifies that the flattening map Fη : Ω → Ω[η] from (1.4.4) is +a smooth diffeomorphism that is sufficiently smooth up to the boundary as to preserve the notion of +classical solutions upon undoing the flattening. This same theorem also gives us classical solutions +to (1.4.9) as an easy consequence of various supercritical Sobolev embeddings. By combining these +facts and undoing the nonlinear changes of unknowns that took us from (1.2.1) to (1.4.9), we obtain +the stated result. +□ +Appendix A. Standard Sobolev space tools +A.1. Extension operators. Recall the Poisson extension operator E0 and the variant E introduced +in (1.4.2) and (1.4.3), respectively. The following lemma records some simple mapping properties. +Lemma A.1 (Mapping properties of the Poisson extension operator variants). The following hold. +(1) E0 : Hs+1/2(Σ) → 0H1(Ω) ∩ H1+s(Ω) is a bounded linear map for each for s ∈ N. +(2) E ◦ Π1 +H = E0, and E ◦ Π1 +L : H0(Σ) → W ∞,∞((0, b); H0 +(1)(Rn−1)) is a continuous linear map, +where H0 +(1) is defined by (B.1.3). +Proof. The first item is clear by standard elliptic theory for the Dirichlet problem. The second item +is immediate. +□ +Throughout the paper is is also frequently useful to consider extension operators mapping functions +defined on domains to functions defined on the entire Euclidean space that are regularity preserving +for the entire Sobolev scale. For these we use the celebrated extension operators of Stein; the +following definition sets our notation for these. +Definition A.2 (Stein-Extension operators and domains). The notion of a Stein extension operator +is given in Section 3.1 in Chapter VI of Stein [96]. A Stein extension domain is an open set U ⊂ Rd +for which there exists a Stein extension operator, which we will denote by EU. We will also employ +this notation more generally for open subsets of finite dimensional real vector spaces of dimension d +via the standard identification with Rd. + +COMPRESSIBLE TRAVELING WAVES +123 +A.2. Korn’s inequalities. Our first result here, which is classical, involves the (not normalized) +symmetric gradient, which we recall is defined for differentiable vector fields f by Df = ∇f + ∇ft. +Proposition A.3 (Korn’s inequality for the symmetric gradient, n ⩾ 2). Let n ⩾ 2 be the ambient +dimension. +Then there exists a constant c ∈ R+, depending only Ω and n, such that for all +f ∈ 0H1(Ω; Rn) we have the inequality ∥f∥H1 ⩽ c ∥Df∥L2. +Proof. The proof can be found in Lemma 2.7 in Beale [8], and is based on Theorem 12.II of +Fichera [34], which gives a Korn inequality in cubes of general dimension. We note that the result +in [8] is only stated for n = 3, but the same argument works in general for n ⩾ 2. +□ +We are also interested in inequalities involving the trace-free part of the symmetric gradient, +also known as the deviatoric gradient. We denote this for differentiable vector fields f by D0f = +∇f + ∇ft − 2 +n(∇ · f)I. In dimensions n ⩾ 3, the analog of Proposition A.3 for D0 holds. +Proposition A.4 (Deviatoric Korn’s inequality, n ⩾ 3). Let n ⩾ 3 be the ambient dimension. +There exists a constant c ∈ R+, depending only on Ω and n, such that for all f ∈ 0H1(Ω; Rn) we +have the inequality ∥f∥H1 ⩽ c∥D0f∥L2. +Proof. See, for instance, Theorem 1.1 in Dain [22] for a proof of the bound +∥f∥H1(Q;Rn) ≲ ∥D0f∥L2(Q;Rn×n) + ∥f∥L2(Q;Rn) for all f ∈ H1(Q; Rn) +(A.2.1) +whenever Q is a cube. By a standard compactness argument, we can then show that ∥f∥H1(Q;Rn) ≲ +∥D0f∥L2(Q;Rn×n) for all f ∈ H1(Q; Rn) vanishing on one side of the cube, where the implicit +constant depends on Q. We then utilize this inequality and the symmetries of Ω, as was done in +Proposition A.3, to obtain the desired result. +□ +Remark A.5 (Failure of deviatoric Korn in dimension n = 2). For dimension n = 2, however, +the analog of Proposition A.4 is false. See, for instance, Section 6.6 of Bauer, Neff, Pauly, and +Starke [6], where it is proved that there exists a Lipschitz domain ∅ ̸= U ⊂ R2 and a proper segment +∅ ̸= S ⊂ ∂U with the property that ∥·∥H1 ̸≲ ∥D0(·)∥L2 on the subspace of H1(U; R2) consisting of +vector fields vanishing on S. +A.3. Refined interpolation of Sobolev spaces. In this subsection we derive refined interpolation +inequalities for linear mappings between Sobolev spaces; in particular, this discussion considers +improvements for the K-method of real interpolation. For more information on the basic K-method +of interpolation, we refer to Chapter 3 in Bergh and L¨ofstr¨om [9]. First, we recall the definition of +the K-functional. Given a pair of Banach spaces X0 and X1 that embed into a Hausdorff topological +vector space Z, i.e. X0, X1 �→ Z, we define the map K(·, ·; X0, X1) : R+ × (X0 + X1) → R, via +K(t, x; X0, X1) = inf{∥x0∥X0 + t∥x1∥X1 : x = x0 + x1, (x0, x1) ∈ X0 × X1} +(A.3.1) +for (t, x) ∈ R+ × (X0 + X1). Our first lemma realizes the K-functional on Sobolev spaces. +Lemma A.6 (Sobolev space K-functional). Let s0, s1, s ∈ R, be such that s > 0 and s0 + s = s1. +For t ∈ R+ define the bounded linear map P s +t : Hs0(Rn) → Hs0+2s(Rn) via P s +t = (1 + t2⟨∇⟩2s)−1. +For all f ∈ Hs0(Rn) we have that +2−1K(t, f; Hs0; Hs1)2 ⩽ ∥(I − P s +t )f∥2 +Hs0 + t2∥P s +t f∥2 +Hs1 ⩽ K(t, f; Hs0; Hs1)2. +(A.3.2) +Proof. Define ˆK(t, f; Hs0, Hs1) = infg∈Hs1 +� +∥f − g∥2 +Hs0 + t2∥g∥2 +Hs1 +� +and note that elementary +estimates show that 2−1K(t, f; Hs0, Hs1)2 ⩽ ˆK(t, f; Hs0, Hs1) ⩽ K(t, f; Hs0, Hs1)2. The benefit of +switching to ˆK is that we may readily employ the direct method in the calculus of variations to see +that the infimum in the definition of ˆK(t, f; Hs0, Hs1) is actually a minimum, achieved by f⋆ ∈ Hs1 +satisfying (by virtue of the Euler-Lagrange equations) +− ⟨⟨∇⟩s0(f − f⋆), ⟨∇⟩s0g⟩L2 + t2⟨⟨∇⟩s0+sf⋆, ⟨∇⟩s0+sg⟩L2 = 0 for every g ∈ Hs1(Rn). +(A.3.3) + +124 +NOAH STEVENSON AND IAN TICE +This is equivalent to saying that f⋆ is a weak solution to the elliptic pseudo-differential equation (I + +t2⟨∇⟩2s)⟨∇⟩s0f∗ = ⟨∇⟩s0f, or alternatively, f⋆ = (I +t2⟨∇⟩2s)−1f = P s +t f. Thus, ˆK(t, f; Hs0, Hs1) = +∥(I − P s +t )f∥2 +Hs0 + t2∥P s +t f∥2 +Hs1. +□ +Our next lemma expresses the norm in Sobolev spaces in terms of the K-functional. The point of +the following computation is to divide the K-functional norm by a suitable constant to remove the +degeneracies near the end-points. +Lemma A.7 (A norm computation). Let s0, s1, s ∈ R+ be such that s > 0 and s0 + s = s1, and let +σ ∈ (0, 1). Define c(σ) = +� ∞ +0 +τ 1−2σ +1+τ 2 dτ ∈ R+. We have the equivalence of norms +∥f∥H(1−σ)s0+σs1 ⩽ +� 1 +c(σ) +� ∞ +0 +t−2σ−1K(t, f; Hs0, Hs1)2 dt +�1/2 +⩽ +√ +2∥f∥H(1−σ)s0+σs1. +(A.3.4) +Proof. Given t ∈ R+, we may equate +∥(I − P s +t )f∥2 +Hs0 + t2∥P s +t f∥2 +Hs1 = +� +Rn⟨2πξ⟩2s0� +t2⟨2πξ⟩2s +1 + t2⟨2πξ⟩2s +�2 +|F[f](ξ)|2 dξ ++ +� +Rn⟨2πξ⟩2(s0+s)� +t +1 + t2⟨2πξ⟩2s +�2 +|F[f](ξ)|2 dξ = +� +Rn⟨2πξ⟩2s0 +t2⟨2πξ⟩2s +1 + t2⟨2πξ⟩2s |F[f](ξ)|2 dξ. +(A.3.5) +Hence, by Tonelli’s theorem and a change of variables, we have that +� ∞ +0 +t−2σ−1(∥(I − P s +t )f∥2 +Hs0 + t2∥P s +t f∥2 +Hs1) dt += +� +Rn⟨2πξ⟩2s0 +� +R+ t−2σ−1 +t2⟨2πξ⟩2s +1 + t2⟨2πξ⟩2s dt |F[f](ξ)|2 dξ += c(σ) +� +Rn⟨2πξ⟩2(s0+σs)|F[f](ξ)|2 dξ = c(σ)∥f∥2 +H(1−σ)s0+σs1. +(A.3.6) +The result then follows by combining this with Lemma A.6. +□ +We now come to the main result of this subsection of the appendix. +Proposition A.8 (Refined interpolation of Sobolev spaces). Let s0, s1, s, r0, r1, r ∈ R with s, r > 0, +s0 + s = s1, and r0 + r = r1. Assume that +T ∈ L(Hr0(Rn); Hs0(Rn)) ∩ L(Hr1(Rn); Hs1(Rn)) +(A.3.7) +is such that for some constants C0, C1, A ∈ R+ we have the bounds +∥Tf∥s0 ⩽ C0∥f∥r0 and ∥Tf∥s1 ⩽ C1∥f∥r1 + A∥f∥r0, +(A.3.8) +for all appropriate f. Set r−1 = r0 − r. Then for all σ ∈ [0, 1] we have the inclusion +T ∈ L(H(1−σ)r0+σr1(Rn); H(1−σ)s0+σs1(Rn)) +(A.3.9) +and the bound +∥Tf∥H(1−σ)s0+σs1 +4C1−σ +0 +Cσ +1 +⩽ ∥f∥H(1−σ)r0+σr1 + σ1/2 A +C1 +∥f∥H(1−σ)r−1+σr0 +(A.3.10) +for all appropriate f. + +COMPRESSIBLE TRAVELING WAVES +125 +Proof. Given t ∈ R+ and f ∈ Hr0, we decompose f = (I − P r +C−1 +0 +C1t)f + P r +C−1 +0 +C1tf, with (I − +P r +C−1 +0 +C1t)f ∈ Hr0 and P r +C−1 +0 +C1tf ∈ Hr1. By the definition of K and the boundedness hypotheses, we +see may then estimate +K(t, Tf; Hs0, Hs1) ⩽ ∥T(I − P r +C−1 +0 +C1t)f∥Hs0 + t∥TP r +C−1 +0 +C1tf∥Hs1 +⩽ C0 +� +∥(I − P r +C−1 +0 +C1t)f∥Hr0 + C−1 +0 C1t∥P r +C−1 +0 +C1tf∥Hr1 +� ++ At∥P r +C−1 +0 +C1tf∥Hr0. +(A.3.11) +We apply Lemma A.6 to see that +K(t, Tf; Hs0, Hs1) ⩽ +√ +2C0K(C−1 +0 C1t, f; Hr0, Hr1) + At∥P r +C−1 +0 +C1tf∥Hr0. +(A.3.12) +Upon squaring, multiplying by t−2σ−1, integrating over R+, and employing Lemma A.7, we acquire +the bound +4−1c(σ)∥Tf∥2 +H(1−σ)s0+σs1 ⩽ c(σ)C2(1−σ) +0 +C2σ +1 ∥f∥2 +H(1−σ)r0+σr1 ++ A2(C−1 +0 C1)2(σ−1) +� ∞ +0 +τ −2σ+1∥P r +τ f∥2 +Hr0 dτ. +(A.3.13) +The final integral above can then be computed explicitly by using Tonelli’s theorem: +� ∞ +0 +τ −2σ+1∥P r +τ f∥2 +Hr0 dτ = +� +Rn⟨2πξ⟩2r0 +� ∞ +0 +τ −2σ+1 +(1 + τ 2⟨2πξ⟩2r)2 dτ |F[f](ξ)|2 dξ += d(σ) +� +Rn⟨2πξ⟩2(r0+(σ−1)r)|F[f](ξ)|2 dξ = d(σ)∥x∥2 +H(1−σ)r−1+σr0, +(A.3.14) +where we define d(σ) = +� ∞ +0 +τ −2σ+1 +(1+τ 2)2 dτ. Together, (A.3.13) and (A.3.14) imply that +∥Tf∥H(1−σ)s0+σs1 +2C1−σ +0 +Cσ +1 +⩽ ∥f∥H(1−σ)r0+σr1 + A +C1 +� +d(σ) +c(σ) ∥f∥H(1−σ)r−1+σr0. +(A.3.15) +The proof is complete upon noting the bounds c(σ) ⩾ +� 1 +0 +τ 1−2σ +2 +dτ + +� ∞ +1 +τ 1−2σ +2τ 2 +dτ = 1 +4( +1 +1−σ + 1 +σ) and +d(σ) ⩽ +� 1 +0 τ −2σ+1 dτ + +� ∞ +1 τ −2σ−3 dτ = 1 +2( +1 +1−σ + +1 +1+σ), which imply that d(σ) +c(σ) ⩽ +4σ +1+σ ⩽ 4σ. +□ +Appendix B. Some nonstandard function spaces +B.1. The anisotropic Sobolev spaces. For R ∋ s ⩾ 0 and d ∈ N+ we define the anisotropic +Sobolev space +Hs(Rd) = {f ∈ S ∗(Rd; R) : F[f] ∈ L1 +loc(Rd; C), ∥f∥Hs < ∞}, +(B.1.1) +equipped with the norm +∥f∥Hs = +� � +Rd +� +|ξ|−2(ξ2 +1 + |ξ|4)1B(0,1)(ξ) + ⟨ξ⟩2s1Rd\B(0,1)(ξ) +� +|F[f](ξ)|2 dξ +�1/2 +. +(B.1.2) +These spaces were introduced in Leoni and Tice [60], where it was shown, in Proposition 5.2 and +Theorem 5.6, that Hs(Rd) is a Hilbert space and Hs(Rd) �→ Hs(Rd) �→ Hs(Rd) + C∞ +0 (Rd), with +equality in the first embedding if and only if d = 1. The ‘anisotropic’ descriptor is justified by the +low frequency multiplier in (B.1.1) as well as Theorem 5.2 in [60], which shows that Hs(Rd) is not +closed under composition with rotations when d ⩾ 2. +We make the following notation for band-limited subspaces. Given κ ∈ R+, we define the space +H0 +(κ)(Rd) ⊆ +� +s⩾0 +Hs(Rd) via H0 +(κ)(Rd) = {f ∈ H0(Rd) : suppF[f] ⊆ B(0, κ)}. +(B.1.3) +We will also sometimes write Hs +(κ)(Rd) = H0 +(κ)(Rd) for any 0 ⩽ s ∈ R. + +126 +NOAH STEVENSON AND IAN TICE +Now let us enumerate the properties of theses spaces pertinent to this work. First, we discuss +a high-low decomposition for which the following notational convention is set. For κ ∈ R+, we +define the linear operators Πκ +L and Πκ +H on the subspace of f ∈ S ∗(Rd; C) such that F[f] is locally +integrable via +Πκ +Lf = F −1[1B(0,κ)F[f]] and Πκ +Hf = (I − Πκ +L)f. +(B.1.4) +Proposition B.1 (Frequency splitting for anisotropic Sobolev spaces). The following hold for +0 ⩽ s ∈ R, f ∈ Hs(Rd), and κ ∈ R+. +(1) We have the equivalence ∥f∥Hs ≍s,κ +� +∥Πκ +Lf∥2 +H0 + ∥Πκ +Hf∥2 +Hs. +(2) We have that Πκ +Lf ∈ C∞ +0 (Rd) with the estimates ∥Πκ +Lf∥W k,∞ ≲k,κ ∥Πκ +Lf∥H0 for every k ∈ N. +Proof. This is Theorem 5.5 in Leoni and Tice [60]. +□ +The following algebra properties of the anisotropic Sobolev spaces are extremely important in +our nonlinear analysis. +Proposition B.2 (Algebra properties of anisotropic Sobolev spaces). Suppose that f1, f2 ∈ H0 +(κ)(Rd) +for some κ ∈ R+. The following hold. +(1) The pointwise product f1f2 belongs to H0 +(2κ)(Rd) and satisfies the estimate ∥f1f2∥H0 ≲κ +∥f1∥H0∥f2∥H0. +(2) Set +rd = +� +1 + +� d+1 +d−1 +� +, +if d > 1, +1 +if d = 1. +(B.1.5) +Assume additionally that f3, . . . , frd ∈ H0 +(κ)(Rd). Then the pointwise product �rd +j=1 fj belongs +to (L2 ∩ H0 +(rd·κ))(Rd) and satisfies +��� +rd +� +j=1 +fj +��� +L2 ≲κ,d +rd +� +j=1 +∥fj∥H0. +(B.1.6) +(3) If g ∈ Hs(Rd), then the pointwise product f1g belongs to Hs(Rd) and satisfies +∥f1g∥Hs ≲κ,s ∥f1∥H0∥g∥Hs. +(B.1.7) +Proof. The first item is proved in Section 2.2 in Koganemaru and Tice [57]. +We now prove the second item in the case that κ = 1 and d ⩾ 2. The general case for d ⩾ 2 +can be handled similarly, and the case d = 1 is trivial. We define the function µ : B(0, 1) → R via +µ(ξ) = |ξ|−2(ξ2 +1 + |ξ|4) and note that µ is the multiplier that encodes the low frequency control +in H0(Rd). We will calculate for which q ∈ [1, ∞) we have 1/µ ∈ Lq(B(0, 1)). Consider the +decomposition B(0, 1) = E0 ⊔ E1 +E0 = {ξ = (ξ1, ξ∗) ∈ B(0, 1) : |ξ1 ± 1/2|2 + |ξ∗|2 < 1/4}, +E1 = B(0, 1) \ E0. +(B.1.8) +An elementary calculation shows that for ξ ∈ B(0, 1) we have that +ξ ∈ E0 ⇔ |ξ|4 < |ξ1|2 +and +ξ ∈ E1 ⇔ |ξ1|2 ⩽ |ξ|4. +(B.1.9) +In turn, these show that +� +ξ2 +1/ |ξ|2 ⩽ µ(ξ) ⩽ 2ξ2 +1/ |ξ|2 +for ξ ∈ E0 +|ξ|2 ⩽ µ(ξ) ⩽ 2 |ξ|2 +for ξ ∈ E1. +(B.1.10) +For 1 ⩽ q < ∞, we then have the equivalence +� +B(0,1) +µ−q ≍ +� +E0 +ξ−2q +1 +|ξ|2q dξ + +� +E1 +|ξ|−2q dξ. +(B.1.11) + +COMPRESSIBLE TRAVELING WAVES +127 +We may use spherical coordinates in the ξ∗ ∈ Rd−1 variable to see that +� +E1 +|ξ|−2q dξ < ∞ +(B.1.12) +if and only if +� 1/2 +0 +rd−2 +� 1/2−√ +1/4−r2 +0 +dx1 +(x2 +1 + r2)q dr < ∞, +(B.1.13) +but for the latter we can bound +� 1/2 +0 +rd−2 +� 1/2−√ +1/4−r2 +0 +dx1 +(x2 +1 + r2)q dr ⩽ +� 1/2 +0 +rd−2−2q�1 +2 − +� +1 +4 − r2 +� +dr, +(B.1.14) +and since the term in parentheses behaves like r2 for r ∼ 0 this will be finite if and only if +d − 2 − 2q + 2 > −1 ⇔ q < (d + 1)/2. By putting this together, we see that if q < (d + 1)/2 +then the integral (B.1.12) is indeed finite. Next we consider the E0 integral, again using spherical +coordinates in ξ∗. We have that +� +E0 +ξ−2q +1 +|ξ|2q dξ < ∞ +(B.1.15) +if and only if +� 1/2 +0 +rd−2 +� 1/2 +1/2−√ +1/4−r2 +� +1 + r2 +x2 +1 +�q +dx1 dr < ∞, +(B.1.16) +and we know that this integral is finite if we know that +� 1/2 +0 +rd−2 +� 1/2 +1/2−√ +1/4−r2 +� r2 +x2 +1 +�q +dx1 dr < ∞. +(B.1.17) +For this latter integral we compute +� 1/2 +0 +rd−2 +� 1/2 +1/2−√ +1/4−r2 +� r2 +x2 +1 +�q +dx1 dr ⩽ +1 +2q − 1 +� 1/2 +0 +rd−2+2q�1 +2 − +� +1 +4 − r2 +�1−2q +dr, +(B.1.18) +and this is finite if and only if d − 2 + 2q + 2(1 − 2q) > −1 ⇔ q < d+1 +2 . So once more we guarantee +the integral (B.1.15) is finite if q < (d + 1)/2. Hence (B.1.11) is finite for the same range of q. +Now suppose that f ∈ H0 +(1)(Rd). Since ∥f∥H0 ≍ ∥√µ(∇/2πi)f∥L2, it follows from H¨older’s +inequality that for 1 ⩽ p < 2 we have +∥F[f]∥Lp ≲ ∥f∥H0 +� � +B(0,1) +(1/µ)p/(2−p)�1/p−1/2 +. +(B.1.19) +The above analysis shows that the coefficient given by the integral on the right hand side is finite +if and only if p/(2 − p) < (d + 1)/2. This is equivalent to p < 2(d+1) +d+3 , and thus F[f] ∈ Lp(Rd) for +1 ⩽ p < 2(d+1) +d+3 . In turn, thanks to the Hausdorff-Young inequality, we have that +∥f∥Lq ≲q ∥f∥H0 for 2(d + 1) +d − 1 +< q ⩽ ∞. +(B.1.20) +Hence, by H¨older’s inequality, for any k ∈ N with k ⩽ 2(d+1) +d−1 +we have that +∥fk∥Lq ≲q,k ∥f∥k +H0 for 2(d + 1) +k(d − 1) < q ⩽ ∞. +(B.1.21) + +128 +NOAH STEVENSON AND IAN TICE +Define rd = min +� +k ∈ N : +2(d+1) +k(d−1) < 2 +� +. Note that the minimum exists since 2 < 2(d+1) +d−1 +⩽ 6. In fact, +it is easy to see that +rd = 1 + +�d + 1 +d − 1 +� += +� +� +� +� +� +4 +if d = 2, +3 +if d = 3, +2 +if d > 3, +(B.1.22) +and that ∥frd∥L2 ≲ ∥f∥rd +H0. The second item then follows by one more application of H¨older’s +inequality. +Finally, we prove the third item. Proposition B.1 and the first item allow us to estimate +∥f1g∥Hs ⩽ ∥f1Πκ +Lg∥Hs + ∥f1Πκ +Hg∥Hs ≲ ∥f1∥H0∥Πκ +Lg∥H0 + ∥f1Πκ +Hg∥Hs +≲ ∥f1∥H0∥g∥Hs + ∥f1∥W s,∞∥Πκ +Hg∥Hs ≲ ∥f1∥H0∥g∥Hs, +(B.1.23) +which completes the proof of the third item. +□ +Remark B.3. From the proof of Proposition B.2, specifically (B.1.20), and Proposition B.1 we +deduce that for R ∋ s ⩾ 0, 2(d + 1)/(d − 1) < q ⩽ ∞ and f ∈ Hs(Rd), the low mode part Π1 +Lf +belongs to Lq(Rd) and the high mode part Π1 +Hf belongs to L2(Rd). In fact, the induced embedding +Hs(Rd) �→ (Lq + L2)(Rd) is continuous. +We now develop a spatial characterization of the specialized Sobolev spaces that will be useful +when employing these spaces in a priori estimates. Recall that the homogeneous spaces ˙H−1 are +defined in (1.7.4). +Proposition B.4 (Characterizations of the anisotropic Sobolev spaces). The following hold for +R ∋ s ⩾ 0. +(1) If f ∈ S ∗(Rd; R) satisfies ∇f ∈ Hs−1(Rd; Rd) and ∂1f ∈ ˙H−1(Rd), then there exists a +constant c ∈ R such that f − c ∈ Hs(Rd). +(2) Assume d ⩾ 2 and write pd = 2(d+1) +d−1 +> 2. Then we have the equality +Hs(Rd) = +� +f ∈ L2(Rd) + +� +pd0 +such that if f ∈ L2(U) and α ∈ W 1,∞(U; Rn) satisfy ∇ · (αf) ∈ L2(U), then we have that +{Sεf}ε>0 ⊆ C∞ +c (U) with supp(Sεf) ⊆ R + 1, Sεf ε→0 +→ f in L2(U), and ∇ · (αSεf) ε→0 +→ ∇ · (αf) in +L2(U). +Note that Ω ⊂ Rn does not have a compact boundary, so we cannot directly apply this result. +However, thanks to Lemma B.7 we can reduce to this case. Let ϕ ∈ H0 +X(Ω) and let κ ∈ R+. +Thanks to the aforementioned lemma there exists a �ϕ ∈ H0 +X(Ω) with bounded support such that +∥ϕ − �ϕ∥H0 +X ⩽ κ/2. For some 3 ⩽ R ∈ R+, we then have that supp(�ϕ) ⊆ UR = BRn(0, R) ∩ Ω. The +set U2R is a Lipschitz domain with compact boundary, and hence it admits smoothing operators +{Sε}ε>0 as described above. By the support conditions on �ϕ and the properties of these smoothing +operators, there exits an ε0 ∈ R+ such that for all 0 < ε ⩽ ε0 we have that supp(Sε �ϕ) ⊆ U3R/2 and +� +∥Sε �ϕ − �ϕ∥2 +L2(U2R) + ∥∇ · (XSε �ϕ) − ∇ · (X �ϕ)∥2 +L2(U2R) ⩽ κ/2. +(B.2.7) +Let ϕ = Sε0 �ϕ. By support considerations on �ϕ and ϕ, we have that actually ϕ ∈ C∞ +c (Ω) and +� +∥ϕ − �ϕ∥L2(U2R) = ∥ϕ − �ϕ∥L2(Ω), +∥∇ · (Xϕ) − ∇ · (X �ϕ)∥L2(U2R) = ∥∇ · (X(ϕ − �ϕ))∥L2(Ω). +(B.2.8) +Hence, by the previous two equations we get ∥ϕ − �ϕ∥H0 +X ⩽ κ/2. The estimate ∥ϕ − ϕ∥H0 +X ⩽ κ now +follows from the triangle inequality, and density is proved. +□ +The next result records some integration by parts formulas. +Proposition B.9 (Some integration by parts). Let X ∈ W 1,∞(Ω; Rn) be such that Tr∂Ω(X ·en) = 0. +The following hold. +(1) If ϕ ∈ H0 +X(Ω), then +� +Ω +∇ · X +2 +ϕ2 = +� +Ω +ϕ∇ · (ϕX). +(B.2.9) +(2) If ϕ ∈ H0 +X(Ω) and ψ ∈ H1(Ω), then +� +Ω ∇ · (ϕX)ψ = − +� +Ω ϕX · ∇ψ. +Proof. We begin by proving the first item. Assume initially that ϕ ∈ H1(Ω). By using the product +rule in two ways, we compute that +2−1∇ · (ϕ2X) = ∇ · (ϕ2X) − 2−1∇ · (ϕ2X) = ϕ∇ · (ϕX) − 2−1(∇ · X)ϕ2. +(B.2.10) +We then integrate this over Ω, apply the divergence theorem, and use the condition Tr∂Ω(X · en) = 0 +to get (B.2.9). The case of general ϕ ∈ H0 +X(Ω) follows from a density argument via Lemma B.8. +The second item is proved via a similar density argument. +□ +Next we prove a useful estimate. +Proposition B.10 (Refined divergence compatibility estimate). Suppose that X ∈ W 1,∞(Ω) and +ϕ ∈ H0 +X(Ω). Then the function +� b +0 ∇ · (Xϕ)(·, y) dy : Rn−1 → R belongs to ˙H−1(Rn−1), satisfies the +equality +� b +0 +∇ · (Xϕ)(·, y) dy = (∇∥, 0) · +� b +0 +(Xϕ)(·, y) dy, +(B.2.11) +and obeys the estimate +�� b +0 +∇ · (Xϕ)(·, y) dy +� +˙H−1 ≲ ∥Xϕ∥L2 +(B.2.12) +for an implicit constant depending only on b and the dimension. + +132 +NOAH STEVENSON AND IAN TICE +Proof. Let ψ ∈ C∞ +c (Rn−1) and view ψ ∈ C∞ +c (Ω) via the trivial extension ψ(x, y) = ψ(x) for +(x, y) ∈ Rn−1 × (0, b). Thanks to the second item of Proposition B.9 and Fubini’s theorem, we have +that +� +Rn−1 +� � b +0 +∇ · (Xϕ)(·, y) dy +� +ψ = +� +Ω +∇ · (Xϕ)ψ = − +� +Ω +ϕX · ∇ψ += − +� +Rn−1 +� � b +0 +(ϕX)(·, y); dy +� +· (∇∥ψ, 0). +(B.2.13) +Since ψ was arbitrary, the identity (B.2.11) is established. The estimate (B.2.12) readily follows +from (B.2.11) and the definition of the norm on ˙H−1(Rn−1). +□ +Appendix C. A selection of PDE tools +C.1. The divergence boundary value problem. Our focus in this subsection is the construction +of right-inverses of the divergence operator in various spaces. We first make the following notation: +for s ∈ [0, ∞), we define the function space +ˆHs(Ω) = +� +φ ∈ Hs(Ω) : +� b +0 +φ(·, y) dy ∈ ˙H−1(Σ) +� +, +(C.1.1) +where the homogeneous spaces ˙H−1 are defined in (1.7.4). This space is Hilbert for the norm +∥φ∥ ˆHs = +� +∥φ∥2 +Hs + +�� b +0 φ(·, y) dy +�2 +˙H−1 +�1/2. +We now provide a useful compatibility estimate related to the divergence and the normal trace. +Proposition C.1 (Divergence-normal trace compatibility condition). If u ∈ H1(Ω; Rn), then +� +TrΣu · en − TrΣ0u · en − +� b +0 +(∇ · u)(·, y) dy +� +˙H−1(Rn−1) ≲ ∥u∥L2. +(C.1.2) +Proof. Thanks to the fundamental theorem of calculus, we have +TrΣu · en − TrΣ0u · en − +� b +0 +(∇ · u)(·, y) dy = −(∇∥, 0) · +� b +0 +u(·, y) dy. +(C.1.3) +Estimate (C.1.2) now follows by definition of the norm on ˙H−1(Rn−1) (see (1.7.4)). +□ +The rest of this subsection is devoted building solution operators to divergence boundary value +problems. +Proposition C.2 (Solution operators to divergence boundary value problems). There exist bounded +linear operators B0 and B1 such that for k ∈ N, +B0 : ˆHk(Ω) → H1+k(Ω; Rn) ∩ H1 +0(Ω; Rn) +(C.1.4) +and +B1 : {ϕ ∈ H1/2+k(Σ; Rn) : ϕ · en ∈ ˙H−1(Σ)} → H1+k(Ω; Rn) ∩ 0H1(Ω; Rn), +(C.1.5) +with the properties +� +∇ · B0ψ = ψ +in Ω, +B0ψ = 0 +on ∂Ω, +and +� +� +� +� +� +∇ · B1ϕ = 0 +in Ω, +B1ϕ = ϕ +on Σ, +B1ϕ = 0 +on Σ0. +(C.1.6) + +COMPRESSIBLE TRAVELING WAVES +133 +Proof. We begin with the construction of B0. Fix k ∈ N and ψ ∈ ˆHk(Ω). According to the Tonelli +and Parseval theorems, we have that +� +Rn−1 +k +� +j=0 +� b +0 +⟨ξ⟩2(k−j) |∂j +nF[ψ](ξ, y)|2 dy dξ ≍ ∥ψ∥2 +Hk , +(C.1.7) +and in particular this implies that for almost every ξ ∈ Rn−1 we have that +k +� +j=0 +� b +0 +⟨ξ⟩2(k−j) |∂j +nF[ψ](ξ, y)|2 dy < ∞. +(C.1.8) +Without loss of generality, we may assume that, in fact, this quantity is finite for every ξ ∈ Rn−1. +We then define the measurable function ζ : Ω → C via +ζ(ξ, y) = +cosh(2π |ξ| y) +2π |ξ| sinh(2π |ξ| b) +� b +0 +F[ψ](ξ, t) cosh(2π |ξ| (b − t)) dt +− +� y +0 +F[ψ](ξ, t)sinh(2π |ξ| (y − t)) +2π |ξ| +dt. +(C.1.9) +It is a simple matter to verify that for each ξ ∈ Rn−1\{0} we have the inclusion ζ(ξ, ·) ∈ H2((0, b); C) +and that +� +−∂2 +nζ(ξ, ·) + 4π2 |ξ|2 ζ(ξ, ·) = F[ψ](ξ, ·) +in (0, b) +∂nζ(ξ, t) = 0 +for t ∈ {0, b}. +(C.1.10) +Multiplying the ODE satisfied by ζ by ζ and integrating by parts over (0, b), we see that +� b +0 +(|∂nζ(ξ, y)|2 + 4π2 |ξ|2 |ζ(ξ, y)|2) dy = +� b +0 +F[ψ](ξ, y)ζ(ξ, y) dy += +� b +0 +F[ψ](ξ, y)(ζ(ξ, y) − ⟨ζ(ξ, ·)⟩(0,b)) dy + ⟨ζ(ξ, ·)⟩(0,b) +� b +0 +F[ψ](ξ) dy. +(C.1.11) +In the above we have written ⟨·⟩(0,b) to be the integral average over (0, b). +By the Poincar´e, +Cauchy-Schwarz, and Cauchy inequalities, we may then bound +��� +� b +0 +F[ψ](ξ, y)ζ(ξ, y) dy +��� ⩽ 1 +2 +� b +0 +(|∂nζ(ξ, y)|2 + 4π2 |ξ|2 |ζ(ξ, y)|2) dy ++ C +� b +0 +|F[ψ](ξ, y)|2 dy + C +��� +� b +0 +F[ψ](ξ, y) +|ξ| +dy +��� +2 +(C.1.12) +for a constant C > 0 depending only on b. On the other hand, a similar argument shows that +� b +0 +(|ξ|2(k+1) |∂nζ(ξ, y)|2 + 4π2 |ξ|2(k+2) |ζ(ξ, y)|2) dy = +� b +0 +|ξ|k F[ψ](ξ, y) |ξ|k+2 ζ(ξ, y) dy +⩽ 1 +2 +� b +0 +4π2 |ξ|2(k+2) |ζ(ξ, y)|2 dy + C +� b +0 +|ξ|2k |F[ψ](ξ, y)|2 dy +(C.1.13) +for a constant C > 0 depending only on b. Upon combining these bounds, we deduce that ζ(ξ, ·) +obeys the estimate +⟨ξ⟩2(k+1) +� b +0 +(|∂nζ(ξ, y)|2 + 4π2 |ξ|2 |ζ(ξ, y)|2) dy ≲ ⟨ξ⟩2k +� b +0 +|F[ψ](ξ, y)|2 dy + +��� +� b +0 +F[ψ](ξ, y) +|ξ| +dy +��� +2 +. +(C.1.14) + +134 +NOAH STEVENSON AND IAN TICE +Returning to (C.1.10) and exploiting (C.1.8) and a simple finite induction argument, we deduce +that ζ(ξ, ·) ∈ H2+k((0, b); C) and that for 0 ⩽ j ⩽ k we have the bound +⟨ξ⟩2(k−j) +� b +0 +|∂2+j +n +ζ(ξ, y)|2 dy ≲ ⟨ξ⟩2(k−j) +� b +0 +|∂j +nF[ψ](ξ, y)|2 dy ++ ⟨ξ⟩2(k−j+1) 4π2|ξ|2 +� b +0 +|∂j +nζ(ξ, y)|2 dy. +(C.1.15) +We may then take appropriate linear combinations of these estimates, for 0 ⩽ j ⩽ k, and (C.1.14), +in order to deduce the bound +⟨ξ⟩2(k+1) +� b +0 +4π2 |ξ|2 |ζ(ξ, y)|2 dy + +2+k +� +j=1 +⟨ξ⟩2(k+2−j) +� b +0 +|∂j +nζ(ξ, y)|2 dy +≲ +k +� +j=0 +� b +0 +⟨ξ⟩2(k−j) |∂j +nF[ψ](ξ, y)|2 dy + +��� +� b +0 +F[ψ](ξ, y) +|ξ| +dy +��� +2 +. +(C.1.16) +Next, we define the vector field Ξ : Ω → Cn via Ξ(ξ, y) = (−2πiξζ(ξ, y), −∂nζ(ξ, y)). Since ψ is +real-valued we know that F[ψ](ξ, y) = F[ψ](−ξ, y) for ξ ∈ Rn−1 and y ∈ (0, b), and this readily +implies that Ξ(ξ, y) = Ξ(−ξ, y) as well. The bound (C.1.16), integrated over ξ ∈ Rn−1, implies that +Ξ ∈ L2(Ω; Cn), and so we may define X ∈ L2(Ω; Rn) via X = F −1[Ξ]. In turn, (C.1.16) and (C.1.7) +further imply that X ∈ H1+k(Ω; Rn) and +∥X∥H1+k ≲ +� +∥ψ∥2 +Hk + +�� b +0 +ψ(·, y) dy +�2 +˙H−1 +�1/2 += ∥ψ∥ ˆHk . +(C.1.17) +We use (C.1.10) to compute F[∇ · X](ξ, y) = (2πiξ, 0) · Ξ(ξ, y) + ∂nΞ · en(ξ, y) = −∂2 +nζ(ξ, y) + +4π2 |ξ|2 ζ(ξ, y) = F[ψ](ξ, y) and F[X](ξ, t) = Ξ · en(ξ, t) = 0 for t ∈ {0, b}. Thus, X satisfies +� +∇ · X = ψ +in Ω, +X · en = 0 +on ∂Ω. +(C.1.18) +We next recall that standard Sobolev trace theory (see for instance Chapter 7 in Adams and +Fournier [2] or Chapter 18 in Leoni [59]) shows that the trace map +H2+k(Ω) ∋ f �→ (Tr∂Ω[f], Tr∂Ω[∂nf]) ∈ H3/2+k(∂Ω) × H1/2+k(∂Ω) +(C.1.19) +is bounded and linear and admits a bounded right inverse, R. +Consequently, we may define +ω ∈ Hk+2(Ω; Rn−1) via ω = (R(0, −X · e1), . . . , R(0, −X · en−1)). Finally, this allows us to define +B0ψ ∈ H1+k(Ω; Rn) via B0ψ = X + (∂nω, ∇∥ · ω). Then from (C.1.18) and the construction of +ω we have that ∇ · B0ψ = ∇ · X + ∇∥ · ∂nω − ∂n∇∥ · ω = ψ in Ω, while on ∂Ω we have for +j ∈ {1, . . . , n − 1} (B0ψ) · ej = X · ej + ∂nω · ej = X · ej − X · ej = 0. Additionally we have +(B0ψ) · en = X · en − ∇∥ · ω = X · en = 0. Thus, B0ψ satisfies the properties stated in (C.1.6). +Moreover, the construction of B0ψ shows that the resulting map ψ �→ B0ψ is linear and satisfies +∥B0ψ∥Hk+1 ≲ ∥ψ∥ ˆHk. This completes the construction of the bounded linear map B0. +We now build B1. To this end, we write L for a bounded right inverse of the trace map +H1+k(Ω; Rn) ∩ 0H1(Ω; Rn) ∋ u �→ TrΣ(u) ∈ H1/2+k(Σ; Rn), +(C.1.20) +which again exists by standard trace theory. Note that if ϕ ∈ H1/2+k(Σ; Rn) and ϕ·en ∈ ˙H−1(Σ; Rn), +then +� b +0 +∇ · (Lϕ)(·, y) dy = (∇∥, 0) · +� b +0 +(Lϕ)(·, y) dy + ϕ · en(·), +(C.1.21) + +COMPRESSIBLE TRAVELING WAVES +135 +and we readily deduce from this that +�� b +0 +∇ · (Lϕ)(·, y) dy +� +˙H−1 ≲ ∥Lϕ∥L2 + [ϕ · en] ˙H−1 ≲ ∥ϕ∥H1/2+k + [ϕ · en] ˙H−1. +(C.1.22) +We may thus define the linear map B1 via B1ϕ = Lϕ − B0∇ · Lϕ. It is then a simple matter to +verify that B1 is bounded and B1ϕ satisfies the conditions stated in (C.1.6). This completes the +construction of B1. +□ +We next record a couple corollaries. The first constructs an operator on Hk(Ω). +Corollary C.3 (Right inverse to the divergence). There exists a bounded linear operator such that +for k ∈ N, B : Hk(Ω) → H1+k(Ω; Rn) ∩ 0H1(Ω; Rn) with ∇ · Bψ = ψ for all ψ ∈ Hk(Ω). +Proof. If for p ∈ N we define Apψ : Ω → R via Apψ(x, y) = yp +b +� b +0 ψ(x, t) dt, then Apψ ∈ Hk(Ω) and +∥Apψ∥Hk ≲p ∥ψ∥Hk. In other words, Ap ∈ L(Hk(Ω)). +Note that for ψ ∈ Hk(Ω) we have that +� b +0 (ψ(·, y)−A0ψ(·, y)) dy = +� b +0 ψ(·, y) dy− +� b +0 ψ(·, y) dy = 0. +Consequently, we may construct B0(ψ − A0ψ) ∈ H1+k(Ω; Rn) ∩ H1 +0(Ω; Rn) by using the operator +B0 from Proposition C.2. We then define B : Hk(Ω) → H1+k(Ω; Rn) ∩ 0H1(Ω; Rn) via Bψ = +B0(ψ − A0ψ) + (A1ψ)en. It is then a simple matter to check that this is bounded and linear and +satisfies ∇ · Bψ = ψ for all ψ ∈ Hk(Ω). +□ +The second corollary to Proposition C.2 constructs an operator on H1/2+k(Σ) ∩ ˙H−1(Σ). +Corollary C.4 (Solenoidal extension operator). There exists a bounded linear operator such that +for k ∈ N, B2 : H1/2+k(Σ) ∩ ˙H−1(Σ) → 0H1(Ω; Rn) with ∇ · B2χ = 0, TrΣ(B2χ · en) = χ, and +(I − en ⊗ en)TrΣ(B2χ) = 0. +Proof. Let B1 be as in Proposition C.2, and define B2 via B2χ = B1(χen). +□ +C.2. Elliptic theory tools. We begin this subsection by recording some results about the bilinear +form Bm from (4.1.1). While Bm is not coercive, the next result shows that it satisfies G˚arding’s +inequality. +Lemma C.5 (Bm G˚arding inequalities). Let m ∈ N and Λ ∈ W ∞,∞(Ω) with Λ > 0 and 1/Λ ∈ +L∞(Ω). There exist constants c, C ∈ R+, depending only on Λ, the dimension, Ω, and m, such that +c∥ϕ∥2 +Hm ⩽ Bm(Λϕ, ϕ) + C∥ϕ∥2 +L2 for all ϕ ∈ Hm(Ω). +Proof. Consider first the case that Λ = 1. We decompose Bm = Bm,∥ + Bm,n where +Bm,∥(ϕ0, ϕ1) = +n−1 +� +j=1 +� +Ω +∂m +j ϕ0∂m +j ϕ1 and Bm,n(ϕ0, ϕ1) = +� +Ω +∂m +n ϕ0∂m +n ϕ1. +(C.2.1) +For Bm,∥, we use Fubini’s theorem followed by Plancherel’s theorem to estimate +Bm,∥(ϕ, ϕ) = +n−1 +� +j=1 +� b +0 +� +Rn−1 |(2πiξj)mF[ϕ(·, y)](ξ)|2 dξ dy += +� b +0 +� +Rn−1 +�n−1 +j=1 |(2πiξj)m|2 +� +|α|=m |(2πiξ)α|2 +� +|α|=m +|(2πiξ)αF[ϕ(·, y)](ξ)|2 dξ dy +≳m +� +β∈Nn−1 +|β|=m +∥∂(β,0)ϕ∥2 +L2 = ∥ϕ∥2 +˙Hm(Rn−1;L2(0,b)). +(C.2.2) + +136 +NOAH STEVENSON AND IAN TICE +Hence, by the fact that ˙Hm ∩ L2 = Hm, we get that +Bm,∥(ϕ, ϕ) + ∥ϕ∥2 +L2 ≳m ∥ϕ∥2 +Hm(Rn−1;L2(0,b)). +(C.2.3) +On the other hand, for the Bm,n piece, we see that +Bm,n(ϕ, ϕ) = ∥ϕ∥2 +L2(Rn−1; ˙Hm(0,b)). +(C.2.4) +Thus, by again using that ˙Hm ∩ L2 = Hm, we get the bound +Bm,n(ϕ, ϕ) + ∥ϕ∥2 +L2 ≳Ω,m ∥ϕ∥2 +L2(Rn−1;Hm(0,b)). +(C.2.5) +Upon combining (C.2.3) and (C.2.5), we arrive at the estimate +Bm(ϕ, ϕ) + ∥ϕ∥2 +L2 ≳m,Ω ∥ϕ∥2 +HmL2∩L2Hm. +(C.2.6) +The proof in the case that Λ = 1 is then complete as soon as we note the slicing characterization +Hm(Ω) = Hm(Rn−1; L2(0, b)) ∩ L2(Rn−1; Hm(0, b)), +(C.2.7) +with norm equivalence, which can be proved as follows. The slicing characterization for Rn in place +of Ω is proved in Lemma A.6 from Leoni and Tice [60]. We then reduce (C.2.7) to this case with the +help of a Stein extension operator EΩ (see Definition A.2), which continuously maps both Hm(Ω) → +Hm(Rn) and Hm(Rn−1; L2(0, b)) ∩ L2(Rn; Hm(0, b)) → Hm(Rn−1; L2(R)) ∩ L2(Rn; Hm(R)). +We next consider the case of general Λ satisfying the stated hypotheses. We introduce commutators +as follows: +Bm(Λϕ, ϕ) = +n +� +j=1 +� +Ω +Λ(∂m +j ϕ)2 + [Λ, ∂m +j ]ϕ∂m +j ϕ. +(C.2.8) +Hence, we have the estimate +cBm(ϕ, ϕ) ⩽ Bm(Λϕ, ϕ) + +� +Bm(ϕ, ϕ) +� +� +� +� +n +� +j=1 +∥[Λ, ∂m +j ]ϕ∥2 +L2. +(C.2.9) +By Cauchy’s inequality and the boundedness of the commutator operator (see Corollary D.8), +cBm(ϕ, ϕ) ⩽ Bm(Λϕ, ϕ) + C∥ϕ∥2 +Hm−1. +(C.2.10) +Invoking the special case Λ = 1 then shows that +c∥ϕ∥2 +Hm ⩽ C∥ϕ∥2 +L2 + Bm(Λϕ, ϕ) + C∥ϕ∥2 +Hm−1. +(C.2.11) +For the final term above, we use the log-convexity of the norm and Young’s inequality: for κ ∈ (0, 1) +we estimate ∥ϕ∥Hm−1 ≲ κ−m∥ϕ∥L2 + κ∥ϕ∥Hm and then choose κ sufficiently small to absorb the +right hand side’s ∥ϕ∥Hm-contribution by the left. +□ +Our next lemma involves integration by parts. +Lemma C.6 (Bm integration by parts). Suppose that ψ ∈ H2m(Ω) and φ ∈ Hm(Ω). Then +Bm(φ, ψ) = +� +Ω +φLmψ + +m−1 +� +j=0 +� � +Σ +− +� +Σ0 +� +(−1)m−1−j∂j +nφ∂2m−j−1 +n +ψ, +(C.2.12) +where Lm is the linear elliptic operator defined in (3.5.6). +Proof. This is a simple exercise in integration by parts. +□ +Now we prove a priori estimates for the natural Neumann problem associated with the operator +Lm. + +COMPRESSIBLE TRAVELING WAVES +137 +Lemma C.7 (A priori estimates for Lm). Suppose that m ∈ N+, k ∈ N, ψ ∈ Hk(Ω), and +ϕ ∈ Hk+2m(Ω) are related via the equations +� +Lmϕ = ψ +in Ω, +∂m +n ϕ = · · · = ∂2m−1 +n +ϕ = 0 +on ∂Ω. +(C.2.13) +Then we have the a priori estimate +∥ϕ∥Hk+2m ≲ ∥ϕ∥L2 + ∥ψ∥Hk, +(C.2.14) +where the implicit constant depends only on k, m, and Ω. +Proof. We begin by proving the case k = 0. By taking the L2-inner product of the equation with ϕ +and utilizing Lemma C.6, we are left with +Bm(ϕ, ϕ) = +� +Ω +ψϕ. +(C.2.15) +For the right hand side we use the Cauchy-Schwarz inequality, while for the left hand side we use +the G˚arding inequality from Lemma C.5. This yields the inequality +∥ϕ∥H2m ≲ ∥ϕ, ψ∥L2×L2. +(C.2.16) +We now induct on k ∈ N. We have already established the base case, k = 0. Suppose that k ∈ N +and that whenever ϕ ∈ Hk+2m(Ω) and ψ ∈ Hk(Ω) are related via the equations (C.2.13), we have +the a priori estimate +∥ϕ∥Hk+2m ≲ ∥ϕ, ψ∥Hk×Hk. +(C.2.17) +We now prove the above necessarily holds for k + 1 as well. +Assume ϕ ∈ H1+k+2m(Ω) and +ψ ∈ H1+k(Ω) satisfy (C.2.13). Let j ∈ {1, . . . , n − 1} and apply ∂j to these equations. We then see +that +� +Lm∂jϕ = ∂jψ +in Ω, +∂m +n ∂jϕ = · · · = ∂2m−1 +n +∂jϕ = 0 +on ∂Ω. +(C.2.18) +Hence, we can apply the inductive hypothesis and sum over j to arrive at the bounds +n−1 +� +j=1 +∥∂jϕ∥Hk+2m ≲ +n−1 +� +j=1 +∥∂jϕ, ∂jψ∥Hk×Hk ⩽ ∥ϕ, ψ∥H1+k×H1+k. +(C.2.19) +On the other hand, we can rearrange the PDE satisfied by ϕ and ψ to see that (−1)m∂2m +n ϕ = +ψ − (−1)m �n−1 +j=1 ∂2m +j +ϕ. Thus, +∥∂2m +n ϕ∥H1+k ⩽ ∥ψ∥H1+k + +n−1 +� +j=1 +∥∂2m +j +ϕ∥H1+k ⩽ ∥ϕ∥H1+k + +n−1 +� +j=1 +∥∂jϕ∥Hk+2m. +(C.2.20) +Combining (C.2.19) and (C.2.20) then shows that +� +B1+k+2m(ϕ, ϕ) ⩽ +n−1 +� +j=1 +∥∂jϕ∥Hk+2m + ∥∂2m +n ϕ∥H1+k ≲ ∥ϕ, ψ∥H1+k×H1+k. +(C.2.21) +Finally, we use G˚arding’s inequality, Lemma C.5, to see that +∥ϕ∥H1+k+2m ≲ ∥ϕ, ψ∥H1+k×H1+k. +(C.2.22) +This completes the induction argument. +From the induction, we know that a weaker version of (C.2.14) holds, namely the same estimate +but with ∥ϕ∥L2 replaced by ∥ϕ∥Hk. However, we can readily derive the desired bound from this +weaker one by using interpolation, Young’s inequality, and absorption: for κ ∈ R+ we have that +∥ϕ∥Hk ≲ ∥ϕ∥ +2m +k+2m +L2 +∥ϕ∥ +k +k+2m +Hk+2m ≲ κ∥ϕ∥Hk+2m + κ− k +2m ∥ϕ∥L2. +(C.2.23) + +138 +NOAH STEVENSON AND IAN TICE +□ +If we add a 0th term to Lm, we obtain an existence theory. +Lemma C.8 (Existence theory for Lm). Let m ∈ N+ and κ ∈ R+. The operator +κ + Lm : {ϕ ∈ H2m(Ω) : Tr∂Ω(∂m +n ϕ) = · · · = Tr∂Ω(∂2m−1 +n +ϕ) = 0} → L2(Ω) +(C.2.24) +is a Banach isomorphism. +Proof. The proof of Lemma C.5, paired with the semi-definiteness of Bm, shows that the symmetric +bilinear map +Hm(Ω) × Hm(Ω) ∋ (ϕ0, ϕ1) �→ +� +Ω +κϕ0ϕ1 + +n +� +j=1 +� +Ω +∂m +j ϕ0∂m +j ϕ1 ∈ R +(C.2.25) +is bounded and coercive, and thus defines an inner-product on Hm(Ω). The Riesz representation +theorem then provides for the existence of unique weak solutions, but then standard elliptic regularity +arguments (which are elementary in Ω since we can use horizontal difference quotients without +cutoffs) allow us to promote the regularity of weak solutions to H2m(Ω), provided the data lie in +L2(Ω). In turn, by integrating by parts, we verify that such weak solutions satisfy the boundary +conditions. From this we readily deduce that the operator is an isomorphism between the stated +spaces. +□ +We next record a result about inverting a particular pseudodifferential operator. +Lemma C.9 (Simple symbol inversion). Fix s, σ ∈ R with σ ⩾ 0 and M, N ∈ N+. Suppose that +ψ ∈ Hs(Rd). There exists a unique ϕ ∈ Hs+σ(Rd) such that +(N−1(M−1 + (−∆)σ/2) − ∂1)ϕ = ψ. +(C.2.26) +Moreover we have the estimate +∥(NM)−1ϕ, N−1|∇|σϕ∥Hs×Hs ≲ ∥ψ∥Hs +(C.2.27) +for implicit constants depending only on σ. +Proof. The symbol of the pseudodifferential operator in (C.2.26) is +Rd ∋ ξ �→ a(ξ) = (MN)−1⟨2π +� +M|ξ|σ⟩2 − 2πiξ · e1 ∈ C. +(C.2.28) +The stated estimate then follows readily from the elementary estimate +1 +|a(ξ)| ⩽ +MN +⟨2π +� +M|ξ|σ⟩2 ⩽ N +2π min +� +M, |ξ|−σ� +for ξ ∈ Rd. +(C.2.29) +□ +C.3. Dissipation calculation for traveling compressible Navier-Stokes. This subsection +explores the role of forcing in the traveling wave problem (1.4.9). We will require the following +density result. +Lemma C.10 (Density of smooth functions with bounded support). For any s ∈ {−1, 0} ∪ R+, +the subspace +{(q, u, η) ∈ C∞ +c (Ω) × C∞ +c (Ω; Rn) × C∞ +c (Rn−1) : TrΣ0u = 0, and TrΣu · en + ∂1η = 0} +(C.3.1) +is dense in Xs, as defined by (3.1.3). + +COMPRESSIBLE TRAVELING WAVES +139 +Proof. First, we note that Theorem 5.2 and the second item of Theorem 5.6 in Leoni and Tice [60] +imply that Hr(Rn−1) ⊂ Hr(Rn−1) is a continuous and dense inclusion for r ⩾ 0. On the other +hand, C∞ +c (Rn−1) is dense in Hr(Rn−1). These facts combine to give the density of C∞ +c (Rn−1) in +Hr(Rn−1). In turn, we deduce that +C∞ +c (Ω) × C∞ +c (Ω; Rn) × C∞ +c (Rn−1) ⊂ Xs +(C.3.2) +is a dense inclusion, where the latter space is defined by (3.1.1). +It remains to handle the boundary conditions that define the subspace Xs ⊂ Xs. For this we will +modify the map ρX from Lemma 3.2. Let ψ ∈ C∞ +c (R) be such that 0 ⩽ ψ ⩽ 1, supp(ψ) ⊆ (−2, 2), +and ψ = 1 on (−1, 1). +For 1 ⩽ ν ∈ R we define ψν ∈ C∞ +c (Ω) via ψν(x, y) = ψ(|x/ν|2) for +(x, y) ∈ Rn−1 × (0, b). Next, we define Πν : Xs → Xs via +Πν(q, u, η) = (q, u − ψνE1(TrΣ0u, TrΣ(u · en) + ∂1η), η), +(C.3.3) +where E1 is the map defined in the proof of Lemma 3.2. In addition to being linear and bounded +(with supν⩾1 ∥Πν∥L(Xs) < ∞), Πν has the property that if (q, u, η) ∈ Xs is smooth and supported in +a ball of radius ν (appropriately interpreted for each element of the tuple), then Πν(q, u, η) belongs +to Xs and remains smooth and supported in a ball of radius 2ν. +Fix (q, u, η) ∈ Xs. From the dense inclusion (C.3.2), we are assured of the existence of a sequence +{(�qN, �uN, �ηN)}∞ +N=1 ⊂ C∞ +c (Ω) × C∞ +c (Ω; Rn) × C∞ +c (Rn−1) with the Nth element of the sequence +consisting of functions supported in a ball of radius 1 ⩽ RN ∈ R+ and with the additional property +that (�qN, �uN, �ηN) → (q, u, η) in Xs as N → ∞. Set (qN, uN, ηN) = ΠRN (�qN, �uN, �ηN). Thanks to the +aforementioned properties of ΠRN, we know that {(qN, uN, ηN)}∞ +N=1 ⊂ Xs and that this sequence +consists of smooth functions with compact support. +By using the boundary conditions implied by the inclusion (q, u, η) ∈ Xs, we now check that +(q, u, η) − (qN, uN, ηN) = (q − �qN, u − �uN, η − �ηN) +− (0, ψRN E1(TrΣ0(�uN − u), TrΣ(�uN − u) · en + ∂1(�ηN − η)), 0). +(C.3.4) +Then, in light of the convergence of {(�qN, �uN, �ηN)}∞ +N=1 to (q, u, η), we conclude from (C.3.4) that +(qN, uN, ηN) → (q, u, η) in Xs as N → ∞. +□ +We now record an important calculation. +Theorem C.11 (Dissipation-power balance for traveling compressible Navier-Stokes). Suppose that +γ ∈ R+, N ∋ s ⩾ 1 + ⌊n/2⌋, (g, f, k) ∈ Ys, and (q, u, η) ∈ BX1+⌊n/2⌋(0, ρprin) ∩ X1+s, where ρprin is +defined in Theorem 3.13. Further suppose that +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∇ · (σq,η(u − Mηe1)) = g +in Ω, +γ2σq,ηM−t +η (((u − Mηe1) · ∇)(M−1 +η u)) + σq,η∇(q + gη) +−γM−t +η ∇ · (Sσq,η +Aη (M−1 +η u)Mt +η) = f +in Ω, +−((P − Pext) ◦ σq,η − γSσq,η +Aη (M−1 +η u))Mt +ηen − ςH (η)Mt +ηen = k +in Σ, +u · en + ∂1η = 0 +on Σ, +u = 0 +on Σ0. +(C.3.5) +In other words, Ψ(q, u, η, γ) = (g, f, k), where Ψ is the operator given in (3.3.4). Then +� +Ω +γ +Jη +�µ(σq,η) +2 +|D0 +Mtη(M−1 +η u)|2 + λ(σq,η)|∇ · u|2� += +� +Ω +f · u + g(γ2|M−1 +η u|2/2 + q) ++ g +� +Rn−1 +� +|∇∥|−1 +� b +0 +g(·, y) dy +� +|∇∥|η + +� +Σ +k · M−1 +η u, +(C.3.6) + +140 +NOAH STEVENSON AND IAN TICE +where for w : Ω → Rn differentiable and M : Ω → Rn×n we write +D0 +Mw = ∇wMt + M∇wt − 2 +n(M∇) · wI. +(C.3.7) +Proof. First, we claim that it suffices to prove (C.3.6) under the additional assumption that (q, u, η) +belongs to the space in (C.3.1). +Indeed, assume that the identity holds in this special case, +and let (q, u, η) ∈ BX1+⌊n/2⌋(0, ρprin) ∩ X1+s be generic. Thanks to Lemma C.10, there exists a +sequence {(qN, uN, ηN)}∞ +N=1, belonging to the space in (C.3.1), such that (qN, uN, ηN) → (q, u, η) +in X1+s as N → ∞. Due to this convergence, we may assume without loss of generality that the +sequence is contained in BX1+⌊n/2⌋(0, ρprin). We then rewrite the identity Ψ(q, u, η, γ) = (g, f, k) +as Ψ(qN, uN, ηN, γ) = (gN, fN, kN), where gN = g + (Ψ1(qN, uN, ηN) − Ψ1(q, u, η)), fN = f + +(Ψ2(qN, uN, ηN, γ) − Ψ2(q, u, η, γ)), and kN = k + (Ψ3(qN, uN, ηN, γ) − Ψ3(q, u, η, γ)). Thanks to +the continuity of the map Ψ established in Theorem 3.13, we have that (gN, fN, kN) → (g, f, k) in +the space Ys as N → ∞. Using the special case, we have the identity +� +Ω +γ +JηN +�µ(σqN,ηN ) +2 +|D0 +MtηN (M−1 +ηN uN)|2+λ(σqN,ηN )|∇·uN|2� += +� +Ω +fN ·uN +gN(γ2|M−1 +ηN uN|2/2+qN) ++ +� +Σ +kN · M−1 +ηN uN + g +� +Rn−1 +� +|∇∥|−1 +� b +0 +gN(·, y) dy +� +|∇∥|ηN +(C.3.8) +for every N. Since (qN, uN, ηN) → (q, u, η) in X1+s and (gN, fN, kN) → (g, f, k) in Ys as N → ∞ and +s ⩾ 1+⌊n/2⌋, it is a simple matter to send N → ∞ in (C.3.8) to obtain the desired equality (C.3.6). +This completes the proof of the claim. +We now establish (C.3.6) in the special case. We begin by taking the inner product of the second +equation in (C.3.5) with u in L2(Ω; Rn). Since the vector field σq,η(u − Mηe1) has divergence g and +vanishing normal trace, the contribution of the advective derivative is +� +Ω +σq,ηM−t +η (((u − Mηe1) · ∇)(M−1 +η u)) · u = − +� +Ω +1 +2∇ · (σq,η(u − Mηe1))|M−1 +η u|2 ++ +� +Σ +1 +2σq,η(u − Mηe1) · en|M−1 +η u|2 = −1 +2 +� +Ω +g|M−1 +η u|2. +(C.3.9) +The contribution of the viscous stress term is +� +Ω +M−t +η ∇ · (Sσq,η +Aη (M−1 +η u)Mt +η) · u = +� +Σ +Sσq,η +Aη (M−1 +η u)Mt +ηen · M−1 +η u +− +� +Ω +Sσq,η +Aη (M−1 +η u)Mt +η : ∇(M−1 +η u) = +� +Σ +Sσq,η +Aη (M−1 +η u)Mt +ηen · M−1 +η u +− +� +Ω +1 +Jη +�µ(σq,η) +2 +|D0 +Mtη(M−1 +η u)|2 + λ(σq,η)|∇ · u|2� +, +(C.3.10) +where in the final identity we have used the fact that ∇ · (JηAη) = 0. For the pressure contribution, +we first write σq,ηu = σq,η(u − Mηe1) + σq,ηMηe1, and then integrate by parts to see that +� +Ω +σq,η∇(q + gη) · u = − +� +Ω +g(q + gη) + +� +Ω +σq,ηMηe1 · ∇(q + gη). +(C.3.11) +To handle the final term above, we use the definition of σq,η, which appears in (1.4.11), to express +q + gη = H ◦ σq,η − H ◦ ϱ(Fη · en). +(C.3.12) +Since ∇(Fη · en) = JηM−t +η en, it follows that Mηe1 · ∇(H ◦ ϱ(Fη · en)) = Mηe1 · ∇(P ◦ ϱ(Fη · en)) = 0. +We may then rewrite +� +Ω +σq,ηMηe1 · ∇(q + gη) = +� +Ω +Mηe1 · ∇(P(σq,η) − P ◦ ϱ(Fη · en)). +(C.3.13) + +COMPRESSIBLE TRAVELING WAVES +141 +Since ∇ · (Mηe1) = 0 and P(σq,η) − P ◦ ϱ(Fη · en) ∈ C∞ +c (Ω), we may integrate by parts once again +to see that +� +Ω +Mηe1·∇(P(σq,η)−P ◦ϱ(Fη·en)) = − +� +Σ +(P −Pext)(σq,η)∂1η+ +� +Σ +(P ◦ϱ(Fη·en)−Pext)∂1η. (C.3.14) +The final term above vanishes since +� +Σ +(P ◦ ϱ(Fη · en) − Pext)∂1η = +� +Σ +∂1 +� � η +0 +(P ◦ ϱ(b + s)) − Pext) ds +� += 0. +(C.3.15) +Combining these, we deduce that the contribution of the pressure term in the momentum equation +is +� +Ω +σq,η∇(q + gη) · u = +� +Σ +(P − Pext)(σq,η)u · en − +� +Ω +g(q + gη). +(C.3.16) +Upon synthesizing these calculations, we conclude that +� +Ω +f · u + g(γ2|M−1 +η u|2/2 + q + gη) +− +� +Σ +((P − Pext)(σq,η) − γSσq,η +Aη (M−1 +η u))Mt +ηen · M−1 +η u += +� +Ω +γ +Jη +�µ(σq,η) +2 +|D0 +Mtη(M−1 +η u)|2 + λ(σq,η)|∇ · u|2� +. +(C.3.17) +We appeal to the dynamic boundary condition to rewrite +− +� +Σ +((P − Pext)(σq,η) − γSσq,η +Aη (M−1 +η u))Mt +ηen · M−1 +η u += +� +Σ +(k + ςH (η)Mt +ηen) · M−1 +η u = +� +Σ +k · M−1 +η u + ς +� +Σ +∂1(⟨∇∥η⟩ − 1) = +� +Σ +k · M−1 +η u. +(C.3.18) +Finally, from Parseval’s theorem we have the identity +� +Ω +gη = +� +Rn−1 +� +|∇∥|−1 +� b +0 +g(·, y) dy +� +|∇∥|η. +(C.3.19) +Then (C.3.6) follows by combining (C.3.17), (C.3.18), and (C.3.19). +□ +Theorem C.11 immediately leads to the next result. +Corollary C.12. Suppose that N ∋ s ⩾ 2 + ⌊n/2⌋, (q, u, η) ∈ X1+s ∩ BX2+⌊n/2⌋(0, ρWD), and +(T , G, F) ∈ Ws satisfy (1.4.9). Then +� +Ω +γ +Jη +�µ(σq,η) +2 +|D0 +Mtη(M−1 +η u)|2 + λ(σq,η)|∇ · u|2� += +� +Σ +T ◦ FηMt +ηen · M−1 +η u ++ +� +Ω +Jη(σq,ηG ◦ Fη + F ◦ Fη) · M−1 +η u. +(C.3.20) +We now combine this result with Korn-type bounds to deduce a useful uniqueness result. Recall +that ρWD is from Theorem 3.17. +Corollary C.13. There exists a ρ ∈ (0, ρWD], depending on b, g, P, µ, λ, and n, such that if +(q, u, η) ∈ X3+⌊n/2⌋ ∩ BX2+⌊n/2⌋(0, ρ) satisfies (1.4.9) with T = 0, G = 0, and F = 0, then q = 0, +u = 0, and η = 0. +Proof. We begin with some general considerations. Propositions 3.10 and 3.11 show that if +(q, u, η) ∈ BX2+⌊n/2⌋(0, ρWD), +(C.3.21) + +142 +NOAH STEVENSON AND IAN TICE +then c ⩽ σq,η ⩽ C for some constants c, C ∈ R+, and 1/2 ⩽ Jη ⩽ 3/2, which in particular implies +that Mη is invertible. Assume (C.3.21) holds. We may then define +Dη = +� +Ω +�µ(ϱ) +2 +|D0 +Mtη(M−1 +η u)|2 + λ(ϱ)|∇ · u|2� +. +(C.3.22) +It is a simple matter to check that +D0 ≲ Dη + g(∥η∥H9/2+⌊n/2⌋) ∥u∥2 +H1 +(C.3.23) +for an continuous and increasing function g : [0, ∞) → [0, ∞) such that g(0) = 0. In light of the +properties of g and the Korn inequalities from Propositions A.3 and A.4, we may choose 0 < ρ ⩽ ρWD +such that if ∥η∥H9/2+⌊n/2⌋ < ρ, then the term g(∥η∥H9/2+⌊n/2⌋) ∥u∥2 +H1 may be absorbed onto the left +side of (C.3.23), resulting in the bound +D0 ≲ Dη. +(C.3.24) +Now assume that (q, u, η) ∈ X3+⌊n/2⌋ ∩ BX2+⌊n/2⌋(0, ρ) satisfies (1.4.9) with T = 0 and G = F = 0. +Corollary C.12 and the above bounds on σq,η and Jη then imply that Dη = 0. Then (C.3.24) implies +that D0 = 0, and so we may again appeal to the Korn inequalities to see that u = 0. The fourth +equation in (1.4.9) then implies that η = 0, but then the second and third require that q = 0. +□ +Appendix D. Fine tools for nonlinear analysis +D.1. Smoothness of superposition nonlinearities. This subsection is concerned with operators +between Sobolev spaces involving composition nonlinearities. +Lemma D.1. Let k = 2 + ⌊n/2⌋. Suppose that Φ : Rn → Rn is a bi-Lipschitz homeomorphism +and a C1 diffeomorphism. Then there exists a δ > 0, depending on n and the Lipschitz seminorm +[Φ]C0,1, such that if g ∈ W k,∞(Rn; Rn) and h ∈ Hk(Rn; Rn) satisfy +max{∥g∥W k,∞ , ∥h∥Hk} < δ, +(D.1.1) +then Φ + g + h is also a bi-Lipschitz homeomorphism and a C1 diffeomorphism, satisfying the bounds +∥D(Φ + g + h)∥C0 +b < 3 · 2−1∥DΦ∥C0 +b and ∥D(Φ + g + h)−1∥C0 +b < 2∥DΦ−1∥C0 +b . +(D.1.2) +Proof. First note that the Banach fixed point theorem implies that if Ξ : Rn → Rn is a Lipschitz +map satisfying the bound [Ξ]C0,1 < [Φ−1]−1 +C0,1, then Φ+Ξ is also a bi-Lipschitz homeomorphism. The +standard Sobolev embeddings provide a constant C > 0, depending on n, such that [g + h]C0,1 ⩽ +∥g + h∥C1 +b ⩽ C(∥g∥W k,∞ + ∥h∥Hk). Thus, if we set δ = (4C[Φ−1]C0,1)−1, then the bound (D.1.1) +implies that [g + h]C0,1 < 1 +2[Φ−1]−1 +C0,1, and so Φ + g + h is a bi-Lipschitz homeomorphism. On +the other hand, [Φ−1]C0,1 = ∥DΦ−1∥C0 +b and [g + h]C0,1 = ∥Dg + Dh∥C0 +b , so the continuous map +D(Φ + g + h) is everywhere invertible, and so Φ + g + h is then a C1 diffeomorphism by the inverse +function theorem. +It remains to prove the bound (D.1.2). To this end first note that ∥D(g + h)(DΦ)−1∥C0 +b ⩽ +[Φ−1]C0,1[g + h]C0,1 < 2−1 From this we deduce that +∥D(Φ + g + h)∥C0 +b ⩽ ∥DΦ∥C0 +b ∥1 + D(g + h)(DΦ)−1∥C0 +b ⩽ 3 · 2−1 ∥DΦ∥C0 +b +(D.1.3) +and +∥D(Φ + g + h)−1∥C0 +b = ∥(DΦ)−1(1 + D(g + h)(DΦ)−1)−1∥C0 +b +⩽ ∥DΦ−1∥C0 +b +∞ +� +m=0 +∥((D(g + h)(DΦ)−1)−1)m∥C0 +b < ∥DΦ−1∥C0 +b +∞ +� +m=0 +1 +2m = 2∥DΦ−1∥C0 +b . +(D.1.4) +These prove (D.1.2). +□ +The following is a modification of the main argument presented in Inci, Kappeler, and Topalov [46]. + +COMPRESSIBLE TRAVELING WAVES +143 +Theorem D.2. Let N ∋ k ⩾ 2 + ⌊n/2⌋ and m ∈ N. Suppose that Φ : Rn → Rn is a bi-Lipschitz +homeomorphism and a C1 diffeomorphism such that DΦ ∈ W k−1,∞(Rn; Rn×n). Let δ > 0 be +determined by n and Φ as in Lemma D.1. Define the map +Λ : Hm+k(Rn; Rd) × (BW 2+⌊n/2⌋,∞(0, δ) ∩ W k,∞(Rn; Rn)) × (BH2+⌊n/2⌋(0, δ) ∩ Hk(Rn; Rn)) +→ Hk(Rn; Rd) +(D.1.5) +via Λ(f, g, h) = f(Φ + g + h). Then the following hold. +(1) Λ is well-defined and continuous. +(2) If m = 1, then Λ is C1 and satisfies +DΛ(f, g, h)(f1, g1, h1) = Df(Φ + g + h)(g1 + h1) + f1(Φ + g + h). +(D.1.6) +(3) If m ⩾ 2, then Λ is Cm and satisfies +DmΛ(f, g, h)[(f1, g1, h1), . . . , (fm, gm, hm)] = Dmf(Φ + g + h)(gi + hi)m +i=1 ++ +m +� +ℓ=1 +Dm−1fℓ(Φ + g + h)(gi + hi)i̸=ℓ. +(D.1.7) +Proof. We divide the proof into steps. +Step 1: Preliminary observations. +For any f, g, and h in the domain of Λ we have that +max{∥g∥W 2+⌊n/2⌋,∞ , ∥h∥H2+⌊n/2⌋} < δ. Consequently, Lemma D.1 implies that Φ + g + h is a C1 +diffeomorphism and that +∥D(Φ + g + h)∥C0 +b + ∥D(Φ + g + h)−1∥C0 +b ⩽ A0, +(D.1.8) +where A0 > 0 is a constant depending on Φ. Throughout the rest of the proof, when we write ≲ we +allow for the implicit constant to depend on A0, and thus on Φ. +Step 2: Well-definedness and continuity. We now aim to prove that Λ is actually well-defined, i.e. +takes values in Hk(Rn; Rd), and is a continuous map. It suffices to prove this when m = 0, as the +cases m ∈ {1, 2} follow from this case. To prove this, we proceed by finite induction on 0 ⩽ j ⩽ k. +For such j let Pj denote the proposition that +Λ : Hm+j(Rn; Rd) × (BW 2+⌊n/2⌋,∞(0, δ) ∩ W k,∞(Rn; Rn)) × (BH2+⌊n/2⌋(0, δ) ∩ Hk(Rn; Rn)) +→ Hj(Rn; Rd) +(D.1.9) +is well-defined and continuous and obeys the estimate +∥Λ(f, g, h)∥Hj ≲ ⟨∥g∥W k,∞ , ∥h∥Hk⟩j ∥f∥Hj . +(D.1.10) +Consider the base case, j = 0. A change of variables and the bound (D.1.8) allow us to estimate +∥Λ(f, g, h)∥H0 = +� � +Rn |f ◦ (Φ + g + h)|2 �1/2 +⩽ ∥det ∇(Φ+g+h)−1∥1/2 +L∞∥f∥H0 ≲ ∥f∥H0. (D.1.11) +This shows that Λ is well-defined and establishes (D.1.10) when j = 0. Next we bound +∥Λ(f, g, h) − Λ( �f, �g,�h)∥H0 ⩽ ∥Λ(f, g, h) − Λ(f, �g,�h)∥H0 + ∥Λ(f − �f, �g,�h)∥H0. +(D.1.12) +For the latter term we use (D.1.11) to see that lim( �f,�g,�h)→(f,g,h)∥Λ(f − �f, �g,�h)∥H0 = 0, while for the +former we use the density of C∞ +c (Rn; Rd) in H0(Rn; Rd) together with the fact that if (�g,�h) → (g, h) +then Φ + �g + �h → Φ + g + h uniformly to deduce that lim( �f,�g,�h)→(f,g,h)∥Λ(f, g, h) − Λ(f, �g,�h)∥H0 = 0. +Thus, Λ is continuous when j = 0, and we find that P0 holds. + +144 +NOAH STEVENSON AND IAN TICE +Proceeding inductively, we now suppose that Pℓ holds for all 0 ⩽ ℓ ⩽ j ⩽ k − 1 and consider the +case j + 1 ⩽ k. Let (f, g, h) be in the domain of Λ in this case. For any 1 ⩽ a ⩽ n we compute +∂aΛ(f, g, h) = +n +� +b=1 +∂bf ◦ (Φ + g + h)∂a(Φ + g + h)b = +n +� +b=1 +Λ(∂bf, g, h)∂a(Φ + g + h)b +(D.1.13) +in order to exploit the induction hypothesis and a product estimate (see Corollary D.7) to bound +∥Λ(f, g, h)∥Hj+1 ≲ ∥Λ(f, g, h)∥H0 + +n +� +a=1 +∥∂aΛ(f, g, h)∥Hj +≲ ∥f∥H0 + +n +� +a,b=1 +∥Λ(∂bf, g, h)∂a(Φ + g + h)b∥Hj ≲ ∥f∥H0 + ⟨∥g∥W k,∞ , ∥h∥Hk⟩ +n +� +b=1 +∥Λ(∂bf, g, h)∥Hj +≲ ∥f∥H0 + ⟨∥g∥W k,∞, ∥h∥Hk⟩⟨∥g∥W k,∞ , ∥h∥Hk⟩j∥f∥Hj+1 ≲ ⟨∥g∥W k,∞ , ∥h∥Hk⟩j+1∥f∥Hj+1, +(D.1.14) +which shows well-definedness and the bound (D.1.10) for j + 1. +Next, we establish the continuity assertion of Pj+1. We initially compute +∂a(Λ(f, g, h) − Λ( �f, �g,�h)) = +n +� +b=1 +(Λ(∂bf, g, h) − Λ(∂b �f, �g,�h))∂a(Φ + g + h)b ++ +n +� +b=1 +Λ(∂b �f, �g,�h)∂a(g − �g + h − �h)b. +(D.1.15) +Again using basic product estimates (see Corollary D.7), we may deduce from this that +∥Λ(f, g, h) − Λ( �f, �g,�h)∥Hj+1 ≲ ∥Λ(f, g, h) − Λ( �f, �g,�h)∥H0 + +n +� +a=1 +∥∂aΛ(f, g, h) − ∂aΛ( �f, �g,�h)∥Hj ≲ +⟨∥g, h∥W k,∞×Hk⟩ +� +|α|⩽1 +∥Λ(∂αf, g, h)−Λ(∂α �f, �g,�h)∥Hj +∥g−�g, h−�h∥W k,∞×Hk +n +� +b=1 +∥Λ(∂b �f, �g,�h)∥Hj, +(D.1.16) +and this bound and the induction hypothesis readily imply the continuity assertion of Pj+1. Thus +Pj+1 holds, and the induction argument is complete. +Step 3: +Continuous differentiability when m = 1. +Next we aim to prove that Λ is C1 +when m = 1. +To this end fix (f, g, h) in the domain of Λ and pick r > 0 such that r + +max{∥g∥W 2+⌊n/2⌋,∞ , ∥h∥H2+⌊n/2⌋} < δ. Then (f, g, h) + t(f1, g1, h1) belongs to the domain of Λ +for every t ∈ [−1, 1] and f1 ∈ H2+k(Rn; Rd), g1 ∈ W k,∞(Rn; Rn), and h1 ∈ Hk(Rn; Rn) such that +max{∥g1∥W 2+⌊n/2⌋,∞ , ∥h1∥H2+⌊n/2⌋} < r. By using the fundamental theorem of calculus, we may +compute +Λ(f + f1, g + g1, h + h1) − Λ(f, g, h) − Λ(f1, g, h) − +n +� +b=1 +Λ(∂bf, g, h)(g1 + h1)b = R1 + R2 (D.1.17) +for remainder terms +R1 = +n +� +b=1 +(g1 + h1)b +� 1 +0 +(Λ(∂bf, g + tg1, h + th1) − Λ(∂bf, g, h)) dt +(D.1.18) + +COMPRESSIBLE TRAVELING WAVES +145 +and +R2 = +n +� +b=1 +(g1 + h1)b +� 1 +0 +Λ(∂bf1, g + tg1, h + th1)dt. +(D.1.19) +Using the results of the previous step, we readily deduce from these expressions that +∥R1 + R2∥Hk +∥f1∥H1+k + ∥g1∥W k,∞ + ∥h1∥Hk → 0 as (f1, g1, h1) → 0. +(D.1.20) +Hence, Λ is differentiable on its domain, and +DΛ(f, g, h)(f1, g1, h1) = Λ(f1, g, h) + +n +� +b=1 +Λ(∂bf, g, h)(g1 + h1)b, +(D.1.21) +which may be rewritten as (D.1.6). On the other hand, the expression for DΛ(f, g, h) in terms of Λ +shows that DΛ is continuous on its domain when m = 1, and so Λ is actually C1 in this case. +Step 4: mth-order continuous differentiability when m ⩾ 2. Now assume that m ⩾ 2. In this +case, the result of the third step still shows that Λ is C1 on its domain with the same expression +for DΛ given in (D.1.21). However, since DΛ can be expressed in terms of standard products and +Λ, we iteratively deduce that Λ is actually C2 when m = 2, C3 when m = 3, etc., and that the +formula (D.1.7) holds for any such m. +□ +We also need a variant of Theorem D.2 in which the outer composition map is a fixed element +of a Ck +b −type space, and we are considering smoothness as a mapping into the space of Sobolev +multipliers. This latter space is defined as follows. For W ⊆ Rn an open set and s ∈ N, we define +the space of Sobolev multipliers M(Hs(W)) to be the set of bounded linear maps L ∈ L(Hs(W)) +for which there exists m ∈ L1 +loc(W) such that Lf = mf for all f ∈ Hs(W). We endow this space +with the natural operator norm: ∥m∥M(Hs) = sup∥f∥Hs⩽1∥mf∥Hs. It turns out that M(Hs(W)) is +a closed subspace of L(Hs(W)), and is thus complete, and that M(Hs(W)) ⊂ Hs +loc(W). For the +proofs of these facts and more information on the spaces of Sobolev multipliers, we refer the reader +to the book by Maz’ya and Shaposhnikova [77]. +Theorem D.3. Let N ∋ k ⩾ 2 + ⌊n/2⌋, ∅ ̸= V ⊆ Rd be open, and ∅ ̸= O ⊆ Rn be a Stein- +extension domain (see Definition A.2). Suppose that ∅ ̸= U ⊆ W k,∞(O; Rd) × Hk(O; Rd) is an +open set with the property that if (g, h) ∈ U, then (g + h)(O) ⊆ V . For any f ∈ Ck +b (V ) define the +function Θ(f; g, h) : O → R via Θ(f; g, h) = f(g + h). Then the following hold for each m ∈ N and +f ∈ Cm+k +b +(V ). +(1) For each (g, h) ∈ U, the function Θ(f; g, h) defines an element of M(Hk(O)). Moreover, +the induced map Θ(f; ·) : U → M(Hk(O)) is Cm. +(2) If m ⩾ 1, then the induced map Θ(f; ·) : U → M(Hk(O)) satisfies +DmΘ(f; g, h)[(g1, h1), . . . , (gm, hm)] = Dmf(g + h)[(g1, h1), . . . , (gm, hm)]. +(D.1.22) +Proof. We divide the proof into three steps. +Step 1: Well-definedness and continuity when m = 0. Suppose that m = 0. For 0 ⩽ j ⩽ k let Pj +denote the proposition that if f ∈ Cj +b(V ), then Θ(f; ·) : U → M(Hj(O)) is well-defined, continuous, +and obeys the estimate +∥Θ(f; g, h)∥M(Hj) ≲ ∥f∥Cj +b ⟨∥g∥W k,∞ , ∥h∥Hk⟩j−1. +(D.1.23) +We will employ a finite induction to show that Pj holds for all 0 ⩽ j ⩽ k, which then proves the +first item when m = 0. +Consider the base case, j = 0. Then for ϕ ∈ H0(O) we can trivially bound ∥Θ(f; g, h)ϕ∥H0 ⩽ +∥f∥C0 +b ∥ϕ∥H0 in order to see that Θ(f; g, h) ∈ M(H0(O)) with ∥Θ(f; g, h)∥M(H0) ⩽ ∥f∥C0 +b . Thus, +Θ(f; ·) is well-defined and obeys the bound (D.1.23) when j = 0. To prove continuity, we note + +146 +NOAH STEVENSON AND IAN TICE +that for (g, h), (g, h) ∈ U we may argue as above to bound ∥Θ(f; g, h) − Θ(f; g, h)∥M(H0) ⩽ +∥f(g + h) − f(g + h)∥C0 +b , but since k > n/2, convergence in W k,∞ × Hk implies uniform convergence, +and we deduce that lim(g,h)→(g,h)∥Θ(f; g, h) − Θ(f; g, h)∥M(H0) = 0. This establishes the continuity +of Θ(f; ·) in U, and so P0 is proved. +Now suppose that Pℓ holds for all 0 ⩽ ℓ ⩽ j ⩽ k − 1 and suppose that f ∈ Cj+1 +b +(V ). Then for +ϕ ∈ Hj+1(O) and (g, h) ∈ U we have that +∥Θ(f; g, h)ϕ∥Hj+1 ≲ ∥Θ(f; g, h)ϕ∥H0 + +n +� +a=1 +∥∂a[Θ(f; g, h)ϕ]∥Hj+1 +≲ ∥Θ(f; g, h)ϕ∥H0 + +n +� +a=1 +∥∂a[Θ(f; g, h)]ϕ∥Hj + +n +� +a=1 +∥Θ(f; g, h)∂aϕ∥Hj . +(D.1.24) +By the induction hypothesis, we have the bounds ∥Θ(f; g, h)ϕ∥H0 ⩽ ∥Θ(f; g, h)∥M(H0) ∥ϕ∥H0 and +�n +a=1 ∥Θ(f; g, h)∂aϕ∥Hj ≲ ∥Θ(f; g, h)∥M(Hj) ∥ϕ∥Hj+1. On the other hand, +∂a[Θ(f; g, h)] = +n +� +b=1 +∂bf(g + h)∂a(g + h)b = +n +� +b=1 +Θ(∂bf, g, h)∂a(g + h)b, +(D.1.25) +so we may again use the induction hypothesis to bound +n +� +a=1 +∥∂a[Θ(f; g, h)]ϕ∥Hj ≲ +n +� +a,b=1 +∥Θ(∂bf, g, h)∥M(Hj) ∥∂a(g + h)bϕ∥Hj +≲ +n +� +b=1 +∥Θ(∂bf, g, h)∥M(Hj) (∥g∥W k,∞ + ∥h∥Hk) ∥ϕ∥Hj . +(D.1.26) +Upon combining these bounds and again using the induction hypothesis, we find that Θ(f; g, h) ∈ +M(Hj+1(O)) with +∥Θ(f; g, h)∥M(Hj) ≲ ∥Θ(f; g, h)∥M(H0) + ∥Θ(f; g, h)∥M(Hj) ++ +n +� +b=1 +∥Θ(∂bf, g, h)∥M(Hj) (∥g∥W k,∞ + ∥h∥Hk) ≲ ∥f∥Cj +b ⟨∥g∥W k,∞ , ∥h∥Hk⟩j−1 ++ +n +� +b=1 +∥∂bf∥Cj +b ⟨∥g∥W k,∞ + ∥h∥Hk⟩j ≲ ∥f∥Cj+1 +b +⟨∥g∥W k,∞ , ∥h∥Hk⟩(j+1)−1 . +(D.1.27) +This shows that Θ(f; ·) is well-defined on U and obeys (D.1.23) in the case j + 1. +To prove Pj+1, it remains only to establish continuity. For (g, h), (g, h) ∈ U and ϕ ∈ Hj+1(Rn) +we argue as above to bound +∥[Θ(f; g, h) − Θ(f; g, h)]ϕ∥Hj+1 ≲ ∥Θ(f; g, h) − Θ(f; g, h)∥M(H0)∥ϕ∥H0 ++ +n +� +a=1 +∥Θ(f; g, h) − Θ(f; g, h)∥M(Hj)∥∂aϕ∥Hj + +n +� +a=1 +∥∂a[Θ(f; g, h) − Θ(f; g, h)]ϕ∥Hj. +(D.1.28) +The first two terms here are easy to deal with, but we must rewrite the third: ∂a[Θ(f; g, h) − +Θ(f; g, h)] = �n +b=1[Θ(∂bf; g, h)−Θ(∂bf; g, h)]∂a(g +h)b +�n +b=1 Θ(∂bf, g, h)∂a(g +h−g −h)b, which + +COMPRESSIBLE TRAVELING WAVES +147 +then allows us to bound +n +� +a=1 +∥∂a[Θ(f; g, h) − Θ(f; g, h)]ϕ∥Hj ≲ +n +� +a,b=1 +∥Θ(∂bf; g, h) − Θ(∂bf; g, h)∥M(Hj)∥∂a(g + h)bϕ∥Hj ++ +n +� +a,b=1 +∥Θ(∂bf; g, h)∥M(Hj)∥∂a(g + h − g − h)bϕ∥Hj +≲ +n +� +b=1 +∥Θ(∂bf; g, h) − Θ(∂bf; g, h)∥M(Hj) (∥g∥W k,∞ + ∥h∥Hk) ∥ϕ∥Hj ++ +n +� +b=1 +��Θ(∂bf; g, h) +�� +M(Hj) +� +∥g − g∥W k,∞ + +��h − h +�� +Hk +� +∥ϕ∥Hj . +(D.1.29) +Hence, +��[Θ(f; g, h) − Θ(f; g, h)] +�� +M(Hj+1) ≲ +��Θ(f; g, h) − Θ(f; g, h) +�� +M(H0) ++ +��Θ(f; g, h) − Θ(f; g, h) +�� +M(Hj) + (∥g∥W k,∞ + ∥h∥Hk) +n +� +b=1 +��Θ(∂bf; g, h) − Θ(∂bf; g, h) +�� +M(Hj) ++ +� +∥g − g∥W k,∞ + +��h − h +�� +Hk +� +n +� +b=1 +��Θ(∂bf; g, h) +�� +M(Hj) , +(D.1.30) +and we readily deduce from this bound and the induction hypothesis that Θ(f; ·) is continuous on +U when f ∈ Cj+1 +b +(V ). Thus, Pj+1 holds, and the finite induction argument is complete. +Step 2: C1 when f ∈ C1+k +b +(V ). Now suppose that m = 1 and f ∈ C1+k +b +(V ). For (g, h) ∈ U and +(g1, h1) ∈ W k,∞(O; Rd) × Hk(O; Rd) small enough that (g + tg1, h + th1) ∈ U for all t ∈ [0, 1], we +have the identity +Θ(f; g + g1, h + h1) − Θ(f; g, h) − +n +� +b=1 +Θ(∂bf; g, h)(g1 + h1)b += +n +� +b=1 +(g1 + h1)b +� 1 +0 +[Θ(∂bf; g + tg1, h + th1) − Θ(∂bf, g, h)] dt. +(D.1.31) +Using this and the results from Step 1, we may argue as in Step 3 of the proof of Theorem +D.2 to deduce from this that Θ(f; ·) is differentiable on U and satisfies DΘ(f; g, h)(g1, h1) = +�n +b=1 Θ(∂bf; g, h)(g1 + h1)b, which may be rewritten as (D.1.22) when m = 1. In turn, this formula +and the results from the previous step show that Θ(f; ·) is C1. +Step 3: Cm when f ∈ Cm+k +b +(V ). With the m = 1 case established and formula (D.1.22) in +hand, we may then employ a simple induction argument, which we omit for the sake of brevity, +to conclude that if f ∈ Cm+k +b +(V ) for m ⩾ 2, then Θ(f; ·) is Cm on U with the formula (D.1.22) +holding for all m ⩾ 1. +□ +D.2. Tools for tame estimates. In this subsection we record some useful tame estimates that +form the basis for Section 3.2 and numerous other areas of our analysis. We begin by recalling two +versions of the log-convexity result known as Gagliardo-Nirenberg interpolation. The first is for +functions on Rn. + +148 +NOAH STEVENSON AND IAN TICE +Theorem D.4 (Gagliardo-Nirenberg interpolation in full space). Let V be a finite dimensional real +vector space, 1 ⩽ t, p, q ⩽ ∞ and s, r ∈ N+ be such that 1 ⩽ r < s and +r +s +1 +p + +� +1 − r +s +�1 +q = 1 +t . +(D.2.1) +For all ϕ ∈ Lq(Rn; V ) ∩ ˙W s,p(Rn; V ) we have the estimate ∥Drϕ∥Lt ≲ ∥ϕ∥1−r/s +Lq +∥Dsϕ∥s/r +Lp , where the +implicit constant depends only the dimension, r, s, p, and q. +Proof. See, for instance, Theorem 12.85 in Leoni [59] for the proof when V = R. The case for +general V follows by choosing a basis and working component-wise. +□ +The second version of Gagliardo-Nirenberg is for functions in certain domains. +Corollary D.5 (Gagliardo-Nirenberg interpolation in domains). Suppose that V is a finite di- +mensional real vector space and U ⊆ Rn is a Stein extension domain (see Definition A.2). Let +1 ⩽ p, q, t ⩽ ∞ and s, r ∈ N+ be such that 1 ⩽ r < s and (D.2.1) holds. For all ϕ ∈ (Lq∩W s,p)(U; V ) +we have the estimate ∥ϕ∥W r,t ≲ ∥ϕ∥1−r/s +Lq +∥ϕ∥s/r +W s,p, where the implicit constant depends on the di- +mension, r, s, p, q, and U. +Proof. Let EU denote a Stein-extension operator for the domain U. Given ϕ as in the hypotheses, +we use the Theorem D.4 followed by the log-convexity of the norm on the Lebesgue spaces to obtain +the desired bound: +∥ϕ∥W r,t(U;V ) ⩽ ∥EUϕ∥W r,t(Rn;V ) ≲ ∥EUϕ∥Lt(Rn;V ) + ∥DrEUϕ∥Lt(Rn;V ) ≲ ∥ϕ∥Lt(U;V ) ++ ∥ϕ∥1−r/s +Lq(U;V )∥ϕ∥s/r +W s,p(U;V ) ⩽ ∥ϕ∥1−r/s +Lq(U;V )∥ϕ∥r/s +Lp(U;V ) + ∥ϕ∥1−r/s +Lq(U;V )∥ϕ∥s/r +W s,p(U;V ) +≲ ∥ϕ∥1−r/s +Lq(U;V )∥ϕ∥s/r +W s,p(U;V ). +(D.2.2) +□ +Armed with the interpolation theorems, we can now prove a series of useful tame estimates. We +begin by studying products. The following principal result, which has a rather lengthy statement, is +the source of numerous useful and simpler corollaries. +Theorem D.6 (Tame estimates on products). Let U ⊆ Rd be a Stein extension domain (see +Definition A.2), and let ℓ, m, k, α, β ∈ N and α1, . . . , αℓ, β1, . . . , βm ∈ Nd satisfy �ℓ +i=1 |αi| = α, +�m +j=1 |βj| = β, and α+β = k. Suppose that r, s, t, p1, . . . , pℓ, q1, . . . , qℓ, a1, . . . , am, b1, . . . , bm ∈ [1, ∞] +satisfy 1/r = 1/s + 1/t as well as +1 +pi +− 1 +qi += k +α +�1 +s − +ℓ +� +λ=1 +1 +qλ +� +for i ∈ {1, . . . , ℓ} and 1 +aj +− 1 +bj += k +β +�1 +t − +m +� +µ=1 +1 +bµ +� +for j ∈ {1, . . . , m}. +(D.2.3) +Suppose additionally that +ui ∈ (W k,pi ∩ Lqi)(U) for i ∈ {1, . . . , ℓ} and wj ∈ (W k,aj ∩ Lbj)(U) for j ∈ {1, . . . , m}. +(D.2.4) +Then we have the product estimate +��� +� +ℓ� +i=1 +∂αiui +�� m +� +j=1 +∂βjwj +���� +Lr ≲ +m+ℓ +� +n=1 +∥vn∥W k,cn +� +ν̸=n +∥vν∥Ldn, +(D.2.5) +where +vn = +� +un +if n ⩽ ℓ, +wn−ℓ +if ℓ < n, +cn = +� +pn +if n ⩽ ℓ, +an−ℓ +if ℓ < n, and dn = +� +qn +if n ⩽ ℓ, +bn−ℓ +if ℓ < n. +(D.2.6) + +COMPRESSIBLE TRAVELING WAVES +149 +Proof. For i ∈ {1, . . . , ℓ} and j ∈ {1, . . . , m}, we define si, tj ∈ [1, ∞] via +1 +si += |αi|/k +pi ++ 1 − |αi|/k +qi +and 1 +tj += |βj|/k +aj ++ 1 − |βj|/k +bj +. +(D.2.7) +We first claim that �ℓ +i=1 1/si = 1/s and �m +j=1 1/tj = 1/t. Indeed, by invoking (D.2.3), we see that +equation (D.2.7) is equivalent to +1 +si +− 1 +qi += |αi| +k +· k +α +�1 +s − +ℓ +� +λ=1 +1 +qλ +� +and 1 +tj +− 1 +bj += |βj| +k +· k +β +�1 +t − +m +� +µ=1 +1 +bµ +� +(D.2.8) +for i ∈ {1, . . . , ℓ} and j ∈ {1, . . . , m}. Now we sum over i and j in these ranges and use the definition +of α and β to see that +ℓ +� +i=1 +� 1 +si +− 1 +qi +� += 1 +s − +ℓ +� +i=1 +1 +qi +and +m +� +j=1 +� 1 +tj +− 1 +bj +� += 1 +t − +m +� +j=1 +1 +bj +, +(D.2.9) +from which the claim immediately follows. +We next note that by hypothesis 1/r = 1/s + 1/t, and so the claim allows us to begin the product +estimate by applying H¨older’s inequality: +��� +� +ℓ� +i=1 +∂αiui +�� m +� +j=1 +∂βjwj +���� +Lr ⩽ +� +ℓ� +i=1 +∥ui∥W |αi|,si +�� m +� +j=1 +∥wj∥W |βj|,tj +� +. +(D.2.10) +We then use (D.2.7) and apply Gagliardo-Nirenberg interpolation, Corollary D.5, to bound +∥ui∥W |αi|,si ≲ ∥ui∥|αi|/k +W k,pi∥ui∥1−|αi|,k +Lqi +and ∥wj∥W |αj|,sj ≲ ∥wj∥|βj|/k +W k,aj ∥wj∥1−|βi|/k +Lbj +. +(D.2.11) +We set γn = αn if n ⩽ ℓ and γn = βn−ℓ if n > ℓ and note that �m+ℓ +n=1 |γn| /k = (α + β)/k = 1. The +latter identity implies that if {Bn}m+ℓ +n=1 ⊆ [0, ∞), then +m+ℓ +� +n=1 +B1−|γn|/k +n += +m+ℓ +� +n=1 +B +� +ν̸=n|γν|/k +n += +m+ℓ +� +n=1 +� +ν̸=n +B|γν|/k +n += +m+ℓ +� +n=1 +� +ν̸=n +B|γn|/k +ν +. +(D.2.12) +We then combine estimates (D.2.10) and (D.2.11), while using the notation of (D.2.6) and the +identity (D.2.12), to arrive at the estimate +��� +� +ℓ� +i=1 +∂αiui +� m +� +j=1 +∂βjwj +��� +Lr ≲ +m+ℓ +� +n=1 +∥vn∥|γn|/k +W k,cn∥vn∥1−|γn|/k +Ldn += +m+ℓ +� +n=1 +� +∥vn∥W k,cn +� +ν̸=n +∥vν∥Ldn +�|γn|/k +. +(D.2.13) +Then (D.2.5) follows from this via an application of Young’s inequality. +□ +We now can derive more familiar-looking high-low type product estimates. +Corollary D.7 (Tame estimates on simple multipliers). Let U ⊆ Rd be a Stein extension domain +(see Definition A.2), k ∈ N, and ϕ ∈ Hk(U). The following hold. +(1) Suppose that α, β ∈ Nd are such that |α| + |β| = k and ψ ∈ Hmax{1+⌊d/2⌋,k}(U). Then the +product ∂αψ∂βϕ belongs to L2(U) and obeys the bound +∥∂αψ∂βϕ∥L2 ≲ ∥ψ∥H1+⌊d/2⌋∥ψ∥Hk + +� +0 +if k ⩽ ⌊d/2⌋, +∥ψ∥Hk∥ϕ∥H1+⌊d/2⌋ +if 1 + ⌊d/2⌋ < k. +(D.2.14) + +150 +NOAH STEVENSON AND IAN TICE +(2) Suppose that ψ ∈ Hmax{1+⌊d/2⌋,k}(U). Then the product ψϕ belongs to Hk(U) and obeys the +bound +∥ψϕ∥Hk ≲ ∥ψ∥H1+⌊d/2⌋∥ψ∥Hk + +� +0 +if k ⩽ 1 + ⌊d/2⌋, +∥ψ∥Hk∥ϕ∥H1+⌊d/2⌋ +if 1 + ⌊d/2⌋ < k. +(D.2.15) +(3) Suppose that α, β ∈ Nd are such that |α| + |β| = k and ψ ∈ W k,∞(U). Then the product +∂αψ∂βϕ belongs to L2(U) and obeys the bound +∥∂αψ∂βϕ∥L2 ≲ ∥ψ∥L∞∥ϕ∥Hk + ∥ψ∥W k,∞∥ϕ∥L2. +(D.2.16) +(4) Suppose that ψ ∈ W k,∞(U). Then the product ψϕ belongs to Hk(U) and obeys the bound +∥ψϕ∥Hk ≲ ∥ψ∥L∞∥ϕ∥Hk + ∥ψ∥W k,∞∥ϕ∥L2. +(D.2.17) +Proof. The second and fourth items follow directly from the Leibniz rule and the first and third +items, respectively. The third item follows as a direct application of Theorem D.6 in the case that +ℓ = 2, m = 0, r = 2, p1 = 2, p2 = ∞, q1 = 2, q2 = ∞. It remains to prove the first item. In the +case 1 + ⌊d/2⌋ < k, the estimate of the first item again follows from Theorem D.6, applied with +ℓ = 2, m = 0, r = 2, p1 = p2 = 2, q1 = q2 = ∞, combined with the supercritical Sobolev embedding +H1+⌊d/2⌋(U) �→ L∞(U). +Now consider the remaining case, k ⩽ ⌊d/2⌋. We consider first what happens when k < d/2. +In this case we can invoke the subcritical Sobolev embedding Hk(U) �→ L +2d +d−2k (U) to see that +ϕ ∈ (W k,p1 ∩ Lq1)(U) for p1 = 2 and q1 = +2d +d−2k. On the other hand, the supercritical Sobolev +embedding, Corollary D.5, and the estimate 1 + ⌊d/2⌋ > d/2 > k show that we have the embeddings +H1+⌊d/2⌋(U) �→ H1+⌊d/2⌋(U) ∩ L∞(U) �→ W k,2(1+⌊d/2⌋)/k(U) �→ W k,d/k(U). +(D.2.18) +This implies that ψ ∈ (L∞ ∩ H1+⌊d/2⌋)(U). We set p2 = d/k and q2 = ∞ and note that the +hypotheses of Theorem D.6 are satisfied with r = 2, ℓ = 2, and m = 0 again; the bound (D.2.14) +when k < d/2 then follows from the product estimate provided by the theorem. +Finally, we consider the case k = d/2. Thanks to Corollary D.5 again, we obtain that ψ ∈ +H1+d/2(U) �→ W k,(4+2d)/d(U), which motivates setting p2 = 4+2d +d +and q2 = ∞. The critical Sobolev +embedding implies that ϕ ∈ Hk(U) �→ L2+d(U), which dictates that we set p1 = 2, q1 = 2 + d, and +r = 2. Using these parameters, we again apply Theorem D.6 to get (D.2.14) when k = d/2, which +completes the proof of the first item. +□ +Next we consider tame bounds on commutator expressions. +Corollary D.8 (Tame estimates on commutators). Let U ⊆ Rd be a Stein extension domain (see +Definition A.2), k, m ∈ N with m ⩾ 1, j ∈ {1, . . . , d}, and ϕ ∈ Hk+m−1(U). The following hold. +(1) Suppose that ψ ∈ Hmax{2+⌊d/2⌋,k+m} +loc +(U) satisfies ∂jψ ∈ Hmax{1+⌊d/2⌋,k+m−1}(U). Then the +commutator [∂m +j , ψ]ϕ belongs to Hk(U) and obeys the bound +∥[∂m +j , ψ]ϕ∥Hk ≲ ∥∂jψ∥H1+⌊d/2⌋∥ϕ∥Hk+m−1 + +� +0 +if k + m ⩽ 2 + ⌊d/2⌋, +∥∂jψ∥Hk+m−1∥ϕ∥H1+⌊d/2⌋ +if 2 + ⌊d/2⌋ < k + m. +(D.2.19) +(2) Suppose that ψ ∈ W k+m,∞ +loc +(U) satisfies ∂jψ ∈ W k+m−1,∞(U). +Then the commutator +[∂m +j , ψ]ϕ belongs to Hk(U) and obeys the bound +∥[∂m +j , ψ]ϕ∥Hk ≲ ∥∂jψ∥L∞∥ϕ∥Hk+m−1 + ∥∂jψ∥W k+m−1,∞∥ϕ∥L2. +(D.2.20) + +COMPRESSIBLE TRAVELING WAVES +151 +Proof. Fix α ∈ Nd with |α| ⩽ k. By the Leibniz rule we have +∂α([∂m +j , ψ]ϕ) = +m−1 +� +ℓ=0 +� +β⩽α +cℓ,m,β,α∂β∂ℓ +j∂jψ∂α−β∂m−1−ℓ +j +ϕ, +(D.2.21) +and hence +∥[∂m +j , ψ]ϕ∥Hk ≲ +� +|α|⩽k +m−1 +� +ℓ=0 +� +β⩽α +∥∂β∂ℓ +j∂jψ∂α−β∂m−1−ℓ +j +ϕ∥L2. +(D.2.22) +To prove the first item, we now apply the first conclusion of Corollary D.7 and use that |α| ⩽ k. To +prove the second item, we instead use the third conclusion of the aforementioned corollary and that +ℓ ⩽ m − 1 in these sums. +□ +Now we derive tame bounds on superposition multipliers. +Corollary D.9 (Tame estimates on superposition multipliers). Let U ⊆ Rd be a Stein extension +domain (see Definition A.2), k ∈ N+, and V be a finite dimensional real vector space. Suppose that +f ∈ W k,∞(V ), g ∈ W k,∞ +loc (U; V ) is such that Dg ∈ W k−1,∞(U; L(Rd; V )), and ψ ∈ Hk +loc(U; V ) is +such that Dψ ∈ (L∞ ∩ Hk−1)(U; L(Rd; V )). Let R ∈ R+ and S ∈ R+ ∪ {∞} satisfy +⟨∥f∥W 1,∞, ∥Dg∥L∞, ∥Dψ∥L2∩L∞⟩ ⩽ R and ⟨∥f∥W 2+⌊d/2⌋,∞, ∥Dg∥W 1+⌊d/2⌋,∞, ∥Dψ∥H1+⌊d/2⌋⟩ ⩽ S. +(D.2.23) +Then the following hold. +(1) If ϕ ∈ (L∞ ∩ Hk)(U), then the product f(g + ψ)ϕ belongs to Hk(U) and obeys the estimate +∥f(g + ψ)ϕ∥Hk ≲ Rk+1∥ϕ∥Hk + Rk∥f, Dg, Dψ∥W k,∞×W k−1,∞×Hk−1∥ϕ∥L2∩L∞. +(D.2.24) +(2) If ϕ ∈ Hk(U) and we assume additionally that S < ∞, then the product f(g + ψ)ϕ belongs +to Hk(U) and obeys the estimate +∥f(g +ψ)ϕ∥Hk ≲ Sk+1∥ϕ∥Hk +Sk +� +0 +if k ⩽ 1 + ⌊d/2⌋, +∥f, Dg, Dψ∥W k,∞×W k−1,∞×Hk−1∥ϕ∥H1+⌊d/2⌋ +if 1 + ⌊d/2⌋ < k. +(D.2.25) +(3) Suppose that ϕ ∈ Hk(U), S < ∞, and there is a Stein-extension domain O ⊆ V such that +f ∈ W k,∞(O). If the image of g + ψ is a subset of O, then the inclusion and estimate from +the second item hold. +Proof. We begin by proving the first item. We start by estimating +∥f(g + ψ)ϕ∥Hk ≲ ∥f(g + ψ)ϕ∥L2 + ∥Dk[f(g + ψ)ϕ]∥L2 +(D.2.26) +and ∥f(g + ψ)ϕ∥L2 ⩽ ∥f∥L∞∥ϕ∥L2. For the second term in (D.2.26) we have to work harder. By +the differentiation rules for products and compositions, we have that +Dk[f(g + ψ)ϕ] = f(g + ψ)Dkϕ + sym +k−1 +� +j=0 +j+1 +� +ℓ=1 +� +j1+···+jℓ=j+1 +cj+1,k,ℓ,j1,...,jℓ +· Dℓf(g + ψ){Dj1(g + ψ), . . . , Djℓ(g + ψ)} ⊗ Dk−1−jϕ = I + II, +(D.2.27) +where sym denotes symmetrization of the multilinear map. We will estimate the L2-norm of I and +II separately. We handle I trivially: +∥I∥L2 ⩽ ∥f∥L∞∥ϕ∥Hk. +(D.2.28) + +152 +NOAH STEVENSON AND IAN TICE +For II, we use the triangle inequality and study each term in the series: +∥II∥L2 ≲ +k−1 +� +j=0 +j+1 +� +ℓ=1 +� +j1+···+jℓ=j+1 +∥Dℓf(g + ψ){Dj1(g + ψ), . . . , Djℓ(g + ψ)} ⊗ Dk−1−jϕ∥L2. (D.2.29) +By the ℓ-multilinearity of Dℓf and symmetry considerations, we have the upper bound +∥II∥L2 ≲ +k−1 +� +j=0 +j+1 +� +ℓ=1 +� +j1+···+jℓ=j+1 +ℓ +� +ν=0 +∥Dℓf(g + ψ){Dj1ψ, . . . , Djνψ, Djν+1g, . . . , Djℓg} ⊗ Dk−1−jϕ∥L2. +(D.2.30) +We would like to apply Theorem D.6 to the summands in this expression, but due to the appearance +of Dℓf(g + ψ) terms we cannot verify the theorem’s hypotheses. Instead, we modify the argument +used to prove the theorem. Given j ∈ {0, . . . , k − 1}, ℓ ∈ {1, . . . , j + 1}, j1 + · · · + jℓ = j + 1, and +ν ∈ {0, . . . , ℓ}, we set α = k−1−j+�ν +ν=1(jν−1), β = ℓ−1+�ℓ +ν=ν+1(jν−1), α+β = k−1, r = s = 2, +t = a1 = b1 = · · · = aℓ−ν+1 = bℓ−ν+1 = ∞, p1 = · · · = pν+1 = 2, q1 = · · · = qν+1 = 2(k−1)(ν+1)−α +k−1−α +, +u1 = · · · = uν = Dψ, uν+1 = ϕ, w1 = Df, and w2 = · · · = wℓ−ν+1 = Dg. The argument used +to prove Theorem D.6 then pushes through for the summands in (D.2.30) with these parameters, +thanks to the trivial bound ∥Dℓf(g + ψ)∥L∞ ⩽ ∥Dℓf∥L∞; this results in the estimate +∥II∥L2 ≲ ⟨∥Dg, Dψ∥L∞×(L2∩L∞)⟩k−1� +∥Df∥L∞∥Dg, Dψ∥L∞×(L2∩L∞)∥ϕ∥Hk−1 ++ ∥Df∥L∞∥Dg, Dψ∥W k−1,∞×Hk−1∥ϕ∥L2∩L∞ + ∥Df∥W k−1,∞∥Dg, Dψ∥L∞×(L2∩L∞)∥ϕ∥L2∩L∞ +� +. +(D.2.31) +Upon combining estimates (D.2.26), (D.2.28), and (D.2.31), we arrive at the desired conclusion, +estimate (D.2.24), of the first item. +The second item in the case that 1 + ⌊d/2⌋ ⩽ k follows from estimate (D.2.24) of the first item, +the fact that 1 ⩽ R ≲ S, and the supercritical Sobolev embedding H1+⌊d/2⌋ �→ L2 ∩ L∞. On the +other hand, we have the trivial estimate +∥f(g + ψ)ϕ∥L2 ⩽ ∥f∥L∞∥ϕ∥L2 ⩽ S∥ϕ∥L2. +(D.2.32) +This shows that the linear map ϕ �→ f(g+ψ)ϕ has S2+⌊d/2⌋ as an upper-bound on the L(H1+⌊d/2⌋(U)) +operator norm and has S as an upper-bound on the L(L2(U)) operator norm. Employing operator +interpolation (see, for instance, Bergh and L¨ofstr¨om [9]) with these bounds, we achieve (D.2.25) in +the cases k ⩽ 1 + ⌊d/2⌋, which completes the proof of the second item. +The third item follows from the second item applied when f is replaced by EOf, where EO is a +Stein extension operator for O, and the observation that the image hypothesis on g + ψ ensures +that (EOf)(g + ψ) = f(g + ψ). +□ +Now we consider superposition on its own, not as a multiplier. +Corollary D.10 (Tame estimates on superposition). Let N ∋ k ⩾ 2 + ⌊n/2⌋. The following hold. +(1) Let V , W be finite dimensional real vector spaces, O ⊆ V be a Stein extension domain (see +Definition A.2) containing 0 that is star shaped with respect to 0, and U ⊆ Rn be a Stein +extension domain. If f ∈ W k,∞ +loc (O; W) is such that f(0) = 0 and Df ∈ W k−1,∞(O; L(V ; W)), +and ϕ ∈ Hk(U; V ) is such that ϕ(U) ⊆ O, then the superposition f(ϕ) belongs to Hk(U; W) +and obeys the estimate +∥f(ϕ)∥Hk ≲ Sk∥ϕ∥Hk, +(D.2.33) +for any S ∈ R+ satisfying ⟨∥Df∥W k−1,∞, ∥ϕ∥H2+⌊n/2⌋⟩ ⩽ S. + +COMPRESSIBLE TRAVELING WAVES +153 +(2) Suppose that f ∈ Hk(Rn; Rd) and g, ψ ∈ C1(Rn; Rn) are such that Dg ∈ W k−1,∞(Rn; Rn×n), +Dψ ∈ Hk−1(Rn; Rn×n), and g + ψ is a bi-Lipschitz and C1-diffeomorphism of Rn. The +superposition f(g + ψ) belongs to Hk(Rn; Rd) and satisfies the estimate +∥f(g + ψ)∥Hk ≲ Sk+1∥f∥Hk + Sk∥Dg, Dϕ∥W k−1,∞×Hk−1∥f∥H2+⌊n/2⌋, +(D.2.34) +for any S ∈ R+ satisfying +⟨∥det D(g + ψ)−1∥L∞, ∥Dg, Dψ∥W 1+⌊n/2⌋,∞×H1+⌊n/2⌋⟩ ⩽ S. +(D.2.35) +Proof. We begin by proving the first item. Since f(0) = 0 and O is star-shaped with respect to +the origin of V , we can use the fundamental theorem of calculus to obtain the equality f(ϕ) = +� 1 +0 Df(tϕ)[ϕ] dt, from which we deduce that ∥f(ϕ)∥L2 ⩽ ∥Df∥L∞∥ϕ∥L2. We next establish an +Hk−1-bound on the derivative of the superposition, D(f(ϕ)) = Df(ϕ)Dϕ. For this we utilize the +third item of Corollary D.9 and obtain the bound ∥D(f(ϕ))∥Hk−1 ≲ Sk∥Dϕ∥Hk−1. Together, these +estimate give the conclusion of the first item. +We next prove the second item, noting initially that +∥f(g + ψ)∥Hk ≲ ∥f(g + ψ)∥L2 + ∥Dk[f(g + ψ)]∥L2. +(D.2.36) +For the first L2-norm, we use that (g + ψ) is a bi-Lipschitz and C1 diffeomorphism to estimate +∥f(g + ψ)∥L2 = +� � +Rn |f|2| det D(g + ψ)−1| +�1/2 +⩽ S1/2∥f∥L2. +(D.2.37) +For the latter term of (D.2.36), we would like to use Theorem D.6, but as in the proof of Corollary +D.9, we cannot quite do so. Instead, we use the differentiation rules for composition and argue in a +manner similar to the proof of the first item in Corollary D.9. Indeed, we have the formula +Dk[f(g + ψ)] = sym +k +� +ℓ=1 +� +j1+···+jℓ=k +cℓ,k,j1,...,jℓDℓf(g + ψ){Dj1(g + ψ), . . . , Djℓ(g + ψ)}, +(D.2.38) +where sym denotes the symmetrization operator for multilinear maps. +By multilinearity and +symmetry considerations, we then deduce the upper bound +∥Dk[f(g+ψ)]∥L2 ≲ +k +� +ℓ=1 +� +j1+···+jℓ=k +ℓ +� +ν=0 +∥Dℓf(g+ψ){Dj1ψ, . . . Djνψ, Djν+1g, . . . , Djℓg}∥L2. (D.2.39) +To each summand on the right hand side above we apply the argument from the proof of Theorem D.6 +as follows. Given ℓ ∈ {1, . . . , k}, j1 + · · · + jℓ = k, and ν ∈ {0, 1, . . . , ℓ}, we define α = ℓ − 1 + +�ν +ν=1(jν − 1), β = �ℓ +ν=ν+1(jν − 1), α + β = k − 1, r = s = 2, t = a1 = b1 = · · · = aℓ−ν = bℓ−ν = ∞, +p1 = · · · = pν+1 = 2, q1 = · · · = qν+1 = 2(k−1)(ν+1)−α +k−1−α +, u1 = Df, u2 = · · · = uν+1 = Dψ, and +w1 = · · · = wℓ−ν = Dg. The argument used in the theorem pushes through so long as we carry an +extra factor of S, thanks to the bounds +∥Dℓf(g + ψ)∥Lχ ⩽ S1/χ∥Dℓf∥Lχ ⩽ S∥Dℓf∥Lχ for all χ ∈ [1, ∞]. +(D.2.40) +Therefore, we deduce the estimate +∥Dℓf(g + ψ){Dj1ψ, . . . Djνψ, Djν+1g, . . . , Djℓg}∥L2 +≲ S⟨∥Dg, Dψ∥L∞×(L2∩L∞)⟩ℓ−1� +∥Df∥Hk−1∥Dg, Dψ∥L∞×(L2∩L∞) ++ ∥Df∥L2∩L∞∥Dg, Dψ∥W k−1,∞×Hk−1 +� +≲ Sℓ+1∥f∥Hk + Sℓ∥Dg, Dψ∥W k−1,∞×Hk−1∥f∥H2+⌊n/2⌋. +(D.2.41) +Upon combining (D.2.36), (D.2.37), (D.2.39), and (D.2.41), we acquire the desired bound, (D.2.34). +□ + +Notation Index +Fluid mechanical terms +Ω[η], 3 +Ω, 3 +Σ[η], 3 +Σ, 3 +Σ0, 3 +P, 3 +Pext, 4 +H , 4 +g, 4 +ς, 4 +Sτ, 4 +Sτ +A, 10 +D, 123 +D0, 123 +H, 5 +Hmin, Hmax, 5 +ϱ, 5 +Function spaces +Xs, 51 +Xs, 51 +q0,u0,η0 +Xs +, 66 +Xs +m,N, 71 +Ys, 51 +Ys, 52 +Ws, 52 +Es, 52 +Fs, 52 +T k +µ,r(U, E; F), 24 +sT k +µ,r(U, E; F), 24 +ˆHs, 132 +0H1, 22 +˙H−1, 22 +Hs, 125 +H0 +(κ), 125 +XXX, 52 +YYY, 52 +EEE, 52 +W +W +W, 52 +FFF, 52 +H, 53 +H, 53 +W, 53 +H(κ), 53 +WH(κ), 53 +H∞, 22 +W ∞,∞, 22 +Ck +b , 22 +Ck +0 , 22 +C∞ +b , 22 +C∞ +0 , 22 +Linear maps +EU, 122 +E0, 10 +E, 10 +˙M[η], 64 +Πκ +L, 126 +Πκ +H, 126 +Lm, 71 +w0,γ0 +A , 63 +w0,γ0 +Am,N, 71 +w0,γ0 +P , 63 +θ0Q, 69 +q0,u0,η0 +R +, 69 +γ0 +I , 72 +w0γ0 +J , 72 +w0γ0 +J τ +m,N, 72 +K , 73 +Sj, 28 +∆j, 28 +Miscellaneous +⟨·⟩, 21 +∇∥, 4 +�N�, 23 +ρWD, 62 +Nonlinear maps +Ψ, 58 +Bm, 74 +Ψ, 58 +Φ, 58 +Nφ, 56 +N (1) +φ , 56 +N (2) +φ , 56 +Mφ, 58 +M(1) +φ , 57 +e1 · M(2) +φ , 58 +Fη, 10 +Aη, 10 +Jη, 10 +Nη, 10 +Mη, 11 +σq,η, 11 +vw0, 64 +154 + +155 +References +[1] R. 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Tice: iantice@andrew.cmu.edu + diff --git a/_NAyT4oBgHgl3EQf3vlI/content/tmp_files/load_file.txt b/_NAyT4oBgHgl3EQf3vlI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..91044936e994fd85d4943c42c0748e1158393209 --- /dev/null +++ b/_NAyT4oBgHgl3EQf3vlI/content/tmp_files/load_file.txt @@ -0,0 +1,10002 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf,len=10001 +page_content='WELL-POSEDNESS OF THE TRAVELING WAVE PROBLEM FOR THE FREE BOUNDARY COMPRESSIBLE NAVIER-STOKES EQUATIONS NOAH STEVENSON AND IAN TICE Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We prove that traveling waves in viscous compressible liquids are a generic phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The setting for our result is a horizontally infinite, finite depth layer of compressible, barotropic, viscous fluid, modeled by the free boundary compressible Navier-Stokes equations in dimension n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The bottom boundary of the fluid is flat and rigid, while the top is a moving free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A constant gravitational field acts normal to the flat bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We allow external forces to act in the fluid’s bulk and external stresses to act on its free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These are posited to be in traveling wave form, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' time-independent when viewed in a coordinate system moving at a constant, nontrivial velocity parallel to the lower rigid boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the absence of such external sources of stress and force, the fluid system reverts to equilibrium, which corresponds to a flat, quiescent fluid layer with vertically stratified density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In contrast, when such sources of stress or force are present, the system admits traveling wave solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We establish a small data well-posedness theory for this problem by proving that for every nontrivial traveling wave speed there exists a nonempty open set of stress and forcing data that give rise to unique traveling wave solutions, and that these solutions depend continuously on the data and the wave speed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When n ⩾ 3 we prove this with surface tension accounted for at the free boundary, while in the case n = 2 we prove this with or without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To the best of our knowledge, this result constitutes the first general construction of traveling wave solutions to any free boundary compressible fluid equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The traveling wave formulation of the equations is a quasilinear system of mixed type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The interaction of the hyperbolic and elliptic parts leads to derivative loss in the linearizations of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As such, we are compelled to construct solutions via an inverse function theorem of Nash-Moser type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our well-posedness proof has a number of novelties and elements of broader interest, including: a new Nash-Moser variant that works in Banach scales but guarantees minimal regularity loss in the existence as well as continuity of the local inverse;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' a host of results about steady transport equations and their elliptic regularizations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' new results about and uses of a scale of anisotropic Sobolev spaces suited for constructing traveling waves;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' and a robust, streamlined, and flexible approach for constructing solutions to our family of linearized free boundary problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Primary 35Q30, 35R35, 35C07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Secondary 47J07, 76N06, 76N30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Key words and phrases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Free boundary compressible Navier-Stokes, traveling waves, Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Stevenson was supported by an NSF Graduate Research Fellowship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Tice was supported by an NSF Grant (DMS #2204912).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='00773v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='AP] 2 Jan 2023 2 NOAH STEVENSON AND IAN TICE Contents 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Introduction 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Dynamics in Eulerian coordinates and stratified equilibria 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Traveling waves and the role of the stress and forces 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Previous work 7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Enthalpy and flattened reformulations 9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Statement of main result 11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Summary of strategy and layout of paper 14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Conventions of notation 21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A variation on the Nash-Moser inverse function theorem 22 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Tame structure abstraction 23 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Mapping hypotheses and statement of the inverse function theorem 30 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Local surjectivity and injectivity 32 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof of the inverse function theorem 42 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Refinements 45 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Nonlinear analysis of traveling free boundary compressible Navier-Stokes 51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Banach scales for the traveling wave problem 51 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of atomic nonlinearities 53 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Smooth tameness of the nonlinear operator 58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Derivative splitting 63 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Prelude to linear analysis 70 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of steady transport equations and their regularizations 74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Preliminary tame estimates 74 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Some results on steady transport equations and elliptic regularizations 79 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of weak solutions to the principal part linear equations 85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Estimates 85 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Existence of solutions to the regularization 93 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of strong solutions to the linearization 99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of tangential derivatives 99 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of normal systems 104 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Estimates and existence for the principal part 115 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Synthesis of linear analysis 117 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Conclusion 119 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Abstract construction 119 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' PDE construction 120 Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Standard Sobolev space tools 122 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Extension operators 122 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Korn’s inequalities 123 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Refined interpolation of Sobolev spaces 123 Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Some nonstandard function spaces 125 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The anisotropic Sobolev spaces 125 B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Adapted Sobolev spaces 130 Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A selection of PDE tools 132 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The divergence boundary value problem 132 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Elliptic theory tools 135 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Dissipation calculation for traveling compressible Navier-Stokes 138 Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fine tools for nonlinear analysis 142 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Smoothness of superposition nonlinearities 142 D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Tools for tame estimates 147 Notation Index 154 References 155 COMPRESSIBLE TRAVELING WAVES 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Introduction The study of traveling wave solutions to the free boundary problems of fluid mechanics has been of fundamental interest in mathematics for nearly two centuries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' During this time, tremendous progress has been made in the analysis of such solutions for incompressible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Throughout most of this period, the primary focus was inviscid, irrotational fluids;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' only in the past two decades has progress been made on models that account for more robust phenomena such as vorticity and viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, to the best of our knowledge, there are no rigorous results in the literature that account for the fundamental fluid mechanical effect of compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It is this effect that we aim to study in the present paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' All real fluids experience some degree of compressibility and viscosity, even if small, so it is physically important to verify that traveling waves remain a generic phenomenon when these effects are accounted for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From a mathematical perspective, the development of the compressible viscous theory also opens the door to studying incompressible or inviscid limits, which may then shed light on the zoo of incompressible inviscid solutions that have been constructed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that in this work we only study compressible fluids for which the density does not vanish at the free boundary;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' these are often referred to as compressible liquids in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The rest of the introduction proceeds as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 we formulate the dynamical equations for a compressible viscous fluid with free boundary and identify the equilibrium solutions, which correspond to stratified layers of quiescent fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 is concerned with the traveling wave ansatz and a discussion of the role played by external stresses and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Previous work on traveling waves is discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Further reformulations of the equations, made in the interest of identifying ‘good unknowns,’ are recorded in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our main results are stated in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 along with some discussion of their implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 contains a high-level summary of the difficulties in the proof and our strategies for overcoming them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 records the notational conventions we employ throughout the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that for convenience we have included a Notation Index at the end of the paper, just before the references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Dynamics in Eulerian coordinates and stratified equilibria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by formulating the free boundary compressible Navier-Stokes equations, which govern the dynamics of a layer of viscous, compressible, barotropic (isentropic) fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First we must set some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We let n ∈ N \\ {0, 1} denote the spatial dimension;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the cases n ∈ {2, 3} are the physically relevant ones, but our analysis works in general dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The parameter b ∈ R+ designates the equilibrium depth of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If η : Rn−1 → R is a continuous function satisfying η + b > 0, then we define the open set Ω[η] = � (x, y) ∈ Rn−1 × R : 0 < y < b + η (x) � (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) as well as the interfacial sets Σ[η] = � (x, y) ∈ Rn−1 × R : y = b + η (x) � and Σ0 = Rn−1 × {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Note that ∂Ω[η] = Σ[η] ⊔ Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See Figure 1 for a depiction of these sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Throughout the paper we will extensively use the shorthand notation Ω = Ω[0] = Rn−1 × (0, b) and Σ = Σ[0] = Rn−1 × {b}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) The fluid is assumed to occupy a semi-infinite layer of finite depth that changes in time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' more precisely, at time t the fluid occupies the set Ω[ζ (t, ·)] ⊂ Rn for an unknown continuous free surface function ζ (t, ·) : Rn−1 → R satisfying ζ (t, ·) + b > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The upper boundary of this set, Σ[ζ (t, ·)], is called the free boundary, while the lower boundary Σ0 is referred to as the fixed boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fluid is described by its velocity vector field w (t, ·) : Ω[ζ (t, ·)] → Rn and its scalar density τ (t, ·) : Ω[ζ (t, ·)] → R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Associated to w and τ are two crucial fluid mechanical quantities: the pressure and the viscous stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The pressure within the fluid is given by P(τ), where P ∈ C∞ (R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) is a given pressure law that is strictly increasing and satisfies P ′ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The 4 NOAH STEVENSON AND IAN TICE b + η Σ[η] Σ0 Ω[η] Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Depiction of the domain Ω[η] and its boundary, Σ[η] and Σ0, when n = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' assumption that the pressure depends only on the density is what makes the fluid barotropic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this can be viewed as a consequence of assuming the fluid flow is isentropic, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' entropy remains constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The viscous stress tensor within the fluid is the symmetric tensor Sτw = µ (τ) � ∇w + ∇wt − 2 n∇ · wI � + λ (τ) (∇ · w)I = µ(τ)D0w + λ(τ)(∇ · w)I, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) where the shear and bulk viscosity coefficients are µ, λ ∈ C∞ (R+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' [0, ∞)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The non-negativity of the viscosity coefficients is a requirement of the Clausius–Duhem inequality from continuum thermodynamics, but we will place more assumptions on these below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that the specific forms of the functions P, µ, and λ can be thought of as characterizing the specific material comprising the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The dynamics of ζ, w, and τ are then coupled to the various forces and stresses acting on the fluid through the free boundary compressible Navier-Stokes equations: � � � � � � � � � � � � � � � ∂tτ + ∇ · (τw) = 0 in Ω[ζ (t, ·)] τ (∂tw + w · ∇w) + ∇(P (τ)) − ∇ · Sτw = −gτen + τG + F in Ω[ζ (t, ·)] −(P (τ) − Sτw)νζ + Pextνζ − ςH (ζ) νζ = Tνζ on Σ[ζ (t, ·)] ∂tζ + w · (∇∥ζ, −1) = 0 on Σ[ζ (t, ·)] w = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Note that in the above we have written ∇∥ = (∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ∂n−1) to refer to the ‘tangential gradient’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now enumerate these forces and stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The term −gτen is the gravitational force acting on the fluid, with gravitational strength g > 0 and unit vector en = (0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , 1) ∈ Rn perpendicular to Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The vector fields F (t, ·) , G(t, ·) : Ω[ζ (t, ·)] → Rn are the applied bulk and specific bulk forces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The parameter Pext ∈ R is a constant external pressure, T (t, ·) : Σ[ζ (t, ·)] → Rn×n is the applied surface stress, and νζ is the outward unit normal to the surface Σ[ζ (t, ·)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We note that in continuum mechanics it is usually the case that T (t, ·) is symmetric, but this condition plays no role in our analysis, so have allowed for the most general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The mean curvature operator is H (ζ) = ∇∥ · ((1 + ��∇∥ζ ��2)−1/2∇∥ζ), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) and the parameter ς ⩾ 0 is called the coefficient of surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For technical reasons that will be discussed later, we make the following assumptions about the viscosity coefficients and the coefficient of surface tension:� µ > 0, λ ⩾ 0 in R+, and ς > 0 if n ⩾ 3 µ > 0, λ > 0 in R+, and ς ⩾ 0 if n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) The first equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) is the continuity equation, which asserts conservation of mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The next is the momentum equation, and it dictates a Newtonian balance of forces in the fluid bulk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' After this is the dynamic boundary condition, which enforces a balance of stresses acting on the free surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The penultimate equation is the kinematic boundary condition, which determines how COMPRESSIBLE TRAVELING WAVES 5 the free surface evolves according to the fluid velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The final equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) is simply the no-slip boundary condition for the velocity on the rigid bottom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For a more thorough introduction to the compressible Navier-Stokes equations, including their derivation, we refer to the books of Wehausen and Laitone [104], Feireisl [33], Lions [66], Novotn´y and Straˇskraba [83], Gurtin, Fried, and Anand [38], and Plotnikov and Soko�lowski [88].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The compressible Navier-Stokes system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) admits a vertically stratified equilibrium solution, provided that the barotropic pressure law P, the external depth b, the external pressure Pext, and the gravitational field strength g satisfy some compatibility conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, suppose that the fluid experiences no external forces or stresses, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F = G = 0 and T = 0, and that the fluid is quiescent and occupies a flat slab of depth b, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' w = 0 and ζ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, suppose that ∂tτ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' then (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) reduces to τ(t, x, y) = ϱ(y), where ϱ : [0, b] → R+ is a smooth function solving the Cauchy problem � (P ◦ ϱ)′ = −gϱ in (0, b) P ◦ ϱ(b) = Pext.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) In order to guarantee that a solution exists, we henceforth assume that the following pair of compatibility conditions are satisfied (these conditions are actually necessary and sufficient): Pext ∈ P(R+) and (0, ∞] ∋ � ∞ P −1(Pext) t−1P ′(t) dt > gb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) With (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) in hand, we can conveniently solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) by introducing the enthalpy H : R+ → R, which is the smooth increasing function defined via H(s) = −gb + � s P −1(Pext) t−1P ′(t) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Note that since P ′ > 0 we have that H′ > 0 as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We can also calculate the image H(R+) = (Hmin, Hmax) ⊆ R, where Hmin = −gb− � P −1(Pext) 0 t−1P ′(t) dt < −gb and Hmax = −gb+ � ∞ P −1(Pext) t−1P ′(t) dt > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Since we now know that H : R+ → (Hmin, Hmax) is a smooth diffeomorphism and [−gb, 0] ⊆ H(R+), we may realize ϱ as the smooth decreasing function defined by ϱ(y) = H−1(−gy) for y ∈ [0, b], (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) which is the unique solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) in light of the construction of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The equilibrium density ϱ will play a crucial role in our subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a concrete example, if the pressure satisfies the well-known polytropic law P(t) = Ktα for α ⩾ 1 and K ∈ R+, then ϱ(y) = � PextK−1 exp(gK−1(b − y)) if α = 1 ((PextK−1)(α−1)/α + (α − 1)α−1K−1g(b − y))1/(α−1) if α > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Traveling waves and the role of the stress and forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The main thrust of this paper is the study of traveling wave solutions to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These are solutions that are time- independent when viewed in an inertial coordinate system obtained from the above Eulerian coordinates through a Galilean transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In order for time-independence to hold, the moving coordinate system must travel at a constant velocity parallel to Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Without loss of generality (we can always apply a rigid rotation that fixes the vector en to change coordinates), we may assume that the traveling coordinate system moves at constant velocity γe1 for a wave speed γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the new coordinates, the stationary free boundary is described by η (x − γte1) = ζ (t, x) for a new unknown free surface function η : Rn−1 → (−b, ∞), which then determines the fluid domain Ω[η] and the free boundary Σ[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then posit that the other quantities are also in traveling 6 NOAH STEVENSON AND IAN TICE wave form: γv (x − tγe1, y) = w (t, x, y), σ(x − γte1, y) = τ(t, x, y), F (x − tγe1, y) = F (t, x, y), G (x − tγe1, y) = G (t, x, y), and T (x − tγe1, y) = T (t, x, y), where v : Ω[η] → Rn, σ : Ω[η] → R, F, G : Ω[η] → Rn, and T : Σ[η] → Rn×n define the stationary velocity field, density, external and specific forces, and external stresses, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under these assumptions, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) is equivalent to the following traveling compressible Navier-Stokes system for unknowns (σ, v, η) and data (T , G, F): � � � � � � � � � � � � � � � −∂1σ + ∇ · (σv) = 0 in Ω[η] γ2σ (v − e1) · ∇v + ∇(P (σ)) − γ∇ · Sσv = −gσen + σG + F in Ω[η] −(P (σ) − γSσv)νη + Pextνη − ςH (η) νη = T νη on Σ[η] −∂1η + v · (∇∥η, −1) = 0 on Σ[η] v = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Note that in changing unknowns, we have rescaled the velocity vector by γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This has the effect of nondimensionalizing the vector field v and, more importantly, removing the γ-dependence from the continuity equation and the kinematic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the traveling wave system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) formulated, we turn to a discussion of the role played by the surface stress, specific bulk force, and bulk force data triple, (T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have chosen to study this general form of (T , G, F) in order to allow these to model a variety of physical effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a specific example, the bulk force term G can be thought of as a localized perturbation of the gravitational field caused by a massive object translating above the fluid (a primitive model of the ocean-moon system).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Similarly, a simple example of the surface stress occurs when T = −ϕIn×n for a given scalar function ϕ : Rn → R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' in this configuration, ϕ can be viewed as a spatially localized source of pressure translating above the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See Figure 2 for depictions of the free surface for this latter case of applied stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If (T , G, F) = 0, then it is a simple matter to verify that the stratified equilibrium solution, v = 0, η = 0, and σ = ϱ (defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12)), provides a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with any value of γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This suggests that we should seek solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) as perturbations of this stratified equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' An elementary formal calculation (we will state and prove a rigorous version later after a further reformulation of the problem: see Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 and, in particular Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) reveals that if σ − ϱ, v, and η are in a Sobolev-type framework, then � Ω[η] µ(σ) 2 |D0v|2 + λ(σ)|∇ · v|2 = � Ω[η] (σG + F) · v + � Σ[η] T νη · v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) The physical interpretation of this identity is that if a traveling wave solution exists, then the power supplied by the forces and stress (the right side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2)) must be in exact balance with the energy dissipation rate due to viscosity (the left side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The identity (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) reveals even more if we assume there are no applied forces or stress, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (T , G, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Without a source of external power, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) requires that the left integral vanishes, and so the assumptions (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) together with the Korn inequality (see Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 for Korn in Ω, but similar results hold in Ω[η] if η is sufficiently regular) imply that v = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, the momentum equation implies that ∇(P(σ)) = −gσen and hence σ = H−1(−gidRn · en + c) for some constant c, but we must have c = 0 since σ − ϱ vanishes at infinity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The normal part of the dynamic boundary condition then requires Pext − P ◦ H−1(−g(b + η)) − ς∇∥ · ((1 + |∇∥η|2)−1/2∇∥η) = 0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) and by multiplying this equation by η and integrating by parts in the mean curvature term, we deduce that � Rn−1(Pext − P ◦ H−1(−g(b + η)))η ⩽ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) COMPRESSIBLE TRAVELING WAVES 7 Since Pext − P ◦ H−1(−g(b + ·)) is a strictly increasing function vanishing at zero, the integrand in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is nonnegative and must thus vanish pointwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, η = 0, and we have deduced that (σ, v, η) = (ϱ, 0, 0) when (T , G, F) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The above formal computation suggests that the dissipative nature of viscosity prohibits the existence of nontrivial traveling wave solutions (in Sobolev-type spaces) without applied stress or forcing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This shows that the triple (T , G, F) plays an essential role in the study of traveling wave solutions, as the stress and forcing data are necessary for solutions to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize, however, that the above argument does not preclude the existence of nontrivial solutions with (T , G, F) = 0 in non-Sobolev functional frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now that the importance of the data triple in the traveling wave theory is evident, we can roughly summarize our goal for the paper: we aim to prove that for every traveling wave speed γ ∈ R+ the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) admits a small-data well-posedness theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That is, we aim to identify a nontrivial open set, Uγ, of stress and forcing data in a Sobolev-type framework such that for every (T , G, F) ∈ Uγ there exists a locally unique solution triple (σ, v, η) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), also in a Sobolev-type framework, that depends continuously on (T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, for technical reasons that we will explain at the beginning of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, the variables (σ, v, η) are not suitable for this task, and we must introduce a further reformulation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with a new set of ‘good unknowns’ in order to achieve our goal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Interestingly, this new formulation has the added benefit of allowing us to establish the continuity of solutions with respect to the wave speed γ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The time dependent free boundary compressible fluid equations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and their variants have received much attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A full review is beyond the scope of the paper, so we will settle for a brief survey of some results closely related to ours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A significant portion of the literature on the dynamic problem concerns the inviscid analog of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the free boundary compressible Euler equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lindblad [62, 63] proved local well-posedness of the liquid droplet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jang and Masmoudi [49, 50] and Coutand and Shkoller [20, 21] proved local well-posedness for the vacuum droplet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Well-posedness of the liquid droplet problem with surface tension was studied by Coutand, Hole, and Shkoller [19] and by Disconzi and Kukavica [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The incompressible limit for the liquid droplet problem was derived by Lindblad and Luo [65] without surface tension and by Disconzi and Luo [29] with surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Trakhinin [102] and Luo and Zhang [69] proved local well-posedness for inviscid liquid layers of infinite depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There are also a number of dynamical studies of the free boundary compressible Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Denisova proved local well-posedness for the compressible bubble in a compressible fluid problem in Sobolev spaces [23] and in weighted H¨older spaces [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Denisova [24] also proved similar results for the compressible bubble in an incompressible fluid problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The viscous liquid droplet problem was studied by Secchi and Valli [92], who proved local well-posedness with heat conduction, Solonnikov and Tani [95], who proved local well-posedness with surface tension, and by Denisova and Solonnikov [26], who developed a well-posedness theory with and without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The well-posedness of the viscous liquid droplet problem has also been proved under various conditions using maximal regularity techniques: see the work of Enomoto, von Below, and Shibata [32], Shibata [94], and Burczak, Shibata, and Zaj¸aczkowski [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The viscous literature also has several studies of layer geometries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jin [53] and Jin and Padula [54] proved local and global well-posedness for a layer of periodic barotropic fluid with surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Tanaka and Tani [100] gave local and global well-posedness results for layers of heat conducting fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jang, Tice, and Wang [51, 52] proved local and global well-posedness for multiple layers of barotropic fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Huang and Luo [45] proved global existence for a layer of heat conducting fluid without surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It is also possible to use compressible fluids with free boundaries as a simple model of stellar structure, in which case the fluid is subject to the force of its own gravitational field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When gravity 8 NOAH STEVENSON AND IAN TICE is assumed to be Newtonian rather than relativistic, this problem is called the Euler-Poisson system for inviscid fluids and the Navier-Stokes-Poisson system for viscous ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Makino [71] proved an early local existence result for Euler-Poisson under special assumptions on the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Makino [84] and Matuˇsu-Neˇcasov´a, Okada, and Makino [76] studied the viscous problem with spherical symmetry outside a solid inner-core.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jang [47] proved local existence for the Navier-Stokes-Poisson problem with vacuum boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Luo, Xin, and Zeng [70] studied local well-posedness for radial solutions to the inviscid problem with vacuum boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Ginsberg, Lindblad, and Luo [36] studied the local well-posedness of a self gravitating compressible liquid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jang and Hadˇzi´c [39] constructed global expanding solutions for Euler-Poisson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The self-gravitating problem admits nontrivial radial steady states known as Lane-Emden solutions (see, for instance, the book of Chandrasekhar [14]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' More recent work has rigorously constructed rotating solutions: for a variational approach we refer to Auchmuty and Beals [4] and Li [61], and for a perturbative approach we refer to Jang and Makino [48] and Strauss and Wu [99].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In stark contrast to the above discussion, the compressible traveling problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) has, to the best of our knowledge, not received any prior attention in the PDE literature either with or without viscosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Perhaps the closest work, though still rather distant, concerns the construction of stationary (but not traveling) solutions to some free boundary problems for viscous compressible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Pileckas and Zaj¸aczkowski [87] found stationary solutions to a bounded viscous compressible fluid droplet with surface tension under symmetry considerations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Jin and Padula [55] considered steady flows of viscous compressible fluids in a bounded rigid container with partially free boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The incompressible analogs of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) have, however, received attention in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The incompressible and inviscid analog of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), which is also known as the traveling water wave problem, has received enormous attention in the mathematics literature for more than a century.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We refer to the surveys of Toland [101], Groves [37], Strauss [98], and Haziot, Hur, Strauss, Toland, Wahl´en, Walsh, and Wheeler [41] and the references therein for a thorough review of this extensive literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, progress on the traveling wave theory for the free boundary incompressible Navier-Stokes equations only began quite recently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A small data well-posedness theory was first developed by Leoni and Tice [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This result was subsequently generalized by Stevenson and Tice [97] and Koganemaru and Tice [57] to multi-layered and inclined geometries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A similar well-posedness and stability theory for traveling wave solutions to the one-phase Muskat problem was developed by Nguyen and Tice [82].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Viscous traveling waves were also empirically identified recently in experiments with a tube of air, translating uniformly above a wave tank, blowing onto a single layer of viscous fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For details, we refer to the works of Akylas, Cho, Diorio, and Duncan [18, 27], Masnadi and Duncan [74], and Park and Cho [85, 86].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As we have already mentioned, there are only a few recent works in the literature that rigorously treat traveling waves with viscosity and none that treat compressibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Viscosity and compressibility are important physical effects to account for in studying traveling waves because all real fluids experience some degree of compressibility and viscosity, even if small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Within the applied literature there are works that study the observable effects of compressibility in simplified fluid models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For example, Longuet-Higgins [68] and Kadri [56] studied how the compressibility of water leads to microseisms in the ocean, and Long and Morton [67] and Miesen, Kamp, and Sluijter [78, 79] used asymptotic expansions and numerics to study the role of compressibility in atmospheric solitary traveling waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our proof of well-posedness for the traveling wave problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) uses a novel variation on the Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A literature review concerning our version is delayed until Sections 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 and 2, but we will conclude this subsection with a sampling of interesting results from the fluids PDE literature that have been proved with Nash-Moser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Beale [7] used it to prove the existence of steady water waves for the irrotational incompressible free boundary Euler equations in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Plotnikov and Toland [89] found time periodic standing water waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lindblad [63, 64] used Nash-Moser to study the droplet problem for incompressible and compressible COMPRESSIBLE TRAVELING WAVES 9 Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Chen and Wang [15] studied the existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Trakhinin [102] considered the infinite depth surface wave problem for compressible Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Makino [72, 73] studied spherically symmetric motions of a planet’s compressible atmosphere and the vacuum boundary problem for gaseous stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Buffoni and Wahl´en [12] used a version of Nash-Moser to produce steady three dimensional rotation flows for incompressible Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Chen, Secchi, and Wang [16] studied relativistic vortex sheets in three- dimensional Minkowski spacetime for compressible, relativistic Euler.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Chen, Hu, Wang, Wang, and Yuan [17] used Nash-Moser to study compressible vortex sheets in two-dimensional elastodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Trakhinin and Wang [103] studied the ideal compressible magnetohydrodynamic equations with surface tension via Nash-Moser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Enthalpy and flattened reformulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The goal of this subsection is to further reformu- late the traveling wave system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) so that it is more convenient to analyze.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The motivations for this are three-fold: two common difficulties in free boundary problems and a third more subtle issue specific to the problem at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first issue is that we wish to establish a well-posedness theory in Sobolev-type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, as we discussed at the end of Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, if (T , G, F) = 0 then the solution reduces to the stratified equilibrium, but we cannot expect σ = ϱ to belong to Sobolev-type spaces on sets of infinite measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This suggests that we should rewrite (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) as a perturbation of the equilibrium solution (ϱ, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second issue is that, even if we rewrite in perturbed form, the resulting equations are still posed in an unknown domain Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In order to conveniently employ standard PDE toolboxes, it is advantageous to recast the system in a fixed, known domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The traveling wave formulation precludes the common choice of Lagrangian coordinates, so we instead employ a flattening into the equilibrium domain Ω based only on the free surface function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The third issue arises because the traveling wave structure ultimately forces the free surface function η to belong to a scale of anisotropic Sobolev-type spaces (see Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' After the perturbation and flattened reformulations, η will end up appearing in various nonlinearities in a form that, due to the strange properties of the anisotropic spaces, is quite difficult or impossible to control in a Sobolev-type framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Roughly speaking, our way around this problem is to make a nonlinear change of unknown, shifting from the density to the perturbed enthalpy h = H(σ) − H(ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The unknown h turns out to serve as a sort of ‘good unknown’ in that it recasts the worst nonlinearity, which comes from the term ∇(P(σ)) + gσen in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), in the form σ∇(H(σ) − H(ϱ)) = H−1(h + H(ϱ))∇h, which shifts the nonlinearity outside of the gradient and permits simple Sobolev estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will also employ a nonlinear change of the velocity in order to similarly linearize the kinematic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The cost of these changes is that the continuity equation becomes more cumbersome, but fortunately we can handle its new form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn to the execution of these reformulations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We start by switching to a perturbed enthalpy reformulation by defining the new unknown h = H(σ) − H(ϱ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that σ can be recovered from h via σ = H−1(h + H(ϱ)), provided h + H(ϱ) takes values in (Hmin, Hmax) from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), which will always hold for the solutions we construct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the perturbative enthalpy reformulation of the traveling free boundary compressible Navier-Stokes equations is the following system for (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' v,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η) with data (T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F): � � � � � � � � � � � � � � � � � � � −∂1(H−1(h + H(ϱ))) + ∇ · (H−1(h + H(ϱ))v) = 0 in Ω[η] γ2H−1(h + H(ϱ))(v − e1) · ∇v + H−1(h + H(ϱ))∇h − γ∇ · SH−1(h+H(ϱ))v = H−1(h + H(ϱ))G + F in Ω[η] −((P − Pext) ◦ H−1(h + H(ϱ)) − γSH−1(h+H(ϱ))v)νη − ςH (η)νη = T νη on Σ[η] −∂1η + v · (∇∥η,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' −1) = 0 on Σ[η] v = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) 10 NOAH STEVENSON AND IAN TICE Next we turn our attention to reformulating (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) in the fixed domain Ω = Rn−1 × (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first define a Poisson-like extension operator that takes functions defined on Σ and extends them to functions defined on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The auxiliary mapping E0 : Hs−1/2(Σ) → Hs(Ω) ∩ 0H1(Ω) (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3)), for s ∈ N+, is defined via ϕ �→ E0ϕ, where E0ϕ is the unique solution to the PDE � � � � � −∆E0ϕ = 0 in Ω, E0ϕ = ϕ on Σ, E0ϕ = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Our Poisson-like extension operator E is then defined for appropriate functions (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) η : Σ → R through the assignment Eη(x, y) = y · b−1Π1 Lη(x) + E0Π1 Hη(x, y) for (x, y) ∈ Rn−1 × (0, b), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) where the Fourier projectors are given by Π1 Lη = F −1[1B(0,1)F[η]] and Π1 Hη = η − Π1 Lη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our flattening map is then built from E as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For η : Σ → R satisfying η > −b and belonging to an appropriate function space (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with s > 1 + d/2 and d = n − 1) we define Fη : Ω → Rn via Fη(x, y) = (x, y + Eη(x, y)) for (x, y) ∈ Ω = Rn−1 × (0, b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) We associate to Fη two crucial quantities: the Jacobian Jη : Ω → R+ and (when Jη is nowhere vanishing) the geometry matrix Aη : Ω → Rn×n, which are defined via Jη = det(∇Fη) = 1 + ∂nEη = ∂n(Fη · en) and Aη = (∇Fη))−t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Then, provided that Jη > 0 and Jη, 1/Jη ∈ L∞(Ω), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Fη(Ω) = Ω[η] and Fη is a bi-Lipschitz homeomorphism from Ω to Ω[η] such that its restriction to Ω defines a smooth diffeomorphism to Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, Fη(Σ) = Σ[η] and Fη is the identity on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For our penultimate change of equations, we set h = h ◦ Fη and u = JηAt ηv ◦ Fη to be our new unknowns defined in the fixed domain Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) then transform to the following equivalent system for unknowns (h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η) with data (T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F): � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � � ∇ · (H−1(h + H ◦ ϱ(Fη · en))(u − JηAt ηe1)) = 0 in Ω H−1(h + H ◦ ϱ(Fη · en))(γ2(A−t η u/Jη − e1) · Aη∇(A−t η u/Jη) + Aη∇h) −γ(Aη∇) · SH−1(h+H◦ϱ(Fη·en)) Aη (A−t η u/Jη) = H−1(h + H ◦ ϱ(Fη · en))G ◦ Fη +F ◦ Fη in Ω −((P − Pext) ◦ H−1(h + H ◦ ϱ(Fη · en)) − γSH−1(h+H◦ϱ(Fη·en)) Aη (A−t η u/Jη))Nη −ςH (η)Nη = T ◦ FηNη on Σ u · en + ∂1η = 0 on Σ u = 0 on Σ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) where for τ = H−1(h + H ◦ ϱ(Fη · en)), M = Aη, and w = A−t η u/Jη we have used the notation Sτ Mw = µ(τ) � ∇wMt + M∇wt − 2 n((M∇) · w)I � + λ(τ)((M∇) · w)I, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and we also denote Nη = (−∇∥η, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that in obtaining (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) we did not only flatten the velocity vector, we also multiplied by the matrix JηAt η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This has the following important consequences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we are able to maintain the continuity equation in ‘perfect divergence’ form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, the kinematic boundary condition transforms to a linear equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These properties are indispensable in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have one more change of unknowns left to make before we reach the desired formulation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The issue is that the formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) will not quite let us ensure that h belongs to COMPRESSIBLE TRAVELING WAVES 11 a standard Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rather than develop more nonstandard Sobolev theory, we circumvent this issue with a final change of unknowns and equations by defining q = h − gη, and multiplying the momentum equation by A−1 η , which leads to considerable simplifications in our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We are then left with the following final equivalent form of our equations for the unknowns (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η) with data (T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F): � � � � � � � � � � � � � � � � � � � ∇ · (σq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='η(u − Mηe1)) = 0 in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' γ2σq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='ηM−t η (((u − Mηe1) · ∇)(M−1 η u)) + σq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='η∇(q + gη) −γM−t η (∇ · (Sσq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='η Aη (M−1 η u)Mt η)) = Jησq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='ηM−t η G ◦ Fη + JηM−t η F ◦ Fη in Ω,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' −((P − Pext) ◦ σq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='η − γSσq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='η Aη (M−1 η u))Mt ηen − ςH (η)Mt ηen = T ◦ FηMt ηen on Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u · en + ∂1η = 0 on Σ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u = 0 on Σ0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) where we have set Mη = JηAt η = �(1 + ∂nEη)I(n−1)×(n−1) 0(n−1)×1 −E(∇∥η) 1 � : Ω → Rn×n (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) and σq,η = H−1(q + gη + H ◦ ϱ(Fη · en)) = H−1(−gidRn · en + q + g(I − E)η) : Ω → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) As we discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, the original system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) admits a stratified equilibrium solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This solution is still encoded in the new formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) in the sense that for any γ ∈ R+, (q, u, η) = (0, 0, 0) is a solution to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) when there are no additional stress or forces present, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' when (T , G, F) = (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Furthermore, we show in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 that these trivial solutions are unique among triples (q, u, η) satisfying � ∥q∥2 H4+⌊n/2⌋ + ∥u∥2 H5+⌊n/2⌋ + ∥η∥2 H11/2+⌊n/2⌋ < ρ, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) where ρ is a constant depending only on the equilibrium depth b, gravity g, the pressure law P, the viscosity coefficients µ and λ, and the dimension n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize, though, that this result is recorded in this form for simplicity but could be improved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This reflects the fact that we only expect traveling wave solutions to exist for viscous fluids if they are generated by stress and force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) also obey a balance of power and dissipation analogous to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this is recorded in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Statement of main result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now state our main results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To do so, we first need to introduce a bit of notation to describe the functional framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In our work, as in the previous work of Leoni and Tice [60], Stevenson and Tice [97], and Koganemaru and Tice [57] on traveling wave solutions to the incompressible analog of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), and Nguyen and Tice [82] on traveling wave solutions to the one-phase Muskat problem, the traveling wave structure forces the free surface functions to belong to a scale of nonstandard anisotropic Sobolev spaces, the properties of which end up playing a crucial role in the analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, in order to properly state our main theorem, we first introduce these spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For R ∋ s ⩾ 0 and d ∈ N+ we define the anisotropic Sobolev space Hs(Rd) = {f ∈ S ∗(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) : F[f] ∈ L1 loc(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' C), ∥f∥Hs < ∞}, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) equipped with the norm ∥f∥Hs = � � Rd � |ξ|−2(ξ2 1 + |ξ|4)1B(0,1)(ξ) + ⟨ξ⟩2s1Rd\\B(0,1)(ξ) � |F[f](ξ)|2 dξ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) We refer to Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 for more information on these function spaces, but for the purposes of stating our main theorem, we note here that Hs(Rd) is a Hilbert space and Hs(Rd) �→ Hs(Rd) �→ Hs(Rd) + C∞ 0 (Rd), with equality in the first embedding if and only if d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 12 NOAH STEVENSON AND IAN TICE For the following statement of the main theorem, we set r = 10 + 2⌊n/2⌋ and for s ∈ N we define the sets Us = H1+s+r(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n) × Hs+r(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Hs+r(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × R+, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) Vs = H1+s+r(Ω) × H2+s+r(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H5/2+s+r(Rn−1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) V 0 = {(q, u, η) ∈ V0 : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) We can now state our main theorem, which establishes the well-posedness of the traveling wave formulation of free boundary compressible Navier-Stokes equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 1 (Proved in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that the parameters µ, λ, and ς satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and that P satisfies P ′ > 0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exist a collection {V (γ)}γ∈R+ of open subsets of V 0 and a nonincreasing sequence {Us}∞ s=0 of open subsets of U0 such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Nondegeneracy: We have that {0} × R+ ⊆ �∞ s=0 Us and 0 ∈ � γ∈R+ V (γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Existence and uniqueness: For all tuples (T , G, F, γ) ∈ U0 of applied stress, specific bulk force, bulk force, and wave speed, there exists a unique solution (q, u, η) ∈ V (γ) such that the traveling wave reformulation for the free boundary compressible Navier-Stokes equations, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), is classically satisfied with data (T , G, F) and wave speed γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Regularity, given low norm smallness: If s ∈ N+ and (T , G, F, γ) ∈ Us ∩ Us, then the corresponding solution satisfies (q, u, η) ∈ V (γ) ∩ Vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) Continuous dependence: For any s ∈ N, the solution map Us ∩ Us ∋ (T , G, F, γ) �→ (q, u, η) ∈ Vs ∩ V 0 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) is continuous with respect to the Us and Vs ∩ V 0 topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5) No vacuum formation: There exists positive constants c, C ∈ R+ such that for all (q, u, η) ∈ � γ∈R+ V (γ) we have that c ⩽ σq,η ⩽ C, where σq,η is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6) Flattening map diffeomorphism: For any s ∈ N and (q, u, η) ∈ Vs ∩ � γ∈R+ V (γ), we have that the flattening map Fη from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is a smooth diffeomorphism from Ω to Ω[η] that extends to a C12+⌊n/2⌋+s diffeomorphism from Ω to Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Before enumerating some corollaries, we pause for a few comments and remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we emphasize that a high-level summary of our theorem is that traveling waves for the free boundary compressible Navier-Stokes system are generic: they exist for all nontrivial wave speeds γ ∈ R+, and for a fixed regularity index s ∈ N the set of stress-force-speed data, Us ∩ Us, is open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In particular, for each fixed wave speed γ ∈ R+ and s ∈ N, the set of stress and force data for which we can solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is an open set containing the origin, so the existence of small-data solutions is a generic phenomenon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The utility of working in Us ∩ Us is that it allows us to prove the joint continuity of our solutions with respect to both the stress-forcing data (T , G, F) and the wave speed γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, we can say a bit more: we show in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 that the solution map (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) enjoys a certain form of continuous differentiability, and even higher regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Stating this precisely entails carefully dealing with more issues related to derivative loss, so we have skipped this technical point in the statement of Theorem 1 for the sake of brevity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, it is worth highlighting that our theorem does slightly different things when n = 2 as compared to when n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the case n ⩾ 3, the hypothesis (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) requires that the coefficient of surface tension is positive, ς > 0, and our analysis crucially uses this and the ellipticity of the mean curvature operator to gain regularity for η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When n = 2 we only require ς ⩾ 0, as in this case there is another mechanism built into the equations for regularity gain, namely the equation ∂1η = −u · en on Σ, which is elliptic when n = 2 since then ∂1 is elliptic as a differential operator on Σ ≃ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When n ⩾ 3 we have to contend with the free surface function η belonging to the anisotropic spaces H5/2+s(Rn−1), which are strictly larger than H5/2+s(Rn−1), but when n = 2 we have that H5/2+s(R) = H5/2+s(R), and so our solutions live in standard Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Interestingly, when n = 2 there is one condition that must be stronger than when n ⩾ 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' indeed, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) requires positive COMPRESSIBLE TRAVELING WAVES 13 Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A depiction of three traveling wave free surfaces generated by the same applied stress tensor T = −ϕI3×3, where ϕ ⩾ 0 is compactly supported, with increasing wave speed γ moving from left to right, indicated by the red arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' bulk viscosity, λ > 0, when n = 2 and only λ ⩾ 0 when n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is ultimately related to technical issues with the deviatoric Korn inequality;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' we refer to Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 for further details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In spite of the above dimensional differences in the precise functional setting, the space V 0 ∩ Vs from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) always satisfies the embedding (thanks to Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and standard Sobolev embeddings) V 0 ∩ Vs �→ Cs+k 0 (Ω) × Cs+k+1 0 (Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Cs+k+2 0 (Rn−1) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) for k = 10 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since our theorem guarantees the inclusion (q, u, η) ∈ V0 ∩ Vs, we see that our solutions always decay to zero at infinity, which means that our solutions are what are known as solitary waves in the parlance of the traveling wave literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' At the level of generality of our well-posedness theory, there is not much more qualitative information that can be deduced about our solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, our result opens the door to more detailed qualitative studies given specific forms of the stress-forcing data tuple (T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We also point out that our techniques only allow us to construct traveling wave solutions (γ ∈ R+) and not stationary (γ = 0) solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The strict sign condition on γ plays a key role in our analysis by, in part, allowing us to use a nondegenerate norm on the collection of free surface functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our selected functional framework simply does not work with γ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We face a similar issue with the gravitational constant in our analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' indeed, we can only construct solutions in the case that g > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn our attention to two corollaries of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first formalizes the above discussion about the open set of stress-force data for a given wave speed γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Corollary 2 (Proved in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With r = 10 + 2⌊n/2⌋ as before, for each γ ∈ R+ there exists a nonempty open set (0, 0, 0) ∈ W (γ) ⊂ H1+r(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n) × Hr(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Hr(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) with the property that for all stress-force data tuples (T , G, F) ∈ W (γ) there exists a unique (q, u, η) ∈ V (γ) (where the latter open set is from the statement of Theorem 1) such that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is satisfied with solution (q, u, η), wave speed γ, and data (T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Although the natural formulation for the traveling wave problem from the perspective of well- posedness is in a flattened domain as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), we can also switch our solutions back to the Eulerian formulation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We record this in our second corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Corollary 3 (Proved in Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Each solution to the flattened perturbative enthalpy formula- tion for the traveling wave problem for free boundary compressible Navier-Stokes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), produced by Theorem 1 gives rise to a classical solution to the traveling Eulerian formulation of the problem given by system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 14 NOAH STEVENSON AND IAN TICE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Summary of strategy and layout of paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we aim to summarize the principal difficulties in proving Theorem 1 and our strategies for overcoming them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This will also serve to outline the structure of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' High level summary of difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is posed in an unbounded domain with infinite measure and non-compact boundary, the equations are quasilinear, and there is no variational structure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' as such, compactness, Fredholm, and variational techniques are unavailable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This, along with our expectation of a robust linear theory, suggests that the construction of solutions should proceed through perturbative techniques such as the implicit function theorem, or more fundamentally, an iteration scheme based on some sort of linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, an implicit function theorem strategy, based on the linearization of the equations around vanishing stress-force data and trivial solution triple for a fixed arbitrary wave speed γ ∈ R+, proved successful in recent work [57, 60, 97] on the incompressible version of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), so it is enlightening to begin our discussion by stating the corresponding linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9): � � � � � � � � � � � � � � � ∇ · (ϱu) + g−1ϱ′∂1(q + gη) = g in Ω, −γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f in Ω, −(ϱq − γSϱu)en − ς∆∥ηen = k on Σ, u · en + ∂1η = 0 on Σ, u = 0 on Σ0, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where the linearized unknowns are still labeled (q, u, η) but the linearized data is now the triple (g, f, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The reader familiar with the elliptic structure of the incompressible Stokes problem will recognize a fundamental difficulty appearing already in the first two equations of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1): even if we ignore η or view it as given, these two equations do not constitute an elliptic system for (q, u) in the sense of Agmon, Douglis, and Nirenberg [3], due to the appearance of ∂1q in the first equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Without this elliptic structure to serve as a base for the analysis, it is not obvious that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) will give rise to an isomorphism between Banach spaces, a necessary ingredient for the perturbation strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Remarkably, in spite of this ellipticity failure, we are able to show in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 that the forward linear map (q, u, η) �→ (g, f, k) defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) actually does induce an isomorphism between the Sobolev-type Hilbert spaces 0,0,0 Xs and Ys for s ∈ N, as defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unpacking the details of the space 0,0,0 Xs reveals the fundamental difficulties lurking in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and motivates our overall strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A cursory glance at the spaces 0,0,0 Xs and Ys shows that the regularity count essentially matches that of the incompressible problem if we formally identify q with the pressure in the incompressible problem: in the domain space, if q gets 1 + s derivatives, then u gets 2 + s, and η gets 5/2 + s, while in the codomain g gets 1 + s derivatives, f gets s, and k gets 1/2 + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, a closer inspection reveals two crucial complications with these spaces and this counting scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first, which was already present in the analysis of the incompressible problem, is that the structure of the operators hitting η in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) only allows for the recovery of estimates of ∆∥η ∈ H1/2+s(Rn−1), ∇∥η ∈ Hs(Rn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn−1), and ∂1η ∈ ˙H−1(Rn−1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where ˙H−1(Rn−1) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), which is not enough to guarantee the inclusion η ∈ H5/2+s(Rn−1) for general n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Instead, as we prove in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, these inclusions essentially characterize the anisotropic inclusion η ∈ H5/2+s(Rn−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The takeaway is that the anisotropic spaces are inextricably linked to the traveling wave problem through the structure of the differential operators (and also through the positivity g > 0 and γ > 0, which give us the latter two estimates of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second complication is more severe and new to the compressible problem: knowing COMPRESSIBLE TRAVELING WAVES 15 that q ∈ H1+s(Ω), u ∈ H2+s(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), and η ∈ H5/2+s(Rn−1) alone is not enough to guarantee that g ∈ H1+s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With this count, the continuity equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) only yields g ∈ Hs(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To achieve the higher regularity inclusion g ∈ H1+s(Ω) we have to build the extra condition that ∂1q ∈ H1+s(Ω) into the domain space 0,0,0 Xs , and conversely, with g ∈ H1+s(Ω) we are able to recover that ∂1q ∈ H1+s(Ω) through the linearized continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is what the 0, 0, 0 adornment actually indicates for the space 0,0,0 Xs : the q elements in this space enjoy some ‘bonus partial regularity’ whose precise form is dictated by the structure of the linearized operator around the trivial triple (0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The bonus partial regularity can also be viewed as another manifestation of anisotropy in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fact that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) induces an isomorphism 0,0,0 Xs ∋ (q, u, η) �→ (g, f, k) ∈ Ys is certainly encouraging, but in reality it exposes a much deeper complication with the nonlinear problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, if we attempt to formulate (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) as a nonlinear mapping problem on a space in which (q, u, η) ∈ 0,0,0 Xs , then we immediately see that the bonus partial regularity is lost by the nonlinearity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In more concrete terms: any attempt to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) through an iteration scheme based on the isomorphism from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) will suffer from derivative loss, rendering the scheme useless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This isomorphism issue is actually more generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We also prove in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 that for any appropriately small triple (q0, u0, η0), the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) around (q0, u0, η0) for (T , G, F) = 0 and γ ∈ R+ induces an isomorphism q0,u0,η0 Xs ∋ (q, u, η) �→ (g, f, k) ∈ Ys, where now the ‘adapted space’ q0,u0,η0 Xs encodes the bonus partial regularity vq0,u0,η0 · ∇q ∈ H1+s(Ω) for a vector field vq0,u0,η0 that is determined by the linearization location (q0, u0, η0) (the field is collinear with e1 at (0, 0, 0)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once more, this can be viewed as a sort of anisotropy, but now it is clear that the favored direction depends on the background triple (q0, u0, η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These general adapted spaces face the same problem described above: the bonus regularity is lost by the nonlinearity, leading to derivative loss in any iteration scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The failure of the nonlinearity to preserve the adapted spaces can ultimately be traced to the fact that the bonus regularity is not perturbative: we cannot use control of vq0,u0,η0 ·∇q to say anything useful about vq1,u1,η1 · ∇q for general distinct triples (qi, ui, ηi) with i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, in general the adapted spaces q0,u0,η0 Xs and q1,u1,η1 Xs are inequivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' While these issues preclude the use of elementary perturbation techniques, they also reveal the potential utility of more sophisticated Nash-Moser techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, for any appropriate triple (q0, u0, η0), we have the natural inclusion Xs+1 �→ q0,u0,η0 Xs , where the former space is defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) and encodes no location-specific bonus regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This suggests that we may pose the nonlinear mapping from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) on a scale of spaces, indexed by s, in which the codomain involves Ys but, to compensate for the derivative loss, the domain scale is shifted and requires (q, u, η) ∈ Xs+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The above isomorphism results then suggest that the maps Ys ∋ (g, f, k) �→ (q, u, η) ∈ q0,u0,η0 Xs �→ Xs will allow us to construct the right and left inverse to the derivative of the nonlinear map, provided we expand our view to scales of Banach spaces and accept the reality of derivative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is precisely the purview of the Nash-Moser technique [81, 80], which we have thus chosen as the engine to prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Nash-Moser framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our goal in studying (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is not just to show the existence and uniqueness of solutions, but to establish a proper well-posedness theory that shows the solutions depend continuously on the data in the optimal topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To the best of our knowledge, the only Nash-Moser inverse function theorems in the literature with this capability are the formulations of Sergeraert [93] and Hamilton [40], which actually yield smoothness of the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The Sergeraert and Hamilton Nash-Moser theorems work in the context of smooth tame maps between Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Roughly speaking, one can think of tameness as a family of structured estimates 16 NOAH STEVENSON AND IAN TICE associated to the derivatives of the maps;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' these bounds play an essential role in the use of a modification of Newton’s method to overcome the derivative loss in proving surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The Fr´echet spaces that serve as the domain and codomain of the nonlinear operator in these theorems can be thought of as the intersection of all of the spaces in a Banach scale (like H∞(Rn) = � k∈N Hk(Rn) vis-`a-vis the Banach scale {Hk(Rn)}k∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The hypotheses of the Sergeraert and Hamilton Nash- Moser variants require, among other things, a family of right inverses to the derivative mapping into the domain Fr´echet space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unfortunately, for reasons that are ultimately attributable to the failure of hypoellipticity for the hyperbolic structure appearing in the traveling wave formulation of the continuity equation, in our context we can only verify that in a given open neighborhood of the trivial solution, the derivatives’ inverses only map into finite regularity spaces in the Banach scale, and so we cannot satisfy the basic hypotheses of Sergeraert’s or Hamilton’s formulation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There are Nash-Moser theorems in the literature that allow for this finite invertibility range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The oldest we are aware of is found in the work of Schwartz [90, 91], but this is formulated only for a very specific Banach scale, produces solutions in a suboptimal space, and has no mechanism for regularity promotion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The version due to H¨ormander [42, 43, 44] works for general Banach scales and produces solutions in the optimal or nearly-optimal space in an associated ‘weak Banach scale,’ which in some cases coincides with the original (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H¨older scales) but in general is slightly larger (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' if the original is the Sobolev scale {Hk(Rn)}k∈N, then the weak scale is the larger Besov scale {Bk 2,∞(Rn)}k∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This result also has no regularity promotion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The recent Nash-Moser theorem of Baldi and Haus [5] produces solutions in the optimal space and also has a regularity promotion mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Famously, the Nash-Moser technique overcomes the problem of derivative loss by employing a family of smoothing operators that satisfy a host of precise quantitative estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Baldi and Haus achieve their significant improvement by placing more strenuous conditions on these smoothing operators than in the other Nash-Moser formulations, which allows them to port techniques from Littlewood-Paley theory and the paradifferential calculus into the abstract setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the context of the function spaces we employ, it is relatively easy to construct smoothing operators that satisfy the hypotheses of, say H¨ormander’s Nash-Moser theorem [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, in our context it is rather delicate to show that these operators satisfy the stronger hypotheses of Baldi and Haus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rather than focus effort on this construction, which would mostly apply to our specific problem, we have chosen to use a more general, abstract approach, which is a Banach scale generalization of the notion of a tame Fr´echet space as defined in Hamilton [40], and which we believe may be of use in other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Roughly speaking, the idea is to view a given scale of Banach spaces, in which it is hard or impossible to construct the smoothing operators, as being a retract of another Banach scale in which the smoothing operators are known to exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the setting of Banach spaces, we can think of the retract property in terms of the former spaces being direct summands, or complemented subspaces, of the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that this idea does not pull the smoothing operators back to the initial scale, but rather pushes the nonlinear map forward to the larger scale, and so in some sense our method shifts the focus from constructing smoothing operators to identifying the direct summand structure, which is more amenable to PDE techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In light of the above discussion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' we have opted to craft another version of the abstract Nash-Moser inverse function theorem,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' synthesizing the desired elements of the Sergeraert,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hamilton,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' and Baldi and Haus formulations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' that is capable of working within our finite range of invertibility context,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' produces solutions in the optimal space,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' provides a regularity promotion mechanism,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' and provides some degree of regularity for the inverse map,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' in particular a continuity assertion in an optimally strong norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our new version employs the direct summand method to sidestep the smoothing operator construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This new Nash-Moser formulation, the precise statement of which is given in Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24, is essential for our proof of Theorem 1, but we believe it is likely to be of broader interest and applicability due to the flexibility of its hypotheses and improved conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We refer to Section 2 for further exposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 17 With our Nash-Moser variant in hand, our strategy for proving Theorem 1 is simple to state: encode the conclusions of the theorem as properties of a nonlinear map associated the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) that can be granted by our Nash-Moser inverse function theorem, and then verify the hypotheses of Nash-Moser.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These hypotheses, which are stated precisely in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, are divided into two categories: nonlinear and linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the former, we need to verify that the nonlinear operator satisfies certain differentiability conditions and that the derivatives obey certain tame estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' At the linear level we need to study the linearization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) around a generic triple (q0, u0, η0) in an open neighborhood of zero, and in particular we need to construct the family of left and right inverses to these derivatives and verify they also obey a set of tame estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The diagram in Figure 3 represents the logical flow of dependencies for our strategy as it is implemented in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The gray boxes correspond to the abstract nonlinear analysis in Section 2, which culminates with the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The green boxes correspond to the main features of our nonlinear analysis, which are found in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The blue boxes show our linear analysis strategy, which is then executed in Sections 4, 5, and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, at the bottom, we have the red box representing our conclusion and final proof of Theorem 1, appearing in Section 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Well posed- ness, Section 7 Existence for regularized principal part, Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 Linear analysis syn- thesis, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 Refined Nash-Moser inverse function theorem, Section 2 Smooth tameness verification for nonlinear operator, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 Tame structure abstraction, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 Derivative loss vector field, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 Analysis of atomic nonlinearities, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 Spitting of the derivative, Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 Analysis of (regularized) steady transport equations, Section 4 Estimates on principal part strong solutions, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 Estimates for regularized principal part strong solutions, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 Estimates on regularized principal part weak solutions, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 Estimates on principal part weak solutions, Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Logical flow of the paper Nonlinear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our abstract nonlinear analysis begins in Section 2 with the study of the tame structure of differentiable maps between scales of Banach spaces, which form the basic 18 NOAH STEVENSON AND IAN TICE framework of our Nash-Moser theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In contrast to the Sergeraert [93] and Hamilton [40] frameworks, we study tameness in a finite regularity context and on possibly finite Banach scales rather than Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We develop a calculus of such maps, showing closure under various operations such as sums, products, and compositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These results turn out to be essential in our subsequent verification of the ‘nonlinear hypotheses’ of our Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is done, we then formulate and prove our version of the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 we precisely define the nonlinear operator associated to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), namely Ψ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), and formulate the Banach scales that serve as its domain and codomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We verify, in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, that the domain Banach scale is tame and that the codomain Banach scale is a tame direct summand of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we endeavor to check that Ψ is tamely twice continuously differentiable and has order one derivative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Within a standard Sobolev framework this could be accomplished, more or less, with off-the-shelf tools.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unfortunately, in our context the free surface functions η belong to the anisotropic spaces H5/2+s(Rn−1), and their ubiquity in the nonlinearities of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) then requires a much more delicate and customized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To make this task less arduous, we first identify within Ψ a number of simpler ‘atomic’ nonlinearities that can be analyzed separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the most part, these atoms are handled via elementary high-low estimates from Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, combined with some results from the abstraction of tame structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that our choice of the enthalpy formulation makes the atoms that comprise the momentum equation all of this form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By contrast, the nonlinearity arising from the continuity equation in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), namely (q, u, η) �→ Ξ(q, u, η) = ∇ · (σq,η(u − Mηe1)), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) is significantly more delicate and requires more sophisticated ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The principal difficulty here is that our functional setting requires that � b 0 Ξ(q, u, η)(·, y) dy ∈ ˙H−1(Rn−1), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) where ˙H−1(Rn−1) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' At first glance, this inclusion appears to follow readily from the identity � b 0 Ξ(q, u, η)(·, y) dy = (∇∥, 0) · � b 0 (σq,η(u − Mηe1) + ϱe1)(·, y) dy, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) but a closer inspection reveals that, because of the anisotropic spaces, the integral argument of the divergence on the right does not belong to L2(Rn−1) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To get around this problem we employ the ‘Taylor expansion trick’ of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 in conjunction with the subtle vector field decomposition from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These rely crucially on various nontrivial algebraic properties of the anisotropic spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then synthesize the analysis of the atomic nonlinearities with our analysis of tame structure to complete the verification of most of the ‘nonlinear hypotheses’ of the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is done in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Principal part identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next pass through some applications of our nonlinear results to set up the linear analysis in a simpler form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We identify the manifestation of the derivative loss at the linear level as the vector field vq0,u0,η0 in the linearized continuity equation, as described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The relevant properties of this derivative loss vector field are enumerated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This understanding allows us to perform a ‘derivative splitting’ for the nonlinear operator Ψ of the form DΨ = DΨprin + DΨrem, where the linear operator DΨprin is the ‘principal part’ and DΨrem is the perturbative remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Associated to the principal part operator DΨprin is the following principal COMPRESSIBLE TRAVELING WAVES 19 part PDE system for linearized unknowns (q, u, η) with given data (g, f, k) and wave speed γ ∈ R+: � � � � � � � � � � � � � � � ∇ · (ϱu) + ∇ · (vq0,u0,η0(q + gη)) = g in Ω, −γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f in Ω, −(ϱq − γSϱu)en − ς∆∥ηen = k on Σ, u · en + ∂1η = 0 on Σ, u = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) This splitting provides two key benefits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, the problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) is as close as possible to the linearization around the trivial background, namely (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), while retaining the entirety of the derivative loss information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, the remainder piece of the linearization has no derivative loss and is effectively small so that it can be handled perturbatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We refer to Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22 for the precise details of this derivative splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a result of this careful splitting, we effectively reduce most of our linear analysis to the study of this family of principal part linear equations, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we discuss the strategy for the verification of the ‘linear hypotheses’ for our Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The majority of the work is devoted to studying the principal part system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) with vq0,u0,η0 obtained from a general background triple (q0, u0, η0) in an open neighborhood of zero, and in particular showing that it induces the aforementioned isomorphism q0,u0,η0 Xs ∋ (q, u, η) �→ (g, f, k) ∈ Ys and obeys related tame estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fundamental difficulty here, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), is the lack of ellipticity in the base Stokes system for q and u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' indeed, one should view (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) as an unhappy marriage of elliptic (Lam´e and mean-curvature type) and hyperbolic (steady transport type) operators whose individual regularity theories are incompatible and appear not to combine without substantial difficulty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To deal with the difficulties inherent in the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), it is convenient to initially decouple the problems of estimates and existence and only combine them at the last moment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The key to this strategy is the introduction of a regularizing term in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) that makes the elliptic parts interface better with the hyperbolic steady transport part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unfortunately, the natural technique of applying a smoothing operator to the steady transport term in the continuity equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) does not work well in Ω: operators that preserve the good elliptic energy structure seemingly lack good commutators and so fail to give high regularity estimates, while operators that have good commutators do not seem to respect the energy structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We are thus led to employ an elliptic regularization by replacing the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) with � � � � � � � � � � � � � � � � � � � ∇ · (ϱu) + τ∇ · (vq0,u0,η0(q + gη)) + N−1Lm(q + gη) = g in Ω, −γ2ϱ∂1u + ϱ∇(q + gη) − γ∇ · Sϱu = f in Ω, −(ϱq − γSϱu)en − ς∆∥ηen = k on Σ, u · en + ∂1η = N−1(−∆∥)m−1/4η on Σ, u = 0 on Σ0, ∂m n q = · · · = ∂2m−1 n q = 0 on ∂Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) where the regularization parameter is N−1 for N ∈ N+, Lm is the 2mth-order linear elliptic differential operator Lm = (−1)m n � j=1 ∂2m j , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) (−∆∥)m−1/4 is a standard fractional power of the Laplace operator on Σ ≃ Rn−1, m ∈ N+ is a tunable regularity parameter, and τ ∈ [0, 1] is an operator homotopy parameter (τ = 1 corresponds to a regularization of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that the Neumann boundary conditions for q recorded in the 20 NOAH STEVENSON AND IAN TICE final equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) are new relative to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' their presence is dictated by our introduction of Lm, but the specific choice of the Neumann conditions plays a crucial role later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The benefits of the elliptic regularization (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) are manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, the domain space for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) is Xs m,N, as defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Crucially, this space is independent of the background triple (q0, u0, η0) but continuously embeds into the background-dependent space q0,u0,η0 Xs .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This makes it an ideal setting for using arguments based on the method of continuity to extend the solvability theory from τ = 0 to τ = 1, which is the form of the problem we actually care about.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, the regularization operators preserve the energy structure of the original problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) while giving a relatively simple regularity gain mechanism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Third, and perhaps most important, they are compatible with the derivation of N−independent estimates at high-regularity, which we will employ to solve (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) in q0,u0,η0 Xs via weak compactness arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that in doing so we will always need m ⩾ 1 + s so that the artificial Neumann conditions for q in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) do not pass to the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Of course, regularization does not come without its downsides, and a good amount of work is needed to deal with technical complications it introduces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn to a somewhat more detailed account of how we use (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) to complete the linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We study a priori estimates for the principal part and its regularization in tandem;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' for the former the goal is to develop the desired tame estimates, but for the latter the goal is high regularity N−independent estimates with only a weaker form of tameness with respect to the background tuple (q0, u0, η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The starting point for the principal part is estimates for weak solutions, which are proved in the usual manner modulo some minor technical complications due to some nonlinear expressions involving members of anisotropic Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the weak estimates in hand, we then turn our attention to high regularity estimates for strong solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The idea here is to exploit the fact that we can apply tangential derivatives to the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) without changing the basic structure of the equations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this allows us to employ the weak solution estimates to get bounds on these tangential derivatives as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' After we achieve this control of the tangential derivatives, it remains to recover control of the normal, or vertical derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Here we implement a version of the classic technique of Matsumura and Nishida [75], originally developed for fixed domains with Dirichlet boundary conditions, to reveal a subtle dissipative structure for the normal derivatives of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This, together with some supplementary analysis of steady transport equations (see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), provides an estimate on the high norms of the solution in terms of the data and the tangentially differentiated solution alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is recorded in the first conclusion of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Synthesizing, we then derive the first conclusion of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16, which is the desired a priori estimates for the principal part equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The strategy for the a priori estimates of the regularized problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) is similar at a bird’s eye view, although the specifics of the argument are rather distinct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We again start by proving a priori estimates for weak solutions and then study the equations satisfied by the tangentially differentiated solution: see Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and the second conclusion of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The argument used to derive estimates for the normal system, the second conclusion of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15, now substantially deviates from that used for the principal part case because of the need to pass through analysis of the regularized steady transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It is here that the regularization Lm serves as a liability rather than an asset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, deriving estimates with good dependence on N is technically delicate and requires a very careful use of the bilinear form associated to Lm and the precise Neumann boundary conditions imposed on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The majority of Section 4 is devoted to this regularized transport analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the end, we obtain the uniform in N estimates as stated in the second conclusion of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The existence theory for the regularized problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) for all τ ∈ [0, 1] is developed by first establishing the existence of weak solutions in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is proved through the method of continuity, based on the a priori estimates from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and an existence result for the problem with τ = 0, which itself requires a further two-parameter regularization and compactness COMPRESSIBLE TRAVELING WAVES 21 argument to establish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The weak solutions are then promoted to strong solutions in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 via elliptic regularity arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that while this result shows that (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) induces an isomorphism, it does not establish N−uniform estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The existence theory for the principal part problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) is then established by way of the regularized existence theory, the N−uniform a priori estimates, and another weak limiting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is our Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the conclusion of the linear analysis, we combine the work that allowed us to identify the principal part equations, namely the splitting of the derivative, with the previously discussed principal part analysis on estimates and existence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is the synthesis of linear analysis result of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We conclude with a couple remarks on our linear strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we emphasize that our existence strategy for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) intentionally bypasses direct regularity promotion of weak solutions in favor of high regularity inherited by weak limits of solutions to the regularized equations (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is advantageous since the mixed elliptic-hyperbolic nature of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) creates substantial technical difficulties in attempting to implement the standard techniques for regularity promotion (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' finite differences or mollification and commutators).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second feature we wish to highlight is that our methods of handling the free surface unknown in the linear existence theory are significantly different and simpler than in the previous work on the incompressible problem [60, 97, 57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In these works the free surface unknown is constructed in terms of the data alone via a pseudodifferential equation, the symbol for which is inverted after a careful asymptotic analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In our work, we entirely circumvent the use of these delicate pseudodifferential techniques in our existence theory, replacing them with regularizations, a priori estimates, weak limits, and a new spatial characterization of the anisotropic Sobolev spaces Hs(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Conclusion and appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once we have completed the nonlinear and linear analysis, the hypotheses of our Nash-Moser theorem are verified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then invoke the result with a few minor additional arguments in Theorems 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 to complete the proof of well-posedness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The remainder of the paper consists of four appendices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These mostly consist of various analytical and PDE tools that are customized or optimized for our particular needs in the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, some of the results there appear to be new and may be of independent interest for use in other problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Appendix A records tools related to standard Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In contrast, Appendix B is concerned with properties of the nonstandard spaces we employ in this paper, namely the anisotropic spaces and the adapted spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Appendix C focuses on PDE tools, with an emphasis on the specified divergence problem in various contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Appendix D contains a selection of nonlinear analysis tools that are essential in our abstract tame analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Conventions of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We recall that we have included a Notation Index at the end of the paper, which catalogs the numerous operators, function spaces, and other symbols used throughout.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We write N for the set of nonnegative integers, N+ = N \\ {0}, and R+ = (0, ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Whenever α ≲ β appears in a result, it means there is a constant C ∈ R+ depending only on the parameters mentioned in the formulation of the result such that α ⩽ Cβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To emphasize this dependence, we will sometimes write α ≲a,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=',b β to indicate the parameters a, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will also write α ≍ β to mean α ≲ β and β ≲ α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will use the bracket notation ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , xp⟩ = (1 + x2 1 + · · · + x2 p)1/2 (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) for p ∈ N+ and x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , xp ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given x ∈ Rd, we will often abbreviate ⟨x⟩ = ⟨x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , xd⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If U and V are some open sets we write U ⋐ V to mean that the closure U is compact and U ⊂ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We denote the gradient and its tangential counterpart by ∇ = (∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ∂n) and ∇∥ = (∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ∂n−1), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The divergence and tangential divergence operators are written ∇ · f = �n j=1 ∂j(f · ej) and (∇∥, 0) · f = �n−1 k=1 ∂k(f · ek) for appropriate Rn-valued functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will also use subscript ‘∥’ to indicate that a differential operator depends only on ∂1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ∂n−1, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∆∥ = �n−1 ℓ=1 ∂2 ℓ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 22 NOAH STEVENSON AND IAN TICE If ∥·∥ is a norm on a product of normed spaces, X1 × · · · × Xq, we will typically write ∥x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , xq∥ in place of ∥(x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , xq)∥, where xi ∈ Xi for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , q}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given Λ ⊆ R, we say a decreasing collection of Banach spaces {Xs}s∈Λ with norms {∥·∥Xs}s∈Λ is log-convex if for all s0, s1, s ∈ Λ such that s0 < s1 and s = (1 − σ)s0 + σs1 for some σ ∈ [0, 1] we have that ∥x∥Xs ≲s0,s1,σ ∥x∥1−σ s0 ∥x∥σ s1 for all x ∈ Xs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Suppose that k ∈ N, U ⊆ Rd is open, and V is a finite dimensional real vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For p ∈ [1, ∞], we write W k,p(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) for the usual Lp-based Sobolev space of order k with functions valued in V , and we write Hk(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = W k,2(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We write Ck b (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = Ck(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) ∩ W k,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ), endowed with the obvious norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The space Ck 0 (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) is the closure of C∞ c (Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) in Ck b (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We also write C∞ b (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = � k∈N Ck b (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) and C∞ 0 (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = � k∈N Ck 0 (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Similarly, we write H∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = � k∈N Hk(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) and W ∞,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = � k∈N W k,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that W ∞,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = C∞ b (U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For S ∈ {Σ0, Σ}, we write TrS for the trace operator that maps appropriate functions defined on Ω to functions on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following closed subspace of H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) is frequently used: 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) = {u ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) : TrΣ0(u) = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) The Fourier transform, which we normalize to be unitary on L2, is denoted by F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will utilize the homogeneous Sobolev space of order −1, which is defined as ˙H−1(Rd) = {f ∈ S ∗(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) : F[f] ∈ L1 loc(Rd \\ {0};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' C) and [f] ˙H−1 < ∞} (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) for the seminorm [f] ˙H−1 = ∥1Rd\\{0}| · |−1F[f]∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we emphasize that we frequently identify Σ and Rn−1 in the canonical way when working with function spaces defined on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A variation on the Nash-Moser inverse function theorem As we discussed in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, our strategy for proving the well-posedness of the system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is to employ a new version of the Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The goal of this section is to prove this theorem and develop its abstract framework, which will be employed generally in all of our subsequent analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Before we begin, we provide a brief overview of the Nash-Moser strategy and some variants of the theorem that have been developed in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The abstract setting of the classical inverse function theorem is C1 maps Ψ : U → F, where E and F are Banach spaces and U ⊆ E is an open set, for which DΨ(u) ∈ L(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) is invertible for some u ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The typical proof leverages the invertibility of DΨ(u) to prove the local bijectivity of Ψ by way of the contraction mapping principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In particular, surjectivity is established via a Picard iteration scheme that crucially employs the map DΨ(u)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The Nash-Moser approach aims to handle the case when DΨ(u) fails to be invertible but remains ‘nearly invertible’ in the sense that it admits a right inverse L(u), defined on F but only mapping into some larger vector space E0 ⊃ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This phenomenon is typically regarded as ‘derivative loss,’ as in practice E0 often consists of functions, and the subspace E consists of functions of higher regularity than those in E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To fully take advantage of this, the Nash-Moser strategy generalizes the setting of the standard inverse function theorem to maps between one-parameter scales of Banach spaces, say {Es}s∈S and {F s}s∈S for some S ⊆ R, where roughly speaking one should think of the parameter s as measuring the regularity of the elements of the space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The derivative loss is then required to be uniform along the scale in the sense that DΨ maps from Es+µ to F s, for some µ > 0 measuring the extent of the derivative loss, but its right inverse L (which is now required to exist in the entirety of an appropriate open set) only maps from F s to Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See Figure 4 for a diagram.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The profound idea of Nash [81], which was expanded upon by Moser [80], was to establish local surjectivity not via Picard iteration, which is not available due to the derivative loss, but instead with an iteration scheme, based on Newton’s method, that employs a family of smoothing operators that increase the regularity along the Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The extreme speed of convergence of Newton’s COMPRESSIBLE TRAVELING WAVES 23 method is needed to make the derivative loss and smoothing operators cooperate, and in order to properly implement this idea various precise quantitative estimates are required for the map and the smoothing operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The above approach is extremely flexible and customizable, and has thus become more of a strategy than a specific theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, many variants of the Nash-Moser theorem have appeared in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Most of these are really local surjectivity theorems rather than full inverse function theorems, which must establish injectivity as well as continuity or higher regularity of the induced local inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We are aware of a few exceptions in the literature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Sergeraert [93] and Hamilton [40] prove smoothness of the inverse map by working in the more restrictive context of smooth tame maps between Fr´echet spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Berti, Bolle, and Procesi [10] study an implicit function theorem with parameters, and they prove continuous differentiability of the local solution map only with respect to a finite dimensional space of parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Ekeland [30] and Ekeland and S´er´e [31] prove an inverse function theorem and deduce some suboptimal Lipschitz estimates of the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unfortunately, the well-posedness problem for (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is not amenable to any of the above results, so we have endeavored to develop a new Nash-Moser inverse function theorem that works well for our problem, produces solutions in optimal spaces, provides a regularity promotion mechanism, and provides an optimal continuity and even some higher regularity results for the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our approach is to synthesize ideas from Hamilton [40] with the iteration scheme of Baldi and Haus [5], which is an improvement on the results of H¨ormander [42, 43, 44] that employs ideas from Littlewood-Paley theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When compared to the standard inverse function theorem, the Nash-Moser variants have hypothe- ses that are much more involved, but are similar at a high level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To guarantee a local inverse for a nonlinear operator, one needs to check that: 1) the nonlinear operator is defined on tame scales of Banach spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 2) the operator is tamely C2, with a fixed derivative loss;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' and, 3) the derivative of the operator admits a tame family of inverses in an open neighborhood of a point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As all of these hypotheses involve the adjective ‘tame,’ we devote Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 to defining tame structures and developing a calculus of tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is done, we use Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 to state the precise hypotheses and conclusions of our version of the Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Afterward, in Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 the theorem is then proved in several parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the reader looking to take the inverse function theorem as a ‘black box’ and more readily proceed to the analysis of the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), we suggest restricting to the abstract tame structure and the statement of our Nash-Moser theorem, but initially skipping over Sections 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Tame structure abstraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we are first concerned with tame mappings between scales of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our presentation here primarily inspired by Hamilton [40] and Baldi and Haus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our initial concern is scales of Banach spaces, starting with some notation for indexing them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (Subsets of N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given N ∈ N ∪ {∞} we define �N� ⊆ N to be the set �N� = {n ∈ N : n ⩽ N}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we define scales of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let N ∈ N+ ∪ {∞} and let E = {Es}s∈�N� be a collection of Banach spaces over a common field, either R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) We say that E is a Banach scale if for each s ∈ �N − 1� we have the non-expansive inclusion Es+1 �→ Es, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥·∥Es ⩽ ∥·∥Es+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) If E is a Banach scale, then we define the scale’s terminal space to be EN = � s∈�N� Es and endow it with the Fr´echet topology induced by the collection of norms {∥·∥Es}s∈�N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that if N < ∞, then EN has the standard Banach topology from its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 24 NOAH STEVENSON AND IAN TICE (3) If E is a Banach scale, then we write BEr(u, δ) ⊆ Er for the Er-open ball of radius δ > 0, centered at u ∈ Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) If E is a Banach scale and EN is dense in Es for each s ∈ �N�, then we say that E is terminally dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We note that finite scales of Banach spaces correspond to the case N ∈ N in this definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we record a quick remark regarding the product of Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Products of Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that Ei = {Es i }s∈�N� is a Banach scale for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n}, each over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then F = �n i=1 Ei = {�n i=1 Es i }s∈�N� is a Banach scale when each �n i=1 Es i is endowed with the norm ∥u1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , un∥�n i=1 Es i = � n � i=1 ∥ui∥2 Es i �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) We now consider tame mappings between Banach scales in our next definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (Tame maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over the same field, µ, r ∈ �N� with µ ⩽ r, U ⊆ Er be an open set, and P : U → F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) We say that P satisfies tame estimates of order µ and base r (with respect to the Banach scales E and F) if for all �N� ∋ s ⩾ r, there exists a constant Cs ∈ R+ such that for all f ∈ U ∩ Es we have the inclusion P(f) ∈ F s−µ as well as the estimate ∥P(f)∥F s−µ ⩽ Cs⟨∥f∥Es⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where ⟨·⟩ is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) We say that P is µ-tame with base r if for each f ∈ U, there exists an Er-open set V ⊆ U such that f ∈ V and the restricted map P ↾ V : V → F 0 satisfies tame estimates of order µ and base r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) We say that P is strongly µ-tame with base r if on every Er-open and bounded subset V ⊆ U, the restricted map P ↾ V satisfies tame estimates of order µ and base r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) We say that P is (strongly) µ-tamely C0 with base r if P is (strongly) µ-tame with base r and if for every �N� ∋ s ⩾ r we have that P : U ∩ Es → F s−µ is continuous as a map from Es to F s−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The collections of such maps will be denoted by T 0 µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) and sT 0 µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5) For k ∈ N+ we say that P is (strongly) µ-tamely Ck with base r if for all �N� ∋ s ⩾ r the map P : U ∩ Es → F s−µ is Ck and for all j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} the jth derivative map, thought of as mapping DjP : U × �j ℓ=1 Er → F 0, is (strongly) µ-tame with base r with respect to the Banach scales E1+j and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, DjP ∈ T 0 µ,r(U × �j ℓ=1 Er, E1+j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) or, in the strong case, DjP ∈ sT 0 µ,r(U × �j ℓ=1 Er, E1+j;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The collections of such maps will be denoted by T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) and sT k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6) We denote the collections of (strongly) µ-tamely C∞ with base r maps by T ∞ µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) = � k∈N T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) and sT ∞ µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) = � k∈N sT k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (7) When U = Er we will often use the abbreviated notation T k µ,r(E, F) and sT k µ,r(E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) in place of T k µ,r(Er, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) and sT k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following result is a useful characterization of tameness when there is multilinear structure present in the map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (Multilinearity and tame maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let k ∈ N+ and r, µ ∈ �N� with µ ⩽ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let F = {F s}s∈�N�, G = {Gs}s∈�N�, and Ej = {Es j}s∈�N�, for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k}, be Banach scales over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let U ⊆ Gr be an open set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the following are equivalent for all maps P : U × �k j=1 Er j → F 0 such that P(g, ·) is k-multilinear for all g ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 25 (1) P ∈ T 0 µ,r(U × �k j=1 Er j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G × �k j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) The restriction of P to (U ∩ Gs) × �k j=1 Es j is continuous as a map into F s−µ for all �N� ∋ s ⩾ r, and for all g ∈ U there exists a Gr-open subset V ⊆ U such that g ∈ V and whenever �N� ∋ s ⩾ r, f ∈ V ∩ Gs, and hi ∈ Es i for i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} we have the estimate ∥P(f, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk)∥F s−µ ≲ ⟨∥f∥Gs⟩ k � i=1 ∥hi∥Er i + k � j=1 ∥hj∥Es j � i̸=j ∥hi∥Er i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) A similar equivalence holds for maps P ∈ sT 0 µ,r(U ×�k j=1 Er j ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G×�k j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) if we change the space in the first item to the space of strongly tame maps and we change the second item’s quantification of V to ‘for all bounded Gr-open sets V ⊆ U’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second item implies the first by noting that if {hi}k i=1 lies within a bounded subset of �k i=1 Er i , then we immediately obtain the required tame estimates on P from inequality (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we look to the converse, fixing g ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first item provides an open set �V ⊆ U × �k j=1 Er j such that (g, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , 0) ∈ �V and if (f, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk) ∈ �V ∩ (Gs × Es 1 × · · · × Es k) for �N� ∋ s ⩾ r, then ∥P(f, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk)∥F s−µ ≲ ⟨∥f∥Gs, ∥h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk∥Es 1×···×Es k⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Now, since �V is open, there exists δ ∈ R+ such that Vδ = BGr(0, δ) × �k j=1 BEr j (0, δ) ⊆ �V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, if f ∈ BGr(0, δ) and h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk ∈ �k j=1(Er k \\ {0}) are such that (g, h1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , hk) ∈ Gs × Es 1 × · · · × Es k, then we have that (f, δh1/∥h1∥Er 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , δhk/∥hk∥Er k) ∈ Vδ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and so we can invoke estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and multiply through by δ−k∥h1∥Er 1 · · · · · ∥hk∥Er k to acquire the desired bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A similar argument applies in the case of strongly tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If P ∈ T 0 µ,r(�k j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) is a k-multilinear mapping, then a simple modification of the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 shows that P is actually strongly tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, by multilinearity, we have that P is automatically a smooth function whose derivatives are also multilinear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This combines with the previous fact to show that actually P ∈ sT ∞ µ,r(�k j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a consequence of the previous lemma, we have structured estimates of the derivatives of tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (Derivative estimates on tame maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over a common field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let r, µ ∈ �N� with µ ⩽ r, U ⊆ Er be an open set, and P ∈ T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then for each f0 ∈ U, there exists an open set f0 ∈ V ⊆ U with the property that for every �N� ∋ s ⩾ r there exists a constant Cs,V ∈ R+, depending on s and V , such that for all f ∈ V ∩ Es and all g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk ∈ Es we have the estimate ∥DkP(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk]∥F s−µ ⩽ Cs,V k � ℓ=1 ∥gℓ∥Es � j̸=ℓ ∥gj∥Er + Cs,V ⟨∥f∥Es⟩ k � j=1 ∥gj∥Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Moreover, if P ∈ sT k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F), then a similar assertion holds for every bounded and open subset V ⊆ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is a direct application of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Our next result studies the interaction of tame maps via composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 (Composition of tame maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N�, F = {F s}s∈�N�, and G = {Gs}s∈�N� be a triple of Banach scales over a common field (R or C), and let k ∈ N and µ, µ′, r, r′ ∈ �N� be such that µ ⩽ r, µ′ ⩽ r′, r′ + µ ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that U ⊆ Er, U ′ ⊆ F r′ are open sets, 26 NOAH STEVENSON AND IAN TICE P ∈ T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F), and Q ∈ T k µ′,r′(U ′, F;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Set V = U ∩ Emax{r,r′+µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If P(V ) ⊆ U ′, then Q ◦ P ∈ T k µ+µ′,max{r,µ+r′}(V, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A similar assertion holds for the composition of strongly tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We proceed by induction on k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consider first the case that k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fix f0 ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since P(f0) ∈ U ′, we can appeal to the tameness of Q to obtain an open set P(f0) ∈ W ′ ⊆ U ′ such that Q satisfies a tame estimate of order µ′ and base r′ in W ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The map P is continuous and hence P −1(W ′) ⊆ V is open and contains f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In light of the tameness of P, there exists an open set f0 ∈ W ⊆ V in which P satisfies a tame estimate of order µ and base r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now, the open set W ∩ P −1(W ′) ⊆ V contains f0 and is such that whenever �N� ∋ s ⩾ max{r, r′ + µ} and f ∈ Es ∩ W ∩ P −1(W ′) we may estimate ∥Q ◦ P(f)∥Gs−(µ+µ′) ≲ ⟨∥P(f)∥F s−µ⟩ ≲ ⟨∥f∥Es⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Hence, Q ◦ P is indeed (µ + µ′)-tamely C0 with base max{r, r′ + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now suppose that for 1 ⩽ k ∈ N the result has been proved at the level k − 1, and P and Q satisfy the hypotheses at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Applying the induction hypothesis handles the tameness of all derivatives up to order k − 1, so it suffices to show that Dk(Q ◦ P) is (µ + µ′)-tame with base max{r, r′ + µ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fix f0 ∈ V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For ℓ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} we have that DℓQ is µ-tame with base r and hence there exists an open set P(f0) ∈ W ′ ℓ ⊆ U ′ such that in W ′ ℓ we have that DℓQ satisfies an ℓ-multilinear tame estimate of order µ′ and base r′ (see Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The map P is continuous and hence �k ℓ=1 P −1(W ′ ℓ) is an open subset of V containing f0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By appealing to the tameness of P and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 again, for ℓ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} there exists an open set f0 ∈ Wℓ ⊆ V such that in Wℓ we have that DℓP satisfies an ℓ-multilinear tame estimate of order µ and base r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For each R > 0, the set Of0,R = �k ℓ=1(Wℓ ∩ P −1(W ′ ℓ)) × B(Emax{r,r′+µ})k(0, R) is an open V × �k p=1 Emax{r,r′+µ} containing (f0, 0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For �N� ∋ s ⩾ max{r, r′ + µ} and (f, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk) ∈ Of0,R ∩ Es, we use the Fa`a di Bruno theorem (see, for instance, Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4A in Abraham, Marsden, and Ratiu [1]) to compute Dk(Q ◦ P)(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk] = � σ∈Sk k � ℓ=1 � j1+···+jℓ=k cσ,ℓ,k,j1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=',jℓDℓQ ◦ P(f){Dj1P(f), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , DjℓP(f)}[gσ(1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gσ(k)], (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) where Sk denotes the permutation group on {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k}, and c∗ denotes some combinatorial constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will estimate the norm in Gs−(µ+µ′) of each term in the sum above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fix ℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} and j1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , jℓ ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k} such that j1 + · · · + jℓ = k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 and the fact that max1⩽i⩽k∥gi∥Er ⩽ R, we are free to estimate � � � � � � � ∥Dj1P(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gj1]∥F s−µ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥DjℓP(f)[gjℓ−k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk]∥F s−µ ≲R ⟨∥f, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk∥Es×···×Es⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) By the same argument, � � � � � � � ∥Dj1P(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gj1]∥F r′ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥DjℓP(f)[gjℓ−k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk]∥F r′ ≲R ⟨∥f, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk∥Emax{r′+µ}×···×E{r′+µ}⟩ ≲R 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) COMPRESSIBLE TRAVELING WAVES 27 Thus, applying the estimate from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 once more yields ∥DℓQ ◦ P(f){Dj1P(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gj1], .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , DjℓP(f)[gjℓ−k+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk]}∥Gs−(µ+µ′) ≲R ⟨∥f, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk∥Es×···×Es⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) combine to give us the estimate ∥Dk(Q ◦ P)(f)[g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' gk]∥Gs−(µ+µ′) ≲R ⟨∥f, g1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gk∥Es×···×Es⟩, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) which is the desired tame bound at level k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The result thus holds for all k ∈ N by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ A convenient application of the previous result is that tame products of tame maps result in a tame map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' More precisely, we have the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 (Tame products of tame Ck maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let ℓ, k ∈ N with ℓ ⩾ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ℓ} let Ej = {Es j}s∈�N� and Fj = {Es j}s∈�N� be Banach scales over a common field, and let G = {Gs}s∈�N� be a Banach scale over the same field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , rℓ ∈ �N�, µj ∈ �rj� for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ℓ}, r′ ∈ �N�, and µ′ ∈ �r′�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, suppose that Uj ⊆ Erj j is open, Pj ∈ T k µj,rj(Uj, Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fj), and B ∈ T 0 µ′,r′(�ℓ j=1 Fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G) is k-multilinear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If we set µ = µ′ + max{µ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , µℓ} and r = max{r1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , rℓ, µ1 + r′, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , µℓ + r′} (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) and assume that r ∈ �N�, then the product map satisfies the inclusion B(P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , Pℓ) ∈ T k µ,r � ℓ� j=1 Uj ∩ Er j , ℓ� j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) A similar assertion holds for products of strongly tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first note that by Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, B in fact belongs to sT ∞ µ′,r′(�ℓ j=1 Fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The inclusion (P1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , Pℓ) ∈ T k max{µ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=',µℓ},max{r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=',rℓ} � ℓ� j=1 Uj ∩ Emax{r1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=',rℓ} j , ℓ� j=1 Ej;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ℓ� j=1 Fj � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) is clear, so the conclusion then follows from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We can also combine families of tame maps by integrating over parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Although more general results hold, we will need only the following simple realization of this fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 (Integrals of one-parameter families of tame Ck maps).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let k ∈ N ∪ {∞}, E = {Es}s∈�N� and F = {F s}s∈�N� be a pair of Banach scales over the same field, and let µ, r ∈ �N� satisfy µ ⩽ r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that U ⊆ Er is an open set, and for all t ∈ [0, 1] let Pt ∈ T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose additionally that the defining inequalities for these tame estimates (as in the first item of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 for the map and its derivatives) are satisfied uniformly in t ∈ [0, 1] and that for every j ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , k}, �N� ∋ s ⩾ r, f ∈ U ∩ Es, and f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , fj ∈ Es the map [0, 1] ∋ t �→ DjPt(f)[f1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , fj] ∈ F s−µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the integral map � 1 0 Pt dt : U → F 0 given by f �→ � 1 0 Pt(f) dt is well-defined and satisfies � 1 0 Pt dt ∈ T k µ,r(U, E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A similar assertion holds for integrals of strongly tame maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof essentially amounts to checking definitions, so we will only sketch the argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hypothesis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) ensures that the map � 1 0 Pt dt is well-defined, while the assumed uniform tame estimates ensure that the integrands are uniformly bounded with respect to t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The map is Ck as a map from U ∩ Es to F s−µ thanks to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and standard dominated convergence arguments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fact that � 1 0 Pt dt obeys the defining inequalities to be µ-tamely Ck with base r follows from uniformity in t and the fact that the norm of the integral is at most the integral of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 28 NOAH STEVENSON AND IAN TICE The latter half of this subsection is concerned with various specialized classes of Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The nicest classes of these are introduced in the subsequent definition, which closely follows Baldi and Haus [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 (Smoothable and LP-smoothable Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� be a Banach scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) We say that E is smoothable if for each j ∈ N+ there exists a linear map Sj : E0 → EN satisfying the following smoothing conditions for every u ∈ E0: ∥Sju∥Es ≲ ∥u∥Es for all s ∈ �N�, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) ∥Sju∥Et ≲ 2j(t−s) ∥Sju∥Es for all s, t ∈ �N� with s < t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) ∥(I − Sj)u∥Es ≲ 2−j(t−s) ∥(I − Sj)u∥Et for all s, t ∈ �N� with s < t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) ∥(Sj+1 − Sj)u∥Et ≲ 2j(t−s) ∥(Sj+1 − Sj)u∥Es for all s, t ∈ �N�, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) where the implicit constants are independent of j and are increasing in s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) We say that E is LP-smoothable if E is smoothable and the smoothing operators satisfy the following Littlewood-Paley condition: for s ∈ �N� and u ∈ Es we have that lim j→∞∥(I − Sj)u∥Es = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) and there exists a constant A > 0, possibly depending on s, such that A−1 ∥u∥Es ⩽ � ∞ � j=0 ∥∆ju∥2 Es �1/2 ⩽ A ∥u∥Es , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) where the operators {∆j}∞ j=0 are defined via ∆j = Sj+1 − Sj with the convention that S0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now give some examples of LP-smoothable Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first is trivial, but instructive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12 (A fixed Banach space).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that X is a Banach space and N ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the scale {X}s∈�N�, generated by X alone, is LP-smoothable in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, provided we take Sj = I for all j ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We also have the less trivial example of Sobolev spaces on all of Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 (Sobolev spaces on Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let V be a finite dimensional real vector space and N ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The real Banach scale of Sobolev spaces {Hs(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V )}s∈�N� is LP-smoothable for the smoothing operators {Sj}∞ j=0 given by Sj = 1B(0,2j)(∇/2πi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Unfortunately, the (LP-)smoothable Banach scales are insufficiently general for our purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As such, we introduce a broader class that captures Banach scales which are essentially closed and complemented subspaces of LP-smoothable Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is analogous to Hamilton’s notion of a tame Fr´echet space, which is given in Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 of [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 (Tame direct summands and tame Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� be a Banach scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Suppose that F = {F s}s∈�N� is a Banach scale over the same field as E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We say that E is a tame direct summand of F if there exist bounded linear maps λ : E0 → F 0 and ρ : F 0 → E0 such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (a) ρλu = u for all u ∈ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (b) For s ∈ �N� we have that λ(Es) ⊆ F s and ρ(F s) ⊆ Es, and the induced maps λ : Es → F s and ρ : F s → Es are bounded and linear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this case, we say λ is the lifting map and ρ is the restriction map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) We say that E is tame if there exists an LP-smoothable Banach scale F such that E is a tame direct summand of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 29 Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We note that if E is a tame direct summand of F, then basic functional analysis shows that λ(Es) ⊆ F s is a closed and complemented subspace of F s, and π = λ ◦ ρ : F s → F s is bounded and linear projection onto λ(Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consequently, for each s ∈ �N� there exists a closed subspace Gs ⊆ F s such that F s = λ(Es)⊕Gs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is the motivation for calling E a direct summand of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, if E is a direct summand of F, then for each s ∈ N we have the equivalence ∥u∥Es ≍ ∥λu∥F s for u ∈ Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have the following example of a tame Banach scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 (Sobolev spaces on domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let U ⊂ Rd be an open set that is a Stein extension domain in the sense of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, and let EU denote the associated extension operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let V be a finite dimensional real vector space and N ∈ N ∪ {∞}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The real Banach scale of Sobolev spaces {Hs(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V )}s∈�N� is tame in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, we realize this scale as a tame direct summand of {Hs(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V )}s∈�N�, which is LP-smoothable by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13, with the lifting operator λ = EU and restriction operator given by standard restriction, ρ = RU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now record a result on products of (LP-)smoothable and tame Banach scales, the proof of which is straightforward and thus omitted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 (Products of Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The product of a finite family of (smoothable, LP- smoothable, tame) Banach scales over a common field is again a (smoothable, LP-smoothable, tame) Banach scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consider now the following useful inequalities about smoothing in tame Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 (Smoothing in tame Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� be a tame Banach scale in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exist a sequence of smoothing operators {Tj}∞ j=0 ⊂ L(E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' EN) and, for s ∈ �N�, sequences of seminorms {ms j}∞ j=0, {ns j}∞ j=0 on Es such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) For all g ∈ Es we have the estimates ∥(Tj+1 − Tj)g∥Es ≲ ms j(g) and ∥(I − Tj)g∥Es ≲ ns j(g), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) where the implicit constants depend only on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For s, t ∈ �N� we have that � ms j ≍ 2j(s−t)mt j for all s, t, ns j ≲ 2j(s−t)nt j for all s ⩽ t, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) with the implicit constants depending only on s and t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) We have the equivalence ∥·∥2 Es ≍ ∞ � j=0 (ms j)2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) with implicit constants depending only on s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) For every g ∈ Es we have that ns j(g) ≲ ∥g∥Es (with implicit constants depending only on s), and ns j(g) → 0 as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5) E is terminally dense in the sense of the fourth item of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let F be an LP-smoothable Banach scale witnessing the definition of tameness for E with lifting λ ∈ L(E0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F 0), restriction ρ ∈ L(F 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' E0), and smoothing operators {Sj}∞ j=0 ⊆ L(F 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We define Tj = ρ ◦ Sj ◦ λ and for s ∈ �N� and g ∈ Es set ms j(g) = ∥(Sj+1 − Sj) ◦ λg∥F s and ns j(g) = ∥(I − Sj) ◦ λg∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) Then the first item follows from the boundedness of ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second, third, and fourth items are immediate consequences of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 and Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fifth assertions follows from the first and the fourth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 30 NOAH STEVENSON AND IAN TICE Next we give an interpolation result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 (Log-convexity in tame Banach scales).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈N be a tame Banach scale in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 over either R or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For r, s, t ∈ �N� with r < s < t we have that ∥u∥Es ≲ ∥u∥ t−s t−r Er ∥u∥ s−r t−r Et (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) for all u ∈ Et, where the implicit constant is increasing r, t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The bound is trivial if u = 0, so assume u ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let F be an LP-smoothable Banach scale witnessing the definition of tameness for E, with lifting, restriction, and smoothing operators λ, ρ, and {Sj}∞ j=0, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15, we have that ∥u∥Ep ≍ ∥λu∥F p for p ∈ {s, r, t}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we invoke the properties of the smoothing operators from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 to see that for any j ∈ N ∥λu∥F s ⩽ ∥(I − Sj)λu∥F s + ∥Sjλu∥F s ≲ 2j(s−t)∥λu∥F t + 2j(s−r)∥λu∥F r (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) and hence ∥u∥Es ≲ 2j(s−t)∥u∥Et + 2j(s−r)∥u∥Er.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now, since 1 ⩽ ∥u∥Et/∥u∥Er, we can choose j = ⌊log(∥u∥Et/∥u∥Er)/((t − r) log 2)⌋ ∈ N to obtain the desired inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Mapping hypotheses and statement of the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now intro- duce a lengthy definition that records a number of conditions that must be placed on the nonlinear map in our version of the Nash-Moser inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 (Mapping hypotheses).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We say that a triple (E, F, Ψ) satisfies the RI mapping hypotheses with parameters (µ, r, R) ∈ �N�3 if E = {Es}s∈�N� and F = {F s}s∈�N� are Banach scales over a common field, 1 ⩽ µ ⩽ r < R < ∞, R + µ ∈ �N�, and there exists 0 < δr ∈ R such that Ψ : BEr(0, δr) → F r−µ is a map satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Ψ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) µ-tamely C2: For every r − µ ⩽ s ∈ �N − µ� we have that Ψ : BEr(0, δr) ∩ Es+µ → F s is C2, and for every u0 ∈ BEr(0, δr) ∩ Es+µ we have the tame estimate ��D2Ψ(u0)[v, w] �� F s ⩽ C1(s)(∥v∥Es+µ ∥w∥Er + ∥v∥Er ∥w∥Es+µ + ⟨∥u0∥Es+µ⟩ ∥v∥Er ∥w∥Er).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Here the constant C1(s) is increasing in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we have the inclusion Ψ ∈ sT 2 µ,r(BEr(0, δr), E;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' F) according to the notation from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Derivative inversion: There exists δR ∈ R satisfying 0 < δR ⩽ δr such that for every u0 ∈ BEr(0, δR) ∩ EN there exists a bounded linear operator L(u0) : F r → Er satisfying the following three conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (a) For every s ∈ N ∩ [r, R] we have that the restriction of L(u0) to F s defines a bounded linear operator with values in Es, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' L(u0) ∈ L(F s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (b) DΨ(u0)L(u0)f = f for every f ∈ F r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (c) We have the tame estimate ∥L(u0)f∥Es ⩽ C2(s)(∥f∥F s + ⟨∥u0∥Es+µ⟩ ∥f∥F r) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) for every f ∈ F s and r ⩽ s ⩽ R, where again the constant C2(s) is increasing in s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Here the use of the prefix RI- is meant to indicate that the maps L(u0) are only required to be right inverses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We say that the triple (E, F, Ψ) satisfies the LRI mapping hypotheses if condition (b) in the third item is augmented by the left-inverse condition (b′) : L(u0)DΨ(u0)v = v for every v ∈ Er+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) See Figure 4 for a diagrammatic depiction of how L(u0) and DΨ(u0) interact with E and F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To conclude this subsection, we state our version of the Nash-Moser inverse function theorem, which is divided into two parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 31 F s+µ Es+µ L(u0) F s Es L(u0) DΨ(u0) F s−µ Es−µ L(u0) DΨ(u0) F 0 E0 F N EN Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Commutative diagram arising from the LRI mapping hypotheses for u0 ∈ EN ∩ BEr(0, δR) and r + µ ⩽ s ⩽ R − µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The ‘�→’ are the inclusion maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 (Inverse function theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over the same field, and assume that E and F satisfy one of the following two conditions: I: E and F are LP-smoothable (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), or II: E is tame and F is a tame direct summand of E (see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume the triple (E, F, Ψ) satisfies the LRI mapping hypotheses of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 with parameters (µ, r, R) satisfying 2(r + µ) + 1 < (r + R)/2, and set β = 2(r + µ) + 1 ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then there exist ε, κ1, κ2 > 0 such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Existence of local inverse: For every g ∈ BF β(0, ε) there exists a unique u ∈ BEβ(0, κ1ε) such that Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Estimates of local inverse: The induced bijection Ψ−1 : BF β(0, ε) → Ψ−1(BF β(0, ε)) ∩ BEβ(0, κ1ε) obeys the estimate ∥Ψ−1(g)∥Eβ ⩽ κ1∥g∥F β for all g ∈ BF β(0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Moreover, if N ∋ ν ⩽ R + r − 2β, then we have that Ψ−1 : BF β(0, ε) ∩ F β+ν → Eβ+ν (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) with the estimate ∥Ψ−1(g)∥Eβ+ν ≲ ∥g∥F β+ν, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) where the implied constant is independent of g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Basic continuous dependence: The map Ψ−1 obeys the Lipschitz bound ∥Ψ−1(g0) − Ψ−1(g1)∥Eβ−µ ⩽ κ2∥g0 − g1∥F β−µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) for all g0, g1 ∈ BF β(0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The inequality 3r + 4µ + 2 < R is equivalent to β = 2(r + µ) + 1 < (r + R)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Furthermore, when �N� is finite, the mapping hypotheses require that R + µ ⩽ N, and so we obtain the necessary relation 3r + 5µ + 2 < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We present the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 after first establishing two other theorems that prove separate components of the theorem under different hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn to the statement of the second part of our inverse function theorem, beginning with some notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Due to the derivative loss in the nonlinear operators under consideration, the higher regularity of the local inverse map is most conveniently phrased in terms of some variation on Gateaux derivatives rather than the usual Fr´echet notion of differentiability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is analogous to what is done in Section I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 and elsewhere in Hamilton [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23 (Continuous Gateaux differentiability).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let X and Y be Banach spaces over a common field, U ⊆ X an open set, and f : U → Y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 32 NOAH STEVENSON AND IAN TICE (1) We say that f is continuously Gateaux differentiable on U if there exists a continuous map Df : U × X → Y such that for all x ∈ U we have that Df(x) ∈ L(X;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Y ) and for all z ∈ X lim t→0 t−1(f(x + tz) − f(x)) = Df(x)z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) (2) For N ∋ ℓ ⩾ 2, we say that f is ℓ-times continuously Gateaux differentiable if f is continu- ously Gateaux differentiable and Df : U × X → Y is (ℓ − 1)-times continuously Gateaux differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We can now state the second part of our Nash-Moser inverse function theorem, the proof of which is in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24 (Further conclusions of the inverse function theorem).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and additionally that the Banach scale E consists of reflexive spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following additional conclusions hold for the local inverse map Ψ−1 : BF β(0, ε) → Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Continuity: For s ∈ [β, R + r − β − µ) ∩ N the map Ψ−1 : BF β(0, ε) ∩ F s → Es (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Continuous differentiability: For s ∈ [β, R + r − β − µ) ∩ N, the map Ψ−1 : BF β(0, ε) ∩ F s → Es−µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) is differentiable in the Fr´echet sense with DΨ−1 = L ◦ Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, when viewing DΨ−1 as map DΨ−1 : (BF β(0, ε) ∩ F s) × F s−µ → Es−µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) we have that it is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Higher regularity: Let N ∋ ℓ ⩾ 2 and assume that the map Ψ is ℓ-times continuously Gateaux differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ [β + (ℓ(ℓ + 1)/2 − 1)µ, R + r − β − µ) ∩ N we have that the map Ψ−1 : BF β(0, ε) ∩ F s → Es− ℓ(ℓ+1) 2 µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) is also ℓ-times continuously Gateaux differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Local surjectivity and injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin this section by proving an existence result modeled on the main theorem of Baldi and Haus [5], but adapted to our specific setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that the following theorem only requires the existence of right inverses in the mapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 (Local surjectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over the same field, and suppose that (E, F, Ψ) satisfies the RI mapping hypotheses of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 with parameters (µ, r, R) such that 2(r + µ) + 1 < (r + R)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Set β = 2(r + µ) + 1 ∈ �N� and let Aβ > 0 be such that ∥∆jf∥F β ⩽ Aβ ∥f∥F β for all j ∈ N and f ∈ F β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let Ci = Ci(R) denote the constants from the mapping hypotheses for i ∈ {1, 2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then there exist K1, K2, K3, K4 ∈ R+ such that for every g ∈ F β satisfying ∥g∥F β < δR 2[(1 + Aβ)K1 + K2 + K3 + K4 + (1 + Aβ)2K1(K1 + K3)] (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) there exists a sequence {(uj, vj, hj, yj, fj, ej)}∞ j=0 ⊆ (BEr(0, δR)∩ER)×EN ×ER ×F N ×F N ×F R−µ satisfying the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) When j = 0 we have the identities u0 = 0, v0 = S0u0 = 0, y0 = 0, f0 = ∆0g = S1g, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) h0 = L(u0)f0 = L(0)S1g, e0 = Ψ(u0 + h0) − Ψ(u0) − f0, COMPRESSIBLE TRAVELING WAVES 33 while for j ⩾ 1 we have the recursive relations uj = uj−1 + hj−1, vj = Sjuj, yj = −Sj j−1 � n=0 en − j−1 � n=0 yn, fj = ∆jg + yj, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) hj = L(vj)fj, ej = Ψ(uj + hj) − Ψ(uj) − fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For j ∈ N we have the estimates ∥hj∥Es ⩽ K1(∥g∥F β 2−j + ∥∆jg∥F β)2j(s−β) for all s ∈ [r, R] ∩ N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) ∥vj∥Es ⩽ K2∥g∥F β2j(s−β) for all β ⩽ s ∈ �N�, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) ∥uj − vj∥Es ⩽ K3∥g∥F β2j(s−β) for all s ∈ [0, R] ∩ N, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) ∥uj∥Eβ ⩽ K4∥g∥F β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and ∥ej∥F s ≲ C1∥g∥F β2j(s+µ+r−2β) for all s ∈ [r − µ, R − µ] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) (3) There exists u ∈ Eβ such that uj → u in Eβ as j → ∞, ∥u∥Eβ ⩽ K4 ∥g∥F β, and Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) If we know additionally that ν ∈ N is such that ν ⩽ R + r − 2β and g ∈ F β+ν, then actually uj → u in Eβ+ν as j → ∞ and we have the estimate ∥u∥Eβ+ν ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Here the implied constant depends on K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , K4, Aβ, Aβ+ν, µ, r, δR, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first three steps establish a trio of crucial claims that will be used in the fourth step to inductively construct the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The convergence result is then proved in the fifth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The higher regularity assertions of the fourth item are then proved in the sixth step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Throughout the proof we will utilize the sequence {ξn}∞ n=0 ⊆ [0, ∞) defined by ξn = ∥g∥F β 2−n + ∥∆ng∥F β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) Step 1: An estimate in the style of Littlewood-Paley theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let ℓ, k ∈ N satisfy 0 ⩽ ℓ ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that if {hj}k j=0 ⊆ ER are given and satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) for 0 ⩽ j ⩽ k, then ��� k � n=ℓ hn ��� Eβ ≲ K1 � k � n=ℓ ξ2 n �1/2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) where ξn is defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) To prove the claim, we begin by noting that for any j ∈ N, 0 ⩽ n ⩽ k, and s = β + sgn(j − n) ∈ [r, R] ∩ N we may bound, via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), ∥∆jhn∥Eβ ≲ 2j(β−s)∥hn∥Es ≲ K1ξn2j(β−s)2n(s−β) = K1ξn2(j−n)(β−s) = K1ξn2−|j−n|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) We estimate the sum on the left hand side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) via the ‘Littlewood-Paley’ characteriza- tion (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22), the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), and Young’s convolution inequality (setting ξn = 0 for n ∈ Z such that 0 < n): ��� k � n=ℓ hn ��� 2 Eβ ≲ ∞ � j=0 ���∆j k � n=ℓ hn ��� 2 Eβ ⩽ ∞ � j=0 � k � n=ℓ ∥∆jhn∥Eβ �2 ≲ K2 1 ∞ � j=0 � k � n=ℓ ξn2−|j−n|�2 ⩽ K2 1 � j∈Z � � n∈Z ξn1[ℓ,k](n)2−|j−n|�2 ⩽ K2 1 � � n∈Z ξ2 n1[ℓ,k](n) �� � n∈Z 2−|n|�2 ≲ K2 1 k � n=ℓ ξ2 n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) This is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 34 NOAH STEVENSON AND IAN TICE Step 2: An estimate from Taylor’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that if g ∈ F β satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and {(uj, vj, hj)}k j=0 ⊆ (BEr(0, δR) ∩ ER) × EN × ER are given and satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) for 0 ⩽ j ⩽ k, as well as the conditions u0 = v0 = 0 and uj = uj−1 + hj−1 if 1 ⩽ j ⩽ k, then {uj}k j=0, {uj + hj}k j=0, {vj}k j=0 ⊆ BEr(0, δR) ⊆ BEr(0, δr) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) and ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1∥g∥F β2j(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) for all s ∈ [r − µ, R − µ] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To prove this claim we begin by employing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) imply that ∥uj∥Er ⩽ ∥uj∥Eβ ⩽ K4 ∥g∥F β < δR 2 ⩽ δr 2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) imply that ∥vj∥Er ⩽ ∥vj∥Eβ ⩽ K2 ∥g∥F β < δR 2 ⩽ δr 2 for 0 ⩽ j ⩽ k, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) imply that ∥hj∥Er ⩽ ∥hj∥Eβ ⩽ K1 (∥g∥F β + Aβ ∥g∥F β) = K1(1 + Aβ) ∥g∥F β < δR 2 ⩽ δr 2 for the same range of j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Together, these three bounds imply the inclusions (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) is established, we know that Ψ(uj + hj), Ψ(uj), and DΨ(vj) are well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may then employ the fundamental theorem of calculus, Taylor’s theorem, and the convexity of BEr(0, δr) to write Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj = (DΨ(uj) − DΨ(vj))hj + � 1 0 (1 − t)D2Ψ(uj + thj)[hj, hj]dt = � 1 0 D2Ψ((1 − t)vj + tuj)[hj, uj − vj]dt + � 1 0 (1 − t)D2Ψ(uj + thj)[hj, hj]dt = Ij + IIj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) By using the tame C2 estimate from the mapping hypotheses of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20, we readily deduce that for r − µ ⩽ s ∈ �N − µ�, ∥Ij∥F s ≲ C1⟨∥uj − vj∥Er⟩ ∥uj − vj∥Es+µ ∥hj∥Er + C1⟨∥vj∥Es+µ⟩ ∥uj − vj∥Er ∥hj∥Er + C1 ∥uj − vj∥Er ∥hj∥Es+µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and ∥IIj∥F s ≲ C1⟨∥hj∥Er⟩ ∥hj∥Es+µ ∥hj∥Er + C1⟨∥uj∥Es+µ⟩ ∥hj∥2 Er .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) Synthesizing these bounds, we find that ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1⟨∥uj − vj∥Er⟩ ∥uj − vj∥Es+µ ∥hj∥Er + C1 ∥uj − vj∥Er ∥hj∥Es+µ + C1⟨∥hj∥Er⟩ ∥hj∥Es+µ ∥hj∥Er + C1⟨∥uj, vj∥Es+µ⟩ ∥hj∥Er (∥hj∥Er + ∥uj − vj∥Er) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) for r − µ ⩽ s ∈ �N − µ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we turn our attention to estimates for uj and vj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that if 1 ⩽ j ⩽ k, then the identity u = �j−1 n=0 hn and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) imply that ∥uj∥ER ⩽ j−1 � n=0 ∥hn∥ER ⩽ K1(1 + Aβ) ∥g∥F β j−1 � n=0 2n(R−β) ≲ K1(1 + Aβ) ∥g∥F β 2j(R−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) In turn, when we couple this with the smoothing estimates from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, the interpolation result from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), this implies that ∥vj∥Es ≲ ∥uj∥Es ≲ ∥uj∥ R−s R−β Eβ ∥uj∥ s−β R−β ER ≲ K R−s R−β 4 (K1(1 + Aβ)) s−β R−β ∥g∥F β 2j(s−β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) for all s ∈ N ∩ [β, R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) implies that K R−s R−β 4 (K1(1 + Aβ)) s−β R−β ∥g∥F β ⩽ 1, so we may further bound ∥vj∥Es ≲ ∥uj∥Es ≲ 2j(s−β) for all s ∈ N ∩ [β, R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 35 On the other hand, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) give us that ∥vj∥Es ≲ ∥uj∥Es ≲ ∥uj∥Eβ ≲ K4 ∥g∥F β ≲ 1 for all s ∈ N ∩ [r, β].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Upon combining these, we find that ∥uj∥Es + ∥vj∥Es ≲ 2j max{0,s−β} for all s ∈ N ∩ [r, R] and 0 ⩽ j ⩽ k, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) where we note that these bounds are trivial for j = 0 since u0 = v0 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we combine (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) to bound, for 0 ⩽ j ⩽ k and s ∈ N ∩ [r − µ, R − µ], ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1⟨K3 ∥g∥F β 2j(r−β)⟩K3 ∥g∥F β 2j(s+µ−β) · K1(1 + Aβ) ∥g∥F β 2j(r−β) + C1K3 ∥g∥F β 2j(r−β) · K1(1 + Aβ) ∥g∥F β 2j(s+µ−β) + C1⟨K1(1 + Aβ) ∥g∥F β 2j(r−β)⟩K1(1 + Aβ)2j(s+µ−β) · K1(1 + Aβ)2j(r−β) + C1⟨2j max{0,s+µ−β}⟩K1(1 + Aβ) ∥g∥F β 2j(r−β) · (K1(1 + Aβ) ∥g∥F β 2j(r−β) + K3 ∥g∥F β 2j(r−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) For the last term we note that � r − µ ⩽ s ⩽ β − µ ⇒ 2r − 2β ⩽ s + µ + r − 2β β − µ ⩽ s ⩽ R − µ ⇒ s + µ − β + 2r − 2β ⩽ s + µ + r − 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) Thus, upon regrouping and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), we then see that ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1(1 + Aβ)2K1(K1 + K3) ∥g∥2 F β 2j(s+µ+r−2β) ≲ C1 ∥g∥F β 2j(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) for all s ∈ N ∩ [r − µ, R − µ] and 0 ⩽ j ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14), and so the proof of the claim is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: A recursive identity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that {(yj, ej)}k j=0 ⊆ F N × F R−µ is given for 2 ⩽ k ∈ N and satisfies y0 = 0 and the recursive condition yj = −Sj j−1 � n=0 en − j−1 � n=0 yn for 1 ⩽ j ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) We claim that yj = −Sjej−1 − ∆j−1 j−2 � n=0 en for 2 ⩽ j ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) To see this, note that from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) we deduce that for any 1 ⩽ j ⩽ k the identity �j n=0 yn = −Sj �j−1 n=0 en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, for 2 ⩽ j ⩽ k we have yj = −Sj �j−1 n=0 en − Sj−1 �j−2 n=0 en, but by the definitions of the ∆n and en, this is the same as yj = −Sjej−1 −∆j−1 �j−2 n=0 en, and therefore (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 4: Inductive construction of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now aim to inductively construct the desired sequence under some assumptions on the constants K1, K2, K3, K4, which we will work out as we proceed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by seeding the sequence, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' constructing its elements with j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Define u0 = 0 ∈ BEr(0, δR) ∩ ER, v0 = S0u0 = 0 ∈ EN, y0 = 0 ∈ F N, f0 = ∆0g = S1g ∈ F N, h0 = L(u0)f0 = L(0)S1g ∈ ER, and e0 = Ψ(h0) − Ψ(0) − f0 ∈ F R−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By construction, the bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) hold trivially when j = 0 for all possible choices of K2, K3, K4 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, 36 NOAH STEVENSON AND IAN TICE we can apply the tame estimate for L(0) from the mapping hypotheses to see that for s ∈ N ∩ [r, R], ∥h0∥Es ⩽ C2 ∥S1g∥F s + C2(1 + ∥0∥Es+µ) ∥S1g∥F r ≲ C2 ∥∆0g∥F β + C2 ∥g∥F β ⩽ K1 � ∥g∥F β 2−0 + ∥∆jg∥F β � 20(s−β), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) provided the constant K1 satisfies the bound C2 ≲ K1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) which we henceforth assume holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then have that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is satisfied for j = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now proceed to the inductive step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let k ∈ N and suppose that {(uj, vj, hj, yj, fj, ej)}k j=0 ⊆ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) are given and satisfy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) for 0 ⩽ j ⩽ k, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) if 1 ⩽ j ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will construct (uk+1, vk+1, hk+1, yk+1, fk+1, ek+1) ∈ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ and show that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) continue to hold for j = k + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we define uk+1 = uk + hk ∈ ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The claim established in Step 2 guarantees that uk+1 ∈ Br(0, δR) ⊆ Br(0, δr).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, we may use a telescoping argument to see that uk+1 = uk+1 − u0 = �k n=0 hn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, the claim established in Step 1 shows that ∥uk+1∥Eβ ≲ K1 � ∞ � n=0 ξ2 n �1/2 ≲ K1 ∥g∥F β , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) where in the last inequality we have used the Littlewood-Paley bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, uk+1 ∈ BEr(0, δR) ∩ ER satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) with j = k + 1, provided that K1 and K4 satisfy K1 ≲ K4, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) which we henceforth assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, we define vk+1 = Sk+1uk+1 ∈ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and the smoothing bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17)– (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) to estimate ∥vk+1 − uk+1∥ER = ∥(I − Sk+1)uk+1∥ER ≲ ∥uk+1∥ER ⩽ k � n=0 ∥hn∥ER ⩽ K1 k � n=0 ξn2n(R−β) ⩽ K1 � ∞ � n=0 ξ2 n �1/2� k � n=0 22n(R−β)�1/2 ≲ K1 ∥g∥F β 2(k+1)(R−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) On the other hand, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) and the smoothing bounds (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) show that ∥vk+1 − uk+1∥E0 = ∥(I − Sk+1)uk+1∥E0 ≲ 2−(k+1)β ∥(I − Sk+1)uk+1∥Eβ ≲ 2−(k+1)β ∥uk+1∥Eβ ≲ K1 ∥g∥F β 2−(k+1)β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) Upon combining these two estimates and interpolating with the help of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, we find that for s ∈ N ∩ [0, R], ∥vk+1 − uk+1∥Es ≲ ∥vk+1 − uk+1∥1−s/R E0 ∥vk+1 − uk+1∥s/R ER ≲ K1 ∥g∥F β 2(k+1)(s−β), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and so vk+1 ∈ EN satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) with j = k + 1 provided that K1 and K3 satisfy K1 ≲ K3, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) which we henceforth assume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 37 Continuing with vk+1, we observe that for β ⩽ s ∈ �N� we can use the smoothing bounds of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 to estimate ∥vk+1∥Es = ∥Sk+1uk+1∥Es ≲ 2(k+1)(s−β) ∥Sk+1uk+1∥Eβ ≲ 2(k+1)(s−β) ∥uk+1∥Eβ ≲ K1 ∥g∥F β 2(k+1)(s−β), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) where we have again used (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, vk+1 obeys the bound (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) for j = k + 1 provided that K1 and K2 satisfy K1 ≲ K2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) which we henceforth assume, and in this case (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) in turn implies that ∥vk+1∥Er ⩽ ∥vk+1∥Eβ ⩽ K2 ∥g∥F β < δR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) Third, we introduce some useful estimates for the terms {ej}k j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For 0 ⩽ j ⩽ k we know that hj = L(vj)fj, and so fj = DΨ(vj)hj by the RI mapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may thus invoke the claim established in Step 2 in order to see that for 0 ⩽ j ⩽ k (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) implies that ∥ej∥F s = ∥Ψ(uj + hj) − Ψ(uj) − fj∥F s = ∥Ψ(uj + hj) − Ψ(uj) − DΨ(vj)hj∥F s ≲ C1 ∥g∥F β 2j(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) for all s ∈ N ∩ [r − µ, R − µ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fourth, we turn our attention to defining yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If k = 0, then we simply set y1 = −S1e0 ∈ F N, while if k ⩾ 1 then we set yk+1 = −Sk+1 �k n=0 en − �k n=0 yn ∈ F N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may then invoke the claim of Step 3 to see that the formula yj = −Sjej−1 − ∆j−1 j−2 � n=0 en (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) holds whenever 1 ⩽ j ⩽ k + 1, provided that we understand sums over empty ranges to mean zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) to estimate yk+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Initially we use the smoothing operator properties together with (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) to estimate ∥Sk+1ek∥F s ≲ � ∥ek∥F s if r − µ ⩽ s ⩽ R − µ 2(k+1)(s−R+µ) ∥ek∥F R−µ if R − µ ⩽ s ∈ �N� ≲ C1 ∥g∥F β 2k(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) for every r − µ ⩽ s ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Similarly, if k ⩾ 1 then we can use the fact that R + r − 2β > 0 to bound ��� k−1 � j=0 ∆kej ��� F s ⩽ k−1 � j=0 ∥∆kej∥F s ≲ 2k(s−R+µ) k−1 � j=0 ∥∆kej∥F R−µ ≲ 2k(s−R+µ) k−1 � j=0 ∥ej∥F R−µ ≲ C1 ∥g∥F β 2k(s−R+µ) k−1 � j=0 2j(R+r−2β) ≲ C1 ∥g∥F β 2k(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) for every r − µ ⩽ s ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Synthesizing (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42), we deduce that ∥yk+1∥F s ≲ C1 ∥g∥F β 2k(s+µ+r−2β) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) for every r − µ ⩽ s ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As the penultimate update we define fk+1 = ∆k+1g + yk+1 ∈ F N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We know that vk+1 satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38), so the operator L(vk+1) exists, and we may make the final update by setting hk+1 = L(vk+1)fk+1 ∈ ER.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The tame estimates for L(vk+1) then provide for the bound ∥hk+1∥Es ⩽ C2(∥∆k+1g∥F s + ∥yk+1∥F s) + C2⟨∥vk+1∥Es+µ⟩(∥∆k+1g∥F r + ∥yk+1∥F r) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) 38 NOAH STEVENSON AND IAN TICE for s ∈ N ∩ [r, R].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5), which we established above holds for j = k + 1, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) we may estimate ∥vk+1∥Es+µ ≲ K4 ∥g∥F β 2(k+1)(s+µ−β) ≲ 2(k+1)(s+µ−β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By plugging this and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44), we then find that ∥hk+1∥Es ≲ C2 � ∥∆k+1g∥F β 2(k+1)(s−β) + C1 ∥g∥F β 2k(s+µ+r−2β)� + C2⟨2(k+1)(s+µ−β)⟩ � ∥∆k+1g∥F β 2(k+1)(r−β) + C1 ∥g∥F β 2k(2r+µ−2β)� ≲ C2(1 + C1)2(k+1)(s−β)(∥∆k+1g∥F β (2(k+1)(r−s) + 2(k+1)(µ+r−β)) + ∥g∥F β (2(k+1)(µ+r−β) + 2(k+1)(µ+2r−s−β) + 2(k+1)(2r+2µ−β)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) To consolidate all of the exponents we recall that that r − s ⩽ 0 and 1 = β − 2r − 2µ > 0, which show that µ + r − β ⩽ 2µ + 2r − β = −1 ⩽ 0 and µ + 2r − s − β ⩽ µ + r − β ⩽ −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the previous estimate implies that ∥hk+1∥Es ≲ C2(1 + C1)2(k+1)(s−β)(∥∆k+1g∥F β + ∥g∥F β 2−(k+1)), and we deduce that hk+1 obeys (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) for j = k + 1, provided that K1 satisfies C2(1 + C1) ≲ K1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) which we assume holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have now established conditions on K1, K2, K3, and K4 that are sufficient for the construction of (uk+1, vk+1, hk+1, yk+1, fk+1, ek+1) ∈ (BEr(0, δR) ∩ ER) × EN × ER × F N × F N × F R−µ, namely (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It is a simple matter to choose parameters satisfying these conditions, and so the inductive step is complete provided these parameters are chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We thus have the desired sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) follows as above from the claim established in Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 5: Convergence of the sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the sequence in hand from Step 4, we use the claim from Step 1 with 0 ⩽ ℓ ⩽ k and the fact that uk+1 − uℓ = �k n=ℓ hℓ to see that ∥uk+1 − uℓ∥Eβ ⩽ k � n=ℓ ∥hn∥Eβ ≲ K1 � ∞ � n=ℓ ∥g∥2 F β 2−2n + ∥∆ng∥2 F β �1/2 → 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) as ℓ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, {uj}∞ j=0 is a Cauchy sequence in Eβ, and hence convergent to some u ∈ Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Sending j → ∞ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) shows that ∥u∥Eβ ⩽ K4 ∥g∥F β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It remains only to show that Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, we again telescope and use that Ψ(0) = Ψ(u0) = 0 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) to write, for k ⩾ 2, Ψ(uk+1) = k � j=0 (Ψ(uk+1) − Ψ(uk)) = k � j=0 (Ψ(uk + hk) − Ψ(uk)) = k � j=0 (ej + fj) = k � j=0 ∆jg + k � j=0 ej + k � j=0 yj = Sk+1g + k � j=0 ej − Sk k−1 � j=0 ej = Sk+1g + ek + (I − Sk) k−1 � j=0 ej.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) Now, we know from the properties of the smoothing operators, which are given in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, that ∥(I − Sk+1)g∥F β → 0 as k → ∞, while (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) implies that ∥ek∥F β−µ ≲ C1 ∥g∥F β 2k(r−β) → 0 as k → ∞ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) COMPRESSIBLE TRAVELING WAVES 39 and (observing that β − µ − 1 ⩾ 2r + µ ⩾ 1) ���(I − Sk) k−1 � j=0 ej ��� F β−µ−1 ⩽ k−1 � j=0 ∥(I − Sk)ej∥F β−µ−1 ≲ 2−k k−1 � j=0 ∥(I − Sk)ej∥F β−µ ≲ C1∥g∥F β2−k k−1 � j=0 2j(r−β) ≲ C1∥g∥F β2−k → 0 as k → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) Upon combining these, we find that ∥Ψ(uk+1) − g∥F β−µ−1 → 0 as k → ∞ but since uk → u in Eβ as k → ∞, the continuity of Ψ guarantees that Ψ(uk+1) → Ψ(u) in F β−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 6: Higher regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let N ∋ ν ⩽ R + r − 2β and now suppose additionally that g ∈ F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We wish to prove that the solution u ∈ Eβ constructed in Step 5 actually belongs to Eβ+ν and satisfies ∥u∥Eβ+ν ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will establish this via finite induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proposition to be proved inductively is the following statement, depending on ν ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ν}: if j ∈ N and s ∈ N ∩ [r, R], then ∥hj∥Es ≲ 2j(s−(β+ν))(∥∆jg∥F β+ν + 2−j∥g∥F β+ν) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) for an implicit constant depending on ν, as well as K1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , K4, Aβ, Aβ+ν, µ, r, δR, and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The case ν = 0 was established in the fourth step, in particular in the verification of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume now that for some ν ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ν − 1} we have that the induction hypothesis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) holds at level ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We wish to prove it at level ν + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by mimicking Step 1 and deriving improved bounds on the sequence {uj}∞ j=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For n ∈ N we set ξν n = ∥∆ng∥F β+ν + 2−n∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By arguing as in the first step, we can show that for any 0 ⩽ j ⩽ k we have the estimate ��� k � n=ℓ hn ��� Eβ+ν ≲ � k � n=ℓ (ξν n)2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) Since g ∈ F β+ν, we deduce from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) that {ξν n}∞ n=0 ∈ ℓ2(N);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' hence, the sequence of partial sums {�j n=0 hn}∞ j=0 ⊂ Eβ+ν is Cauchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have already established convergence of the scheme, giving that u = �∞ n=0 hj in Eβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, u ∈ Eβ+ν and, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52), we deduce that for all j ∈ N ∥uj∥Eβ+ν ≲ � j−1 � n=0 (ξν n)2�1/2 ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) Now we derive improved bounds on the sequence {vj}j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From properties (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) of the smoothing operators and the improved bounds of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53), we deduce for β + ν ⩽ s ∈ �N� that ∥vj∥Es = ∥Sjuj∥Es ≲ 2j(s−(β+ν))∥uj∥Eβ+ν ≲ 2j(s−(β+ν))∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) Next, we bound the sequence {uj − vj}j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, for s ∈ N ∩ [0, R] we have the bound ∥uj − vj∥Es ≲ ∥uj − vj∥1−s/R E0 ∥uj − vj∥s/R ER .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55) The E0-norm term we bound using properties (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) of the smoothing operators and estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53): ∥uj − vj∥E0 = ∥(I − Sj)uj∥E0 ≲ 2−j(β+ν)∥uj∥Eβ+ν ≲ 2−j(β+ν)∥g∥Eβ+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) 40 NOAH STEVENSON AND IAN TICE On the other hand, the ER term is handled via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17), the telescoping identity uj = �j−1 n=0 hn, induction hypothesis (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51), and the bound ξν n ≲ ∥g∥F β+ν: ∥uj − vj∥ER ⩽ j−1 � n=0 ∥(I − Sj)hn∥ER ≲ j−1 � n=0 ∥hn∥ER ≲ ∥g∥F β+ν j−1 � n=0 2n(R−(β+ν)) ≲ 2j(R−(β+ν))∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57) We synthesize (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57) to get ∥uj − vj∥Es ≲ 2j(s−(β+ν))∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58) Our next endeavor is to derive improved bounds on the sequence {ej}j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We turn to esti- mate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) for s ∈ [r − µ, R − µ] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To handle the right hand side, we invoke the following bounds: max{∥uj − vj∥Er, ∥hj∥Er} ≲ 2j(r−β) min{1, 2−jν∥g∥F β+ν}, max{∥uj − vj∥Es+µ, ∥hj∥Es+µ} ≲ 2j(s+µ−β) min{1, 2−jν∥g∥F β+ν}, ∥uj, vj∥Es+µ ≲ 2j max{0,s+µ−β}, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59) which are consequences of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58), and finally (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this way we acquire the estimate ∥ej∥F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) Now we estimate the sequence {yj}j∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we recall identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By arguing as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41), but using the bounds established in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) for {ej}j∈N, we learn that for j ∈ N and r − µ ⩽ s ∈ �N� ∥Sj+1ej∥F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61) Similarly, by arguing as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42), but instead using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) with s = R − µ and the fact that R + r − 2β − ν > 0 (since ν < ν), we gain the bound ��� j−1 � n=0 ∆jen ��� F s ≲ 2j(s+µ+r−2β−ν)∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) We combine (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) to see that for j ∈ N and r − µ ⩽ s ∈ �N�, ∥yj∥F s ≲ 2(j−1)(s+µ+r−2β−ν)∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63) At last, we are ready to obtain an improved estimate on the sequence {hj}j∈N and close the induc- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By using the identity hj = L(vj)(∆jg + yj) with the right inverse estimates of equation (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), we find that for j ∈ N, ∥hj∥Es ≲ ∥∆jg∥F s + ∥yj∥F s + ⟨∥vj∥Es+µ⟩(∥∆jg∥F r + ∥yj∥Er) for s ∈ [r, R] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) We estimate ∥yj∥F s and ∥yj∥F r according to the improved estimates of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63), the ∆jg norms are handled via smoothing estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥∆jg∥F k = 2j(k−(β+ν+1))∥∆jg∥F β+ν+1 for k ∈ {r, s}, and the vj-term is estimated according to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) in the case s ⩽ β and via (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) yields ∥hj∥Es ≲ 2j(s−(β+ν+1))∥∆jg∥F β+ν+1 + 2j(s+µ+r−2β−ν)∥g∥F β+ν + ⟨2j(s−β)⟩(2j(r−(β+ν+1))∥∆jg∥F β+ν+1 + 2j(µ+2r−2β−ν)∥g∥F β+ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='65) Upon regrouping, we acquire the bound ∥hj∥Es ≲ 2j(s−(β+ν+1))(1 + ⟨2j(s−β)⟩2j(r−s))∥∆jg∥F β+ν+1 + 2j(s−(β+ν+1))2−j(2j(2+µ+r−β) + ⟨2j(s−β)⟩2j(2+µ+2r−β−s)))∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66) COMPRESSIBLE TRAVELING WAVES 41 Since r ⩽ min{s, β}, we have that 1 + ⟨2j(s−β)⟩2j(r−s) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67) The inequalities 1 ⩽ µ + r and 2(µ + r) < β imply that 2j(2+µ+r−β) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='68) The inequality µ ⩾ 1 implies that min{2β, s + β} ⩾ β ⩾ 1 + 2(µ + r) ⩾ 2 + µ + 2r and hence ⟨2j(s−β)⟩2j(2+µ+2r−β−s) ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='69) We combine inequalities (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='68), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='69) to see that ∥hj∥Es ≲ 2j(s−(β+ν+1))(∥∆jg∥F β+ν+1 + 2−j∥g∥F β+ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='70) This means that the inductive proposition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) has been verified for ν + 1, and hence the induction is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We therefore know that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) holds for ν = ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then argue as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) with the sequence {ξν n}n∈N to deduce that u ∈ Eβ+ν with the estimate ∥u∥Eβ+ν ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now complement the previous local surjectivity result with the following local injectivity result, which has no analog in Baldi and Haus [5] but is analogous to a result of Hamilton [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize that in the following we only need left inverses and weaker forms of the tame estimates than in the LRI mapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26 (Local injectivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let E = {Es}s∈�N� and F = {F s}s∈�N� be Banach scales over the same field, and suppose that E is terminally dense in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let σ, µ ∈ N be such that σ + µ ∈ �N�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that there exists γσ ∈ R+ and a C2 map Ψ : BEσ+µ(0, γσ) → F σ such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) For every u0 ∈ BEσ+µ(0, γσ) we have the bound ��D2Ψ(u0)[v, w] �� F σ ≲ ⟨∥u0∥Eσ+µ⟩ (∥v∥Eσ+µ ∥w∥Eσ + ∥v∥Eσ ∥w∥Eσ+µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='71) (2) For every u0 ∈ BEσ+µ(0, γσ) ∩ EN there exists a bounded linear operator L(u0) : F σ → Eσ satisfying the following two conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (a) L(u0)DΨ(u0)v = v for every v ∈ Eσ+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (b) We have the estimate ∥L(u0)f∥Eσ ≲ ⟨∥u0∥Eσ+µ⟩ ∥f∥F σ for every f ∈ F σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then there exists 0 < δinj,σ ⩽ γσ/4 such that if u0 − u1 ∈ BEσ+µ(0, 2δinj,σ), with ui ∈ BEσ+µ(0, γσ) for i ∈ {0, 1}, then ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72) In particular, the restriction Ψ : BEσ+µ(0, δinj,σ) → F σ is injective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose initially that u0 − u1 ∈ BEσ+µ(0, 2δ) ∩ EN and ui ∈ BEσ+µ(0, γσ) for some 0 < δ ⩽ γσ/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may use Taylor’s theorem to write Ψ(u1) − Ψ(u0) = DΨ(u0)(u1 − u0) + � 1 0 (1 − t)D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] dt, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='73) with the understanding that this equality holds in the space F σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then apply the bounded linear map L(u0) : F σ → Eσ and rearrange to see that u1 − u0 = L(u0) � Ψ(u1) − Ψ(u0) − � 1 0 (1 − t)D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] dt � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='74) 42 NOAH STEVENSON AND IAN TICE Next we couple the identity (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='74) to the estimate for L(u0) and the trivial bound ⟨∥u0∥Eσ+µ⟩ ≲ 1 to see that ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ + � 1 0 (1 − t) ��D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] �� F σ dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='75) For the latter term we use the C2 estimate to bound ��D2Ψ((1 − t)u0 + tu1)[u1 − u0, u1 − u0] �� F σ ≲ ⟨∥(1 − t)u0 + tu1∥Eσ+µ⟩ ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ ≲ ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='76) for every t ∈ [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then plug this bound into (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='75) to see that ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ + ∥u1 − u0∥Eσ+µ ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ + δ ∥u1 − u0∥Eσ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='77) From this we readily deduce the existence of 0 < δinj ⩽ γσ/4 such that if δ ⩽ δinj then we can absorb the right-most term onto the left to conclude that ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ for all u0 − u1 ∈ Bσ+µ(0, 2δinj) ∩ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='78) Estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='78) is not quite the desired result since it requires u0, u1 ∈ EN, but we can use the fact that E is terminally dense to promote this result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, given u0, u1 ∈ BEσ+µ(0, γσ) such that u0 − u1 ∈ BEσ+µ(0, 2δinj) we can pick {un i }n∈N ⊆ EN such that un i → ui in Eσ+µ for i ∈ {0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since BEσ+µ(0, 2δinj) and BEσ+µ(0, γσ) are open, we may assume without loss of generality, that {un 0 − un 1}n∈N ⊆ BEσ+µ(0, 2δinj) and for i ∈ {0, 1}, {un i }n∈N ⊂ BEσ+µ(0, γσ) ∩ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then apply (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='78) to the sequence to see that ∥un 1 − un 0∥Eσ ≲ ∥Ψ(un 1) − Ψ(un 0)∥F σ for all n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By sending n → ∞ and using the continuity of Ψ : BEσ+µ(0, γσ) → F σ, we deduce that ∥u1 − u0∥Eσ ≲ ∥Ψ(u1) − Ψ(u0)∥F σ for all u0 − u1 ∈ BEσ+µ(0, 2δinj), with u0, u1 ∈ BEσ+µ(0, γσ) which is (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof of the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We are now ready for the proofs of the inverse function theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proofs are given under conditions I and II separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, assuming I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that condition I is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' E and F are LP- smoothable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First note that Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 implies that E is terminally dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This and the LRI mapping hypotheses from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 imply the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26 with σ = β − µ and γσ = δR ⩽ δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let δinj,σ > 0 be the constant from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the theorem tells us that Ψ : BEβ(0, δinj,σ) → F β−µ is injective and obeys the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, by hypothesis, we know that the inequality 2(r + µ) < β < (r + R)/2 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let K1, K2, K3, and K4 be the constants from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 and let εsurj > 0 denote the constant appearing on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 guarantees that for every g ∈ BF β(0, εsurj) there exists u ∈ Eβ satisfying Ψ(u) = g as well as the bound ∥u∥Eβ ⩽ K4 ∥g∥F β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Set ε = min{εsurj, δinj,σ/K4}, κ1 = K4, and κ2 > 0 to be the constant on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since κ1ε ⩽ δinj,σ, the above analysis shows that for every g ∈ BF β(0, ε) there exists a unique u ∈ BEβ(0, κ1ε) such that Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, the induced map Ψ−1 : BF β(0, ε) → Ψ−1(BF β(0, ε)) ∩ BEβ(0, κ1ε) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) in light of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now suppose that N ∋ ν ⩽ R + r − 2β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From the fourth item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25, we deduce (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' that the inverse Ψ−1 sends BF β(0, ε) ∩ F β+ν to Eβ+ν with the tame estimate ∥Ψ−1(g)∥Eβ+ν ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ COMPRESSIBLE TRAVELING WAVES 43 A similar, but slightly more involved argument is needed to prove the inverse function theorem under assumption II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, assuming II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that condition II is satisfied, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' E is tame and F is a direct summand of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since E is tame, there exists an LP-smoothable Banach scale G = {Gs}s∈�N� such that E is a tame direct summand of G with lifting and restriction operators λE : E0 → G0 and ρE : G0 → E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Similarly, since F is a tame direct summand of E we can pick associated lifting and restriction operators λF : F 0 → E0 and ρF : E0 → F 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Before proceeding, we need to introduce three bits of notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we define H = G3, endowed with the 2−norm for the sake of definiteness, which makes H into an LP-smoothable Banach scale thanks to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, for s ∈ �N� write Qs = ∥ρE∥L(Gs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Es).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then by construction we have that ρE(BGs(0, δ/Qs)) ⊆ BEs(0, δ) for all s ∈ �N�, 0 < δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Third, we define the map Φ : BHr(0, δr/Qr) → Gr−µ via Φ(u, v, w) = λE(λFΨ(ρEu) + (I − λFρF)ρEv) + (I − λEρE)w, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) which is well-defined in light of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and the fact that Ψ : BEr(0, δr) → F r−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now claim that (H, G, Φ) satisfy the RI mapping hypotheses with parameters (µ, r, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Clearly, Φ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now let r − µ ⩽ s ∈ �N − µ�.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By the mapping properties of Ψ and the lifting and restriction operators, we readily deduce that Φ : BHr(0, δr/Qr) ∩ Hs+1 → Gs is C2 and satisfies DΦ(u0, v0, w0)(a, b, c) = λE(λFDΨ(ρEu0)ρEa + (I − λFρF)ρEb) + (I − λEρE)c (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and D2Φ(u0, v0, w0)[(a1, b1, c1), (a2, b2, c2)] = λEλFD2Ψ(ρEu0)[ρEa1, ρEa2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) In turn, the tame C2 estimate for D2Ψ and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) imply that ��D2Φ(u0, v0, w0)[(a1, b1, c1), (a2, b2, c2)] �� Hs ⩽ C′ 1(s)[∥(a1, b1, c1)∥Hs+µ ∥(a2, b2, c2)∥Hr + ∥(a1, b1, c1)∥Hr ∥(a2, b2, c2)∥Hs+µ] + C′ 1(s) ⟨∥(u0, v0, w0)∥Hs+µ⟩ ∥(a1, b1, c1)∥Hr ∥(a2, b2, c2)∥Hr (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) for C′ 1(s) = C1(s) ∥λEλF∥L(F s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Gs) (1 + Qs+µ)(Qr + Q2 r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) This proves the first and second items of the RI mapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To complete the proof of the claim, it remains to show that the third item of the RI mapping hypotheses is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < δR ⩽ δr be given by the RI mapping hypotheses for (E, F, Ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For (u0, v0, w0) ∈ BHr(0, δR/QR) ∩ HN we then define the bounded linear operator Λ(u0, v0, w0) : Gs → Hs, for r ⩽ s ⩽ R, via Λ(u0, v0, w0)ξ = (λEL(ρEu0)ρFρEξ, λEρEξ, ξ), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) which is well-defined since ρEu0 ∈ BEr(0, δR) ∩ EN whenever (u0, v0, w0) ∈ BHr(0, δR/QR) ∩ HN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) to verify that Λ(u0, v0, w0) is a right inverse of DΦ(u0, v0, w0), using the identities ρFλF = 1 and ρEλE = 1: DΦ(u0, v0, w0)L(u0, v0, w0)ξ = λE(λFDΨ(ρEu0)ρEλEL(ρEu0)ρFρEξ + (I − λFρF)ρEλEρEξ) + (I − λEρE)ξ = λE(λFρFρEξ + (I − λFρF)ρEξ) + (I − λEρE)ξ = λEρEξ + (I − λEρE)ξ = ξ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) 44 NOAH STEVENSON AND IAN TICE for all ξ ∈ Gr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, for ξ ∈ Gs, the tame estimate for L(u0) allows us to bound ∥Λ(u0, v0, w0)ξ∥Hs = ∥λEL(ρEu0)ρFρEξ∥Gs + ∥λEρEξ∥Gs + ∥ξ∥Gs ⩽ C2(s) ∥λE∥L(Es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Gs) (∥ρFρEξ∥F s + ⟨∥ρEu0∥Es+µ⟩ ∥ρFρEξ∥F r) + ⟨∥λEρE∥L(Gs)⟩ ∥ξ∥Gs ⩽ C′ 2(s) ∥ξ∥Gs + ⟨∥(u0, v0, w0)∥Hs+µ⟩ ∥ξ∥F r , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) where the constant C′ 2(s) depends on C2(s) as well as on the quantities ∥λEρE∥L(Gs), ∥λE∥L(Es;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Gs), ∥ρFρE∥L(Gs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='F s), and ∥ρE∥L(Gs+µ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Es+µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, the third item of the RI hypotheses is satisfied by the triple (H, G, Φ) with parameters (µ, r, R), and the claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We emphasize, though, that we are not asserting that the LRI hypotheses are satisfied, as the left inverse condition fails in general for DΦ and Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the claim in hand, we now consider β = 2(r + µ) + 1 ∈ �N� and invoke Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 for the triple (H, G, Φ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let K1, K2, K3, and K4 be the constants from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 and let ε′ surj > 0 denote the constant appearing on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 guarantees that for every ξ ∈ BGβ(0, ε′ surj) there exists (u′, v′, w′) ∈ Hβ = (Gβ)3 satisfying Φ(u′, v′, w′) = ξ as well as the bound ∥u′∥Gβ + ∥v′∥Gβ + ∥w′∥Gβ = ∥(u′, v′, w′)∥Hβ ⩽ K4∥ξ∥Gβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Set εsurj = ε′ surj/∥λEλF∥L(F β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Gβ) and note that λEλF(BF β(0, εsurj)) ⊆ BGβ(0, ε′ surj) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) by construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consequently, for any g ∈ BF β(0, εsurj) there exists (u′, v′, w′) ∈ Hβ such that Φ(u′, v′, w′) = λEλFg, which unravels to λE(λFΨ(ρEu′) + (I − λFρF)ρEv′) + (I − λEρE)w′ = λEλFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) Applying ρE and using the identity ρEλE = 1, this implies the identity λFΨ(ρEu′) + (I − λFρF)ρEv′ = λFg, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) to which we apply ρF and using the identity ρFλF = 1 to see that Ψ(ρEu′) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) Thus, if we set u = ρEu′ ∈ Eβ, then Ψ(u) = g, and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) implies that ∥u∥Eβ ⩽ κ1 ∥g∥F β (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) for κ1 = K4 ∥ρE∥L(Gβ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Eβ) ∥λEλF∥L(F β;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Gβ), which in particular means that u ∈ BEβ(0, κ1εsurj).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 implies that E is terminally dense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This and the LRI mapping hypotheses imply that the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26 are satisfied by the triple (E, F, Ψ) with σ = β − µ and γσ = δR ⩽ δr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let δinj,σ > 0 be the constant from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the theorem tells us that Ψ : BEβ(0, δinj,σ) → F β−µ is injective and obeys the estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Set ε = min{εsurj, δinj,σ/κ1} and κ2 > 0 to be the constant on the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since κ1ε ⩽ δinj,σ, the above analysis shows that for every g ∈ BF β(0, ε) there exists a unique u ∈ BEβ(0, κ1ε) such that Ψ(u) = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, the induced map Ψ−1 : BF β(0, ε) → Ψ−1(BF β(0, ε)) ∩ BEβ(0, κ1ε) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) in light of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, if we assume that N ∋ ν ⩽ R + r − 2β and g ∈ BF β(0, ε) ∩ F β+ν, then we are assured by the fourth item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 that there exists (u′, v′, w′) ∈ Hβ+ν such that Φ(u′, v′, w′) = λEλFg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By unraveling as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13)–(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15), we find that for u = ρEu′ ∈ Eβ+ν we have that Ψ(u) = g and ∥u∥Eβ+ν ≲ ∥g∥F β+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ COMPRESSIBLE TRAVELING WAVES 45 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we aim to strengthen the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In particular, we will study the continuity and higher order smoothness of the inverse map Ψ−1 provided by the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we analyze the right and left linear inverse map L by making an extension to backgrounds outside of the terminal space EN, and then proving various continuity and differentiability assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, we return to the map Ψ−1 and show a more refined continuity estimate than the basic assertion of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we prove differentiability of Ψ−1 and relate the derivative to the operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is done, we conclude by reading off higher regularity assertions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now enumerate further properties of the family of inverses L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27 (Extension and regularity of the right and left inverse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under the LRI mapping hypotheses set forth in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 and the additional assumptions that r + µ ⩽ R and that the Banach scale {Es}s∈�N� consists of reflexive spaces, we have that the following properties of L hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Existence of extension: There exists a family of bounded linear maps L : BEr(0, δR)∩Er+µ → L(F r;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Er) such that the following extension properties are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (a) L = L on the subset BEr(0, δR) ∩ EN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (b) If g ∈ F r and u0 ∈ BEr(0, δR) ∩ Er+µ, then we have that DΨ(u0)L(u0)g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Addi- tionally, if u ∈ Er+µ then L(u0)DΨ(u0)u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (c) For s ∈ [r, R] ∩ N we have that L : BEr(0, δR) ∩ Es+µ → L(F s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Es) with the tame estimate ∥L(u0)g∥Es ≲ ∥g∥F s + ⟨∥u0∥Es+µ⟩∥g∥F r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Continuity: For s ∈ [r, R−µ]∩N, if we view L as mapping L : (BEr(0, δR)∩Es+µ)×F s → Es, then this map is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Higher regularity: Assume that, for some N ∋ ℓ ⩾ 2, the map Ψ is ℓ-times continuously Gateaux differentiable in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If for s ∈ [r, R − ℓµ] ∩ N we view L as a mapping L : (BEr(0, δR) ∩ Es+ℓµ) × F s+(ℓ−1)µ → Es, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) then L is (ℓ − 1)-times continuously Gateaux-differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Constructing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we provide a continuity estimate on the right and left inverse map L (a priori only defined for EN-backgrounds).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that u0, w0 ∈ BEr(0, δR) ∩ EN, s ∈ [r, R − µ] ∩ N, and g ∈ F s+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will estimate the difference L(u0)g − L(w0)g = −L(u0)(DΨ(u0) − DΨ(w0))L(w0)g = − � 1 0 L(u0)D2Ψ((1 − t)w0 + tu0)(u0 − w0, L(w0)g) dt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) in Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By applying the tame estimates of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) repeatedly and the embedding inequalities of the first item of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we get the somewhat crude estimate ∥L(u0)g − L(w0)g∥Es ≲ ⟨∥u0, w0∥Es+µ⟩3∥u0 − w0∥Es+µ∥g∥F r + ⟨∥u0, w0∥Es+µ⟩∥u0 − w0∥Er(∥g∥F s+µ + ⟨∥w0∥Es+2µ⟩∥g∥F r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) While forgoing tameness, this estimate is strong enough to allow us to define our extension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, let u0 ∈ BEr(0, δR) ∩ Er+µ and fix some g ∈ F r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that the sequence {L(Tju0)Tjg}∞ j=ℓ ⊂ Er is Cauchy, where the operators {Tj}∞ j=0 are from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 and ℓ ∈ N is the first index for which Tju0 ∈ BEr(0, δR) for all N ∋ j ⩾ ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We verify this by first estimating ∥L(Tj+ku0)Tj+kg − L(Tju0)Tjg∥Er ⩽ ∥L(Tj+ku0)(Tj+kg − Tjg)∥F r + ∥(L(Tj+ku0) − L(Tju0))Tjg∥Er = Ij,k + IIj,k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) 46 NOAH STEVENSON AND IAN TICE For Ij,k we simply apply estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18: Ij,k ≲ ⟨∥Tj+ku0∥Er+µ⟩∥Tj+kg − Tjg∥F r ≲ ⟨∥u0∥Er+µ⟩(nr j(g) + nr j+k(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Hence limj,k→∞ Ij,k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, for IIj,k, we first employ estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3): IIj,k ≲ ⟨∥Tj+ku0, Tju0∥Er+µ⟩3∥Tj+ku0 − Tju0∥Er+µ∥Tjg∥F r + ⟨∥Tj+ku0, Tju0∥Er+µ⟩∥Tj+ku0 − Tju0∥Er(∥Tjg∥F r+µ + ⟨∥Tju0∥Er+2µ⟩∥Tjg∥F r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Thanks again to Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18, we are free to make the following bounds: ∥Tj+ku0∥Er+µ, ∥Tju0∥Er+µ ≲ ∥u0∥F r+µ, ∥Tjg∥F r ≲ ∥g∥F r, ∥Tj+ku0 − Tju0∥Er ≲ 2−jµ(nr+µ j (u0) + nr+µ j+k(u0)), ∥Tjg∥F r+µ ≲ 2jµ∥g∥F r, ∥Tj+ku0 − Tju0∥Er+µ ≲ nr+µ j (u0) + nr+µ j+k(u0) ∥Tju0∥Er+2µ ≲ 2jµ∥u0∥Er+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Upon combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), we acquire the estimate IIj,k ≲ ⟨∥u0∥Er+µ⟩3∥g∥F r(nr+µ j (u0) + nr+µ j+k(u0)), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and hence limj,k→∞ IIj,k = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We deduce that the sequence {L(Tju0)Tjg}∞ j=ℓ ⊂ Er is Cauchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence there exists L(u0)g = lim j→∞ L(Tju0)Tjg, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and this defines L as a family of linear maps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 2: Properties of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now examine the restriction of L to higher regularity (larger s) spaces in the scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that u0 ∈ BEr(0, δR) ∩ Es+µ and that g ∈ F s for some s ∈ [r, R] ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For N ∋ j sufficiently large we can apply estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and again Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 to see that ∥L(Tju0)Tjg∥Es ≲ ∥Tjg∥F s + ⟨∥Tju0∥Es+µ⟩∥Tjg∥F r ≲ ∥g∥F s + ⟨∥u0∥Es+µ⟩∥g∥F r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Hence, the sequence {L(Tju0)Tjg}∞ j=ℓ ⊂ Es is bounded by the right hand expression above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The space Es is reflexive, and this sequence already converges in Er, so the limit L(u0)g belongs to Es and has norm bounded above by the right side of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) thanks to the weak sequential lower semicontinuity of the norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next prove that L is a family of right and left inverses for DΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First suppose that g ∈ F r and u0 ∈ BEr(0, δR) ∩ Er+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For j ∈ N sufficiently large, we have that DΨ(Tju0)L(Tju0)Tjg = Tjg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Since Ψ : BEr(0, δr) ∩ Er → F r−µ is C2, {L(Tju0)Tjg}∞ j=0 ⊂ Er converges to L(u0)g, and we have the convergences Tju0 → u0 in Er+µ �→ Er and Tjg → g in F r as j → ∞, we may send j → ∞ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) to see that DΨ(u0)L(u0)g = g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, if we assume that u ∈ Er+µ and u0 ∈ BEr(0, δR) ∩ Er+µ, then we have L(Tju0)TjDΨ(u0)u = u + L(Tju0)(TjDΨ(u0) − DΨ(Tju0))u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) The left hand side converges in Er to L(u0)DΨ(u0)u as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the right hand side we may estimate ∥L(Tju0)(TjDΨ(u0) − DΨ(Tju0))u∥Er ≲ ⟨∥u0∥Er+µ⟩∥(TjDΨ(u0) − DΨ(Tju0))u∥F r ≲ ⟨∥u0∥Er+µ⟩(∥(I − Tj)DΨ(u0)u∥F r + ∥(DΨ(u0) − DΨ(Tju0))u∥F r), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) and the right hand side of this evidently converges to zero as j → ∞ thanks again to the properties of I − Tj from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18, the continuity properties of DΨ, and the fact that u, u0 ∈ Er+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence by sending j → ∞ in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) we learn that L(u0)DΨ(u0)u = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 47 Now we are ready to prove that L is actually an extension of L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that u0 ∈ BEr(0, δR)∩EN and that g ∈ F r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then by the right and left inverse properties, for every j ∈ N we have the identity L(u0)Tjg = L(u0)DΨ(u0)L(u0)Tjg = L(u0)Tjg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) Upon sending j → ∞ and using that L(u0) and L(u0) are bounded, we deduce that L(u0)g = L(u0)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof of all of the assertions of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: Continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now study continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have established that for s ∈ [r, R − µ] ∩ N, u0, w0 ∈ BEr(0, δR) ∩ Es+2µ, and g ∈ F s+µ the sequence {L(Tju0)Tjg − L(Tjw0)Tjg}∞ j=0 ⊂ Es+µ converges weakly up to a subsequence in Es+µ and strongly in Er to the limit L(u0)g − L(w0)g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, upon invoking the log-convexity of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, we obtain strong convergence in Es up to a subsequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By passing along this subsequence in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) and then taking the supremum over ∥g∥F s+µ ⩽ 1, we obtain the Lipschitz estimate ∥L(u0) − L(w0)∥L(F s+µ,Es) ≲ ⟨∥u0, w0∥Es+2µ⟩3∥u0 − w0∥Es+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) We use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) as an intermediate step in deriving the stated continuity in the second item of the Theorem statement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let u0, w0 ∈ BEr(0, δR) ∩ Es+µ and g, h ∈ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For j ∈ N sufficiently large, we consider the difference L(u0)g − L(w0)h = (L(u0)g − L(Tju0)Tjg) + (L(Tju0)Tjg − L(Tjw0)Tjh) + (L(Tjw0)Tjh − L(w0)h) = Ij + IIj + IIIj (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and estimate the three terms individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For Ij, we note that (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) hold with r replaced by s (by the same proof), and hence we may send k → ∞ to acquire the bound ∥Ij∥Es ≲ ⟨∥u0∥Es+µ⟩3� ∥g∥F sns+µ j (u0) + ns j(g) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) For IIj we simply apply the local Lipschitz estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) and use properties of the operators Tj: ∥IIj∥Es ≲ ⟨2jµ∥g∥F s⟩⟨2jµ∥u0, w0∥Es+µ⟩3∥u0 − w0, g − h∥Es+µ×F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) For IIIj, we begin by bounding as in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17): ∥IIIj∥Es ≲ ⟨∥w0∥Es+µ⟩3� ∥h∥F sns+µ j (w0) + ns j(h) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) Now, by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18, we may estimate ns+µ j (w0) ≲ ns+µ j (u0) + ∥u0 − w0∥Es+µ and ns j(h) ≲ ns j(g) + ∥g − h∥F s , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) and hence deduce that ∥L(u0)g − L(w0)h∥Es ≲ 23µj⟨∥u0, w0∥Es+µ, ∥g, h∥F s⟩3∥u0 − w0, g − h∥Es+µ×F s + ⟨∥u0, w0∥Es+µ, ∥g, h∥F s⟩3(ns+µ j (u0) + ns j(g)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) By taking j large relative to u0 and g and then taking w0 and h sufficiently close u0 and g to we see that L is continuous as a map from Es+µ × F s to Es, but not necessarily uniformly so.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof of the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 4: Higher regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we prove the third assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By arguing as in the derivation of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), we learn that for s ∈ [r, R − 2µ] ∩ N, w0, h0, w0 + h0 ∈ BEr(0, δR) ∩ Es+2µ, g ∈ F s+µ, and τ ∈ (0, 1) we have the decomposition τ −1(L(w0 + τh0)g − L(w0)g) + L(w0)D2Ψ(w0)(h0, L(w0)g) = Iτ + IIτ (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) where Iτ = −(L(w0 + τh0) − L(w0)) � 1 0 D2Ψ(w0 + τh0)(h0, L(w0)g) dt (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) 48 NOAH STEVENSON AND IAN TICE and IIτ = −L(w0) � 1 0 (D2Ψ(w0 + tτh0) − D2Ψ(w0))(h0, L(w0)g) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) Since Ψ is µ-tamely C2, L satisfies the continuity assertions of the second item, and L obeys the same tame estimates as L, it holds that ∥Iτ, IIτ∥Es → 0 as τ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This proves that the map L : (BEr(0, δR) ∩ Es+2µ) × F s+µ → Es (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) is Gateaux differentiable with derivative given by DL(u0, g0)[w, h] = −L(u0)D2Ψ(u0)(w, L(u0)g0) + L(u0)h (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) for u0 ∈ BEr(0, δR) ∩ Es+2µ, w ∈ Es+2µ, and g0, h ∈ F s+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now suppose that N ∋ ℓ ⩾ 3 and that Ψ is ℓ-times continuously Gateaux differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By a simple induction argument using the formula (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26), we find the remaining conclusions of the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Now that we have a refined understanding of the mapping properties of the family of right and left inverses L, we return to studying the local inverse map Ψ−1, which we recall is granted by the conclusions of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, we now prove Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Throughout the proof we will use the operator L from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27, which is an extension of the operator L from the LRI mapping hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By a very mild abuse of notation, we write L in place of L in what follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under either hypothesis I or II, the Banach scales E and F are tame, and so there exist smoothing operators {Tj}∞ j=0 as in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We aim to establish that if s ∈ [β, R + r − β − µ) ∩ N and g ∈ BF β(0, ε) ∩ F s, then Ψ−1(Tjg) → Ψ−1(g) in the space Es as j → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By Taylor expanding Ψ at Tjg to second order as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26 and using the left invertibility of DΨ(Tjg) by L ◦ Ψ−1(Tjg), we arrive at the identity Ψ−1(Tj+1g) − Ψ−1(Tjg) = L ◦ Ψ−1(Tjg)(Tj+1 − Tj)g − L ◦ Ψ−1(Tjg) � 1 0 (1 − t)D2Ψ((1 − t)Ψ−1(Tjg) + tΨ−1(Tj+1g))(Ψ−1(Tj+1g) − Ψ−1(Tjg))⊗2 dt = Ij + IIj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) We will estimate the right hand side of the above in the spaces Es+σ for σ ∈ {−1, 0, 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From the assumed tame structure, we have the estimate ∥L ◦ Ψ−1(Tjg)h∥Es+σ ≲ ⟨∥Ψ−1(Tjg)∥Es+µ+σ⟩∥h∥F r + ∥h∥F s+σ ≲ ⟨2j(µ+σ)∥g∥F s⟩∥h∥F r + ∥h∥F s+σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) For h = (Tj+1 − Tj)g we estimate � ∥(Tj+1 − Tj)g∥F r ≲ 2j(r−s)ms j(g), ∥(Tj+1 − Tj)g∥F s+σ ≲ 2jσms j(g), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) and hence (since µ + r ⩽ s) ∥Ij∥Es+σ ≲ 2jσ⟨∥g∥F s⟩ms j(g), where we recall that the seminorms ms j are from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, for some t ∈ [0, 1], we take ht = D2Ψ((1 − t)Ψ−1(Tjg) + tΨ−1(Tj+1g))(Ψ−1(Tj+1g) − Ψ−1(Tjg))⊗2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) and estimate for ℓ ∈ {r, s + σ} ∥ht∥Eℓ ≲ ⟨∥Tjg, Tj+1g∥F ℓ+µ⟩∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥2 Er + ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Eℓ+µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) COMPRESSIBLE TRAVELING WAVES 49 If j is sufficiently large, say j ⩾ J(g), then we have that Tjg, Tj+1g ∈ BF β(0, ε), and hence we can apply the estimate from the third conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, to bound � ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Er+µ ≲ ∥(Tj+1 − Tj)g∥Eβ−µ ≲ 2j(β−µ−s)ms j(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) On the other hand, we trivially bound ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Es+σ+µ ≲ ∥Tj+1g, Tjg∥F s+σ+µ ≲ 2j(µ+σ)∥g∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) Therefore, we get that ∥ht∥Es+σ ≲ 22j(β−µ−s)+j(µ+σ)⟨∥g∥F s⟩ms j(g)2 + 2j(β+σ−s)∥g∥F sms j(g) ≲ 2jσ⟨∥g∥F s⟩2ms j(g) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and ⟨2j(µ+σ)∥g∥F s⟩∥ht∥Er ≲ 2jσ⟨∥g∥F s⟩3ms j(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Upon synthesizing these bounds we arrive at the estimate ∥IIj∥Es+σ ≲ 2jσ⟨∥g∥F s⟩3ms j(g), and hence (for σ ∈ {−1, 0, 1} and N ∋ j ⩾ J(g)) we have ∥Ψ−1(Tj+1g) − Ψ−1(Tjg)∥Es+σ ≲ 2jσ⟨∥g∥F s⟩3ms j(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) We now use (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) to prove that the sequence {�n j=J(g)(Ψ−1(Tj+1g) − Ψ−1(Tjg))}∞ n=J(g) is Cauchy in Es.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Again, the Banach scales E and F are tame in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14, and so there exist LP-smoothable Banach scales E and F with lifting and restriction pairs λE, ρE and λF, ρF witnessing the definition of tameness for E and F, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let ξj(g) = λE(Ψ−1(Tj+1g)− Ψ−1(Tjg)) ∈ E R+r−β and ηj(g) = ⟨∥g∥F s⟩3(Sj+1 − Sj) ◦ λFg ∈ F R+r−β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By the continuity of λE, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35), and properties of the LP-smoothing operators from Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, we deduce that for j, k ∈ N with j ⩾ J(g) we have the bound ∥∆kξj(g)∥Es ≲ 2−|j−k|∥ηj(g)∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) Hence, we obtain the following estimates for ℓ, n ∈ N with ℓ ⩾ J(g): ��� ℓ+n � j=ℓ ξj(g) ��� 2 Es ≲ ∞ � k=0 � ℓ+n � j=ℓ ∥∆kξj(g)∥Es �2 ≲ ∞ � k=0 � ℓ+n � j=ℓ 2−|j−k|∥ηj(g)∥F s �2 ≲ ℓ+n � j=ℓ ∥ηj(g)∥2 F s, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) where in the last inequality above we have employed Young’s convolution inequality, as was done in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since g ∈ F s, we have the inclusion {∥ηj(g)∥F s}∞ j=0 ∈ ℓ2(N) and hence from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37), we deduce that {�n j=J(g) ξj(g)}∞ n=J(g) ⊂ E s is Cauchy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The map ρE is continuous and linear and thus by taking the image of this sequence we learn that � n � j=J(g) (Ψ−1(Tj+1g) − Ψ−1(Tjg)) �∞ n=J(g) = {Ψ−1(Tn+1g) − Ψ−1(TJ(g)g)}∞ n=0 ⊂ Es (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) is Cauchy, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, the third conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 yields: ∥Ψ−1(Tn+1g) − Ψ−1(g)∥Eβ−µ ≲ ∥(I − Tn+1)g∥F β−µ → 0 as n → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) Therefore, the limit in Es of the above sequence, which exists by the satisfaction of the Cauchy condition, necessarily is Ψ−1(g).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By tracing back through the estimates, we find the following quantitative rate of convergence for n ⩾ J(g): ∥Ψ−1(Tn+1g) − Ψ−1(g)∥Es ≲ ⟨∥g∥F s⟩3� ∞ � j=n ms j(g)2�1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) We now have all the tools we need to establish that the map Ψ−1 : BF β(0, ε) ∩ F s → Es is continuous for every s ∈ [β, R + r − β − µ) ∩ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let g, h, g + h ∈ BF β(0, ε) ∩ F s, and assume that ∥h∥F s is sufficiently small so that J(g + h) ⩽ J(g) + 1, where once more we let J(g + h) denote the 50 NOAH STEVENSON AND IAN TICE first index for which j ⩾ J(g + h) implies that Tj(g + h) ∈ BF β(0, ε).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For any N ∋ n ⩾ J(g) + 1, we may then estimate ∥Ψ−1(g) − Ψ−1(g + h)∥Es ⩽ ∥Ψ−1(Tn+1g) − Ψ−1(g)∥Es + ∥Ψ−1(Tn+1g) − Ψ−1(Tn+1(g + h))∥Es + ∥Ψ−1(Tn+1(g + h)) − Ψ−1(g + h)∥Es = In + IIn + IIIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) For In and IIIn we employ the quantitative rate of convergence (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) along with the inequality (which is true by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) � ∞ � j=n ms j(g + h)2�1/2 ≲ ∥h∥F s + � ∞ � j=n ms j(g)2�1/2 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) to see that In + IIIn ≲ ⟨∥g, h∥F s⟩3� ∥h∥F s + � ∞ � j=n ms j(g)2�1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) It remains to handle IIn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As before, we have Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g) = L ◦ Ψ−1(Tn+1g) � Tn+1h− � 1 0 (1 − t)D2Ψ((1 − t)Ψ−1(Tn+1g) + tΨ−1(Tn+1(g + h)))(Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g))⊗2 dt � (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) as a consequence of Taylor’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence we have the estimate IIn ≲ ⟨2µn∥g∥F s⟩ � ∥h∥F s+ ⟨2µn∥g, h∥F s⟩∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Es+µ∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Er� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) We then crudely estimate ∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Es+µ ≲ 2nµ∥g, h∥F s (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) and also apply the third conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 to see that ∥Ψ−1(Tn+1(g + h)) − Ψ−1(Tn+1g)∥Er ⩽ ∥Tn+1h∥F β−µ ≲ ∥h∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) Synthesizing these bounds shows that IIn ≲ ⟨2µn∥g, h∥F s⟩3∥h∥F s, and hence ∥Ψ−1(g)−Ψ−1(g+h)∥Es ≲ ⟨∥g, h∥F s⟩3� ∥h∥F s+ � ∞ � j=n ms j(g)2�1/2� +⟨2µn∥g, h∥F s⟩3∥h∥F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) By taking n large relative to g and then taking h small, we see that the above estimate proves that Ψ−1 is continuous at g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We continue by studying the differentiability of the inverse map Ψ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that s ∈ [β, R + r − β − µ) ∩ N, g, h, g + h ∈ BF β(0, ε) ∩ F s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By expanding Ψ to second order via Taylor’s theorem with integral remainder and utilizing the left invertibility of L (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44)), we derive the equality Ψ−1(g + h) − Ψ−1(g) − L ◦ Ψ−1(g)h = − L ◦ Ψ−1(g) � 1 0 (1 − t)D2Ψ((1 − t)Ψ−1(g) + tΨ−1(g + h))(Ψ−1(g + h) − Ψ−1(g))⊗2 dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) COMPRESSIBLE TRAVELING WAVES 51 By taking the norm of both sides in the space Es−µ, employing tame estimates, and utilizing the third item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, we find that ∥Ψ−1(g + h) − Ψ−1(g) − L ◦ Ψ−1(g)h∥Es−µ ≲ ⟨∥g, h∥F s⟩2∥Ψ−1(g + h) − Ψ−1(g)∥Es∥Ψ−1(g + h) − Ψ−1(g)∥Er ≲ ⟨∥g, h∥F s⟩2∥Ψ−1(g + h) − Ψ−1(g)∥Es∥h∥F β−µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) Since F s �→ F β−µ and we have already established that Ψ−1 is continuous with respect to the F s and Es topologies, estimate (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) shows that Ψ−1 is differentiable at g when viewed as a map from F s to Es−µ, and we have the derivative formula DΨ−1(g)h = L ◦ Ψ−1(g)h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, since Ψ−1 is continuous relative to the F s and Es topologies, and L is continuous relative to the Es × F s−µ and Es−µ topologies, we find, by composition of continuity, that DΨ−1 is continuous with respect to the F s × F s−µ and Es−µ topologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This proves the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The third item now follows by pairing the final item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27 and the derivative formula DΨ−1 = L ◦ Ψ−1 with a simple induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Nonlinear analysis of traveling free boundary compressible Navier-Stokes With our tools from nonlinear analysis now established in abstract form, we return to our main goal: the analysis of the traveling wave free boundary compressible Navier-Stokes equations, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this section we verify most of the ‘nonlinear’ hypotheses for our Nash-Moser inverse function theorem and make preparations for the linear analysis that follows in subsequent sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We select a nonlinear mapping with tame Banach scales for the domain and codomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 we verify condition II of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To check that our rather complicated operator is 1-tamely C2, we analyze each of the atomic nonlinearities individually in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and then synthesize in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 via the calculus of tame maps from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is done, in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 we then use our newly developed understanding of the nonlinear map to decompose its derivative into a principal part and a remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then conclude in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 with some preliminary results for our linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Banach scales for the traveling wave problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by defining some function spaces that will comprise our Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we introduce spaces that will play a role in the domain of our nonlinear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ {−1, 0} ∪ R+ we define Xs = H1+s(Ω) × H2+s(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H5/2+s(Σ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where H5/2+s(Σ) denotes the anisotropic Sobolev space defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), with Σ identified with Rn−1, and we endow Xs with the Hilbert norm ∥q, u, η∥Xs = � ∥q∥2 H1+s + ∥u∥2 H2+s + ∥η∥2 H5/2+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) We single out an important closed subspace of Xs: Xs = {(q, u, η) ∈ Xs : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) Second, we define some spaces that will play a role in the codomain of our nonlinear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ {−1, 0} ∪ R+ we define the space Ys = � L2(Ω) × (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ if s = −1, H1+s(Ω) × Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) if s ⩾ 0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and endow it with the norm � � � ∥g, F∥Ys = � ∥g∥2 L2 + ∥F∥2 (0H1)∗ if s = −1, ∥g, f, k∥Ys = � ∥g∥2 H1+s + ∥f∥2 Hs + ∥k∥2 H1/2+s if s ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) 52 NOAH STEVENSON AND IAN TICE We also single out an important subspace of Ys: Ys = � � � � (g, F) ∈ Ys : � b 0 g(·, y) dy ∈ ˙H−1(Σ) � if s = −1, � (g, f, k) ∈ Ys : � b 0 g(·, y) dy ∈ ˙H−1(Σ) � if s ⩾ 0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) which we endow with the norm � � � ∥g, F∥Ys = � ∥g, F∥2 Y−1 + [ � b 0 g(·, y) dy]2 ˙H−1 if s = −1, ∥g, f, k∥Ys = � ∥g, f, k∥2 Ys + [ � b 0 g(·, y) dy]2 ˙H−1 if s ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) The spaces Ys and Ys are clearly complete and are Hilbert spaces in the case s ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the notation of (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), we have that Ys = ˆH1+s(Ω) × Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) for R ∋ s ⩾ 0, while Y−1 = ˆH0(Ω) × (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Third, we introduce spaces that will contain the stress and forcing data tuple (T, G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For R ∋ s ⩾ 0 we define the space Ws = H1+s(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n) × Hs(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Hs(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and endow it with the norm ∥T , G, F∥Ws = � ∥T ∥2 H1+s + ∥G∥2 Hs + ∥F∥2 Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) Finally, for s ∈ N we set Es = Xs × W1+s × R and Fs = Ys × W1+s × R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) and endow these with the norm from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From these spaces we now define the following Banach scales: XXX = {Xs}s∈N, YYY = {Ys}s∈N, W W W = {W1+s}s∈N, EEE = {Es}s∈N, FFF = {Fs}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) It is a simple matter to check that these all satisfy the conditions required by Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 to be Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the next result we check the second condition in the statement of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 for these scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (Tameness of domain and codomain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The Banach scale EEE is tame, and the Banach scale FFF is a tame direct summand of EEE in the sense of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving that EEE is a tame Banach scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The scale W W W is tame in light of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13, as it is the product of scales of L2-based Sobolev spaces on all of Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since W W W is tame, this lemma also shows that it suffices to prove that R and XXX = {Xs}s∈N are tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R is handled by Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12, so it remains only to handle XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ N write Xs = H1+s(Rn) × H2+s(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H5/2+s(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, Exam- ple 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13, and Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 the Banach scale {Xs}s∈N is LP smoothable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now show that {Xs}s∈N is tame by showing it is a tame direct summand of {Xs}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, we define an auxiliary mapping E1 : H1/2+s(Σ0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2+s(Σ) → H1+s(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) via E1(φ, ϕ) = w, where w is the unique solution to the PDE � � � � � � � � � −∆w = 0 in Ω, w = φ on Σ0, (I − en ⊗ en)w = 0 on Σ, w · en = ϕ on Σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) COMPRESSIBLE TRAVELING WAVES 53 Also, let EΩ denote a Stein extension operator, in the sense of Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, and write RΩ for the linear operator corresponding to restriction from Rn to Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then define the lifting operators λX : Xs → Xs and the restriction operators ρX : Xs → Xs via λX(q, u, η) = (EΩq, EΩu, η) and ρX(q, u, η) = (RΩq, RΩu−E1(TrΣ0u, TrΣ(u·en)+∂1η), η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) In light of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, it is straightforward to check that λX and ρX indeed define bounded linear maps between their stated domains and codomains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Additionally, we have that ρXλX = idX0, which completes the proof that {Xs}s∈N is a tame direct summand of {Xs}s∈N, and hence is tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, this completes the proof that EEE is tame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We continue by showing that FFF is a tame direct summand of EEE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this, it is clearly sufficient to prove that YYY is a tame direct summand of XXX.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, we have the stronger result that these spaces are isomorphic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To see this, consider the map Γ : Xs → Ys defined by Γ(q, u, η) = (g, f, k), where � � � � � � � � � � � � � � � ∇ · u = g in Ω, −∂1u + ∇(q + η) − ∆u = f in Ω, −(q − Du)en − ∆∥ηen = k on Σ, u · en + ∂1η = 0 on Σ, u = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) This is well-defined by virtue of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 in Leoni and Tice [60] establishes that this map is a Banach isomorphism, although the theorem itself states the isomorphism between slightly different spaces, which is needed in [60] due to a change of unknown made by taking a particular linear combination of q and η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of atomic nonlinearities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we put to use the abstract analysis of smooth tame structures from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and the tools for verifying smoothness and tame estimates from Appendices D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in the study of the nonlinear expressions appearing in the equations under consideration, namely system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The majority of the results in this subsection are a recasting of the tame calculus estimates from Appendix D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in the language of Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As such, some of the proofs are not much more than mere referrals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' it is the statements that are important as we move forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once these are established, we conclude this subsection by considering two nonlinearities that play a distinguished role in our analysis of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These aid in understanding the mapping properties of the continuity equation, which is rather subtle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The Banach scales we work with in this subsection are built from combinations of the following atomic scales: H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = {Hs(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V )}s∈N, H(Rd) = {Hs(Rd)}s∈N, W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) = {W s,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V )}s∈N, WH(κ) = {W s,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs (κ)(Rd))}s∈N, H(κ)(Rd) = {Hs (κ)(Rd)}s∈N, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where κ ∈ R+, d ∈ N+, U ⊆ Rd is a Stein extension domain (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), V is a finite dimensional vector space over R, and the spaces Hs, Hs (κ) are defined in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will often suppress the domain and codomain in this notation when they are clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It is a simple matter to check that these are indeed Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by looking at products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall the spaces of (strongly)-tame maps introduced in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Smooth tameness of products, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let V1, V2 and W be real finite dimensional vector spaces, B : V1 × V2 → W be a bilinear map, and U ⊆ Rd be a Stein-extension domain (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following inclusions hold when viewing B as a product for functions taking values in V1 and V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 54 NOAH STEVENSON AND IAN TICE (1) B ∈ sT ∞ 0,1+⌊d/2⌋(H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) B ∈ sT ∞ 0,0(H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) × W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By using the universality of tensor products to factor B = L ◦ ⊗ for a linear map L : V1 ⊗ V2 → W and then working component-wise, we see that this is a direct application of the high-low product estimates of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now handle more complicated products also involving sums.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (Smooth tameness of products, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m ∈ N+, V , W, and V1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , Vm be real finite dimensional vector spaces, B : V1 × · · · × Vm × V → W be (m + 1)-linear, and U ⊆ Rd be a Stein extension domain (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The (m + 1)-linear map PB defined via PB((gj, ψj)m j=1, ϕ) = B(g1 + ψ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , gm + ψm, ϕ) satisfies the inclusion PB ∈ sT ∞ 0,1+⌊d/2⌋ � m � j=1 (W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Vj) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Vj)) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We proceed via induction on m ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The base case, m = 1, is a simple application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now suppose that m ∈ N+ and the result holds for m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We wish to prove it for m + 1, so let B : V1 × · · · Vm × Vm+1 × V → W be an (m + 2)-linear map on some finite dimensional real vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that there exists a bilinear map B1 : V1 × (V2 ⊗ · · · ⊗ Vm+1 ⊗ V ) → W and an (m + 1)-linear map B2 : V2 × · · · × Vm+1 × V → V2 ⊗ · · · ⊗ Vm+1 ⊗ V such that B(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , vm+1, v) = B1(v1, B2(v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , vm+1, v)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) Indeed, by the universality property of tensor products there exists a linear map L : V1 ⊗ · · · ⊗ Vm+1 ⊗ V → W defined via L(v1 ⊗ · · · ⊗ vm+1 ⊗ v) = B(v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , vm+1, v), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) so we set B1(v1, τ) = L(v1 ⊗ τ) and B2(v2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , vm+1, v) = v2 ⊗ · · · ⊗ vm+1 ⊗ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then apply the induction hypothesis to the operator PB2 to see that PB2 ∈ sT ∞ 0,1+⌊d/2⌋ � m+1 � j=2 (W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Vj) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Vj)) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2 ⊗ · · · ⊗ Vm+1 ⊗ V ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Similarly, we apply the base case to the operator PB1 and acquire the inclusion PB1 ∈ sT ∞ 0,1+⌊d/2⌋(W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2 ⊗ · · · ⊗ Vm+1 ⊗ V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) implies that PB((gj, ψj)m+1 j=1 , ϕ) = PB1((g1, ϕ1), PB2((gj, ψj)m+1 j=2 , ϕ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the inclusion PB ∈ sT ∞ 0,1+⌊d/2⌋ is a consequence of the composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We also consider products with members of the anisotropic Sobolev spaces, which are defined in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (Smooth tameness of products, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Let rd ∈ N+ be as defined in the second item of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have that the map �rd j=1 H0 (1)(Rd) ∋ (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ηrd) �→ �rd j=1 ηj ∈ H0(Rd) belongs to sT ∞ 0,0(�rd j=1 H(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For any χ ∈ W ∞,∞(0, b) and ν ∈ N+ we have that the map �ν j=1 H0 (1)(Rd) ∋ (η1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , ην) �→ χ �ν j=1 ηj ∈ W 0,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H0 (ν)(Rd)) belongs to sT ∞ 0,0(�ν j=1 H(1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' WH(ν)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fact that the maps in the statement are well-defined and smooth follows from mul- tilinearity and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that for s ∈ N, the product map of the first item actu- ally smoothly maps �rd j=1 H0 (1)(Rd) → Hs(Rd) and the map of the second item smoothly maps �ν j=1 H0 (1)(Rd) → W s,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs (ν)(Rd)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the strong tame estimates on all of the derivatives follow from Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 and these improved mapping properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ COMPRESSIBLE TRAVELING WAVES 55 The next key nonlinear structure we handle is superposition as a Sobolev multiplier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 (Smooth tameness of superposition multipliers).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let V1, V2, W, and Z be real finite dimensional vector spaces, and B : V1 × V2 → W be a bilinear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let U ⊆ Rn and O ⊆ Z be Stein extension domains (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), and f ∈ C∞ b (O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, suppose that Y ⊆ W 2+⌊n/2⌋,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) × H2+⌊n/2⌋(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) is an open set with the property that if (g, ψ) ∈ Y , then (g + ψ)(U) ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the nonlinear operator P defined via P(g, ψ, ϕ) = B(f(g + ψ), ϕ) belongs to sT ∞ 0,2+⌊n/2⌋(Y × H2+⌊n/2⌋(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2), W(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) × H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By working component-wise and using the universality of tensor products, we see that as a consequence of Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 and the smoothness of the Sobolev multiplier times Sobolev to Sobolev pairing, for any s ⩾ 2 + ⌊n/2⌋ the map P : (Y × H2+⌊n/2⌋(U, V2)) ∩ (W s,∞(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) × Hs(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Z) × Hs(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V2)) → Hs(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) is well-defined and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, we need only check that the derivatives of P obey the required tame estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consider first the case of zero derivatives, which we refer to as the base case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Employing the third item of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9, working component-wise, and using the universality of tensor products, we readily deduce that the map in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) belongs to the space sT 0 0,2+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now for k ∈ N+ we consider the case of k-derivatives of P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A short computation reveals the derivative formula DkP(g, ψ, ϕ)(gj, ψj, ϕj)k j=1 = B(Dkf(g + ψ)(gj + ψj)k j=1, ϕ) + k � i=1 B(Dk−1f(g + ψ)(gj + ψj)j̸=i, ϕi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) Again by the universality of tensor products, there are linear maps Lk : Lk(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) ⊗ � k � j=1 Z � ⊗ V2 → W and Lk−1 : Lk−1(Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V1) ⊗ � k−1 � j=1 Z � ⊗ V2 → W (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) (depending only on k and B) such that B(Dkf(g + ψ)(gj + ψj)k j=1, ϕ) = Lk� Dkf(g + ψ) ⊗ k � j=1 (gj + ψj) ⊗ ϕ � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and B(Dk−1f(g + ψ)(gj + ψj)j̸=i, ϕi) = Lk−1 � Dk−1f(g + ψ) ⊗ � j̸=i (gj + ψj) ⊗ ϕi � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) We then find that the operators in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) belong to sT 0 0,2+⌊n/2⌋ by applying the base case, the tameness of composition from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, and the second version of tameness of products, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This shows that, indeed, DkP belongs to sT 0 0,2+⌊n/2⌋ for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The next nonlinear structure we examine is superposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (Smooth tameness of superposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let N ∋ k ⩾ 2 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Let V , W be real finite dimensional vector spaces, 0 ∈ O ⊆ V be a Stein extension domain (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) that is star shaped with respect to the origin, U ⊆ Rn be a Stein extension domain, and f ∈ C∞(O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W) be such that f(0) = 0 and Df ∈ C∞ b (O;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' L(V ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let Y ⊆ H2+⌊n/2⌋(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V ) be an open set with the property that if ϕ ∈ Y , then ϕ(U) ⊆ O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the operator ϕ �→ f(ϕ) belongs to sT ∞ 0,2+⌊n/2⌋(Y, H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' V );' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 56 NOAH STEVENSON AND IAN TICE (2) Suppose that Φ ∈ C∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) is a bi-Lipschitz homeomorphism and a C1 diffeomorphism such that DΦ ∈ C∞ b (Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists an open set 0 ∈ UΦ ⊆ W 2+⌊n/2⌋,∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2+⌊n/2⌋(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), depending only on Φ and the dimension, such that for any m ∈ N, the operator P defined via P(f, g, h) = f(Φ + g + h) satisfies P ∈ sT m 0,2+⌊n/2⌋(Hm+2+⌊n/2⌋(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd) × UΦ, Im;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) where Im = {Hm+s(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd) × W s,∞(Rn, Rn) × Hs(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that by using the fundamental theorem of calculus, the map from the first item has the equivalent formula ϕ �→ f(ϕ) = � 1 0 Df(tϕ)(ϕ) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) For each t ∈ [0, 1], the map ϕ �→ Df(tϕ)(ϕ) is seen to be sT ∞ 0,2+⌊n/2⌋ as a consequence of our previous result on the smooth tameness of superposition multipliers, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, and the composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, it is a simple matter to verify that we actually have uniformity of the defining inequalities with respect to t as well as the satisfaction of the remaining hypotheses of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the first item follows by applying the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next consider the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 establishes the existence of an open set UΦ ⊆ W 2+⌊n/2⌋,∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2+⌊n/2⌋(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) such that for every m ∈ N and s ⩾ 2 + ⌊n/2⌋ the map P : Hm+s(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd) × (UΦ ∩ (W s,∞(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Hs(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))) → Hs(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) is well-defined and Cm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus we need only verify that the derivatives of P, which are enumerated in (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), are sT 0 0,2+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The case of the 0th derivative (m = 0) follows immediately from the second item of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 and the bound ∥det D(g+h)−1∥L∞ ⩽ 2n∥DΦ−1∥n L∞ for all (g, h) ∈ UΦ, which holds as a consequence of Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When m > 0, the formula (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) shows that the mth derivative is built from simple products and the 0th derivative structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consequently, the result follows in this case by supplementing this observation with Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The next two results, which happen to be the most subtle part of the nonlinear analysis, deal with superposition-like nonlinearities whose argument contains a member of the anisotropic Sobolev space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 gives the notation and an enumeration of basic properties for these specialized spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the following statement rd ∈ N+, for d = n − 1, refers to the number defined in the second item of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 (Taylor expansion trick).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let I ⊆ R be an open interval containing [−gb, 0] and φ ∈ C∞(I).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρI ∈ R+, N (1) φ ∈ sT ∞ 0,2+⌊n/2⌋(BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI), H(Ω) × H(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Ω)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and N (2) φ ∈ sT ∞ 0,0(H(1)(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' WH(rn−1−1)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) such that N (1) φ (0, 0) = 0, N (2) φ (0) = 0, and Nφ(q, η) − Nφ(0, 0) = N (1) φ (q, η) + N (2) φ (Π1 Lη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) where the projectors Π are defined in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and we write Nφ(q, η) = φ(−gidRn · en + q + g(I − E)η) and Nφ(0, 0) = φ(−gidRn · en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) COMPRESSIBLE TRAVELING WAVES 57 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by choosing ρI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Set RI = dist(R\\I, [−gb, 0])/2 ∈ (0, ∞].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the supercritical Sobolev embeddings, properties of anisotropic Sobolev from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, and the boundedness of the extension operator E from Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we see that the map H2+⌊n/2⌋(Ω) × H2+⌊n/2⌋(Σ) ∋ (q, η) �→ q + g(I − E)η ∈ L∞(Ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) is well-defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the preimage of BL∞(0, RI) under this map is an open set UI containing the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We take ρI = dist(0, ∂UI)/ √ 2 so that BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI) ⊆ UI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) Now let (q, η) ∈ BH2+⌊n/2⌋(0, ρI) × BH2+⌊n/2⌋(0, ρI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that Nφ(q, η), as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19), is well-defined as a function from Ω to R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We write Nφ(q, η) = φ(g0 + g + ϕ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) where g0 = −gidRn · en, g = g(I − idRn · en/b)Π1 Lη, and ϕ = q + g(I − E)Π1 Hη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, Nφ(q, η) − Nφ(0, 0) = (φ(g0 + g + ϕ) − φ(g0 + g)) + (φ(g0 + g) − φ(g0)) = I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) For I we use the fundamental theorem of calculus to express I = ϕ � 1 0 φ′(g0 + g + τϕ) dτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) whereas for II we use Taylor’s formula with integral remainder to write II = rd−1 � j=1 φ(j)(g0) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' gj + grd � 1 0 (1 − τ)rd−1 (rd − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' φ(rd)(g0 + τg) dτ, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) where rd = rn−1 ⩾ 1 is given by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and sums over empty intervals are understood as zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This leads us to define N (1) φ (q, η) = ϕ � 1 0 φ′(g0 + g + τϕ) dτ + grd � 1 0 (1 − τ)rd−1 (rd − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' φ(rd)(g0 + τg)dτ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) and N (2) φ (Π1 Lη) = rd−1 � j=1 φ(j)(g0) j!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' gj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) The computation leading up to these definitions shows that decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) holds and that N (1) φ and N (2) φ vanish at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we analyze the tame smoothness of N (1) φ and N (2) φ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The former is sT ∞ 0,2+⌊n/2⌋ thanks to the results on addition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, smooth tameness of superposition multipliers, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, and the third version of smooth tameness of products, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, the map N (2) φ is sT ∞ 0,0 as a consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now use the previous result to understand the mapping properties of the vector field argument of the divergence in the continuity equation of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 (A vector field decomposition).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let I ⊆ R be an open interval containing [−gb, 0], φ ∈ C∞(I), and ρI be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let the open set OI be given by BH2+⌊n/2⌋(0, ρI) × H2+⌊n/2⌋(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × BH3+⌊n/2⌋(0, ρI), and consider the Banach scale J = {Hs(Ω) × Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1+s(Σ)}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) There exist M(1) φ ∈ sT ∞ 0,2+⌊n/2⌋(OI, J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) 58 NOAH STEVENSON AND IAN TICE and e1 · M(2) φ ∈ sT ∞ 0,0(H(1)(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' WH(rn−1)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) such that M(1) φ (0, 0, 0) = 0, e1 · M(2) φ (0) = 0, Tr∂Ω(M(1) φ (q, u, η) · en) = Tr∂Ω(Nφ(q, η)) � (TrΣ(u · en) + ∂1η)1Σ + TrΣ0(u · en)1Σ0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) and Mφ(q, u, η) − Mφ(0, 0, 0) = M(1) φ (q, u, η) + e1 · M(2) φ (Π1 Lη)e1, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) where we write Mφ(q, u, η) = Nφ(q, η)(u − Mηe1) and Mφ(0, 0, 0) = −Nφ(0, 0)e1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) for Nφ(q, η), Mη, and Π given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19), (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The precise form of the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32), which uses Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 as well as the identities Mηe1 = (1 + ∂nEη)e1 − ∂1Eηen and ∂nEΠ1 Lη = Π1 Lη/b, is given by M(1) φ (q, u, η) = Nφ(q, η)(u − ∂nEΠ1 Hηe1 + ∂1Eηen) − N (1) φ (q, η)(1 + Π1 Lη/b)e1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and e1 · M(2) φ (Π1 Lη) = −(Nφ(0, 0)Π1 Lη/b + N (2) φ (Π1 Lη)(1 + Π1 Lη/b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) That M(1) φ is sT ∞ 0,2+⌊n/2⌋ is a consequence of the composition of tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, the second product result, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, the mapping properties of Nφ, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, and the properties of anisotropic Sobolev spaces enumerated in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That e1 · M(2) φ is sT ∞ 0,0 is a consequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Smooth tameness of the nonlinear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by defining a nonlinear operator associated with the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will employ the Banach scales defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For some r ⩾ 2 + ⌊n/2⌋, 0 < ρ ⩽ ρWD, with ρWD determined by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and s ⩾ r, we set Ψ : (BXr(0, ρ) ∩ X1+s) × Ws × R+ → Ys × Ws × R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) via Ψ(q, u, η, T , G, F, γ) = (Ψ(q, u, η, γ) + Φ(q, u, η, T , G, F), T , G, F, γ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where Ψ : (BXr(0, ρ) ∩ X1+s) × R+ → Ys and Φ : (BXr(0, ρ) ∩ Xs) × Ws → Ys (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) are the principal and auxiliary parts of Ψ, which are defined via Ψ(q, u, η, γ) = (Ψ1(q, u, η), Ψ2(q, u, η, γ), Ψ3(q, u, η, γ)) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and Φ(q, u, η, T , G, F) = (0, Φ2(q, η, G, F), Φ3(η, T )), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) where Ψ1(q, u, η) = ∇ · (σq,η(u − Mηe1)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Ψ2(q, u, η, γ) = γ2σq,ηM−t η ((u − γMηe1) · ∇(M−1 η u)) + σq,η∇(q + gη) − γM−t η ∇ · (Sσq,η Aη (M−1 η u)Mt η), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Ψ3(q, u, η, γ) = −((P − Pext) ◦ σq,η − γSσq,η Aη (M−1 η u))Mt ηen − ςH (η)Mt ηen, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) Φ2(q, η, G, F) = −JηM−t η (σq,ηG ◦ Fη + F ◦ Fη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and Φ3(η, T ) = −T ◦ FηMt ηen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Here we recall that Fη is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), Aη and Jη by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5), Sτ A by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8), Mη by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), and σq,η by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We also recall the assumptions on ς, µ, and λ stated in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 59 Armed with the results from the previous two subsections, we now endeavor to study the smooth tameness of the map Ψ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our strategy is to handle the Ψ and Φ pieces separately and component-wise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our first result studies the continuity equation piece, Ψ1, which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This Ψ1 term is the source of the derivative loss and also requires the most careful analysis among the smooth tameness verification results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 (Smooth tameness of the continuity equation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρcont ∈ (0, ∞), depending only on the domain of the inverse enthalpy (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11)) such that the following hold (1) There exists constants c, C ∈ R+ such that for all (q, u, η) ∈ BX1+⌊n/2⌋(0, ρcont) we have the estimate c ⩽ σq,η ⩽ C where σq,η is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Ψ1 ∈ sT ∞ 1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρcont),XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' { ˆH1+s(Ω)}s∈N), where we recall that the spaces ˆHs(Ω) are defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) (see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 to set ρcont = ρ(Hmin,Hmax), where Hmin and Hmax are defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), to see that ∇ · MH−1(0, 0, 0) = 0, and to equate Ψ1(q, u, η) = ∇ · (M(1) H−1(q, u, η)) + ∂1(e1 · M(2) H−1(Π1 Lη)) = ΨI 1(q, u, η) + ΨII 1 (Π1 Lη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) In the above H−1 refers to the inverse enthalpy function (see 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a direct consequence of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9, we obtain the inclusion ΨI 1 ∈ sT ∞ 1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρcont),XXX;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' {H1+s(Ω)}s∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) On the other hand, the boundary conditions built into the definition of the space X1+⌊n/2⌋ in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), together with condition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31), imply that � b 0 ΨI 1(q, u, η)(·, y) dy = (∇∥, 0) · � b 0 M(1) H−1(q, u, η)(·, y) dy, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) and hence we deduce that ΨI 1 also maps smoothly into the space ˆH0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This fact combined with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) proves that ΨI 1 is sT ∞ 1,1+⌊n/2⌋ with respect to the Banach scales stated in the hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next handle the ΨII 1 piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By the norm in the anisotropic Sobolev spaces given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), we readily see that the embedding W 0,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂1H0 (rn−1)(Σ)) �→ H0(Ω) holds and hence we deduce from conclusion (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 that ΨII 1 ∈ sT ∞ 0,0(H(1)(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By inspection of the anisotropic norm from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) again, we also deduce the embedding W 0,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂1H0 (rn−1)(Σ)) �→ W 0,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ˙H−1(Σ)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and hence the map H0 (1)(Σ) ∋ ζ �→ � b 0 ΨII 1 (ζ)(·, y) dy ∈ ˙H−1(Σ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) is well-defined and smooth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These facts merge to show that ΨII 1 is sT ∞ 0,0 with respect to the Banach scales H(1)(Σ) and { ˆHs(Ω)}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We synthesize our results for ΨI 1 and ΨII 1 to complete the proof of the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now deduce the bounds stated in the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By unpacking the meaning of ρcont = ρI from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, for I = (Hmin, Hmax), we find that for (q, u, η) ∈ BX1+⌊n/2⌋(0, ρcont) we have the inclusion (−gidRn · en + q + g(I − E)η)(Ω) ⊆ (h⋆, h⋆) ⋐ (Hmin, Hmax), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) for some universal h⋆, h⋆ ∈ (Hmin, Hmax).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, by applying the inverse enthalpy, we find that H−1(h⋆) ⩽ σq,η ⩽ H−1(h⋆), which is the first item with c = H−1(h⋆), C = H−1(h⋆) ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Next we examine the Ψ2 piece of the momentum equation, which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 (Smooth tameness of the momentum equation, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exits a ρmome ∈ (0, ρcont] such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 60 NOAH STEVENSON AND IAN TICE (1) For (q, u, η) ∈ BX1+⌊n/2⌋(0, ρmome), we have the bounds 1/2 ⩽ Jη ⩽ 3/2, where Jη is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Ψ2 ∈ sT ∞ 0,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρmome) × R+,XXX × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by selecting ρmome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The map H3/2+⌊n/2⌋(Σ) ∋ η �→ Jη − 1 = ∂nEη ∈ L∞(Ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) is bounded thanks to the supercritical Sobolev embedding and the continuity properties of E from Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Denote the preimage of the set BL∞(0, 1/2) under this map by O ⊆ H3/2+⌊n/2⌋(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' O is open and we set �ρmome = dist(0, ∂O) ∈ R+ and ρmome = min{�ρmome, ρcont}, where the latter is defined in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With this choice of ρmome we have that 1/2 ⩽ Jη ⩽ 3/2, and hence the inverse matrix M−1 η exists whenever ∥η∥H3/2+⌊n/2⌋ < ρmome.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consequently, all of the expressions appearing in Ψ2 are pointwise well-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next decompose Ψ2(q, u, η, γ) = γ2ΨI 2(q, u, η) + ΨII 2 (q, η) + γΨIII 2 (q, u, η), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) where � � � � � ΨI 2(q, u, η) = σq,ηM−t η [(u − Mηe1) · ∇(M−1 η u)], ΨII 2 (q, η) = σq,η∇(q + gη), ΨIII 2 (q, u, η) = −M−t η ∇ · (Sσq,η Aη (M−1 η u)Mt η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) The γ dependence in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) is simple, and it is sufficient to study the pieces of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) individually.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the first piece, we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 to write σq,η = ϱ + N (1) H−1(q, η) + N (2) H−1(Π1 Lη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) where H−1 is the inverse of the enthalpy (see (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The embeddings of the second item of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 allow us to view the latter map as N (2) H−1 ∈ sT ∞ 0,0(H(1)(Σ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' W(Ω)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the smooth tameness of superposition multipliers proved in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, the map (u, η) �→ ∇(M−1 η u) is sT ∞ 0,1+⌊n/2⌋ with respect to the Banach scales {H2+s(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H5/2+s(Σ)}s∈N and {Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By applying the product of tame maps result from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and the composition of smooth tame maps result from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, we then see that the map (u, η) �→ (u − Mηe1) · ∇(M−1 η u) is also sT ∞ 0,1+⌊n/2⌋ with respect to the aforementioned Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By invoking Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 again, we get that the same conclusion is true for the map (u, η) �→ M−t η [(u − Mηe1) · ∇(M−1 η u)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, by using Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, we deduce that (q, u, η) �→ (ϱ + N (1) H−1(q, η) + N (2) H−1(Π1 Lη))M−t η [(u − Mηe1) · ∇(M−1 η u)], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ΨI 2, is sT ∞ 0,1+⌊n/2⌋ with respect to the Banach scales {Xs}s∈N and {Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we examine the ΨII 2 piece.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We again employ the density decomposition in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we see that ΨII 2 is sT ∞ 0,1+⌊n/2⌋ for the Banach scales {H1+s(Ω) × H5/2+s(Σ)}s∈N and {Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N as a consequence of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we examine the ΨIII 2 piece, decomposing further: ΨIII 2 (q, u, η) = ΨIII1 2 (q, u, η) + ΨIII2 2 (q, u, η) + ΨIII3 2 (q, u, η) + ΨIII4 2 (q, u, η), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) where � � � � � � � � � ΨIII1 2 (q, u, η) = −M−t η ∇ · � µ(σq,η)J−1 η ∇(M−1 η u)MηMt η � , ΨIII2 2 (q, u, η) = −M−t η ∇ · (µ(σq,η)J−1 η Mt η∇(M−1 η u)tMt η), ΨIII3 2 (q, u, η) = 2M−t η ∇ · (µ(σq,η)J−1 η (∇ · u)Mt η), ΨIII4 2 (q, u, η) = −M−t η ∇ · (λ(σq,η)J−1 η (∇ · u)Mt η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) COMPRESSIBLE TRAVELING WAVES 61 We handle these four terms in more or less the same way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The µ and λ viscosity coefficients are decomposed as � µ(σq,η) = µ(ϱ) + N (1) µ◦H−1(q, η) + N (2) µ◦H−1(Π1 Lη), λ(σq,η) = λ(ϱ) + N (1) λ◦H−1(q, η) + N (2) λ◦H−1(Π1 Lη), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, while the Mη, M−1 η , Mt η, M−t η , and J−1 η terms are viewed as superposition mul- tipliers and thus are handled via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the fact that each ΨIIIj 2 , j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , 4}, is sT ∞ 0,1+⌊n/2⌋ with respect to the Banach scales {Xs}s∈N and {Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N is a consequence of the aforementioned lemmas, the second result on products, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, and the result on composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Now we examine the Ψ3 piece of the dynamic boundary condition, which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12 (Smooth tameness of the dynamic boundary condition, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With ρdyna = ρmome ∈ R+ as in Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, we have Ψ3 ∈ sT ∞ 0,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρdyna) × R+,XXX × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' {H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As before, we begin the proof by decomposing Ψ3(q, u, η, γ) = ΨI 3(q, η) + γΨII 3 (q, u, η) + ΨIII 3 (η) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) where � � � � � ΨI 3(q, η) = −(P ◦ σq,η − Pext)Mt ηen, ΨII 3 (q, u, η) = Sσq,η Aη (M−1 η u)Mt ηen, ΨIII 3 (η) = −ςH (η)Mt ηen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) For the first piece we recall that H−1 is the inverse of the enthalpy, as in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), and use the fact that σq,η = H−1(−gb + q) and Pext = P ◦ ϱ(b) = P ◦ H−1(−gb) on Σ to rewrite ΨI 3(q, η) = (P ◦ H−1(· − gb) − P ◦ H−1(−gb))(q)Mt ηen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) From this, the operator ΨI 3 is readily seen to belong to sT ∞ 0,1+⌊n/2⌋ with respect to the Banach scales {H1+s(Ω) × H5/2+s(Σ)}s∈N and {H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N, as a consequence of the first item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 regarding the tame smoothness of superposition, the result on tame smoothness of superposition multipliers, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, and the result on the composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fact that ΨII 3 is sT ∞ 0,1+⌊n/2⌋ for the scales {Xs}s∈N and {H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N follows from an argument similar to the analysis of the ΨIII 2 -term from the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we handle ΨIII 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The mean curvature operator has the expression H (η) = ∇∥ ·( � H (∇∥η)), where the map � H ∈ C∞(Rn−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn−1) is given by � H (v) = ⟨v⟩−1v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From this we deduce that ΨIII 3 is sT ∞ 0,1+⌊n/2⌋ by combining the conclusions of Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (the first item), 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now synthesize our previous results to deduce the smooth tameness of the principal part nonlinear operator from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 (Smooth tameness of the principal part nonlinear operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρprin ∈ R+ such that Ψ ∈ sT ∞ 1,1+⌊n/2⌋(BX1+⌊n/2⌋(0, ρprin) × R+,XXX × R;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='YYY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We set ρprin = min{ρcont, ρmome, ρdyna} and apply Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The remainder of this subsection is devoted to the study of the auxiliary piece Φ of the nonlinear operator Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The next result handles the Φ2 piece, which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 62 NOAH STEVENSON AND IAN TICE Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 (Smooth tameness of the momentum equation, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρbulk ∈ R+ such that for every m ∈ N, Φ2 ∈ sT m 0,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρbulk) × Wm+2+⌊n/2⌋, {Xs × Wm+s}s∈N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the smooth tameness of superposition multipliers, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, the decomposition of σq,η = N (1) H−1(q, η) + N (2) H−1(Π1 Lη) + ϱ from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, and the result on composition of smooth tame maps, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, we see that it is sufficient to prove that the nonlinear operator Λ : Hm+2+⌊n/2⌋(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × BH9/2+⌊n/2⌋(Σ)(0, ρ) → H0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) given by Λ(I, η) = I ◦ Fη (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) is, for some ρ > 0, well-defined and sT m 0,2+⌊n/2⌋ for the scales {Hm+s(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H5/2+s(Σ)}s∈N and {Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we let EΩ and RΩ denote the Stein extension (see Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and the restriction operators for Ω, respectively, and note that we have the equivalent formula Λ(I, η) = RΩI(idRn + EΩE(Π1 Lη + Π1 Hη)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) Hence, according to Lemma D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and properties of the maps E and EΩ (see Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16), there exists a ρcomp ∈ R+ (depending only on the dimension and Ω) such that whenever ρ ⩽ ρcomp the map (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) is sT m 0,2+⌊n/2⌋ with respect to the aforementioned Banach scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The penultimate result of this subsection considers the Φ3 piece of Ψ, which is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 (Smooth tameness of the dynamic boundary condition, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For ρsurf = ρbulk ∈ R+, where the latter is from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14, we have that for every m ∈ N, Φ3 ∈ sT m 0,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρsurf) × Wm+2+⌊n/2⌋, {Xs × Wm+s}s∈N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' {H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)}s∈N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We argue is in the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and reduce to studying the map �Λ : Hm+2+⌊n/2⌋(Rn;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × BH9/2+⌊n/2⌋(Σ)(0, ρ) → H0(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) given by �Λ(T , η) = TrΣ(T ◦ Fη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then use the equivalent formula �Λ(T , η) = TrΣRΩT (idRn + EΩE(Π1 Lη + Π1 Hη)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and conclude as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The parameter m ∈ N appearing in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 (and subsequently in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) gives the data terms (T , G, F) m-extra derivatives to ensure that the composition type nonlinearities of Φ2 and Φ3 are Cm into the correct codomain space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' One sees that this extra regularity is necessary upon inspection of the derivative formulae (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) and (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' At last, we are ready to deduce the smooth tameness of the nonlinear operator Ψ, which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the following statement WD is an acronym for ‘well-defined’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 (Smooth tameness of the nonlinear operator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρWD ∈ R+ such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) For N ∋ s ⩾ 2 + ⌊n/2⌋ and (q, u, η) ∈ Xs ∩ BX2+⌊n/2⌋(0, ρWD) we have that the flattening map Fη defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is a smooth diffeomorphism from Ω to Ω[η] that extends to a Cs+2−⌊n/2⌋ diffeomorphism from Ω to Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For every m ∈ N, Ψ ∈ sT m 1,2+⌊n/2⌋(BX2+⌊n/2⌋(0, ρWD) × Wm+1+⌊n/2⌋ × R+, Om;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Pm), where we have denoted Om = {Xs × Wm−1+s × R}s∈N and Pm = {Ys × Wm+s−1 × R}s∈N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 63 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We set ρWD = min{ρprin, ρbulk, ρsurf} and apply Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 and Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This immediately gives the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the first item, we note that Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11 guarantees that Jη > 0 and Jη, 1/Jη ∈ L∞(Ω), which means that Fη is a continuous bijection from Ω to Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, the identity ∂n(Fη · en) = Jη = det(∇Fη) and the inverse function theorem guarantee that Fη : Ω → Ω[η] is a smooth diffeomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The regularity of Fη : Ω → Ω[η] and its inverse now follow from Sobolev embeddings and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the case m = 2, the second item of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 is stating that Ψ ∈ sT 2 1,2+⌊n/2⌋ for the Banach scales EEE and FFF, which are introduced in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Derivative splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn our attention to the study of the derivative of the map Ψ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' When written in full, it is rather complicated, so our focus now is to identify a principal part and handle the remainder terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρWD), (T0, G0, F0) ∈ W3+⌊n/2⌋, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) (q, u, η) ∈ X2+⌊n/2⌋, (T , G, F) ∈ W2+⌊n/2⌋, γ0 ∈ R+, γ ∈ R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and write the 6−tuple θ0 = (q0, u0, η0, T0, G0, F0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) The derivative of the map Ψ has the formula DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = � DΨ(q0, u0, η0, γ0)(q, u, η, γ) + DΦ(θ0)(q, u, η, T , G, F), T , G, F, γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) We will decompose the above into a principal part, A, and remainder terms, P, Q, and R (these symbols will come equipped with various adornments, but for brevity we will often refer to them without these in the main text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The derivative DΨ is split as follows: DΨ(w0, γ0)(q, u, η, γ) = w0,γ0 A (q, u, η) + w0,γ0 P (q, u, η, γ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) where for brevity we define the triple w0 = (q0, u0, η0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) The A piece is meant to be as close as possible to DΨ(0, 0, 0, γ0), but to retain the entirety of the structure responsible for derivative loss in the continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To that end, we set w0,γ0 A (q, u, η) = � w0 A1(q, u, η), γ0 A2(q, u, η), γ0 A3(q, u, η) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) where w0 A1(q, u, η) = ∇ · (ϱu) + ∇ · (vw0(q + gη)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) γ0 A2(q, u, η) = DΨ2(0, 0, 0, γ0)(q, u, η, 0) = −γ2 0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and γ0 A3(q, u, η) = DΨ3(0, 0, 0, γ0)(q, u, η, 0) = −(ϱq − γ0Sϱu)en − ς∆∥ηen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Here vw0 : Ω → Rn is the vector field vw0 = M(H−1)′(q0, u0, η0) = (H−1)′(−gidRn · en + q0 + g(I − E)η0)(u0 − Mη0e1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) where the M notation is from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 and (H−1)′ refers to the derivative of the inverse enthalpy (see equation (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that since ϱ(y) = H−1(−gy) for y ∈ [0, b] we have that v0 = −(H−1)′(−gidRn · en)e1 = g−1ϱ′e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) The P piece is, of course, the remainder of the derivative of Ψ and has the formula w0,γ0 P (q, u, η, γ) = � w0 P 1(q, u, η), w0,γ0 P 2 (q, u, η, γ), w0,γ0 P 3 (q, u, η, γ) � (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) 64 NOAH STEVENSON AND IAN TICE where w0 P 1(q, u, η) = ∇ · ((σq0,η0 − ϱ)(u − ˙M[η]e1)) − g∇ · ((vw0 − ϱ′e1/g)Eη) − ∇ · (ϱ ˙M[η]e1) − ∇ · (ϱ′Eηe1), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) ˙M[η] : Ω → Rn×n is the matrix field given by ˙M[η] = �∂nEηI(n−1)×(n−1) 0(n−1)×1 −E(∇∥η) 0 � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) and w0,γ0 P j (q, u, η, γ) = (DΨj(w0, γ0) − DΨj(0, 0, 0, γ0))(q, u, η, 0) + DΨj(w0, γ0)(0, 0, 0, γ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) for j ∈ {2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Before we define a similar decomposition of the DΦ piece of DΨ, we will prove some basic properties about the A + P decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we have the following lemma which, aside from the final item, is a reprise of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 (Properties of the derivative loss vector field).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and w0 = (q0, u0, η0) ∈ BX1+⌊n/2⌋(0, ρ) ∩ X∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Define the vector field vw0 : Ω → Rn as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a decomposition vw0 = g−1ϱ′e1 + v(1) q0,u0,η0 + v(2) η0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) The vector field v(1) q0,u0,η0 has vanishing normal trace, Tr∂Ω(v(1) q0,u0,η0 · en) = 0, satisfies the inclusion v(1) q0,u0,η0 ∈ H∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), and obeys the estimates ∥v(1) q0,u0,η0∥H1+s ≲ � ρ if s = 1 + ⌊n/2⌋, ∥q0, u0, η0∥Xs if s > 1 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) (2) The vector field v(2) η0 is parallel to e1, satisfies the inclusion v(2) η0 · e1 ∈ W ∞,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H0 (rn−1)(Σ)) �→ W ∞,∞(Ω), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) and obeys the estimates ∥v(2) η0 · e1∥W s,∞((0,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='H0 (rn−1)(Σ)) ≲ ρ for s ⩾ 1 + ⌊n/2⌋, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) where rn−1 ∈ N is from the second item of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 with d = n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) We have the inclusion ∇ · vw0 ∈ H∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) along with the estimates ∥∇ · vw0∥Hs ≲ � ρ if s = 1 + ⌊n/2⌋, ∥q0, u0, η0∥Xs if s > 1 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) (4) If s ∈ N, q ∈ H1+s(Ω), and η ∈ H1+s(Σ), then we have that ∇ · (vw0(q + gη)) ∈ ˆH1+s(Ω), where the latter space is defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) (see also Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), as well as the estimates ∥∇ · (vw0(q + gη))∥ ˆHs ≲ ∥q, η∥H1+s×H1+s + � 0 if s ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) In the above, the implicit constants depend on the dimension, the physical parameters, s, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 65 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We apply Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 to obtain identity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) with ϱ′e1/g = M(H−1)′(0, 0, 0), v(1) q0,u0,η0 = M(1) (H−1)′(q0, u0, η0), and v(2) η0 = e1 · M(2) (H−1)′(Π1 Lη0)e1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The qualitative smoothness assertions in the first, second, and third items now follow immediately the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will prove the quantitative bounds via the fundamental theorem of calculus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since M(1) (H−1)′(0, 0, 0) = 0, we have that v(1) q0,u0,η0 = � 1 0 DM(1) (H−1)′(tq0, tu0, tη0)(q0, u0, η0) dt, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) and hence, by using the strong smooth tameness assertions in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 and the derivative estimates on smooth tame maps from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, we find that for s ⩾ 1 + ⌊n/2⌋, ∥v(1) q0,u0,η0∥H1+s ⩽ � 1 0 ∥DM(1) (H−1)′(tq0, tu0, tη0)[q0, u0, η0]∥H1+s dt ≲ ∥q0, u0, η0∥Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) The same technique proves the quantitative bounds asserted in the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we justify the divergence estimates of the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17), we compute that ∇ · vw0 = ∇ · v(1) q0,u0,η0 + ∂1(v(2) η0 · e1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) From this identity, the quantitative estimates of the first and second items, and the properties of band limited anisotropic Sobolev spaces from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we deduce that for s ⩾ 1 + ⌊n/2⌋, ∥∇ · v(1) q0,u0,η0∥Hs ≲ ∥v(1) q0,u0,η0∥H1+s ≲ ∥q0, u0, η0∥Xs (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) and ∥∂1(v(2) η0 · e1)∥H1+s ≲ ∥v(2) η0 · e1∥W 1+s,∞((0,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='H0 (rn−1)(Σ)) ≲ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) These complete the proof of the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, to prove the fourth item we first note that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) yields the formula ∇ · (vw0(q + gη)) = ∇ · (vw0(q + gΠ1 Hη)) + g∇ · (v(1) q0,u0,η0Π1 Lη) + g∂1(e1 · v(2) η0 Π1 Lη) + ϱ′∂1Π1 Lη = I + II + III + IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) For I we use Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, the first and second items above, and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 to estimate ∥I∥Hs ≲ ∥vw0(q + gΠ1 Hη)∥H1+s ⩽ ∥v(1) q0,u0,η0(q + gΠ1 Hη)∥H1+s + ∥(v(2) η0 + ϱ′e1/g)(q + gηH)∥H1+s ≲ ∥q, η∥H1+s×H1+s + � 0 if s ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) On the other hand, since Tr∂Ω(vq0,u0,η0 · en) = 0, we use estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 to bound �� b 0 I(·, y) dy � ˙H−1 ≲ ∥vw0(q + gΠ1 Hη)∥L2 ≲ ∥q, η∥L2×H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) we deduce that ∥I∥ ˆHs ≲ ∥q, η∥H1+s×H1+s + � 0 if s ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥Xs⟩∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) For II, we use the first item along with Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 again to estimate ∥II∥Hs ≲ ∥v(1) q0,u0,η0∥L2∥Π1 Lη∥W 1+s,∞ + ∥v(1) q0,u0,η0∥H1+s∥Π1 Lη∥L∞ ≲ ∥q0, u0, η0∥Xs∥Π1 Lη∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) On the other hand, since Tr∂Ω(v(1) q0,u0,η0 · en) = 0, we have �� b 0 II(·, y) dy � ˙ H−1 ≲ ∥v(1) q0,u0,η0Π1 Lη∥L2 ≲ ∥Π1 Lη∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) 66 NOAH STEVENSON AND IAN TICE From (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) we deduce that ∥II∥ ˆHs is controlled by the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now estimate III, using the second item and the algebraic properties of the anisotropic Sobolev spaces from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 to get ∥III∥Hs ≲ ∥e1 · v(2) η0 Π1 Lη∥W s,∞((0,b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='H0 (rn−1+1)(Σ)) ≲ ∥e1 · v(2) η0 ∥W s,∞H0 (rn−1)∥Π1 Lη∥H0 ≲ ∥Π1 Lη∥H0 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and �� b 0 III(·, y) dy � ˙H−1 ⩽ ���Π1 Lη � b 0 e1·v(2) η0 (·, y) dy ��� H0 ≲ � b 0 ∥e1·v(2) η0 (·, y)∥H0 dy·∥Π1 Lη∥H0 ≲ ∥Π1 Lη∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) The bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) imply that ∥III∥ ˆHs is controlled by the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The norm ∥IV∥ ˆHs is trivially controlled by the same quantity thanks to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof of the fourth item is then complete upon synthesizing these bounds on I, II, III, and IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now introduce a new scale of adapted spaces pertinent to the analysis of the A piece of DΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As we will see, these are the natural domains on which A is an isomorphism of Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To define the spaces, we simply build an extra condition into the spaces {Xs}s∈N, which we recall were defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), to create a new family in which the effect of the derivative loss is mitigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' What is notable is that these domains depend on the background point w0 = (q0, u0, η0) in a non-trivial way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ {−1, 0} ∪ R+ and (q0, u0, η0) as in the hypotheses of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 we define the space q0,u0,η0 Xs = {(q, u, η) ∈ Xs : ∇ · (vw0q) ∈ H1+s(Ω)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) and equip it with the norm ∥q, u, η∥q0,u0,η0 Xs = � ∥q, u, η∥2 Xs + ∥∇ · (vw0q)∥2 H1+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A simple modification of the proof of the first item of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 reveals that the spaces q0,u0,η0 Xs are Hilbert.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following result captures that the A + P decomposition of DΨ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) is such that A contains all of the derivative loss and P is a small correction term without derivative loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that the conclusions in what follows are stronger than what can be deduced from a direct application of the smooth-tameness verification result, namely Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 (Properties of the A+P decomposition of DΨ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let ρ ∈ R+ and w0 = (q0, u0, η0) be as in the hypotheses of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 and let I ⋐ R+ be an interval with γ0 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) For every s ∈ N, the map w0,γ0 A : q0,u0,η0 Xs → Ys is well-defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For every N ∋ s ⩾ 1 + ⌊n/2⌋, the map w0,γ0 P : Xs × R → Ys is well-defined, continuous, and obeys the estimates �� w0,γ0 P (q, u, η, γ) �� Ys ≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥X1+s∥q, u, η, γ∥X⌊n/2⌋×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) The implicit constants depend on the dimension, the physical parameters, s, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Upon inspecting the components of w0,γ0 A in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), we see that the second and third are trivially well-defined and continuous, so to prove the first item we heed to Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and reduce to showing that the map q0,u0,η0 Xs ∋ (q, u, η) �→ w0 A1(q, u, η) ∈ ˆH1+s(Ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) COMPRESSIBLE TRAVELING WAVES 67 is well-defined and continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a consequence of the fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, the identity � b 0 ∇ · (ϱu)(·, y) dy = (∇∥, 0) · � b 0 (ϱu)(·, y) dy − ϱ(b)∂1η (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) (which is true by virtue of the boundary condition build into the Xs spaces, as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3)), and the norm equivalence (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, we see that the operator in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) continuously maps into ˆH0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' On the other hand, by employing the first, second, and third items of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, the equality ∇ · (vw0(q + gη)) = g∇ · (vw0η) + ∇ · (vw0q), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) and the definition of the norm in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37), we readily deduce that w0 A1 maps boundedly into H1+s(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn to the proof of the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) that w0,γ0 P has components w0,γ0 P i for i ∈ {1, 2, 3}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the remainder of the proof into three steps, each of which handles one of these components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Estimates on w0 P 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first aim to exploit some hidden cancellation by observing that the sum of the final two terms in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, since ˙M[η]e1 = ∂nEηe1 − ∂1Eηen we have that ∇ · ( ˙M[η]e1) = 0, and therefore − ∇ · (ϱ′Eηe1) − ∇ · (ϱ ˙M[η]e1) = −ϱ′∂1Eη − ϱ′(∂nEηe1 − ∂1Eηen) · en = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) Thus, we have the more accessible formula w0 P 1(u, η) = ∇ · ((σq0,η0 − ϱ)(u − ˙M[η]e1)) − g∇ · ((vw0 − ϱ′e1/g)Eη) = I + II, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) and we will deal with I and II separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The term I is handled with the Taylor expansion trick of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' we decompose σq0,η0 − ϱ = σ(1) q0,η0 + σ(2) η0 , where σ(1) q0,η0 = N (1) H−1(q0, η0) and σ(2) η0 = N (2) H−1(Π1 Lη).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) Employing the above decomposition and recalling (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15), we may further rewrite I = ∇ · ((σq0,η0 − ϱ)(u + ∂1Eηen) − σ(1) q0,η0∂nEηe1 − σ(2) η0 ∂nE0Π1 Hηe1) − ∂1(σ(2) η0 Π1 Lη)/b = ∇ · I1 + ∂1I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) To handle ∇·I1 we will first derive a Sobolev estimate for I1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, thanks to multiple applications of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 and Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 , we are free to bound, for any s ∈ N, ∥I1∥Hs ≲ ∥σ(1) q0,η0, σ(2) η0 ∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥u, η∥Hs×H1/2+s + � 0 if s ⩽ 1 + ⌊n/2⌋, ∥σ(1) q0,η0, σ(2) η0 ∥Hs×W s,∞∥u, η∥H1+⌊n/2⌋×H3/2+⌊n/2⌋ if 1 + ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) Then we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 combined with the fundamental theorem of calculus as in the first part of the proof of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to acquire the bounds ∥σ(1) q0,η0, σ(2) η0 ∥Hs×W s,∞ ≲ � ∥q0, u0, η0∥X1+⌊n/2⌋ if s ⩽ 2 + ⌊n/2⌋, ∥q0, u0, η0∥Xs−1 if 2 + ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) Upon synthesizing the bounds (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47), we find that ∥I1∥Hs ≲ ρ∥q, u, η∥Xs−2 + � 0 if s ⩽ 2 + ⌊n/2⌋, ∥q0, u0, η0∥Xs−1∥q, u, η∥X⌊n/2⌋−1 if 2 + ⌊n/2⌋ < s, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) 68 NOAH STEVENSON AND IAN TICE which is the aforementioned Sobolev estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then, in light of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48), the fact that Tr∂Ω(I1·en) = 0, and divergence - normal trace compatibility estimates from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we arrive at the bound ∥∇ · I1∥ ˆH1+s ⩽ ∥I1∥H2+s ≲ ρ∥q, u, η∥Xs + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋−1 if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) Having dispatched I1, we turn our attention to the I2 term in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The product σ(2) η0 Π1 Lη is handled via Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 and the algebra properties of band limited members of H0(Σ) enumerated in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, applied to σ(2) η0 (·, y)Π1 Lη for y ∈ [0, b]: ∥∂1I2∥ ˆH1+s ≲ � ∂1 � Π1 Lη � b 0 σ(2) η0 (·, y) dy �� ˙H−1 + max 0⩽j⩽1+ν sup y∈[0,b] ∥∂1∂j nI2(·, y)∥H1+s(Rn−1) ≲ ��� � b 0 σ(2) η0 (·, y) dy ��� H0∥Π1 Lη∥H0 + max 0⩽j⩽1+s sup y∈[0,b] [∂1∂j n(σ(2) η0 (·, y)Π1 Lη)] ˙H−1 ≲ max 0⩽j⩽1+s sup y∈[0,b] ∥∂j nσ(2) η0 (·, y)∥H0∥Π1 Lη∥H0 ≲ ρ∥Π1 Lη∥H0, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) where in deducing the final inequality we again use the tame mapping properties of σ(2) η0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, combined with the fundamental theorem of calculus as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the handling of I2, and hence of I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we consider II from equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the decomposition of vw0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, II has the further decomposition II = −g∇ · ((v(1) q0,u0,η0 + v(2) η0 )E0Π1 Hη + v(1) q0,u0,η0EΠ1 Lη) − g∂1(idRn · env(2) η0 · e1Π1 Lη)/b = ∇ · II1 + ∂1II2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) As in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48), we use Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 and the mapping properties of v(1) q0,u0,η0 and v(2) η0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to see that for any s ∈ N, ∥II1∥Hs ≲ ∥v(1) q0,u0,η0, v(2) η0 ∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥η∥Hs−1/2 + � 0 if s ⩽ 1 + ⌊n/2⌋, ∥v(1) q0,u0,η0, v(2) η0 ∥Hs×W s,∞∥η∥H1/2+⌊n/2⌋ if 1 + ⌊n/2⌋ < s, ≲ ρ∥q, u, η∥Xs−3 + � 0 if s ⩽ 2 + ⌊n/2⌋, ∥q0, u0, η0∥Xs−1∥q, u, η∥X⌊n/2⌋−2 if 2 + ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) In turn, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52), the fact that Tr∂Ω(II1 · en) = 0, and Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 provide the bound ∥∇ · II1∥H1+s ≲ ∥II1∥H2+s ≲ ρ∥q, u, η∥Xs−1 + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋−1 if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) For II2, we argue as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50), estimating the product (idRn ·en)(v(2) η0 ·e1)ηL by invoking the second item from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 as well as the algebra properties of band-limited members of H0(Rn−1) from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this results in the bound ∥∂1II2∥ ˆH1+s ≲ ρ∥Π1 Lη∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) Finally, we synthesize (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) to get the w0 P 1 estimate �� q0,u0,η0 P 1 (u, η) �� ˆH1+s ≲ ρ∥q, u, η∥Xs + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋−1 if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55) COMPRESSIBLE TRAVELING WAVES 69 Step 2: Estimates on w0,γ0 P 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is simpler than the previous step, as we can use the tame calculus conclusions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that w0,γ0 P 2 (q, u, η, γ) = (DΨ2(w0, γ0) − DΨ2(0, 0, 0, γ0))(q, u, η, 0) + DΨ2(w0, γ0)(0, 0, 0, γ) = J1 + J2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) where Ψ2 is the nonlinear map defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may use that Ψ2 is C2 paired with the fundamental theorem of calculus to write J1 = � 1 0 D2Ψ2(tw0, γ0)((q0, u0, η0, 0), (q, u, η, 0)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57) Hence, by the strong tame estimates on the second derivative from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11, the log-convexity of the norm of the Xs-spaces (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), and Young’s inequality we get that for ν ⩾ 1+⌊n/2⌋: ∥J1∥Hs ≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥Xs∥q, u, η∥X1+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58) On the other hand, by inspection we see that DΨ2(0, γ0)(0, 0, 0, γ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus by a similar fundamental theorem of calculus argument, we have that J2 = � 1 0 D2Ψ2(tw0, γ0)((w0, 0), (0, 0, 0, γ)) dt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59) This lets us use the strong tame estimates on the second derivative again to bound ∥J2∥Hs ≲ |γ|∥q0, u0, η0∥Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) By combining equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60), we obtain the stated bounds for w0,γ0 P 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: Estimates on w0,γ0 P 3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We perform the same analysis, utilizing the fundamental theorem of calculus and the tame C2-estimates as in the second step, but this time we employ Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12 in place of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The remainder of this subsection is devoted to the decomposition of the DΦ piece of DΨ, which is derived from equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), into subcomponents Q and R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The key observation here is that Φ : BX2+⌊n/2⌋(0, ρWD) × W3+⌊n/2⌋ → Y0 is linear in the second factor, and hence for θ0 as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) we can write, DΦ(θ0)(q, u, η, T , G, F) = θ0Q(q, u, η) + q0,u0,η0 R (T , G, F) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61) where θ0Q(q, u, η) = DΦ(θ0)(q, u, η, 0, 0, 0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) and q0,u0,η0 R (T , G, F) = DΦ(q0, u0, η0, 0, 0, 0)(0, 0, 0, T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63) The following result reveals the point of the Q + R decomposition of DΦ: Q is small when the background is small, and R is independent of (q, u, η) and thus enjoys a useful decoupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In contrast with the A + P decomposition (Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21), the next result is less delicate and only relies on the smooth-tameness results from Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22 (Properties of the Q + R decomposition of DΦ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where ρWD is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and θ0 = (q0, u0, η0, T0, G0, F0) ∈ (BX2+⌊n/2⌋(0, ρ) × BW3+⌊n/2⌋(0, ρ)) ∩ (X∞ × W∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) The following hold for N ∋ s ⩾ 2 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 70 NOAH STEVENSON AND IAN TICE (1) The map θ0Q : Xs → Ys is well-defined, continuous, and obeys the estimate �� θ0Q(q, u, η) �� Ys ≲ ρ∥q, u, η∥Xs + ∥θ0∥Xs×W1+s∥q, u, η∥X2+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='65) (2) The map q0,u0,η0 R : W1+s → Ys is well-defined, continuous, and obeys the estimate �� q0,u0,η0 R (T , G, F) �� Ys ≲ ∥T , G, F∥W1+s + ⟨∥q0, u0, η0∥Xs⟩∥T , G, F∥W3+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66) In the above the implicit constants depend on the dimension, the physical parameters, s, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That θ0Q is well-defined, continuous, and, when ρ = ρWD and s ⩾ 2 + ⌊n/2⌋, obeys a tame estimate of the form �� θ0Q(q, u, η) �� Ys ≲ ∥q, u, η∥Xs + ⟨∥θ0∥Xs×W1+s⟩∥q, u, η∥X2+⌊n/2⌋ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67) follows from the formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62), the tame smoothness assertions of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15, and the tame estimates on derivatives from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We can improve the above estimate by leveraging the fact that for fixed (q0, u0, η0) the map (T0, G0, F0) �→ θ0Q is linear;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' indeed, by arguing as in the proof of Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, we deduce from estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67) that actually �� θ0Q(q, u, η) �� Ys ≲ ∥T0, G0, F0∥W3+⌊n/2⌋∥q, u, η∥Xs + ∥θ0∥Xs×W1+s∥q, u, η∥X2+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='68) The first item now follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The assertions and estimate of the second item are a direct consequence of formula (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63), the tame smoothness assertions of Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15,and the tame estimates on derivatives from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Prelude to linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that Ψ is the nonlinear operator associated with the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our goal now is to prove that this map satisfies the hypotheses of the inverse function theorem, which are enumerated in Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20, Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, and Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The previous analysis of this section, in particular Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, shows that the ‘nonlinear hypotheses,’ other than a trivial issue with the domain and the first item of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20, of this inverse function theorem are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It only remains, then, to show that the ‘linear hypotheses’ given in the third item of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we aim to prove the following assertion about the derivative, DΨ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given N ∋ s ⩾ 2 + ⌊n/2⌋ and an interval I ⋐ R+, there exists an existence and estimates parameter ρEE(s) ∈ (0, ρWD] (also depending on I), where ρWD ∈ R+ is from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, with the property that whenever (recall that θ0 is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3)) (θ0, γ0) ∈ (BX2+⌊n/2⌋(0, ρEE(s)) × BW3+⌊n/2⌋(0, ρEE(s)) × I) ∩ (X1+s × W2+s × R+) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (g, f, k, T , G, F, γ) ∈ Ys × W1+s × R, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) there exists a unique (q, u, η) ∈ Xs solving DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' moreover, if we set Ξ = (g, f, k, T , G, F, γ) and Θ = (q, u, η, T , G, F, γ) then the solution tuple obeys the tame estimate ∥Θ∥Xs×W1+s×R ≲ ∥Ξ∥Ys×W1+s×R + ⟨∥θ0∥X1+s×W2+s⟩∥Ξ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) for an implicit constant depending only on the dimension, the various physical parameters, s, I, and ρEE(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the analysis of the previous subsection, namely Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22, we have that the behavior of DΨ is governed by the principal part operator A defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus our above goal is essentially achieved, modulo minor supplementary analysis, as soon as we prove the following assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For any s ∈ N and γ0 ∈ I ⋐ R+an interval, there exists an existence and COMPRESSIBLE TRAVELING WAVES 71 estimates principal part parameter ρEEP(s) ∈ (0, ρWD] (also depending on I) with the property that whenever w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρEEP(s)) ∩ X∞ and (g, f, k) ∈ Ys, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) there exists a unique (q, u, η) ∈ Xs solving w0,γ0 A (q, u, η) = (g, f, k);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' moreover, the solution obeys the tame estimates ∥q, u, η∥Xs ≲ ∥g, f, k∥Ys + � 0 if s ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+s⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Again we allow for implied constants to depend on the dimension, the physical parameters, s, I, and ρEEP(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will see that a necessary condition for the estimate (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) to hold is that the operator w0,γ0 A : q0,u0,η0 Xs → Ys is a Banach isomorphism, where we recall the adapted spaces are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This expresses one of the core difficulties of the linear analysis: we either have that the family of operators { w0,γ0 A } ⊂ L(X1+s;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Ys) are defined on the common Banach space X1+s but not isomorphisms, or else each individual operator w0,γ0 A is defined on a larger adapted Banach space q0,u0,η0 Xs making it an isomorphism onto Ys, but the spaces { q0,u0,η0 Xs } are inequivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consequently, we cannot port invertibility from one operator to the next via the method of continuity, even if we were to establish the estimates of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) a priori.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We overcome this difficulty via an elliptic regularization procedure which we now describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For m ∈ N+ we define the 2mth-order linear elliptic differential operator Lm = (−1)m n � j=1 ∂2m j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) We will consider a sequence of operators obtained by adding a vanishing contribution of this operator to the continuity equation component of A and also adding a vanishing contribution of (−∆∥)m−1/4 to the kinematic boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These new operators have the following domains: for m, N ∈ N+ and s ∈ {−1} ∪ N we define the space Xs m,N = {(q, u, η) ∈ Xs : TrΣ0(u) = 0, TrΣ(u · en) + ∂1η = N−1(−∆∥)m−1/4η (q, η) ∈ H1+s+2m(Ω) × H1+s+2m(Σ), Tr∂Ω(∂m n q) = · · · = Tr∂Ω(∂2m−1 n q) = 0} (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and endow it with the norms ∥q, u, η∥Xs m,N = � ∥q, u, η∥2 Xs + N−2∥q, η∥2 H1+s+2m×H1+s+2m, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and ∥q, u, η∥q0,u0,η0 Xs m,N = � ∥q, u, η∥2 Xs m,N + ∥∇ · (vq0,u0,η0q)∥2 H1+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) We note that, in light of the fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, ∥·∥q0,u0,η0 Xs m,N is a norm equivalent to ∥·∥Xs m,N , with equivalence constants depending on s, m, N, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ N we then define the regularized principal part operator w0,γ0 Am,N : Xs m,N → Ys (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) via w0,γ0 Am,N(q, u, η) = (N−1Lm(q + gη) + w0,γ0 A1 (q, u, η), γ0 A2(q, u, η), γ0 A3(q, u, η)), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) where the Ai terms are the same as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 72 NOAH STEVENSON AND IAN TICE The upshot is that the operators { w0,γ0 Am,N} are all contained in the space L(Xs m,N, Ys) and, as we will see, obey a family of nice a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus the method of continuity is available for the regularized operators’ existence theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once we have existence for the regularized operators, we would like to show that we can pass to the limit as N → ∞ and obtain existence for A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is achieved by carefully proving N-independent a priori estimates for Am,N by inductively working up from the base case of a priori estimates for weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This inductive estimate procedure will also be done in tandem with the operator A to obtain the sought-after a priori estimates of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a result, we will need to set notation for weak formulations of the operators A and Am,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First we recall the definition of the space 0H1 from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The operator associated with the weak formulation of the momentum equation is γ0 I : X−1 → (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) defined by ⟨ γ0 I (q, u, η), w⟩(0H1)∗,0H1 = � Ω −γ2 0ϱ∂1u · w − q∇ · (ϱw) + gϱ∇η · w + γ0Sϱu : ∇w − ς⟨∆∥η, TrΣ(w · en)⟩H−1/2,H1/2 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) for (q, u, η) ∈ X−1 and w ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next define a family of operators associated with the weak formulation of the full problem by setting w0,γ0 J : q0,u0,η0 X−1 → Y−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) via w0,γ0 J (q, u, η) = � ∇ · (ϱu) + ∇ · (vw0(q + gη)), γ0 I (q, u, η) � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) where we recall that the vector field vw0 is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Furthermore, given τ ∈ [0, 1] and m, N ∈ N+ with m ⩾ 2 we define the map w0,γ0 J τ m,N : X−1 m,N → Y−1 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) via w0,γ0 J τ m,N(q, u, η) = (∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη), γ0 I (q, u, η)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) We now give a basic result on the well-definedness and continuity of these operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23 (Well-definedness check for linear analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let ρ ∈ R+ and w0 = (q0, u0, η0) be as in the hypotheses of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let I ⋐ R+ be an interval with γ0 ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) The weak formulation operator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) is well-defined and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) For N ∋ m ⩾ 2, N ∈ N+, and τ ∈ [0, 1], the regularized weak formulation operator of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) is well-defined and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) For N ∋ m ⩾ 2 and N ∈ N+ the regularization of the principal part operator defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) is well-defined and bounded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the first item we see that it suffices to check that the first component of γ0,w0 J is well- defined and maps boundedly into ˆH0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we can argue in a manner similar to the first part of the proof of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, where we studied (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The only difference is in handling the ∇ · (vw0q) term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That this term maps boundedly into L2(Ω) is immediate from the definition of the norm on q0,u0,η0 X−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then use Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, paired with the fact that vw0 ∈ L∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), to obtain that q �→ � b 0 ∇ · (vw0q)(·, y) dy maps boundedly into ˙H−1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 73 The second and third items follow as soon as we check that X−1 m,N ∋ (q, u, η) �→ ∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) ∈ ˆH0(Ω) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) is a well-defined and bounded linear map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We handle the τ∇ · (vw0(q + gη)) term in the same way that we handled (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) from Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the N−1Lm(q + gη) term we see that it obviously maps into L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To acquire the divergence compatibility condition, we use the Neumann boundary conditions built into the domain X−1 m,N to compute � b 0 1 N Lm(q + gη)(·, y) dy = 1 N n−1 � j=1 ∂2m j � gbη + � b 0 q(·, y) dy � , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) but since m ⩾ 1, this provides the bound �� b 0 1 N Lm(q + gη)(·, y) dy � ˙H−1 ≲ 1 N (∥∇∥η∥H2m−2 + ∥q∥H2m−1) ≲ ∥q, u, η∥X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) Finally we handle the ∇ · (ϱu) term of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Evidently, this maps into L2(Ω), leaving us to check the divergence compatibility condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By integrating, we learn that � b 0 ∇ · (ϱu)(·, y) dy = (∇∥, 0) · � b 0 (ϱu)(·, y) dy − ϱ(b)∂1η + ϱ(b) N (−∆∥)m−1/4η, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) and since m ⩾ 2 this yields the estimate �� b 0 ∇ · (ϱu)(·, y) dy � ˙H−1 ≲ ∥u∥L2 + ∥η∥H0 + 1 N ∥∇∥η∥H2m−5/2 ≲ ∥q, u, η∥X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) □ We now consider the relationship between the weak and strong operators, J , J τ m,N and A, Am,N, by introducing a functional that maps the strong form of the data to the weak formulation of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For s ∈ N we define the strong-to-weak data map K : Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) given by ⟨K (f, k), w⟩(0H1)∗,0H1 = � Ω f · w + � Σ k · w, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) where w ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), (f, k) ∈ Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2+s(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), and s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24 (Strong and weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under the hypotheses of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23, the following hold for (g, f, k) ∈ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If (q, u, η) ∈ q0,u0,η0 X0 , then w0,γ0 A (q, u, η) = (g, f, k) if and only if w0,γ0 J (q, u, η) = (g, K (f, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) If m, N ∈ N+, m ⩾ 2, and (q, u, η) ∈ X0 m,N, then w0,γ0 Am,N(q, u, η) = (g, f, k) if and only if w0,γ0 J 1 m,N(q, u, η) = (g, K (f, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These follow directly from integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ This section is concluded with the following simple lemma on log-convexity, which we recall is defined in Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 74 NOAH STEVENSON AND IAN TICE Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25 (Log-convexity of the norms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following Banach scales are log-convex for the stated norms: � Xs, ∥·∥Xs� s∈N, �q0,u0,η0 Xs , ∥·∥q0,u0,η0 Xs � s∈N, � Xs m,N, ∥·∥Xs m,N � s∈N, � Xs m,N, ∥·∥q0,u0,η0 Xs m,N � s∈N, � Ys, ∥·∥Ys� s∈N, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) where we take m, N ∈ N and (q0, u0, η0) be as in the hypotheses of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 shows that the anisotropic Sobolev spaces {Hs(Rd)}s∈N are log-convex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The result then follows from the log-convexity of standard Sobolev spaces (see Theorem D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and the evident preservation of log-convexity under products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of steady transport equations and their regularizations In the previous section we introduced the regularized principal part operator w0,γ0 Am,N (defined in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10)), which perturbs the operator w0,γ0 A (defined in equation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7)) by adding a multiple of the elliptic operator Lm (defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6)) to the part of w0,γ0 A corresponding to the linearized continuity equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this section we focus our attention on solutions to such regularized steady transport equations, with the aim of deriving precise estimates that are independent of the regularization parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These will play an essential role in our subsequent linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Along the way, we also develop some estimates of solutions to the standard steady transport equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To be precise, we now study equations of the type αf + ∇ · (vf) + εLmf = g in Ω, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where α, g : Ω → R are given functions, v : Ω → Rn is a given vector field, ε ⩾ 0 is small, Lm is the linear elliptic operator given by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), and f : Ω → R is the unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The key results of this section are Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In what follows, a key player is a vector field X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) such that Tr∂Ω(X · en) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume X0 ∈ H∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), X1 ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), ρmax ∈ R+ is arbitrary but fixed, 0 < ρ ⩽ ρmax, r ∈ N, X = X0 + X1, and (DX0, DX1) ∈ BHr(0, ρ) × BW r,∞(0, ρ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) We will frequently reference this equation when we need to quantify a vector field of this type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Preliminary tame estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given m ∈ N+, we define a bilinear form associated to Lm via Bm : Hm(Ω) × Hm(Ω) → R via Bm(ϕ0, ϕ1) = n � j=1 � Ω ∂m j ϕ0∂m j ϕ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) We have recorded a number of basic properties of Bm in Appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our first result here considers an estimate for Bm in which we are able to a save a derivative thanks to integration by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (A bilinear estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ϕ ∈ Hm(Ω), X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies Tr∂Ω(X · en) = 0 as well as (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, and that ∇ · (Xϕ) ∈ Hm(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we have the estimates |Bm(ϕ, ∇ · (Xϕ))| ≲ ρ∥ϕ∥2 Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and |Bm(ϕ, ∇ · (Xϕ))| ≲ ρ∥ϕ∥2 Hm + ∥ϕ∥Hm � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) COMPRESSIBLE TRAVELING WAVES 75 Here the implicit constants depend on m, ρmax, and the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since ϕ, ∇ · (Xϕ) ∈ Hm(Ω) and X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), we can apply the Leibniz rule to see that ∇ · (X∂m j ϕ) = ∂m j ∇ · (Xϕ) − m � k=1 �m k � ∇ · (∂k j X∂m−k j ϕ) ∈ L2(Ω), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and so ∂m j ϕ ∈ H0 X(Ω), the space defined by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) in Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By introducing commutators and employing Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9, we obtain the identity Bm(∇ · (Xϕ), ϕ) = n � j=1 � Ω �∇ · X 2 ∂m j ϕ + ∇ · ([∂m j , X]ϕ) � ∂m j ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Hence, we have the estimate |Bm(∇ · (Xϕ), ϕ)| ≲ � ∥DX∥L∞∥ϕ∥Hm + n � j=1 ∥[∂m j , X]ϕ∥H1 � ∥ϕ∥Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) According to Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, we have the estimate ∥[∂m j , X0]ϕ∥H1 ≲ ∥∂jX0∥H1+⌊n/2⌋∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥∂jX0∥Hm∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and ∥[∂m j , X1]ϕ∥H1 ≲ ∥∂jX1∥L∞∥ϕ∥Hm + ∥∂jX1∥W m,∞∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) Estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) combine to show that n � j=1 ∥[∂m j , X]ϕ∥H1 ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) This establishes (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now continue to prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the latter case of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), we use interpolation and Young’s inequality to bound ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ ≲ ρ− 1+⌊n/2⌋ m−1−⌊n/2⌋ ∥DX0, DX1∥ m m−1−⌊n/2⌋ Hm×W m,∞∥ϕ∥L2 + ρ∥ϕ∥Hm ≲ρmax ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2 + ρ∥ϕ∥Hm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Together, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) provide the bound n � j=1 ∥[∂m j , X]ϕ∥H1 ≲ ρ∥ϕ∥Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Now we return to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) and plug in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) to derive the estimate |Bm(∇ · (Xϕ), ϕ)| ≲ ρ∥ϕ∥2 Hm + ρ−1−⌊n/2⌋⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) which is (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ For the next two results we use the notation ∇Xϕ = X · ∇ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following rather technical lemma considers traces of the normal derivatives of ∇Xϕ when X has vanishing normal trace and ϕ has some vanishing normal derivative traces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The point is that the higher norms on normal derivative traces depend only on X and a lower norm of ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 76 NOAH STEVENSON AND IAN TICE Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (A trace estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies Tr∂Ω(X · en) = 0 and decomposes as X = X0 + X1 with DX0 ∈ H∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n) and DX1 ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume additionally that ϕ ∈ H2m+1(Ω) and satisfies the Neumann conditions ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) The we have the following normal derivative estimates for ℓ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m − 1}: ∥Tr∂Ω(∂m+ℓ n ∇Xϕ)∥H−1/2 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and ∥Tr∂Ω(∂m+ℓ n ∇Xϕ)∥H−1/2 ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞⟩∥ϕ∥Hm + ⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) Here the implicit constants depend on the domain, the dimension, m and ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by computing the argument of the trace in the stated estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Fix ℓ ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to the Leibniz rule, we have ∂m+ℓ n ∇Xϕ = m+ℓ � k=0 �m + ℓ k � ∂m+ℓ−k n X · ∇∂k nϕ = m+ℓ � k=0 �m + ℓ k � ∂m+ℓ−k n X · (∇∥, 0)∂k nϕ + m+ℓ � k=0 �m + ℓ k � ∂m+k−ℓ n (X · en)∂k+1 n ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) Applying Tr∂Ω and using that ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 and X · en = 0 on ∂Ω, we compute Tr∂Ω(∂m+ℓ n ∇Xϕ) = m−1 � k=0 �m + ℓ k � Tr∂Ω(∂m+ℓ−k n X · (∇∥, 0)∂k nϕ) + m−2 � k=0 �m + ℓ k � Tr∂Ω(∂m+ℓ−k n (X · en)∂k+1 n ϕ) + � 0 if ℓ < m − 1, Tr∂Ω(X · en∂2m n ϕ) if ℓ = m − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' = m−2 � k=0 �m + ℓ k � Tr∂Ω(∂m+ℓ−k n X · ∇∂k nϕ) + �m + ℓ m − 1 � Tr∂Ω(∂ℓ+1 n X · (∇∥, 0)∂m−1 n ϕ) = I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) We then take H−1/2-norms and handle I and II separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We bound I via the embedding H1/2 �→ H−1/2 and the H1 → H1/2 continuity of the trace map: ∥I∥H−1/2 ≲ m−2 � k=0 ∥∂m+ℓ−k n X · ∇∂k nϕ∥H1 ≲ m−2 � k=0 ∥Dm+ℓ−kX ⊗ Dk+1ϕ∥L2 + m−1 � k=0 ∥Dm+1+ℓ−kX ⊗ Dk+1ϕ∥L2 = III + IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) COMPRESSIBLE TRAVELING WAVES 77 For III and IV we employ the splitting X = X0 + X1 and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7: III ⩽ m−2 � k=0 � ∥Dm−1−k(D1+ℓX0) ⊗ Dk+1ϕ∥L2 + ∥Dm−1−k(D1+ℓX1) ⊗ Dk+1ϕ∥L2 � ≲ ∥D1+ℓX0, D1+ℓX1∥H1+⌊n/2⌋,W 1+⌊n/2⌋,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥D1+ℓX0, D1+ℓX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) and similarly, IV ⩽ m−1 � k=0 � ∥Dm−1−k(D2+ℓX0) ⊗ Dk+1ϕ∥L2 + ∥Dm−1−k(D2+ℓX1) ⊗ Dk+1ϕ∥L2 � ≲ ∥D2+ℓX0, D2+ℓX1∥H1+⌊n/2⌋,W 1+⌊n/2⌋,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥D2+ℓX0, D2+ℓX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) Hence, the estimate on I we obtain is ∥I∥H−1/2 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) For II we make a perfect divergence to exploit the negative norm in the H−1/2 space, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂ℓ+1 n X · (∇∥, 0)∂m−1 n ϕ = (∇∥, 0) · (∂ℓ+1 n X∂m−1 n ϕ) − (∇∥, 0) · (∂ℓ+1 n X)∂m−1 n ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) Then again by the continuity of the trace map and Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, we find that ∥II∥H−1/2 ≲ ∥∂ℓ+1 n X∂m−1 n ϕ∥H1 + ∥(∇∥, 0) · (∂ℓ+1 n X)∂m−1 n ϕ∥H1 ≲ ∥Dℓ+1X ⊗ Dm−1ϕ∥H1 + ∥Dℓ+2X ⊗ Dm−1ϕ∥H1 ≲ ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) Combining the estimates for I and II completes the proof of estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the remaining estimate, we first take 1 + ⌊n/2⌋ < m and use interpolation and Young’s inequality to bound ∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥H1+⌊n/2⌋ ≲ ∥ϕ∥Hm + ∥DX0, DX1∥ m m−(1+⌊n/2⌋) H1+m+ℓ×W 1+m+ℓ,∞∥ϕ∥L2 ≲ ∥ϕ∥Hm + ⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) Hence, for any m we see that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ As an application of the previous result, we develop the following estimate for the map Bm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As was the case for Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, the point of this estimate is that the right hand side depends on fewer derivatives that one might expect from a na¨ıve inspection of the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Another bilinear estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that m ∈ N+ and X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies Tr∂Ω(X · en) = 0 and decomposes as X = X0 + X1, where DX0 ∈ H∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn×n) and DX1 ∈ 78 NOAH STEVENSON AND IAN TICE W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that ϕ ∈ H3m(Ω) satisfies the Neumann conditions ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then |Bm(∇Xϕ, Lmϕ)| ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩∥ϕ∥H3m∥ϕ∥Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩6∥ϕ∥Hm∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) Here the implicit constant depends on the domain, the dimension, and m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by integrating by parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, we have the identity Bm(∇Xϕ, Lmϕ) = � Ω Lm(∇Xϕ)Lmϕ + m−1 � ℓ=0 � � Σ − � Σ0 � (−1)ℓ∂m+ℓ n (∇Xϕ)∂m−1−ℓ n Lmϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) Hence, by trace theory, we have the estimate |Bm(∇Xϕ, Lmϕ)| ≲ ��� � Ω Lm(∇Xϕ)Lmϕ ��� + m−1 � ℓ=0 ∥Tr∂Ω(∂m+ℓ n (∇Xϕ))∥H−1/2∥Lmϕ∥Hm−ℓ = I + II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) We will handle I and II separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For I we first use ∇Xϕ = ∇ · (ϕX) − ϕ∇ · X to expand � Ω Lm(∇Xϕ)Lmϕ = � Ω ∇ · Lm(ϕX)Lmϕ − Lm(ϕ(∇ · X))Lmϕ = � Ω ∇ · (XLmϕ)Lmϕ + ∇ · ([Lm, X]ϕ)Lmϕ − Lm(ϕ(∇ · X))Lmϕ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) and then employ Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 on the first term on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This yields the bound I ⩽ ��� � Ω ∇ · X 2 (Lmϕ)2��� + ��� � Ω ∇ · ([Lm, X]ϕ)Lmϕ ��� + ��� � Ω Lm(ϕ(∇ · X))Lmϕ ��� ≲ � ∥DX∥L∞∥ϕ∥H2m + n � j=1 ∥[∂2m j , X]ϕ∥H1 + ∥(∇ · X)ϕ∥H2m � ∥ϕ∥H2m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) We next use Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, followed by interpolation and Young’s inequality, to estimate ∥(∇ · X)ϕ∥H2m ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥H2m + � 0 if 2m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥H2m×W 2m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < 2m, ≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) On the other hand, for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n}, we use Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 and another interpolation and Young inequality argument to estimate ∥[∂2m j , X]ϕ∥H1 ≲ ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞∥ϕ∥H2m + � 0 if 2m ⩽ 1 + ⌊n/2, ∥DX0, DX1∥H2m×W 2m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < 2m, ≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) Upon synthesizing (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29),(4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31), we learn that I ≲ ⟨∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞⟩∥ϕ∥2 H2m + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H2m∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) COMPRESSIBLE TRAVELING WAVES 79 Now we turn our attention to II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First we input the estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2: II ⩽ ∥ϕ∥Hm m−1 � ℓ=0 ⟨∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞⟩∥Lmϕ∥Hm−ℓ + ∥ϕ∥L2 m−1 � ℓ=0 ⟨∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞⟩2+⌊n/2⌋∥Lmϕ∥Hm−ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) Next we utilize the interpolation inequalities ∥Lmϕ∥Hm−ℓ ≲ ∥Lmϕ∥ ℓ m−1 H1 ∥Lmϕ∥ m−1−ℓ m−1 Hm , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) ∥DX0, DX1∥H2+⌊n/2⌋+ℓ×W 2+⌊n/2⌋+ℓ,∞ ≲ ∥DX0, DX1∥ m−1−ℓ m−1 H2+⌊n/2⌋×W 2+⌊n/2⌋,∞∥DX0, DX1∥ ℓ m−1 H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) and ∥DX0, DX1∥H1+m+ℓ×W 1+m+ℓ,∞ ≲ ∥DX0, DX1∥ m−1−ℓ m−1 H1+m×W 1+m,∞∥DX0, DX1∥ ℓ m−1 H2m×W 2m,∞, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) together with Young’s inequality to bound II ⩽ ∥ϕ∥Hm� ⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩∥Lmϕ∥Hm + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩∥Lmϕ∥H1 � + ∥ϕ∥L2 � ⟨∥DX0, DX1∥H1+m×W 1+m,∞⟩2+⌊n/2⌋∥Lmϕ∥Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥Lmϕ∥H1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) Upon combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37), we see that I + II ≲ ⟨∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞⟩(∥ϕ∥2 H2m + ∥ϕ∥Hm∥ϕ∥H3m) + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩∥ϕ∥Hm∥ϕ∥H2m+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) We then interpolate again: ∥ϕ∥2 H2m ≲ ∥ϕ∥Hm∥ϕ∥H3m, and if m ⩾ 2 then we use Young’s inequality to bound ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩∥ϕ∥H2m+1 ≲ ∥ϕ∥H3m + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩ 3m m−1 ∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) By combining these with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38), we prove (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Some results on steady transport equations and elliptic regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the lemmas from Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 in hand, we may now begin to derive the required precise estimates for steady transport equations and their regularizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The former is much simpler, and our first result gives all of the necessary a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (A priori estimates for steady transport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ϕ ∈ Hm(Ω), X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 1+⌊n/2⌋ and 0 < ρ ⩽ ρmax, and that ∇·(Xϕ) ∈ Hm(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose additionally that Λ ∈ W ∞,∞(Ω) is such that Λ > 0, 1/Λ ∈ L∞(Ω), and Λϕ + ∇ · (Xϕ) = ψ in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) 80 NOAH STEVENSON AND IAN TICE There exists a ρ(m) ∈ R+, depending only on m, Λ, and the dimension, such that if 0 < ρ ⩽ ρ(m), then ∥ϕ∥Hm ≲ ∥ψ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Here the implicit constant depends only on m, ρ(m), Λ, and the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by multiplying (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) by ϕ and applying Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 to obtain the identity � Ω � Λ + ∇ · X 2 � ϕ2 = � Ω ϕψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) Selecting ρ(0) sufficiently small, depending on ∥1/Λ∥L∞, and using Cauchy-Schwarz on the right hand side, we obtain the a priori estimate ∥ϕ∥L2 ≲ ∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Now suppose that m ⩾ 1 and 0 < ρ ⩽ ρ(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Plugging (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) into the bilinear form Bm from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), we have the identity Bm(Λϕ + ∇ · (Xϕ), ϕ) = Bm(ψ, ϕ), and hence Bm(Λϕ, ϕ) ⩽ ∥ψ∥Hm∥ϕ∥Hm + |Bm(∇ · (Xϕ), ϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) By applying G˚arding’s inequality, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, we thus obtain the bound ∥ϕ∥2 Hm ≲ ∥ψ∥Hm∥ϕ∥Hm + ∥ϕ∥2 L2 + |Bm(∇ · (Xϕ), ϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Now we invoke Cauchy’s inequality for the ∥ψ∥Hm∥ϕ∥Hm term, estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) for the ∥ϕ∥2 L2 term, and estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 for the final term above;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' these then yield the bound ∥ϕ∥2 Hm ≲ ∥ψ∥2 Hm + ρ∥ϕ∥2 Hm + ∥ϕ∥Hm � 0 if m ⩽ 1 + ⌊n/2, ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) We define ρ(m) ∈ (0, ρ(0)] to be sufficiently small so that when taking ρ ⩽ ρ(m) we can absorb the right hand side’s ∥ϕ∥2 Hm-contribution with the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is done, estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) follows from one last application of Cauchy’s inequality to the final term in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The remainder of this subsection develops corresponding a priori estimates for regularized steady transport equations, with the right uniformities with respect to the approximation parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is more complicated than Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 since the theory now has to account for a rather unhappy marriage of elliptic and hyperbolic structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our first result in this direction handles the case of low regularity, namely L2, data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (A priori estimate for regularized steady transport with data in L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m ∈ N+ and Lm be the operator given in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ψ ∈ L2(Ω), ϕ ∈ H2m(Ω), X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, Λ ∈ W ∞,∞(Ω) satisfies Λ > 0 and 1/Λ ∈ L∞(Ω), and N ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume additionally that the equations � N−1ΛLmϕ + ∇ · (Xϕ) = ψ in Ω, ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If N ≳ ⟨∥DX0, DX1∥Hm×W m,∞⟩4+2⌊n/2⌋, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) then we have the a priori estimate ∥ϕ∥H2m ≲ N∥ϕ, ψ∥L2×L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) where the implied constants depend on m, the dimension, Λ, and ρmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 81 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We multiply the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) by N−1Lmϕ and integrate over Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By applying Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 on the ∇ · (Xϕ)-term, we acquire the identity 1 N2 � Ω Λ(Lmϕ)2 + 1 N Bm(∇ · (Xϕ), ϕ) = � Ω ψ 1 N Lmϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) Now we use the hypotheses on Λ paired with Cauchy’s inequality and absorption: 1 N2 ∥Lmϕ∥2 L2 ≲ ∥ψ∥2 L2 + 1 N |Bm(∇ · (Xϕ), ϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) We now handle the Bm(∇·(Xϕ), ϕ) term by using estimate (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) from Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (with ρ = ρmax) and then interpolating: 1 N |Bm(∇ · (Xϕ), ϕ)| ≲ 1 N ∥ϕ∥2 Hm + 1 N ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥Hm∥ϕ∥L2 ≲ 1 N ∥ϕ∥H2m∥ϕ∥L2 + 1 N1/2 ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋ ∥ϕ∥1/2 H2m N1/2 ∥ϕ∥3/2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) Therefore, if (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is satisfied, then 1 N |Bm(∇ · (Xϕ), ϕ)| ≲ 1 N ∥ϕ∥H2m∥ϕ∥L2 + ∥ϕ∥1/2 H2m N1/2 ∥ϕ∥3/2 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) Upon returning to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12), we see that ∥Lmϕ∥L2 ≲ N∥ψ∥L2 + N1/2∥ϕ∥1/2 H2m∥ϕ∥1/2 L2 + N3/4∥ϕ∥1/4 H2m∥ϕ∥3/4 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) As the boundary conditions in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) are satisfied, we may then invoke the a priori estimate for Lm from Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 to acquire the bound ∥ϕ∥H2m ≲ ∥ϕ, Lmϕ∥L2×L2 ≲ N∥ϕ, ψ∥L2×L2 + N1/2∥ϕ∥1/2 H2m∥ϕ∥1/2 L2 + N3/4∥ϕ∥1/4 H2m∥ϕ∥3/4 L2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) The proof is complete upon applying Young’s inequality in order to absorb the ∥ϕ∥H2m terms from the right onto the left.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Our next two results consider what happens when the data for a regularized steady transport equation is of higher regularity than L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first of these analyzes the bonus regularity of the solution gained from the vanishing elliptic term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 (A priori estimate for regularized steady transport with high regularity data, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ψ ∈ Hm(Ω), ϕ ∈ H3m(Ω), X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 2 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax, Λ ∈ W ∞,∞(Ω) satisfies Λ > 0 and 1/Λ ∈ L∞(Ω), and N ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume additionally that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If N ≳ ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩4+2⌊n/2⌋, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) then we have the a priori estimate ∥ϕ∥H3m ≲ N∥ψ, ϕ∥Hm×Hm + N⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) Here the implied constants depend only on m, the dimension, Λ, and ρmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We rearrange the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) as N−1ΛLmϕ = ψ − (∇ · X)ϕ − ∇Xϕ (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) and then take the Bm-product of the above equation with N−1Lmϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' After applying G˚arding’s inequality, Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, followed by the L2-a priori estimate of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, we find that N−2∥Lmϕ∥2 Hm ≲ ∥ϕ, ψ∥2 L2×L2 + N−1∥ψ, (∇ · X)ϕ∥Hm×Hm∥Lmϕ∥Hm + N−1|Bm(∇Xϕ, Lmϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) 82 NOAH STEVENSON AND IAN TICE Hence, by Cauchy’s inequality we get the bound N−2∥Lmϕ∥2 Hm ≲ ∥ϕ, ψ∥2 L2×L2 + ∥ψ, (∇ · X)ϕ∥2 Hm×Hm + N−1|Bm(∇Xϕ, Lmϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) Now we apply the a priori estimates for Lm of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 to learn that N−2∥ϕ∥2 H3m ≲ ∥ϕ, ψ∥2 L2×L2 + ∥ψ, (∇ · X)ϕ∥2 Hm×Hm + N−1|Bm(∇Xϕ, Lmϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) By Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 and a familiar argument using interpolation and Young’s inequality, we deduce that ∥(∇ · X)ϕ∥Hm ≲ ∥ϕ∥Hm + � 0 if m ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥Hm×W m,∞∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < m, ≲ ∥ϕ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) On the other hand, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 shows that |Bm(∇Xϕ, Lmϕ)| ≲ ∥ϕ∥H3m∥ϕ∥Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥H3m∥ϕ∥L2 + ⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m⟩6∥ϕ∥Hm∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) Inserting (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) into (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) and using Cauchy’s inequality then shows that N−2∥ϕ∥2 H3m ≲ ∥ϕ, ψ∥2 Hm×Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩4+2⌊n/2⌋∥ϕ∥2 L2 + N−1⟨∥DX0, DX1∥H1+⌊n/2⌋+m×W 1+⌊n/2⌋+m,∞⟩6∥ϕ∥2 Hm, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) and upon combining this with hypothesis (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) we deduce the bound N−2∥ϕ∥2 H3m ≲ ∥ϕ, ψ∥2 Hm×Hm + ⟨∥DX0, DX1∥H2m×W 2m,∞⟩4+2⌊n/2⌋∥ϕ∥2 L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) This is the stated estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Next, we aim to estimate the norm of the solution to a regularized steady transport equation in the same space as the regular data, but independently of the approximation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (A priori estimate for regularized steady transport with high regularity data, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m, N ∈ N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ψ ∈ Hm(Ω), ϕ ∈ H3m(Ω), X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 1 + ⌊n/2⌋ and 0 < ρ ⩽ ρmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Further suppose that, for i ∈ {0, 1}, Λi ∈ W ∞,∞(Ω) satisfies Λi > 0 and 1/Λi ∈ L∞(Ω), and that the equations � Λ0ϕ + N−1Λ1Lmϕ + ∇ · (Xϕ) = ψ in Ω, ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a �ρ(m) ∈ R+, depending only on the dimension, m, Λ0, and Λ1 such that if 0 < ρ ⩽ �ρ(m), then we have the a priori estimate ∥ϕ∥Hm ≲ ∥ψ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) where the implicit constant depends only on m, �ρ(m), Λ0, Λ1, and the dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We test the first equation in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) with Lmϕ in the L2(Ω) inner product and integrate by parts via Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 to obtain the identity Bm(Λ0ϕ, ϕ) + N−1∥ � Λ1Lmϕ∥2 L2 = Bm(ψ, ϕ) + Bm(∇ · (Xϕ), ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) Hence, Bm(Λ0ϕ, ϕ) ⩽ ∥ψ∥Hm∥ϕ∥Hm + |Bm(∇ · (Xϕ), ϕ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is exactly inequality (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) from Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We can therefore argue in exactly the same way here to reach the desired conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ COMPRESSIBLE TRAVELING WAVES 83 The final result of this section is the culmination of our steady transport analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Essentially, we interpolate between the low regularity estimate of Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 and the high regularity estimates of Propositions 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In doing so, we aim to preserve a key structure of the previous estimates: all norms involving the vector field are multiplied by the lowest regularity norms of the data or solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This requires some fine interpolation results, which are proved in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 (Estimates for regularized steady transport).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let m, N ∈ N+ and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ψ ∈ Hj(Ω), ϕ ∈ Hj+2m(Ω), and X ∈ W ∞,∞(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) satisfies (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) with r = 2+⌊n/2⌋ and 0 < ρ ⩽ ρmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Further suppose Λi ∈ W ∞,∞(Ω) satisfies Λi > 0 and 1/Λi ∈ L∞(Ω) for i ∈ {0, 1}, and that the equations � Λ0ϕ + N−1Lmϕ + ∇ · (Λ1Xϕ) = ψ in Ω, ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρ(m) ∈ R+, depending only on m, Λ0, and Λ1, such that if max{ρ, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} ⩽ ρ(m) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) and N ≳ ⟨∥X0, X1∥H2+⌊n/2⌋+m×W 2+⌊n/2⌋+m,∞⟩4+2⌊n/2⌋, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) then we have the a priori estimate ∥ϕ, ∇ · (Λ1Xϕ), N−1ϕ∥Hj×Hj×Hj+2m ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) Here the implied constants depend only on Λ0, Λ1, m, the dimension, and ρ(m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We let ρ ∈ R+ be such that max{ρ, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} ⩽ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) Throughout the proof we will take ρ to be ever smaller to meet various criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Ultimately, we then define ρ(m) to be the value for ρ we have at the end of the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For ℓ ∈ N, we will also make use of the Banach space Z(m, ℓ) = {φ ∈ Hℓ+2m(Ω) : Tr∂Ω(∂m n φ) = · · · = Tr∂Ω(∂2m−1 n φ) = 0} (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) and the bounded linear map L : Z(m, ℓ) → Hℓ(Ω) defined via Lϕ = Λ0ϕ + N−1Lmϕ + ∇ · (Λ1Xϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the remainder of the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Establishing invertibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that L is a Banach isomorphism for every ℓ ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This follows from standard elliptic theory arguments once we establish the fact that the bilinear form B defined by Hm(Ω) × Hm(Ω) ∋ (ϕ0, ϕ0) B �→ � Ω Λ0ϕ0ϕ1 + ∇ · (Λ1Xϕ0)ϕ1 + N−1Bm(ϕ0, ϕ1) ∈ R (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) is coercive when ρ is sufficiently small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To prove coercivity, we first use Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 to compute B(ϕ, ϕ) = � Ω � Λ0 + 2−1∇ · (Λ1X) � ϕ2 + N−1Bm(ϕ, ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) In light of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34), and the Sobolev embeddings, we may bound ∥∇ · (Λ1X)∥L∞(Ω) ⩽ ∥Λ1∇ · X∥L∞(Ω) + ∥∇Λ1 · X∥L∞(Ω) ≲ ρ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) where the implicit constant depends on Λ0 and Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, if we take 0 < ρ ⩽ ρ(0) sufficiently small, we get that B(ϕ, ϕ) ≳ ∥ϕ∥2 L2 + N−1Bm(ϕ, ϕ), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) and hence Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 shows that B is coercive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 84 NOAH STEVENSON AND IAN TICE Step 2: Low-norm estimates on the inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that ψ ∈ L2(Ω), ϕ ∈ Z(m, 0), and that Lϕ = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39), we have the estimate ∥ϕ∥2 L2 ≲ B(ϕ, ϕ) = � Ω ψϕ, and hence ∥L−1ψ∥L2 = ∥ϕ∥L2 ≲ ∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) Now we may invoke Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (the hypotheses of which are satisfied thanks to (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32)) followed by (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) to see that ∥L−1ψ∥H2m = ∥ϕ∥H2m ≲ N∥ψ − Λ0ϕ, ϕ∥L2×L2 ≲ N∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, we have shown that L−1 maps L2(Ω) into Z(m, 0) ⊂ H2m(Ω) with the operator bounds ∥L−1ψ, N−1L−1ψ∥L2×H2m ≲ ∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) Step 3: High-norm estimates on the inverse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now assume that ψ ∈ Hm(Ω) and ϕ ∈ Z(m, m) ⊂ H3m(Ω) satisfy Lϕ = ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that this equation is equivalent to �Λ0ϕ + ∇ · (Xϕ) + N−1�Λ1Lmϕ = �ψ, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) where �Λ0 = Λ0/Λ1, �Λ1 = 1/Λ1, and �ψ = (ψ − ϕX · ∇Λ1)/Λ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We take ρ ⩽ �ρ(m), where the latter is given by Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, and then apply the proposition to gain the bound ∥L−1ψ∥Hm ≲ ∥ �ψ∥Hm + ⟨∥DX0, DX1∥Hm×W m,∞⟩2+⌊n/2⌋∥ �ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) Thanks to Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, interpolation, and Young’s inequality, we have that ∥ �ψ∥Hm ≲ ∥ψ∥Hm + ρ∥ϕ∥Hm + ⟨∥X0 · ∇Λ1, X1 · ∇Λ1∥Hm×W m,∞⟩2+⌊n/2⌋∥ϕ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) On the other hand, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) provides the bound ∥ �ψ∥L2 ≲ ∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) We combine (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) and then take ρ ⩽ ρ(m) ⩽ ρ(0) to be sufficiently small so that the right hand side’s ∥ϕ∥Hm-contribution can be absorbed by the left;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this results in the bound ∥L−1ψ∥Hm ≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+m×W 1+m,∞⟩2+⌊n/2⌋∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) We next apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 followed by estimates (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41): N−1∥ϕ∥H3m ≲ ∥ψ − Λ0ϕ, ϕ∥Hm + ⟨∥D(Λ1X0), D(Λ1X1)∥H2m×W 2m,∞⟩2+⌊n/2⌋∥ϕ∥L2 ≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) The culmination of this analysis is that we have shown that L−1 maps Hm(Ω) into Z(m, m) ⊂ H3m(Ω) with the operator bounds ∥L−1ψ, N−1L−1ψ∥Hm×H3m ≲ ∥ψ∥Hm + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) Step 4: Interpolation and conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let EΩ denote a Stein extension operator for Ω (see Defini- tion A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and let RΩ denote the operator given by restriction to Ω of functions defined on Rn (see Ex- ample 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We define a map T ∈ L(L2(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H2m(Rn))∩L(Hm(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H3m(Rn)) via the formula T = EΩL−1RΩ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the previous steps, we know that for A = ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋, T satisfies the operator bounds � ∥Tψ∥L2 ≲ ∥ψ∥L2 for all ψ ∈ L2(Rn), ∥Tψ∥Hm ≲ ∥ψ∥Hm + A∥ψ∥L2 for all ψ ∈ Hm(Rn), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) and � ∥Tψ∥H2m ≲ N∥ψ∥L2 for all ψ ∈ L2(Rn), ∥Tψ∥H3m ≲ N∥ψ∥Hm + NA∥ψ∥L2 for all ψ ∈ Hm(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) We are therefore in a position to apply Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 to deduce that T ∈ L(Hj(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hj+2m(Rn)) for j ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m}, with the estimates ∥Tψ, N −1Tψ∥Hj×Hj+2m ≲ ∥ψ∥Hj + A∥ψ∥Hj−m for all ψ ∈ Hj(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) COMPRESSIBLE TRAVELING WAVES 85 By utilizing that L−1 = RΩTEΩ, we can port (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) to an estimate on L−1, namely: ∥L−1ψ, N −1L−1ψ∥Hj×Hj+2m ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) for all ψ ∈ Hj(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It remains to obtain an estimate on ∇ · (Λ1XL−1ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we rearrange the equation as ∇ · (Λ1Xϕ) = ψ − Λ0ϕ − N−1Lmϕ, take the norm in Hj(Ω) of both sides, and apply the established estimates of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This yields the bound ∥∇ · (Λ1Xϕ)∥Hj ≲ ∥ψ∥Hj + ⟨∥X0, X1∥H1+2m×W 1+2m,∞⟩2+⌊n/2⌋∥ψ∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) The proof is complete upon combining (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of weak solutions to the principal part linear equations In this section we study weak solutions to the PDE � � � � � � � � � � � � � � � ∇ · (ϱu) + ∇ · (vw0(q + gη)) = g in Ω, −γ2 0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu = f in Ω, −(ϱq − γ0Sϱu)en − ς∆∥ηen = k on Σ, u · en + ∂1η = 0 on Σ, u = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Here the given data are g : Ω → R, f : Ω → Rn, and k : Σ → Rn, as well as γ0 ∈ R+ and a vector field vw0 : Ω → Rn defined via a fixed triple w0 = (q0, u0, η0) as in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) (see also Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The unknowns are q : Ω → R, u : Ω → Rn, and η : Σ → R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we are interested in the weak formulation principal part linear operator w0,γ0 J , which we recall is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The above system is not elliptic in the sense of Agmon, Douglis, and Nirenberg [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Because of this and various other linear effects of the derivative loss, we are led to consider the following regularized version of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with parameters τ ∈ [0, 1], m, N ∈ N+, and m ⩾ 2: � � � � � � � � � � � � � � � � � � � ∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) = g in Ω, −γ2 0ϱ∂1u + ϱ∇(q + gη) − γ0∇ · Sϱu = f in Ω, −(ϱq − γ0Sϱu)en − ς∆∥ηen = k on Σ, u · en + ∂1η = N−1(−∆∥)m−1/4η on Σ, u = 0 on Σ0, ∂m n q = · · · = ∂2m−1 n q = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where the linear elliptic operator Lm is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we are also considering in this section the weak formulation regularized principal part linear operators w0,γ0 J τ m,N, which are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The strategy is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 by proving a priori estimates for weak solutions to the systems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) that are appropriately uniform with respect to the background solution, N, and τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 we develop the existence theory for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) from our a priori estimates and the method of continuity, which is the reason for including the homotopy parameter τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As it turns out, the problem with τ = 0 can be solved by taking a two parameter limit of solutions to similar equations, which we solve with the help of the Lax-Milgram lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we prove a priori estimates for weak solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first require the following technical lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 86 NOAH STEVENSON AND IAN TICE Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and let w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that η ∈ H0(Σ) has Fourier support in the punctured ball BRn−1(0, 1) \\ {0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we have the estimate ��� � Ω (∇ · vw0)η2��� ≲ ρ∥η∥2 H0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where the implied constant depends only on the dimension, the various physical parameters, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First note that the support hypotheses on η imply that η ∈ H∞(Σ) (see, for instance, Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, we compute ∇ · vw0 = ∇ · (vw0 − ϱ′/ge1), and use this, integration by parts, Fubini-Tonelli, the fundamental theorem of calculus, and the fact that Tr∂Ω(vw0 · en) = 0 to rewrite 1 2 � Ω ∇·(vw0 −ϱ′/ge1)η2 = � Ω ∇·((vw0 −ϱ′/ge1)η)η = � Rn−1 � (∇∥, 0)· � b 0 (vw0 −ϱ′/ge1)η � η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) In turn, this readily implies that ���1 2 � Ω (∇ · vw0)η2��� ⩽ � (∇∥, 0) · � b 0 (vw0 − ϱ′/ge1)η � ˙H−1[η] ˙H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) By Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, specifically (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), [η] ˙H1 ≲ ∥η∥H0, so it remains to estimate the ˙H−1 term on the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we use the decomposition (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, which allows us to rewrite (∇∥, 0) · � b 0 (vw0 − ϱ′/ge1)η = (∇∥, 0) · � b 0 v(1) q0,u0,η0η + ∂1 � η � b 0 v(2) η0 (·, y) · e1 dy � = I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) We bound I1 by using the fact that the integrand belongs to L2(Ω) (and tacitly using Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, and the first item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19): [I1] ˙H−1 ≲ ∥v(1) q0,u0,η0η∥L2 ≲ ∥v(1) q0,u0,η0∥L2∥η∥L∞ ≲ ρ∥η∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) We bound I2 using the algebra properties of the specialized Sobolev spaces (see Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and the second item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19: [I2] ˙H−1 ⩽ ���η � � b 0 v(2) η0 (·, y) dy ���� H0 ≲ ��� � b 0 v(2) η0 (·, y) dy ��� H0∥η∥H0 ≲ ρ∥η∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Combining these bounds yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ With the lemma in hand, we are ready to study estimates of weak solutions to the principal part equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that the spaces q0,u0,η0 X−1 and Y−1 are defined in equations (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), while the weak formulation operator w0,γ0 J is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Also recall the surface tension ς and viscosity µ, λ hypotheses set forth in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (A priori estimates for weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, let w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, and let γ0 ∈ I, where I ⋐ R+ is some interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ q0,u0,η0 X−1 and (g, F) ∈ Y−1 satisfy the equation w0,γ0 J (q, u, η) = (g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) There exists a ρweak ∈ R+ such that if 0 < ρ ⩽ ρweak, then we have the a priori estimate ∥q, u, η∥q0,u0,η0 X−1 ≲ ∥g, F∥Y−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) The implicit constants and ρweak depend on the physical parameters, the dimension, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 87 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the proof into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Reduction to g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim first that it suffices to prove the result under the specialized assumption that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, suppose this special case has been proved and let (q, u, η), w0 = (q0, u0, η0), γ0, and (g, F) be related as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the help of the operator B0 from Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we define w ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) via w = u − ϱ−1B0g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since B0g ∈ H1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), we have that TrΣ(w · en) = TrΣ(u · en) = −∂1η, and hence (q, w, η) ∈ q0,u0,η0 X−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recalling that γ0 I is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12), we calculate that γ0 I (q, w, η) = F − γ0 I (0, B0g/ϱ, 0) and ∇ · (ϱw) + ∇ · (vw0(q + gη)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) Thus, we can apply the special case to obtain the estimate ∥q, w, η∥q0,u0,η0 X−1 ≲ ∥F − γ0 I (0, B0g/ϱ, 0)∥(0H1)∗ ≲ ∥g, F∥Y−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) To switch from w to u in this bound we use the estimate provided by Proposition C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, namely ∥u∥H1 ≲ ∥w∥H1 + ∥g∥ ˆH0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By chaining this together with the previous estimate we prove the result in general, which completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the remaining steps we will prove the result in the special case that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 2: A priori bound on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that the bound ∥u∥2 H1 ≲ ∥F∥2 (0H1)∗ + ρ(∥q∥2 L2 + ∥η∥2 H3/2) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We make the following notational simplification for the Fourier space decompositions of η from (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4): ηκ,L = Πκ Lη and ηκ,H = Πκ Hη.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) According to Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we have that ηκ,H ∈ L2(Σ) and ηκ,L ∈ H∞(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We test the equation γ0 I (q, u, ηκ,H) = F − γ0 I (0, 0, ηκ,L) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) with u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the left hand side of the resulting identity reads ⟨ γ0 I (q, u, ηκ,H), u⟩ = � Ω γ0 �µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2� − (q + gηκ,H)∇ · (ϱu) + ⟨(gϱ − ς∆∥)ηκ,H, TrΣ(u · en)⟩H−1/2,H1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and we next aim to estimate the latter two terms on the right side this expression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, we recall that TrΣ(u · en) = −∂1η and − ∇ · (ϱu) = ∇ · (vw0(q + gη)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) These allow us to compute ⟨(gϱ − ς∆∥)ηκ,H, TrΣ(u · en)⟩H−1/2,H1/2 = −⟨(gϱ − ς∆∥)ηκ,H, ∂1ηκ,H⟩H−1/2,H1/2 = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and (employing the integration by parts trick of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) � Ω −(q + gηκ,H)∇ · (ϱu) = � Ω ∇ · vw0 2 (q + gηκ,H)2 + ∇ · (vw0ηκ,L)(q + gηκ,H) = I1 + I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) We handle I1 first by expanding I1 = � Ω ∇ · vw0 2 (q + gη1,H)2 + g � Ω ∇ · vw0(q + gη1,H)(ηκ,H − η1,H) + � Ω g2∇ · vw0 2 (ηκ,H − η1,H)2 = I1,1 + I1,2 + I1,3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) According to the third item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we may estimate |I1,1| ⩽ ∥∇ · vw0∥L∞∥q + gη1,H∥2 L2 ≲ ρ(∥q∥2 L2 + ∥η∥2 H0) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) 88 NOAH STEVENSON AND IAN TICE and |I1,2| ≲ ∥∇ · vw0∥L2∥q + gη1,H∥L2∥ηκ,H − η1,H∥L∞ ≲ ρ(∥q∥2 L2 + ∥η∥2 H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) For I1,3, we instead use Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1: |I1,3| ≲ ρ∥η∥2 H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) Upon piecing together the previous three estimates, we deduce that |I1| ≲ ρ(∥q∥2 L2 + ∥η∥2 H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) Now we turn our attention to the term I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Again, we first decompose I2 = � Ω ∇ · (vw0ηκ,L)(q + gη1,H) + g � Ω ∇ · (vw0ηκ,L)(ηκ,H − η1,H) = I2,1 + I2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) I2,1 is handled via fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19: |I2,1| ⩽ ∥∇ · (vw0ηκ,L)∥L2∥q + gη1,H∥L2 ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) For I2,2 we instead integrate by parts and use Fubini-Tonelli: |I2,2| = g ��� � Rn−1 � (∇∥, 0) · � b 0 vw0ηκ,L � (ηκ,H − η1,H) ��� ≲ � (∇∥, 0) · � b 0 vw0ηκ,L � ˙H−1[ηκ,H − η1,H] ˙ H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) By decomposing vw0 = ϱ′g−1e1 + v(1) q0,u0,η0 + v(2) η0 (see (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17)) and arguing as in the proof of Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we acquire the bound � (∇∥, 0) · � b 0 vw0ηκ,L � ˙H−1 ≲ ∥ηκ,L∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) Hence, |I2| ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) and upon combining this with the I1 estimate we deduce that ��� � Ω −(q + gηκ,H)∇ · (ϱu) ��� ≲ ρ(∥q∥2 L2 + ∥η∥2 H0) + ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) With (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) in hand, we return to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) to obtain the inequality � Ω γ0 �µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2� ≲ ∥F − I (0, 0, ηκ,L)∥(0H1)∗∥u∥H1 + ρ(∥q∥2 L2 + ∥η∥2 H0) + ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) This holds for all κ ∈ (0, 1) and the implicit constant is independent of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus we may send κ → 0 and use the fact that, as a consequence of the dominated convergence theorem and the definition of the norm on the anisotropic Sobolev spaces (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), ∥ηκ,L∥H0 → 0 to arrive at the bound γ0 � Ω µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2 ≲ ∥F∥(0H1)∗∥u∥H1 + ρ(∥q∥2 L2 + ∥η∥2 H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) Recall that the assumptions on µ, λ ∈ C∞(R+) are that µ > 0 and λ > 0 if n = 2, while µ > 0 and λ ⩾ 0 if n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) follows from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30), the inclusion γ0 ∈ I ⋐ R+, and either Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 in the case n ⩾ 3 or else Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 in the case n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: A priori bound on ∇∥η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we claim that we have the a priori bound ∥∇∥η∥H1/2 ≲ ∥u∥H1 + ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) We first consider the case that surface tension is positive: ς > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To prove this we will utilize the operator B2 from Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall from the previous step that for κ ∈ (0, 1) we have the decomposition η = ηκ,L + ηκ,H defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We test identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) with wκ ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), defined via wκ = −ϱ−1B2(ϱ(b)⟨∇∥⟩−1∆∥ηκ,H), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) COMPRESSIBLE TRAVELING WAVES 89 noting that ∇ · (ϱwκ) = 0, TrΣ(wκ · en) = −⟨∇∥⟩−1∆∥ηκ,H, and ∥wκ∥H1 ≲ ∥⟨∇∥⟩−1∆∥ηκ,H∥ ˙H−1∩H1/2 ≍ ∥⟨∇∥⟩1/2|∇∥|ηκ,H∥L2 ≲ ∥∇∥ηκ,H∥H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) The result is the identity ⟨ γ0 I (q, u, ηκ,H), wκ⟩ = � Ω γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇wκ + gϱ∇ηκ,L · wκ + ⟨(gϱ − ς∆∥)ηκ,H, −⟨∇∥⟩−1∆∥ηκ,H⟩H−1/2,H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) As ∥∇∥ηκ,H∥2 H1/2 ≲ ⟨(gϱ − ς∆∥)ηκ,H, −⟨∇∥⟩−1∆∥ηκ,H⟩H−1/2,H1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) we obtain the estimate ∥∇∥ηκ,H∥2 H1/2 ≲ (∥u∥H1 + ∥F∥(0H1)∗ + ∥ηκ,L∥H3/2)∥wκ∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) follows from this and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) upon sending κ → 0, which is valid since the implicit constants are independent of κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This proves the claim in the case of positive surface tension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now we consider the case of vanishing surface tension, ς = 0, in dimension n = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we simply look to the boundary condition satisfied by u, namely ∂1η = −TrΣ(u · en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since n = 2 and η is defined on Σ ≃ R, we have ∂1η = ∇∥η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore we have ∥∇∥η∥H1/2 = ∥TrΣ(u·en)∥H1/2 ≲ ∥u∥H1, so (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 4: A priori bound on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that we have the a priori bound ∥q∥L2 ≲ ∥u∥H1 + ∥∇∥η∥H1/2 + ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) To see this, we first let w = ϱ−1Bq ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), where B is constructed in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By construction, we have that ∇ · (ϱw) = q and ∥w∥H1 ≲ ∥q∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) Then we we test the identity γ0 I (q, u, η) = F with w to see that � Ω γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w + gϱ∇η · w − q2 − ς⟨∆∥η, TrΣ(w · en)⟩H−1/2,H1/2 = ⟨F, w⟩(0H1)∗,0H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) Estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) readily follows from this, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38), and the estimates established in the previous steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This proves the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 5: A priori bound on ∂1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we claim that [∂1η] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) First, we note that by using the decomposition of vw0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, the continuity equation is equivalently written as 0 = ∇ · (ϱu + vw0q) + g∇ · (v(1) q0,u0,η0η) + g∂1(v(2) η0 · e1η) + ϱ′∂1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) We integrate this in the nth coordinate over (0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' after recalling the identities TrΣ(u · en) + ∂1η = 0 and (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, this results in the equality ϱ(0)∂1η = (∇∥, 0) · � b 0 (ϱu + vw0q + gv(1) q0,u0,η0η) + g∂1 � η � � b 0 v(2) η0 (·, y) · e1 dy �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) Hence, we may use the estimates from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, the fact that � b 0 v(2) η0 (·, y) dy has rn−1 as a band limit, the algebra properties of the anisotropic Sobolev spaces in Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, and 90 NOAH STEVENSON AND IAN TICE estimate (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) from Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 to bound ϱ(0)[∂1η] ˙H−1 ≲ ∥ϱ + vw0q + gv(1) q0,u0,η0η∥L2 + g ��� � � b 0 v(2) η0 (·, y) dy � η ��� H0 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) This proves the claim since ϱ(0) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 6: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now synthesize the claims of the previous steps to conclude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we take the bound from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and plug it into the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this yields the inequality ∥∇∥η∥H1/2 ≲ ∥F∥(0H1)∗ + √ρ(∥q∥L2 + ∥η∥H3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) Second, we take (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and insert them into the right hand side of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) to get ∥q∥L2 ≲ ∥F∥(0H1)∗ + √ρ(∥q∥L2 + ∥η∥H3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) Now, while heeding to (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), we sum the estimates (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45), and then use (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) on the right hand side;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the resulting estimate is ∥q, u, η∥X−1 ≲ ∥F∥(0H1)∗ + √ρ∥q, u, η∥X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) We choose ρweak ∈ R+ sufficiently small so that when taking ρ ⩽ ρweak we can absorb the X−1 contribution onto the left side and obtain the clean a priori bound ∥q, u, η∥X−1 ≲ ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) It remains to only estimate the L2(Ω)-norm of ∇ · (vw0q) in terms of the data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is now a simple matter since we can isolate it in the continuity equation via −∇ · (vw0q) = ∇ · (ϱu) + g∇ · (vw0η) and then note that the right hand side is controlled by ∥q, u, η∥X−1 (see in particular the fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Before our next a priori estimates result, we need a lemma from the theory of regularized steady transport equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Regularized steady transport lemma).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, m, N ∈ N+, τ ∈ [0, 1], ϕ ∈ H2m(Ω), and ψ ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that � N−1Lmϕ + τ∇ · (vw0ϕ) = ψ in Ω, ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) where Lm is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If N ≳ ⟨τ∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, then we have the a priori estimate ∥ϕ∥H2m ≲ N∥ϕ, ψ∥L2×L2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) where the implicit constant depends only on the physical parameters, the dimension, m, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Most of the work in verifying this result has already been executed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We invoke Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 with Λ = 1 and the decomposed vector field X = X0 + X1, where X = τvw0, X0 = τv(1) q0,u0,η0, and X1 = τ(v(2) η0 + g−1ϱ′e1), with v(1) q0,u0,η0 and v(2) η0 as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) then follows from properties of the vector field vq0,u0,η0 stated in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, namely: for s ∈ {1 + ⌊n/2⌋, m} we have ∥DX0, DX1∥Hs×W s,∞ ⩽ ∥X0, X1∥H1+s×W 1+s,∞ ≲ τ⟨∥q0, u0, η0∥Xs⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Our next result studies estimates on weak solutions to the regularized equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that the spaces X−1 m,N, the norms ∥·∥q0,u0,η0 X−1 m,N , and the mappings w0,γ0 J τ m,N are defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 91 Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (A priori estimates for regularized weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, and γ0 ∈ I, where I ⋐ R+ is some interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that m, N ∈ N+, τ ∈ [0, 1], (q, u, η) ∈ X−1 m,N, and that (g, F) ∈ Y−1 satisfy the equation w0,γ0 J τ m,N(q, u, η) = (g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρweak,reg ∈ R+ such that if 0 < ρ ⩽ ρweak,reg and N ≳ ⟨∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) then we have the a priori estimate ∥q, u, η∥q0,u0,η0 X−1 m,N ≲ ∥g, F∥Y−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) The implicit constants and ρweak,reg depend on the dimension, physical parameters, m, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We proceed in much the same way as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, breaking to steps that mirror the structure of the argument used there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Reduction to the case g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that it suffices to prove the result in the special case that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, the exact same argument used in the first step of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 proves the claim here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In the remaining steps we will prove the result in the special case that g = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 2: A priori bound on u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that we have the a priori bound ∥u∥2 H1 ≲ ∥F∥2 (0H1)∗ + ρ(∥q∥2 L2 + ∥η∥2 H3/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) To prove the claim we again use the Fourier space decomposition of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12), which again yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then test (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) with u to arrive at (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) as in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note, though, that in the present context we have the identities TrΣ(u · en) = −∂1η + N−1(−∆∥)m−1/4η (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) and − ∇ · (ϱu) = τ∇ · (vw0(q + gη)) + N−1Lm(q + gη), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) which are somewhat different from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We insert (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and integrate by parts to see that � Ω γ0 �µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2� + τ∇ · vw0 2 (q + gηκ,H)2 + 1 N n � j=1 (∂m j (q + gηκ,H))2 + g � Ω (q + gηκ,H) � τ∇ · (vw0ηκ,L) + 1 N Lm,∥ηκ,L � + 1 2∥(gϱ − ς∆∥)1/2(∆∥)m/2−1/8ηκ,H∥2 L2 = ⟨F − I (0, 0, ηκ,L), u⟩(0H1)∗,0H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55) By combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55), the Korn inequalities from Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 as well as the end of the second step of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, and the fact that τ ∈ [0, 1], we obtain the estimate ∥u∥2 H1 ≲ ��� � Ω ∇ · vw0 2 (q + gηκ,H)2��� + g ��� � Ω (q + gηκ,H) � τ∇ · (vw0ηκ,L) + 1 N Lm,∥ηκ,L ���� + (∥F∥(0H1)∗ + ∥ηκ,L∥H3/2)∥u∥H1 = I1 + I2 + I3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) The term I1 is identical to the term I1 that appeared in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17), so we may use the same argument used in the second step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 to arrive at the estimate I1 ≲ ρ(∥q∥2 L2 + ∥η∥2 H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57) For the term I2 we bound I2 ⩽ g ��� � Ω (q + gηκ,H)τ∇ · (vw0ηκ,L) ��� + g N ��� � Ω (q + gηκ,H)Lm,∥ηκ,L ��� = I2,1 + I2,2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58) 92 NOAH STEVENSON AND IAN TICE Arguing as in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23)–(5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27), we find that I2,1 ≲ ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59) For the remaining piece we exploit the fact that ηκ,L and ηκ,H have disjoint Fourier supports, together with (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) to see that I2,2 = g N ��� � Ω qLm,∥ηκ,L ��� ≲ 1 N ∥q∥L2∥Lm,∥ηκ,L∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ≲ 1 N ∥q∥L2∥ηκ,L∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) Finally, we plug (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) into (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) and then use Cauchy’s inequality on I3 to deduce that ∥u∥2 H1 ≲ ∥F∥2 (0H1)∗ + ρ(∥q∥2 L2 + ∥η∥2 H0) + ∥ηκ,L∥H0(∥q∥L2 + ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61) The implicit constant is independent of κ, so we may send κ → 0 to obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) from this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: A priori bound on ∇∥η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that we have the a priori bound ∥∇∥η∥H1/2 ≲ ∥u∥H1 + ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) Indeed, the claim is proved in exactly the same way as the claim from third step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 4: A priori bound on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that we have the a priori bound ∥q∥L2 ≲ ∥u∥H1 + ∥∇∥η∥H1/2 + ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63) Again, this is proved in exactly the same way as the fourth step in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 5: High norm a priori bounds on η and q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next claim that we have the bound ∥η1,H∥H2m + ∥q∥H2m ≲ N(∥u∥H1 + ∥q∥L2 + ∥∇∥η∥H1/2 + ρ∥η∥H0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) To see this, we begin by rewriting the equation ∇ · (ϱu) + τ∇ · (vw0(q + gη)) + N−1Lm(q + gη) = 0 (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='65) with the help of the splitting (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) with κ = 1: N−1Lm(q + gη1,H) + τ∇ · (vw0(q + gη1,H) = −∇ · (ϱu) − gτ∇ · (vw0η1,L) − gN−1Lm,∥η1,L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66) As ∂m n (q + gη1,H) = · · · = ∂2m−1 n (q + gη1,H) = 0, we are in a position to apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, provided that N ≳ ⟨∥q0, u0, η0∥Xm⟩4+2⌊n/2⌋, which we are free to assume;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this yields the bound ∥q + gη1,H∥H2m ≲ N(∥q∥L2 + ∥u∥H1 + ρ∥η∥H0 + ∥∇∥η∥H1/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67) Next we obtain a bound on η1,H in H2m(Σ) through the normal trace boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, we test equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) with N−1(−∆∥)m+1/4η1,H to see that N−2∥(−∆∥)mη1,H∥2 L2 ⩽ N−1∥TrΣ(u · en)∥H1/2∥(−∆∥)m+1/4η1,H∥H−1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='68) and hence ∥η1,H∥H2m ≲ N∥u∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='69) We then obtain (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='67) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='69).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 6: A priori bounds on ∂1η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that we have the a priori bound [∂1η] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + N−1∥q∥H2m + ρ∥η∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='70) COMPRESSIBLE TRAVELING WAVES 93 As in the fifth step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we integrate equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) in the nth coordinate from 0 to b and isolate the non-small ∂1η contributions: ((1 − τ)ϱ(b) + τϱ(0))∂1η = (∇∥, 0) · � b 0 (ϱu + τvw0q + τgv(1) q0,u0,η0η) + τg∂1 � η � b 0 v(2) η0 (·, y) · e1 dy � + 1 N Lm,∥ � b 0 (q + gη) + ϱ(b) N (−∆∥)m−1/4η = I1 + I2 + I3 + I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='71) Fix κ ∈ (0, 1) and set ηκ = (Π1/κ L − Πκ L)η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We take the L2-inner product of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='71) with |∇∥|−2∂1ηκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The terms involving I1 and I2 are estimated via Cauchy-Schwarz ⟨I1 + I2, |∇∥|−2∂1ηκ⟩L2,L2 ≲ [I1 + I2] ˙H−1[∂1ηκ] ˙H−1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='72) but then the argument used in the fifth step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 shows that [I1] ˙H−1 + [I2] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + ρ∥η∥H0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='73) so ⟨I1 + I2, |∇∥|−2∂1ηκ⟩L2,L2 ≲ (∥u∥L2 + ∥q∥L2 + ρ∥η∥H0) [∂1ηκ] ˙H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='74) On the other hand, for the term involving I3 we have ⟨I3, |∇∥|−2∂1ηκ⟩L2,L2 = 1 N � |∇∥|−1Lm,∥ � b 0 q, |∇∥|∂1ηκ � L2,L2 ≲ 1 N ∥q∥H2m−1[ηκ] ˙ H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='75) while for the I4 term we compute ⟨I4, |∇∥|−2∂1ηκ⟩L2,L2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' All together, these combine to show that [∂1ηκ] ˙H−1 ≲ ∥u∥L2 + ∥q∥L2 + N−1∥q∥H2m + ρ∥η∥H0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='76) and since the implicit constant is independent of κ, we can send κ → 0 to arrive at (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 7: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we derive the desired estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) by combining the bounds from the previous steps and taking ρ to be sufficiently small to absorb, exactly as in the proof of the sixth step of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Existence of solutions to the regularization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we prove the existence of weak solutions to the regularized problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and then quickly deduce qualitative regularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First we have our main existence result for weak solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (Existence of regularized weak solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, m, N ∈ N+ with m ⩾ 2, τ ∈ [0, 1], (g, F) ∈ Y−1, and γ0 ∈ I for some interval I ⋐ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) is satisfied, then there exists a unique (q, u, η) ∈ X−1 m,N satisfying w0,γ0 J τ m,N(q, u, η) = (g, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In functional analytic terms, we aim to prove that for every τ ∈ [0, 1] the operator w0,γ0 J τ m,N ∈ L(X−1 m,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Y−1), which is well-defined thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23, is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We divide the proof of this into several steps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 1: Reduction to proving existence with g = 0 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will achieve the reduction to τ = 0 through the method of continuity (see, for instance, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in Gilbarg and Trudinger [35]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, the convex homotopy of operators [0, 1] ∋ τ �→ w0,γ0 J τ m,N ∈ L(X−1 m,N;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Y−1) satisfies τ-uniform a priori estimates thanks to Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, the method of continuity guarantees that w0,γ0 J τ m,N is 94 NOAH STEVENSON AND IAN TICE an isomorphism for every τ ∈ [0, 1] if and only if w0,γ0 J 0 m,N is an isomorphism, so we reduce to proving that w0,γ0 J 0 m,N is an isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we note that the a priori estimate of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 guarantees that w0,γ0 J 0 m,N is injective, so we further reduce, by way of the bounded inverse theorem, to proving that this map is surjective.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, we reduce to proving the existence of solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with g = 0 and τ = 0 with the help of the operator B0, exactly as in the first step in the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have now shown that it suffices to establish the existence of solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) with g = 0 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, in the remainder of the proof we set g = 0 and τ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 2: Existence of a two parameter family of approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We introduce the approximation parameters M, K ∈ N+ and fix data F ∈ (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then claim that there exists a collection {(qM,K, uM,K, ηM,K)}M,K∈N+ ⊂ H2m(Ω) × 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2m(Σ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) satisfying γ0 I (qM,K, uM,K, ηM,K) = F, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) � N−1� K−1 + Lm � (qM,K + gηM,K) = −∇ · (ϱuM,K) in Ω, ∂m n (qM,K + gηM,K) = · · · = ∂2m−1 n (qM,K + gηM,K) = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and TrΣ(uM,K · en) + ∂1ηM,K = N−1(M−1 + (−∆∥)m−1/4)ηM,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) The high-level idea for producing these approximate solutions is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we note that equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) determines ηM,K as a function of uM,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) determines qM,K as a function of uM,K and ηM,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These allow us to rewrite (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) as an equation relating F and uM,K alone, and it turns out that this can be solved by utilizing the Lax-Milgram lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To prove the claim we begin by recalling that the space 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) is defined by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) and defining the bounded linear maps pK : 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → H2m(Ω) and ηM : 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → H2m(Σ) via pK(u) = −N(K−1 + Lm)−1(∇ · (ϱu)) and ηM(u) = N(M−1 + (−∆∥)m−1/4 − N∂1)−1TrΣ(u · en).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) The map ηM is well-defined and bounded in light of the symbol inversion result in Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9, while pK is well-defined and bounded by virtue of Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These maps allow us to define the bilinear form BM,K : 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → R via BM,K(u, w) = ⟨ γ0 I (pK(u) − gηM(u), u, ηM(u)), w⟩(0H1)∗,0H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Thanks to integration by parts and the definitions of pK and ηM, we have the equivalent formulation BM,K(u, w) = � Ω γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w − pK(u)∇ · (ϱw) + ⟨(gϱ − ς∆∥)ηM(u), TrΣ(w · en)⟩H−1/2,H1/2 = � Ω γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w + 1 N � Ω 1 K pK(u)pK(w) + n � j=1 ∂m j pK(u)∂m j pK(w) + ⟨(gϱ − ς∆∥)ηM(u), −∂1ηM(w) + N−1(M−1 + (−∆∥)m−1/4)ηM(w)⟩H−1/2,H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) The boundedness of BM,K is straightforward to check, but it is also coercive since by anti-symmetry we have � Ω γ2 0ϱu ⊗ e1 : ∇u = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) COMPRESSIBLE TRAVELING WAVES 95 by orthogonality we have � Ω Sϱu : ∇u = � Ω µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) by squaring we have 1 N � Ω 1 K pK(u)2 + n � j=1 (∂m j pK(u))2 ⩾ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) and by anti-symmetry and symmetry considerations again we have ⟨(gϱ − ς∆∥)ηM(u), −γ∂1ηM(w) + N−1(M−1 + (−∆∥)m−1/4)ηM(w)⟩H−1/2,H1/2 = ⟨(gϱ − ς∆∥)ηM(u), N−1(M−1 + (−∆∥)m−1/4)ηM(u)⟩ ⩾ 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) which together with the Korn inequalities of Propositions A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 imply the coercivity estimate BM,K(u, u) ⩾ γ0 � Ω µ(ϱ) 2 |D0u|2 + λ(ϱ)(∇ · u)2 ≳ ∥u∥2 H1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) Recall that our conditions on µ, λ ∈ C∞(R+) are that µ, λ > 0 if n = 2, and µ > 0, λ ⩾ 0 if n ⩾ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The hypotheses of the Lax-Milgram lemma (see, for instance Theorem 6 in Chapter 6 of Lax [58]) are satisfied, so we are therefore granted uM,K ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) with the property that BM,K(uM,K, w) = ⟨F, w⟩(0H1)∗,0H1 for all w ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) We then set ηM,K = ηK(uM,K) ∈ H2m(Σ) and qM,K = pK(uM,K) − gηM(uM,K) ∈ H2m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By construction, the collection {(qM,K, uM,K, ηM,K)}M,K∈N+ satisfies the claimed inclusions and satisfies (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 3: Estimates on the two parameter family of approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that the approximate solutions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) obey the K-independent bounds ∥qM,K, uM,K, ηM,K∥H2m×H1×H2m ≲m,M,N ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) To prove the claim, we first take w = uM,K in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and then use the coercive inequality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) to get the control sup M,K∈N+∥uM,K∥H1 ≲ ∥F∥(0H1)∗, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) where the implied constant only depends on the physical parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we employ the continuity of the map ηM to see that ∥ηM,K∥H2m ≲M N∥uM,N∥H1 ≲M N∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) Obtaining a K-independent bound on the qM,K is slightly more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We test the weak formulation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) with w = ϱ−1B(qM,K + gηM,K) ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), where B is the right inverse to the divergence constructed in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since ∇ · (ϱw) = qM,K + gηM,K = pK(uM,K), the identity BM,K(uM,K, w) = ⟨F, w⟩ and the bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) imply that ∥qM,K + gηM,K∥L2 ≲ ∥uM,K∥H1 + ∥ηM,K∥H3/2 + ∥F∥(0H1)∗ ≲N,M ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) To get a higher-regularity estimate, we rewrite the first equation in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) as (1 + Lm)(qM,K + gηM,K) = K−1 K (qM,K + gηM,K) − N∇ · (ϱuM,K) and then apply Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 with κ = 1 (and the already established K-independent bounds) to arrive at the estimate ∥qM,K + gηM,K∥H2m ≲m (1 − K−1)∥qM,K + gηM,K∥L2 + N∥uM,K∥H1 ≲m,M,N ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) imply that ∥qM,K∥H2m ≲m,M,N ∥F∥(0H1)∗, and we then combine this with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) to complete the verification of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15), and hence the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 96 NOAH STEVENSON AND IAN TICE Step 4: Existence of a one parameter family of approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the fixed data F ∈ (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ as in the previous steps, we claim that there exists a sequence {(qM, uM, ηM)}M∈N+ ⊂ H2m(Ω) × 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2m(Σ) (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) such that γ0 I (qM, uM, ηM) = F, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) � N−1Lm(qM + gηM) = −∇ · (ϱuM) in Ω, ∂m n (qM + gηM) = · · · = ∂2m−1 n (qM + gηM) = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) and TrΣ(uM · en) + ∂1ηM = N−1(M−1 + (−∆∥)m−1/4)ηM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) The existence of this sequence follows by taking a weak subsequential limit in the K parameter in our previously constructed collection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' More precisely, for each fixed M ∈ N+ we have established in the previous step that the K-independent bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, by weak compactness, there exist (qM, uM, ηM) ∈ H2m(Ω) × 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2m(Σ) that are a weak subsequential limit of the sequence {(qM,K, uM,K, ηM,K)}K∈N+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Routine weak convergence arguments applied to the identities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) then show that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the construction and the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 5: Estimates on the one parameter family of approximate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We claim that the one parameter family of approximate solutions from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) obeys the M-independent bounds ∥qM, uM, ηM∥H2m×H1×H2m ≲m,N ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) To see this, we first employ the weak sequential lower semicontinuity of the norm and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) to obtain the estimate sup M∈N+∥uM∥H1 ≲ ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) Next, as in the third step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, in the case that ς > 0, we test (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) against the function wM = −ϱ−1B2(ϱ(b)⟨∇∥⟩−1∆∥ηM) ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), where B2 is defined in Corol- lary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This yields the identity ⟨F, wM⟩(0H1)∗,0H1 = � Ω γ0(γ0ϱuM ⊗e1 +SϱuM) : ∇wM +⟨(gϱ−ς∆∥)ηM, −⟨∇∥⟩−1∆∥ηM⟩H−1/2,H1/2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) which in turn yields the estimate ∥∇∥ηM∥2 H1/2 ≲ (∥F∥(0H1)∗ + ∥uM∥H1)∥wM∥H1 ≲ ∥F∥(0H1)∗∥wM∥H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) The continuity of B2 shows that ∥wM∥H1 ≲ ∥∇∥ηM∥H1/2, and hence ∥∇∥ηM∥H1/2 ≲ ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) On the other hand, if ς = 0 and n = 2, estimate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) follows by testing (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) with ∂1⟨∇∥⟩ηM = ∇∥⟨∇∥⟩ηM ∈ H−1/2(Σ) and using orthogonality, Cauchy-Schwarz, boundedness of traces, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now, as in the fourth step of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we employ wM = ϱ−1Bq ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), where B is constructed in Corollary C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, as a test function in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) in order to see that ⟨F, wM⟩(0H1)∗,0H1 = � Ω γ0(γ0ϱuM ⊗ e1 + SϱuM) : ∇wM + gϱ∇ηM · wM − � Ω q2 M − ς⟨∆∥ηM, TrΣ(wM · en)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) From this we readily deduce the bound ∥qM∥2 L2 ≲ (∥F∥(0H1)∗ +∥uM∥H1 +∥∇∥ηM∥H1/2)∥wM∥H1, but this combines with our already established bounds and the continuity estimate ∥wM∥H1 ≲ ∥qM∥L2 to show the low regularity bound ∥qM∥L2 ≲ ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 97 We now have low regularity estimates on qM and ηM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To promote these to high regularity bounds, we proceed as in the fifth step of the proof of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first identity in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) is equivalent to N−1Lm(qM + gηH M) = −∇ · (ϱuM) − gN−1Lm,∥ηL M, where we have decomposed ηL M = Π1 LηM and ηH M = Π1 HηM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since ∂m n (qM + gηH M) = · · · = ∂2m−1 n (qM + gηH M) = 0, we can apply Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 with τ = 0 and exploit the Fourier supports of ηL M and ηH M to obtain the estimate ∥qM + gηH M∥H2m ≲ N(∥uM∥H1 + ∥Lm,∥ηL M∥L2 + ∥qM + gηH M∥L2) ≲ N(∥uM∥H1 + ∥∇∥ηM∥H1/2 + ∥qM∥L2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) Now we test the identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) against N−1(−∆∥)m+1/4ηH M ∈ H−1/2(Σ) and employ the Fourier support of ηH M as well as the bound (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) to arrive at the estimate ∥ηH M∥H2m ≲ N∥uM∥H1 ≲ N∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) Together, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) imply that ∥qM, ηH M∥H2m×H2m ≲ N∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) In light of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and equation (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), it remains only to obtain M-uniform bounds on [∂1ηM] ˙H−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we integrate (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) over (0, b) in the nth-coordinate and recall (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) to acquire the equality ϱ(b)∂1ηM = 1 N Lm,∥ � b 0 (qM + gηM) + ϱ(b) N � 1 M + (−∆∥)m−1/4� ηM + (∇∥, 0) · � b 0 ϱuM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) For κ ∈ (0, 1) we write ηκ M = (Π1/κ L − Πκ L)ηM ∈ H∞(Σ) and then take the L2 inner product of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) with |∇∥|−2∂1ηκ M ∈ H∞(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The right hand side ηM terms all vanish due to the ∂1 operator, and we arrive at the equality ϱ(b)[∂1ηκ M]2 ˙H−1 = �|∇∥|−1 N Lm,∥ � b 0 qM + |∇∥|−1(∇∥, 0) · � b 0 ϱuM, |∇∥|−1∂1ηκ M � L2,L2, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) from which we readily deduce that [∂1ηM] ˙H−1 = lim κ→0[∂1ηκ M] ˙H−1 ≲m N−1∥qM∥H2m−1 + ∥uM∥L2 ≲m ∥F∥(0H1)∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) Then (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) follows by combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) and recalling the equivalent norm on H2m(Σ) given in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The claim is proved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Step 6: Conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The M-uniform bounds (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) guarantee the existence of a weak subse- quential limit for the sequence (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20), say (q, u, η) ∈ H2m(Ω) × 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H2m(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Routine weak convergence arguments applied to the identities (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) then reveal that (q, u, η) satisfy γ0 I (q, u, η) = F, � N−1Lm(q + gη) = −∇ · (ϱu) in Ω, ∂m n (q + gη) = · · · = ∂2m−1 n (q + gη) = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) and TrΣ(u·en)+∂1η = N−1(−∆∥)m−1/4η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore (q, u, η) ∈ X−1 m,N satisfy w0,γ0 J 0 m,N(q, u, η) = (0, F), and so the proof is complete in light of the first step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ As a consequence of the existence of weak solutions to the regularization, we now show that the operators associated to the strong formulation of the regularization, namely w0,γ0 Am,N defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), are automatically isomorphisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The catch is that at this point we cannot guarantee the inverses come with estimates independent of N, so will will have to work harder in subsequent sections to verify this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 98 NOAH STEVENSON AND IAN TICE Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 (Isomorphisms induced by the regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, m, N ∈ N+ with m ⩾ 2, and γ0 ∈ I for an interval I ⋐ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then for ν ∈ N+, the map w0,γ0 Am,N : Xν m,N → Yν (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) is a Banach isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That the map (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) is well-defined is a consequence of the third item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving injectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ ker w0,γ0 Am,N ⊆ Xν m,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24 shows that strong solutions are weak solutions, and hence w0,γ0 J 1 m,N(q, u, η) = (0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then deduce that (q, u, η) = 0 by invoking the a priori estimates of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now turn to the proof of surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (g, f, k) ∈ Yν, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) and, by utilizing Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 and the map from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23), define (q, u, η) via (q, u, η) = � w0,γ0 J 1 m,N �−1(g, K (f, k)) ∈ X−1 m,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) We claim that we have the higher-regularity inclusion (q, u, η) ∈ Xν m,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once this is shown, another application of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24 reveals that w0,γ0 Am,N(q, u, η) = (g, f, k), which establishes surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To prove the claim we will employ a finite induction argument to promote the regularity of the triple (q, u, η) one step at a time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, it is useful to unpack the definition of weak solution in a more helpful way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Equation (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) is equivalent to: (q, u, η) ∈ X−1, � N−1Lmq = g − ∇ · (vw0(q + gη)) − ∇ · (ϱu) − gN−1Lm,∥η in Ω, ∂m n q = · · · = ∂2m−1 n q = 0 on ∂Ω, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) N−1(−∆∥)m−1/4Π1 Hη = −N−1(−∆∥)m−1/4Π1 Lη + ∂1η + TrΣ(u · en), TrΣ0(u) = 0, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) and for all w ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) it holds that � Ω γ0(γ0ϱu ⊗ e1 + Sϱu) : ∇w = ⟨ �F, w⟩(0H1)∗,0H1, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) where �F = K ( �f, �k) for �f = f − ϱ∇(q + gη) and �k = k + (ϱq + ς∆∥η)en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The identity (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) means, in other words, that u is a weak solution to the elliptic boundary value problem � � � � � −γ2 0ϱ∂1u − γ0∇ · Sϱu = �f in Ω, γ0Sϱuen = �k on Σ, u = 0 on Σ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) From the inclusions (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) and (q, η) ∈ H2m(Ω) × H2m(Σ) and the norm (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) we deduce that �f ∈ Hmin{ν,2m−1}(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) and �k ∈ Hmin{1/2+ν,2m−2}(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) and so the standard elliptic regularity gain for the problem (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) (see, for instance, Agmon, Douglis, and Nirenberg [3]) guarantees the inclusion u ∈ Hmin{2+ν,2m−1/2}(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) �→ H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) where the embedding holds since m ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With the improved u regularity from (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) in hand, we return to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) to see that Π1 Hη ∈ H1+2m(Σ), and hence, by Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, η ∈ H1+2m(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, we use use the improved u and η regularity together with the fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40), appealing to Lemma C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 to deduce the improvement q ∈ H1+2m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 99 Proceeding by finite induction, we assume now that for some 0 ⩽ ν ⩽ ν − 1 we have the inclusion (q, u, η) ∈ Hν+1+2m(Ω) × Hν+2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × Hν+1+2m(Σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) This implies that �f ∈ Hmin{ν,ν+2m}(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) and �k ∈ Hmin{1/2+ν,ν−1+2m}(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), and so elliptic regularity for (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) implies that u ∈ Hmin{2+ν,ν+1/2+2m}(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) �→ Hν+3(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then argue exactly as above to use the improved u regularity to promote to η ∈ Hν+2+2m(Σ) and q ∈ Hν+2+2m(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) holds with ν replaced by ν + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By finite induction, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) then also holds for ν = ν, and so (q, u, η) ∈ Xν m,N, which completes the proof of the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of strong solutions to the linearization Previously, we have established estimates for weak solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), and for the latter we proved existence and qualitative regularity in Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The majority of this section is devoted to building a series of tools to aid in the estimation of the higher regularity norms of these weak solutions when given sufficiently regular data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we require a specific quantitative understanding of the regularity for systems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' for the former we will develop tame estimates of the solution norms with respect to the background w0 = (q0, u0, η0) and γ0, while for the latter we will prove high regularity bounds that are independent of the approximation parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We proceed as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 analyzes the equations satisfied by the tangential derivatives of the solutions to systems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 reduces the regularity promotion of solutions to estimates on tangential derivatives via analysis of the so-called normal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 combines results from the tangential derivative and normal system analysis to derive the sought-after precise a priori estimates for the principal part equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then conclude in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 by deducing the existence of strong solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and then to the full linearization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of tangential derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In order to study the tangential derivatives of solutions to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), we must first understand the commutators of the operators w0,γ0 A , w0,γ0 Am,N, w0,γ0 J , and w0,γ0 J 1 m,N with the tangential derivatives ∂j for j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These operators are nearly tangentially translation invariant, and so nearly commute with the tangential derivatives;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the failure of each to commute is precisely due to the appearance of ∇ · (vw0(q + gη)) in their continuity equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following definition and subsequent lemma capture this almost tangential translation invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We recall that the ˆHs(Ω) spaces are defined in (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (The principal part commutator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and let w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1} we define the map w0 C j : H1+s(Ω) × H1+s(Σ) → ˆHs(Ω) via w0 C j(q, η) = ∇ · (∂jvw0(q + gη)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now check that the C maps are well-defined and then explore their mapping properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (Mapping properties of C ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under the hypotheses of Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, we have the following estimates for s ∈ N: �� w0 C j(q, η) �� ˆHs ≲ ρ∥q, η∥H1+s×H1+s + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) Here the implicit constant depends only on s, the physical parameters, the dimension, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 (in particular the splitting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17)) to compute ∂jvw0 = ∂jv(1) q0,u0,η0 + ∂jv(2) η0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Then, given N ∋ s ⩾ 1 + ⌊n/2⌋, we use the first item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to estimate ∥∂jv(1) q0,u0,η0∥Hs ⩽ ∥v(1) q0,u0,η0∥H1+s ≲ ∥q0, u0, η0∥Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) 100 NOAH STEVENSON AND IAN TICE For the other piece, we may use the second item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to deduce that v(2) η0 = (v(2) η0 · e1)e1 and that v(2) η0 has rn−1 as a band-limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This, in addition to the second item of the aforementioned lemma, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, allow us to estimate ∥∂jv(2) η0 ∥Hs ≲ sup 0⩽p⩽s sup y∈[0,b] ∥∂j∂p nv(2) η0 (·, y)∥Hs−p(Rn−1) ⩽ sup 0⩽p⩽s sup y∈[0,b] ∥∂p nvη0(·, y) · e1∥H0 ≲ ∥Π1 Lη0∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Together, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) imply the bound ∥∂jvw0∥Hs ≲ ∥q0, u0, η0∥Xs for s ⩾ 1 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' With this established, proving the stated estimates is a simple application of Corollary D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, it shows that for any s ∈ N and (q, η) ∈ H1+s(Ω) × H1+s(Σ), we have �� w0 C j(q, η) �� Hs ≲ ∥∂jvw0(q + gη)∥H1+s ≲ ρ∥q, Π1 Hη, Π1 Lη∥H1+s×H1+s×W 1+s,∞ + � 0 if s < ⌊n/2⌋, ∥∂jvq0,u0,η0∥H1+s∥q, Π1 Hη, Π1 Lη∥H1+⌊n/2⌋×H1+⌊n/2⌋×W 1+⌊n/2⌋,∞ if ⌊n/2⌋ ⩽ s, ≲ ρ∥q, η∥H1+s×H1+s + � 0 if s < ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, η∥H1+⌊n/2⌋×H1+⌊n/2⌋ if ⌊n/2⌋ ⩽ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Additionally, since Tr∂Ω(vw0 · en) = 0 and ∂jvw0 ∈ (L∞ ∩ L2)(Ω), we have the estimate �� b 0 w0 C j(q, η)(·, y) dy � ˙H−1 ≲ ∥∂jvw0(q + gη)∥L2 ≲ ρ∥q, Π1 Hη, Π1 Lη∥L2×L2×L∞ ≲ ρ∥q, η∥L2×H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) follows by combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now apply the mapping properties of C to obtain low regularity estimates for first order tangential derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Low norm estimates on tangential derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under the hypotheses of Defini- tion 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, the following hold for γ0 ∈ I with I ⋐ R+ an interval, (g, f, k) ∈ Y0, and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n−1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If (q, u, η) ∈ q0,u0,η0 X0 satisfies w0,γ0 A (q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ q0,u0,η0 X−1 and obeys the estimate �� w0,γ0 J (∂jq, ∂ju, ∂jη) �� Y−1 ≲ ρ∥q, u, η∥X0 + ∥g, f, k∥Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) (2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ X0 m,N satisfy w0,γ0 Am,N(q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ X−1 m,N and obeys the estimate �� w0,γ0 J 1 m,N(∂jq, ∂ju, ∂jη) �� Y−1 ≲ ρ∥q, u, η∥X0 + ∥g, f, k∥Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) Here the implicit constants depend on the dimension, the physical parameters, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will prove only the first item;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the proof of the second follows from a nearly identical argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin with three observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, recall the bounded linear map K : L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1} we construct a related bounded linear map Kj : L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) × H1/2(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For (f, k) ∈ L2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)×H1/2(Σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) we initially define the linear map Kj(f, k) : 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn)∩H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → R via ⟨Kj(f, k), w⟩ = ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 = − � Ω f · ∂jw − � Σ k · ∂jw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For such f, k, and w, we may bound |⟨Kj(f, k), w⟩| ⩽ ∥f∥L2∥∂jw∥L2 + ∥k∥H1/2∥∂jTrΣw∥H−1/2 ≲ ∥f, k∥L2×H1/2∥w∥H1, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) COMPRESSIBLE TRAVELING WAVES 101 and from this and the density of 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) ∩ H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) in 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) we deduce that Kj(f, k) uniquely extends to an element Kj(f, k) ∈ (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ such that ⟨Kj(f, k), w⟩(0H1)∗,0H1 = ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) for all w ∈ H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) ∩ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) and ∥Kj(f, k)∥(0H1)∗ ≲ ∥f, k∥L2×H1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In particular, the latter estimate shows that the induced map Kj is bounded and linear with the domain and codomain stated above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, we recall the map γ0 I : X−1 → (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗ defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ X0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1} we have that (∂jq, ∂ju, ∂jη) ∈ X−1, which means that γ0 I (∂jq, ∂ju, ∂jη) defines an element of (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may compute the action of this functional on any w ∈ H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) ∩ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) by integrating by parts: ⟨ γ0 I (q, u, η), −∂jw⟩(0H1)∗,0H1 = − � Ω −γ2 0ϱ∂1u · ∂jw − q∇ · (ϱ∂jw) + gϱ∇η · ∂jw + γ0Sϱu : ∇∂jw + ς⟨∆∥η, TrΣ(∂jw · en)⟩H−1/2,H1/2 = � Ω −γ2 0ϱ∂1∂ju · w − ∂jq∇ · (ϱw) + gϱ∇∂jη · w + � Ω γ0Sϱ∂ju : ∇w − ς⟨∆∥∂jη, TrΣ(w · en)⟩H−1/2,H1/2 = ⟨I (∂jq, ∂ju, ∂jη), w⟩(0H1)∗,0H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) For the third observation, suppose that q ∈ H1(Ω) satisfies ∇ · (vw0q) ∈ H1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Testing against members of C∞ c (Ω) and appealing to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we see that for each j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1} we have the distributional identity ∇ · (vw0∂jq) = ∂j∇ · (vw0q) − w0 C j(q, 0) ∈ L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) In particular, this identity establishes that ∂jq ∈ H0 vw0(Ω), where this space is defined in Appen- dix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Having established these three observations, we are now ready to prove the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ q0,u0,η0 X0 and (g, f, k) ∈ Y0 satisfy the strong from equation w0,γ0 A (q, u, η) = (g, f, k), which (due to the assumed level of regularity) is equivalent to the weak form equation w0,γ0 J (q, u, η) = (g, K (f, k)) (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, the weak formulation unpacks into the pair of equations ∇ · (ϱu) + ∇ · (vw0(q + gη)) = g in L2(Ω) and I (q, u, η) = K (f, k) in (0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn))∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) Let j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the first equation in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) we apply ∂j and use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) from the third observation to see that ∇ · (ϱ∂ju) + ∇ · (vw0∂j(q + gη)) = ∂jg − w0 C j(q, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) For the second equation in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) we let w ∈ H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) ∩ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), test against −∂jw, and use (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) from the first and second observations to see that ⟨ γ0 I (∂jq, ∂ju, ∂jη), w⟩(0H1)∗,0H1 = ⟨ γ0 I (q, u, η), −∂jw⟩(0H1)∗,0H1 = ⟨K (f, k), −∂jw⟩(0H1)∗,0H1 = ⟨Kj(f, k), w⟩(0H1)∗,0H1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) Since H2(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) ∩ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) is dense in 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn), we deduce that γ0 I (∂jq, ∂ju, ∂jη) = Kj(f, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) By combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16), we deduce that w0,γ0 J (∂jq, ∂ju, ∂jη) = (∂jg − w0 C j(q, η), Kj(f, k)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) 102 NOAH STEVENSON AND IAN TICE Therefore, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) follows by taking the norm in Y−1 of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and applying both Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and the boundedness of Kj established in the first observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The next result is a higher regularity version of the previous one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (High norm estimates on tangential derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Under the hypotheses of Defi- nition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, the following hold for s ∈ N+, j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}, γ0 ∈ I ⋐ R+ for I an interval, and (g, f, k) ∈ Ys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If (q, u, η) ∈ q0,u0,η0 Xs satisfies w0,γ0 A (q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ q0,u0,η0 Xs−1 and obeys the estimate �� w0,γ0 A (∂jq, ∂ju, ∂jη) �� Ys−1 ≲ ρ∥q, u, η∥Xs + ∥g, f, k∥Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) (2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ Xs m,N satisfy w0,γ0 Am,N(q, u, η) = (g, f, k), then (∂jq, ∂ju, ∂jη) ∈ Xs−1 m,N and obeys the estimate �� w0,γ0 Am,N(∂jq, ∂ju, ∂jη) �� Ys−1 ≲ ρ∥q, u, η∥Xs + ∥g, f, k∥Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) Here the implicit constants depend on s, the dimension, the physical parameters, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will only prove the first item, as the second follows from a nearly identical argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Employing the identity ∇ · (vw0∂jq) = ∂j∇ · (vw0q) − w0 C j(q, 0) and Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, it is evident that the differentiated triple (∂jq, ∂ju, ∂jη) belongs to the space q0,u0,η0 Xs−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Applying ∂j to the equations in w0,γ0 A (q, u, η) = (g, f, k) and rearranging provides the identity w0,γ0 A (∂jq, ∂ju, ∂jη) = � ∂jg − w0 C j(q, η), ∂jf, ∂jk � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) follows by taking the Ys−1 norm of both sides of this identity and applying the estimates from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ To conclude this subsection, we iterate Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and combine with Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 to obtain low norm estimates on high-order tangential derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (Synthesis of tangential derivative analysis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is as in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, s ∈ N, γ0 ∈ I ⋐ R+ for an interval I, (g, f, k) ∈ Ys, and α ∈ Nn \\ {0} be a multiindex such that α · en = 0 and |α| ⩽ 1 + s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If (q, u, η) ∈ q0,u0,η0 Xs satisfies w0,γ0 A (q, u, η) = (g, f, k), then (∂αq, ∂αu, ∂αq) ∈ w0,γ0 X−1 and we have the estimate �� w0,γ0 J (∂αq, ∂αu, ∂αη) �� Y−1 ≲ ρ∥q, u, η∥Xs + ∥g, f, k∥Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) COMPRESSIBLE TRAVELING WAVES 103 (2) If m, N ∈ N+ with m ⩾ 2 and (q, u, η) ∈ Xs m,N satisfy w0,γ0 Am,N(q, u, η) = (g, f, k), then (∂αq, ∂αu, ∂αη) ∈ X−1 m,N and we have the estimate �� w0,γ0 J 1 m,N(∂αq, ∂αu, ∂αη) �� Y−1 ≲ ρ∥q, u, η∥Xs + ∥g, f, k∥Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) The implicit constants depend on s, the dimension, the physical parameters, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Again we only prove the first item, as the second follows from a nearly identical argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First we claim that if (q, u, η) ∈ q0,u0,η0 Xs and α ∈ Nn satisfies 1 ⩽ |α| ⩽ s and α · en = 0, then (∂αq, ∂αu, ∂αη) ∈ q0,u0,η0 Xs−|α| and �� w0,γ0 A (∂αq, ∂αu, ∂αη) �� Ys−|α| ≲ ρ∥q, u, η∥Xs + �� w0,γ0 A (q, u, η) �� Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) We establish this via strong induction on |α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The base case of the claim, |α| = 1, was already established in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose now that for a fixed 1 ⩽ ν ⩽ s − 1 the claim holds for all α ∈ Nn such that α · en = 0 and 1 ⩽ |α| ⩽ ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let α ∈ Nn with α·en = 0 be such that |α| = ν+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let β, γ ∈ Nn\\{0} be such that α = β+γ and |γ| = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By repeated applications of the induction hypothesis, it follows that (∂βq, ∂βu, ∂βη) ∈ q0,u0,η0 Xs−|β| and, in turn, that (∂αq, ∂αu, ∂αη) ∈ q0,u0,η0 Xs−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the induction hypothesis also provides the estimates �� w0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='γ0 A (∂αq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂αu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂αη) �� Ys−|α| ≲ ρ∥∂βq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βη∥Xs−|β| + �� w0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='γ0 A (∂βq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βη) �� Ys−|β| + � 0 if s − |β| ⩽ ⌊n/2⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥q0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η0∥X1+s−|β|∥∂βq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βη∥X⌊n/2⌋ if ⌊n/2⌋ < s − |β|,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ≲ ρ � ∥q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η∥Xs + ∥∂βq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βη∥Xs−|β| � + �� w0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='γ0 A (q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η) �� Ys + � 0 if s ⩽ ⌊n/2⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥q0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η0∥X1+s∥q,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η∥X⌊n/2⌋ if ⌊n/2⌋ < s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' + � 0 if s − |β| ⩽ ⌊n/2⌋,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∥q0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' u0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' η0∥X1+s−|β|∥∂βq,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' ∂βη∥X⌊n/2⌋ if ⌊n/2⌋ < s − |β|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) To morph this estimate into the correct form, we note the following two facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, the ∂β-continuity estimates: ∥∂βq, ∂βu, ∂βη∥Xs−|β| ≲ ∥q, u, η∥Xs and ∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ ≲ ∥q, u, η∥X|β|+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Sec- ond, by the log-convexity of the norm in the X-spaces (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) and Young’s inequality, we have that if ⌊n/2⌋ < s − |β|, then ∥q0, u0, η0∥X1+s−|β|∥∂βq, ∂βu, ∂βη∥X⌊n/2⌋ ≲ ∥q0, u0, η0∥X1+s−|β|∥q, u, η∥X|β|+⌊n/2⌋ ≲ � ∥q0, u0, η0∥X1+⌊n/2⌋∥q, u, η∥Xs � |β| s−⌊n/2⌋ � ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ �1− |β| s−⌊n/2⌋ ≲ ρ∥q, u, η∥Xs + ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) 104 NOAH STEVENSON AND IAN TICE Upon combining these facts with (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23), we verify that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) holds, which completes the proof of the inductive step and hence the claim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now, if α ∈ Nn is such that α · en and 1 ⩽ |α| ⩽ s, then the sought-after estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) is true thanks to the estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) established in the above claim and the trivial bounds �� w0,γ0 J (q, u, η) �� Y−1 ≲ �� w0,γ0 A (q, u, η) �� Y0 ⩽ �� w0,γ0 A (q, u, η) �� Ys−|α|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) It then only remains to prove (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) in the case that |α| = s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this case we may write ∂α = ∂j∂β for some j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1} and |β| = s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Applying Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, followed by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22), we arrive at the estimate �� w0,γ0 J (∂αq, ∂αu, ∂αη) �� Y−1 ≲ ρ∥∂βq, ∂βu, ∂βη∥X0 + ∥ w0,γ0 A (∂βq, ∂βu, ∂βη)∥Y0 ≲ ρ∥q, u, η∥Xs + ∥ w0,γ0 A (q, u, η)∥Ys + � 0 if s ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+s∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) which completes the proof in the case |α| = s + 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Analysis of normal systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A useful technique in the study of the dynamic compressible Navier-Stokes system, originally developed by Matsumura and Nishida [75], is to take linear combinations of a normal, or vertical, derivative of the continuity equation with certain components of the momentum equation in order to reveal a subtle dissipative structure for the normal derivative of the density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our goal now is to implement a version of this technique for our traveling wave problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The result will essentially be a bound on various high norms of a solution in terms of the data and norms of tangential derivatives alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin with a computation that motivates the definition of the normal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that w0,γ0 A1 (q, u, η) = g and γ0 A2(q, u, η) = f, where the operators Ai are as defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), and consider the linear combination ∂n w0 A1(q, u, η) + ϱ γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) γ0 A2(q, u, η) · en = ∂ng + ϱ γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))f · en.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) On the left hand side the ∂2 n(u · en)-terms cancel each other out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We may then rearrange the above equation to create a (differentiated) steady transport equation in q, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) is equivalent to ∂n � ϱ2 γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))q + ∇ · (vw0q) � = w0,γ0 N 0 (q, u, η, g, f), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where w0,γ0 N 0 (q, u, η, g, f) = � ϱ2 γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) �′ q + ∂ng − ∂n(ϱ′u · en) − ϱ′∂nu · en − (∇∥, 0) · ∂n(ϱu) − g∂n∇ · (vw0η) + ϱ γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) � f · en + γ2 0∂1u · en + γ0µ(ϱ)∆∥u · en + γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))(∇∥, 0) · ∂nu + γ0ϱ′(µ′(ϱ)D0uen · en + γ0λ′(ϱ)∇ · u) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) The key point is that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) depends only on tangential derivatives of q, u, ∂nu, η along with ∂ng and f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If we perform the same manipulations under the regularized hypotheses N−1Lm(q + gη) + w0 A1(q, u, η) = g and γ0 A2(q, u, η) = f, where we recall that Lm is defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), then we obtain COMPRESSIBLE TRAVELING WAVES 105 the equation ∂n � ϱ2 γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ))q + ∇ · (vw0q) + 1 N Lmq � = w0,γ0 N 0 (q, u, η, g, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) The identities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) are only half of the normal system in that they only allow us to gain control of the normal derivative of q in terms of lower order and tangential derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we see how to obtain similar control of the normal derivatives u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this we only need to examine the equation γ0 A2(q, u, η) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}, then we take the jth-component of this equation and isolate the ∂2 n(u · ej) term: ∂2 n(u · ej) = γ0 N j(q, u, η, f), where γ0 N j(q, u, η, f) = 1 γ0µ(ϱ) � − γ2 0ϱ∂1u · ej + ϱ∂j(q + gη) − γ0µ(ϱ)∆∥u · ej − γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))∂j∇ · u − γ0D0uen · ej(µ(ϱ))′ − f · ej � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) On the other hand, if we take the nth-component of γ0 A2(q, u, η) = f and isolate the ∂2 n(u · en) contribution, then we get the equation ∂2 n(u · en) = γ0 N n(q, u, η, f), where γ0 N n(q, u, η, f) = 1 γ0(2(1 − 1/n)µ(ϱ) + λ(ϱ)) � − γ2 0ϱ∂1u · en + ϱ∂nq − γ0µ(ϱ)∆∥u · en − γ0(µ(ϱ)(1 − 2/n) + λ(ϱ))(∇∥, 0) · ∂nu − γ0ϱ′(µ′(ϱ)D0uen · en + λ′(ϱ)∇ · u) − f · en � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) We record the output of these calculations in the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 (Existence of the normal systems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, γ0 ∈ I for I ⋐ R+ an interval, and (g, f, k) ∈ Y0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold upon setting Λγ0(ϱ) = γ−1 0 ϱ2(2(1 − 1/n)µ(ϱ) + λ(ϱ))−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If (q, u, η) ∈ q0,u0,η0 X0 satisfies w0,γ0 A (q, u, η) = (g, f, k), then ∂n(Λγ0(ϱ)q + ∇ · (vw0q)) = w0,γ0 N 0 (q, u, η, f) (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and ∂2 nu = ( γ0 N 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , γ0 N n)(q, u, η, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) (2) If m, N ∈ N+, with m ⩾ 2, and (q, u, η) ∈ X0 m,N satisfy the equations w0,γ0 Am,N(q, u, η) = (g, f, k), then ∂n(Λγ0(ϱ)q + ∇ · (vw0q) + N−1Lmq) = w0,γ0 N 0 (q, u, η, f), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next study the mapping properties of the N mappings, starting with N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (Boundedness of N 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, γ0 ∈ I for I ⋐ R+ an interval, and s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The linear map w0,γ0 N 0 : Xs × H1+s(Ω) × Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → Hs(Ω) from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) is well-defined, bounded, and obeys the 106 NOAH STEVENSON AND IAN TICE following estimate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' �� w0,γ0 N 0 (q, u, η, g, f) �� Hs ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xs−1 + ∥g, f∥H1+s×Hs + � 0 if s < ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+s⟩∥η∥H1+⌊n/2⌋ if ⌊n/2⌋ ⩽ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) Here the implicit constant depends on the dimension, the physical parameters, ρWD, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We decompose w0,γ0 N 0 (q, u, η, g, f) = 0,γ0 N 0(q, u, η, g, f) − g∂n∇ · ((vw0 − ϱ′e1/g)η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) By inspection, it is clear that for any s ⩾ 0 we have the estimate �� 0,γ0 N 0(q, u, η, g, f) �� Hs ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xs−1 + ∥g, f∥H1+s×Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) For the remaining piece we appeal to the fourth item of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to estimate ∥∂n∇ · ((vw0 − ϱ′e1/g)η)∥Hs ⩽ ∥∇ · (vw0η)∥H1+s + ∥∂1η∥Hs ≲ ∥η∥H2+s + � 0 if s < ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+s⟩∥η∥H1+⌊n/2⌋ if ⌊n/2⌋ ⩽ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) We then conclude by noting that ∥η∥H5/2+s ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j η∥H3/2+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) □ Next, we study the N j maps for 1 ⩽ j ⩽ n, as defined by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 (Boundedness of N j, for 1 ⩽ j ⩽ n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that γ0 ∈ I for I ⋐ R+ an interval, s ∈ N, and i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the linear map γ0 N i : Xs × Hs(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) → Hs(Ω) from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) when j < n or (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) when j = n is well-defined, bounded, and satisfies the following estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' If 1 ⩽ i ⩽ n − 1, then ∥ γ0 N i(q, u, η, f)∥Hs ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xs−1 + ∥f∥Hs, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) and if j = n, then ∥ γ0 N n(q, u, η, f)∥Hs ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xs−1 + ∥∂nq∥Hs + ∥f∥Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' These estimates are clear by inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The next result is an application of some of the analysis from Section 4 to the specific steady transport structure appearing here in the normal system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 107 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 (Steady transport estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, γ0 ∈ I for I ⋐ R+ an interval, and ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ϕ, ψ ∈ Hν(Ω) satisfy ∇ · (vw0ϕ) ∈ Hν(Ω) and Λγ0(ϱ)ϕ + ∇ · (vw0ϕ) = ψ in Ω, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) where Λγ0(ϱ) is defined as in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρST,ν ∈ R+, depending only on the physical parameters, ν, and the dimension, and I, such that if ρ ⩽ ρST,ν then we have the estimate ∥ϕ, ∇ · (vw0ϕ)∥Hν×Hν ≲ ∥ψ∥Hν + � 0 if ν ⩽ 1 + ⌊n/2⌋, ∥q0, u0, η0∥Xν∥ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) The implicit constant depends on ν, the dimension, the physical parameters, ρST,ν, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Most of the work was already carried out in Section 4 in the sense that we endeavor to apply Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 with the decomposed vector field X = g ϱ′ vw0 = X0 + X1, where X0 = g ϱ′ v(1) q0,u0,η0 and X1 = e1 + g ϱ′ v(2) η0 , which is split according to the decomposition of vw0 from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Write Λ = gΛγ0(ϱ)/γϱ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By hypothesis, �ϕ = ϱ′ϕ/g ∈ Hν(Ω) satisfies Λ�ϕ + ∇ · (X �ϕ) = ψ in Ω and ∇ · (X �ϕ) = ∇ · (vw0ϕ) ∈ Hν(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) Thus, we may employ Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 to see that there exists ρ(ν) ∈ R+ (depending only ν, the physical parameters, and I) such that if ∥DX0, DX1∥H1+⌊n/2⌋×W 1+⌊n/2⌋,∞ ⩽ ρ(ν), then we have the estimate ∥�ϕ∥Hν ≲ ∥ψ∥Hν + � 0 if ν ⩽ 1 + ⌊n/2⌋, ∥DX0, DX1∥Hν×W ν,∞∥�ϕ∥H1+⌊n/2⌋ if 1 + ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) Next we note that, with implicit constants depending only on the physical parameters and s ∈ {ν, 1 + ⌊n/2⌋}, we have the estimates ∥ϕ∥Hs ≍ ∥�ϕ∥Hs and ∥DX0, DX1∥Hs×W s,∞ ≲ ∥v(1) q0,u0,η0, v(2) η0 ∥H1+s×W 1+s,∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) By invoking the first and second items of Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, we also see that ∥v(1) q0,u0,η0, v(2) η0 ∥H1+s×W 1+s,∞ ≲ ∥q0, u0, η0∥Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) The claimed bound on ϕ in the Hν(Ω) norm now follows by combining these bounds and taking ρST,ν small enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It remains to establish a bound on ∇ · (vw0ϕ) in the same space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The equation (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) is equivalent to ∇ · (vw0ϕ) = ψ − Λ(ρ)ϕ, so by taking the norm in Hν(Ω) and utilizing the established bounds on ∥ϕ∥Hν, we complete the proof of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We are now ready to identify a recursive estimate for the norm of a solution to the principal part equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The previous normal system identification and boundedness results merge in this next proposition and allow us to control the solution’s norm in terms of the data and a lower norm of the tangentially differentiated solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 (Synthesis of normal system results, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) be as in Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19, γ0 ∈ I for I ⋐ R+ an interval, and ν ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ q0,u0,η0 Xν and (g, f, k) ∈ Yν satisfy w0,γ0 A (q, u, η) = (g, f, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρnormal,ν ∈ R+ , depending only on the physical parameters, the dimension, ν, and I, such 108 NOAH STEVENSON AND IAN TICE that that if ρ ⩽ ρnormal,ν then we have the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 + ∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) The implicit constant depends on ν, ρnormal,ν, the dimension, the physical parameters, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose first that ρ ⩽ ρST,1+ν, where this parameter is from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving the stated bounds on q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Denote ψ = Λγ0(ϱ)q + ∇ · (vw0q), where Λγ0(ϱ) is from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to the steady transport estimate, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9, we have that ∥q, ∇ · (vw0q)∥H1+ν×H1+ν ≲ � ∥ψ∥H1+ν if ν ⩽ ⌊n/2⌋, ∥ψ∥H1+ν + ⟨∥q0, u0, η0∥X1+ν⟩∥q∥H1+⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24) To go from this to the desired bounds on q we will carefully estimate ∥ψ∥H1+ν by splitting into three pieces: ∥ψ∥H1+ν ≲ ∥ψ∥L2 + ∥∇ψ∥Hν ≲ ∥ψ∥L2 + n−1 � j=1 ∥∂jψ∥Hν + ∥∂nψ∥Hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) For the L2-norm in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) we bound via the definition of the norm on q0,u0,η0 X−1 : ∥ψ∥L2 ≲ ∥q∥L2 + ∥∇ · (vw0q)∥L2 ≲ ∥q, u, η∥q0,u0,η0 X−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26) Next, we consider the tangential (1 ⩽ j ⩽ n − 1) derivative terms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) by employing the commutator operator from Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 to write ∂jψ = Λγ0(ϱ)∂jq + ∇ · (vw0∂jq) + w0 C j(q, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) The first and second terms are readily dealt with using the definition of the space q0,u0,η0 Xν−1 : ∥Λγ0(ϱ)∂jq + ∇ · (vw0∂jq)∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 Xν−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) We next use Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 on the w0 C j(q, 0)-term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='27) and then combine with estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28) to see that ∥∂jψ∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 Xν−1 + ρ∥q∥H1+ν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q∥H1+⌊n/2⌋ if ⌊n/2⌋ < ν, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29) for all 1 ⩽ j ⩽ n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In order to handle the normal (j = n) derivative in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), we use the existence of the normal system, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, which shows that ∂nψ = w0,γ0 N 0 (q, u, η, g, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we use the boundedness of the linear map N 0, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, to bound ∥∂nψ∥Hν ≲ 1 � σ=0 n � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 + ∥g, f∥H1+ν×Hν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30) COMPRESSIBLE TRAVELING WAVES 109 We have now handled all of the terms on the right side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' by combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='28), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='29), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='30), we deduce the estimate ∥q, ∇ · (vw0q)∥H1+ν×H1+ν ≲ ρ∥q∥H1+ν + 1 � σ=0 n−1 � j=1 ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 Xν−1 + ∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31) Next, we turn our attention to the estimate of u in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that ∥u∥H2+ν ≲ ∥u∥L2 + n−1 � j=1 ∥∂ju∥H1+ν + ∥∂nu∥H1+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32) The L2-norm and the sum of tangential (1 ⩽ j ⩽ n − 1) derivatives are trivially controlled by the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the normal (j = n) derivative, we split as before: ∥∂nu∥H1+ν ⩽ ∥∂nu∥L2 + n−1 � j=1 ∥∂j∂nu∥Hν + ∥∂2 nu∥Hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) The first two terms in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33) are again trivially bounded by the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, ∂2 nu satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8), so we may employ the boundedness of the linear maps N 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , N n, proved in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, to estimate ∥∂2 nu∥Hν ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 + ∥∂nq∥Hν + ∥f∥Hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) For the ∥∂nq∥Hν term on the right we insert the already established bounds for ∥q∥H1+ν from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then synthesize these bounds to deduce the sought-after estimate of u in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In remains only to handle η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, the estimate for η in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) follows directly from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We are free to choose the function N ∋ ν �→ ρnormal,ν ∈ R+ to be nonincreasing without altering the statement or conclusion of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next, we consider the normal system corresponding to the regularized principal part operator w0,γ0 Am,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We require the following preliminary result, which is analogous to Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9 from the unregularized case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12 (Regularized steady transport estimate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, m, N ∈ N+, and ν ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that ϕ ∈ Hν+2m(Ω) and ψ ∈ Hν(Ω) satisfy � Λγ0(ϱ)ϕ + ∇ · (vw0ϕ) + N−1Lmϕ = ψ, in Ω, ∂m n ϕ = · · · = ∂2m−1 n ϕ = 0 in ∂Ω, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='35) where Λγ0(ϱ) and Lm are defined in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρRST,m ∈ R+, depending only on the physical parameters, the dimension, m, and I, such that if ρ ⩽ ρRST,m and N ≳ ⟨∥q0, u0, η0∥X1+⌊n/2⌋+m⟩4+2⌊n/2⌋, then we have the estimate ∥ϕ, ∇ · (vw0ϕ), N−1ϕ∥Hν×Hν×Hν+2m ≲ ∥ψ∥Hν + ⟨∥q0, u0, η0∥X2m⟩2+⌊n/2⌋∥ψ∥L2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='36) where the implied constants depend on m, the dimension, the physical parameters, ρRST,m, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 110 NOAH STEVENSON AND IAN TICE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Once more, the goal is to invoke the work from Section 4 by applying Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, we set Λ0 = Λγ0(ϱ) and Λ1 = g−1ϱ′, and we use Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19 to define the decomposed vector field X = X0 + X1, where X0 = g ϱ′ v(1) q0,u0,η0 and X1 = e1 + g ϱ′ v(2) η0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to the first and second items of the lemma, we have that ∥X0, X1∥H1+s×W 1+s,∞ ≲ ⟨∥q0, u0, η0∥Xs⟩ for N ∋ s ⩾ 1 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='37) By the additional fact that ∇Λ1 = γg−1ϱ′′en, we see that max{∥DX0, DX1∥H2+⌊n/2⌋×W 2+⌊n/2⌋,∞, ∥X0 · ∇Λ1∥H1+⌊n/2⌋, ∥X1 · ∇Λ1∥W 1+⌊n/2⌋,∞} ≲ ∥q0, u0, η0∥X2+⌊n/2⌋ ⩽ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='38) Thus, we may take 0 < ρRST,m ≲ ρ(m), where the latter is defined in Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8, in order to reach the asserted conclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The next result is the analog of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 for the m, N-regularized principal part operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There are a few key differences, the most glaring of which is the finite range, depending on m, in which the estimate holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Another more minor distinction is that while the following estimates are no longer tame (due to the appearance of the ⟨·⟩2+⌊n/2⌋ term), they still place all of the high norms of the background onto the low norms of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 (Synthesis of normal system results, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, m, N ∈ N+ with m ⩾ 2, and ν ∈ {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that (q, u, η) ∈ Xν m,N and (g, f, k) ∈ Yν satisfy w0,γ0 Am,N(q, u, η) = (g, f, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρreg,normal,ν,m ∈ R+, depending only on the physical parameters, the dimension, ν, m, and I such that if ρ ⩽ ρreg,normal,ν,m and N ≳ ⟨∥q0, u0, η0∥X1+⌊n/2⌋+m⟩4+2⌊n/2⌋, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39) then we have the estimate ∥q, u, η∥q0,u0,η0 Xν m,N ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N + ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40) The implicit constants depend on the dimension, the physical parameters, m, ρreg,normal,ν,m, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof largely mirrors that of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, but since we have to appeal to estimates for solutions to the regularized steady transport equation, the estimates we get are somewhat different in a few key places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As such, we work through most of the argument in detail, appealing to the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 when possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose first that ρ ⩽ ρRST,m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin with the bounds on q by setting ψ = Λγ0(ϱ)q + ∇ · (vw0q) + N−1Lmq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to the regularized steady transport estimate, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12, we have the bound ∥q, ∇ · (vw0q), N−1q∥H1+ν×H1+ν×H1+ν+2m ≲ ∥ψ∥H1+ν + ⟨∥q0, u0, η0∥X2m⟩2+⌊n/2⌋∥ψ∥L2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) and to parlay this into our desired estimate we split ∥ψ∥H1+ν ⩽ ∥ψ∥L2 + n−1 � j=1 ∥∂jψ∥Hν + ∥∂nψ∥Hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) For the L2-norm we bound as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='26): ∥ψ∥L2 ≲ ∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) COMPRESSIBLE TRAVELING WAVES 111 We study the tangential (1 ⩽ j ⩽ n − 1) derivatives in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42) via the identity ∂jψ = Λγ0(ϱ)∂jq + ∇ · (vw0∂jq) + N−1Lm∂jq + w0 C j(q, 0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='44) where w0 C j is as in Definition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, which shows that ∥∂jψ∥Hν ≲ ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 Xν−1 m,N + �� w0 C j(q, 0) �� Hν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) By the boundedness of C j established in Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we obtain the estimate ∥ w0 C j(q, 0)∥Hν ≲ ρ∥q∥H1+ν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q∥H1+⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='46) In the case that ⌊n/2⌋ < ν, we use interpolation (see Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25) and Young’s inequality to further estimate ∥q0, u0, η0∥X1+ν∥q∥H1+⌊n/2⌋ ≲ ρ∥q∥H1+ν + ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, for any ν we have the bound �� w0 C j(q, 0) �� Hν ≲ ρ∥q∥H1+ν + ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q∥L2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) Upon combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='45) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='47) and then summing over 1 ⩽ j ⩽ n − 1, we acquire the tangential bound n−1 � j=1 ∥∂jψ∥Hν ≲ ρ∥q∥H1+ν + n−1 � j=1 ∥∂jq, ∂ju, ∂jη∥q0,u0,η0 Xν−1 m,N + ∥q0, u0, η0∥X2+⌊n/2⌋+ν∥q, u, η∥X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48) It remains to estimate the ∥∂nψ∥Hν-term in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For this, we recall the regularized normal system, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, which says that ∂nψ = w0,γ0 N 0 (q, u, η, g, f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, we may apply the continuity properties of N 0, Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, to see that ∥∂nψ∥Hν ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 + ∥g, f∥H1+ν×Hν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='49) Again, in the latter case of ⌊n/2⌋ < ν we use interpolation and Young’s inequality to bound ⟨∥q0, u0, η0∥X1+ν⟩∥η∥H1+⌊n/2⌋ ≲ ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2⟩∥η∥H3/2+ν + ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2+ν⟩∥η∥H3/2, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='50) and hence in all cases ∥∂nψ∥Hν ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 + ∥g, f∥H1+ν×Hν + ⟨∥q0, u0, η0∥X⌊n/2⌋+1/2+ν⟩∥q, u, η∥X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51) Upon piecing together (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='42), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='48), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='51), we arrive at the bound ∥ψ∥H1+ν ≲ ρ∥q∥H1+ν + 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N + ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥X2+⌊n/2⌋+ν⟩∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) 112 NOAH STEVENSON AND IAN TICE Now we insert (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='43) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='52) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='41) and choose ρreg,normal,ν,m ∈ R+ such that ρreg,normal,ν,m ⩽ ρRST,m and sufficiently small to allow absorption of the ∥q∥H1+ν term on the right by the left hand side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This yields ∥q, ∇ · (vw0q), N−1q∥H1+ν×H1+ν×H1+ν+2m ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N + ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 X−1 m,N , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) and this implies the asserted bound on q in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, the proof of the estimates for u and η is mostly a reprise of the latter part of the proof of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We get equations (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='32), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='33), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='34) exactly as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we insert our new bound for q from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='53) into the ∥∂nq∥Hν-term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This yields the u bound of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The η bound is again trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For fixed m, we are free to choose the function {0, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , m − 1} ∋ ν �→ ρreg,normal,ν,m ∈ R+ to be nonincreasing without altering the statement or conclusion of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The final result of this subsection iterates the conclusions of Propositions 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 to alter the form of the estimate’s right hand side to one in which higher order tangential derivatives appear in a low norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 (Synthesis of normal system results, 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, r ∈ {1 + ⌊n/2⌋, 2 + ⌊n/2⌋}, w0 = (q0, u0, η0) ∈ BXr(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, ν ∈ N, and (g, f, k) ∈ Yν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If r = 1 + ⌊n/2⌋, ρ ⩽ ρnormal,ν, and (q, u, η) ∈ q0,u0,η0 Xν satisfy w0,γ0 A (q, u, η) = (g, f, k), then we have the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ ρ∥q, u, η∥Xν + � |α|⩽1+ν α·en=0 ∥∂αq, ∂αu, ∂αη∥q0,u0,η0 X−1 + ∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) The implied constant depends on ν, the dimension, the physical parameters, ρnormal,ν, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) If r = 2 + ⌊n/2⌋, N ∋ m ⩾ max{2, 1 + ν}, N ∈ N satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39), ρ ⩽ ρreg,normal,ν,m, and (q, u, η) ∈ Xν m,N satisfy w0,γ0 Am,N(q, u, η) = (g, f, k), then there exists χν,m ∈ R+, depending only on ν, m, and the dimension, and C1/ρ ∈ R+, depending only on 1/ρ, ν, m, such that we have the estimate ∥q, u, η∥q0,u0,η0 Xν m,N ≲ ρ∥q, u, η∥q0,u0,η0 Xν m,N + � |α|⩽1+ν α·en ∥∂αq, ∂αu, ∂αη∥q0,u0,η0 X−1 m,N + ∥g, f, k∥Yν + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55) Here the implied constant depends on m, the dimension, the various physical parameters, ρreg,normal,ν,m, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving the first item via an induction argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proposition to be proved inductively is as follows: if s ∈ N+ satisfies s ⩽ ν + 1 for some ν ∈ N, ρ ⩽ ρnormal,ν, and COMPRESSIBLE TRAVELING WAVES 113 (q, u, η) ∈ q0,u0,η0 Xν , then we have the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ ρ∥q, u, η∥Xν + � |α|⩽s α·en=0 ∥∂αq, ∂αu, ∂αη∥q0,u0,η0 Xν−s + �� w0,γ0 A (q, u, η) �� Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) The base case, s = 1, was established in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose now that N ∋ s ⩾ 2 is such that the proposition holds at the level s − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will now verify it at level s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that s ⩽ ν + 1 for some ν ∈ N, ρ ⩽ ρnormal,ν, and (q, u, η) ∈ q0,u0,η0 Xν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We first invoke Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 to obtain the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 + �� w0,γ0 A (q, u, η) �� Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57) Fix σ ∈ {0, 1} and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To handle the ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 term appearing in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57), we invoke the induction hypothesis at level s − 1 (while heeding to Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) in order to learn that ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 ≲ ρ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 + � |α|⩽s−1 α·en=0 ∥∂α∂σ j q, ∂α∂σ j u, ∂α∂σ j η∥q0,u0,η0 Xν−s + �� w0,γ0 A (∂σ j q, ∂σ j u, ∂σ j η) �� Yν−1 + � 0 if ν ⩽ 1 + ⌊n/2⌋, ⟨∥q0, u0, η0∥Xν⟩∥∂σ j q, ∂σ j u, ∂σ j η∥X⌊n/2⌋ if 1 + ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58) We estimate the first term on the right hand side trivially via ∥∂σ j q, ∂σ j u, ∂σ j η∥Xν−1 ≲ ∥q, u, η∥Xν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the term involving the operator w0,γ0 A , we invoke the first item of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (noting that ν > 0) to bound �� w0,γ0 A (∂σ j q, ∂σ j u, ∂σ j η) �� Yν−1 ≲ ρ∥q, u, η∥Xν + �� w0,γ0 A (q, u, η) �� Yν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59) Upon combining inequalities (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='57), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='58), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='59) with the fact that � 0 if ν ⩽ 1 + ⌊n/2⌋, ⟨∥q0, u0, η0∥Xν⟩∥∂σ j q, ∂σ j u, ∂σ j η∥X⌊n/2⌋ if 1 + ⌊n/2⌋ < ν, ≲ ρ∥q, u, η∥Xν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < s, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='60) we establish (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) for the level s, and thus prove the proposition at the s level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proposition then holds for all s ∈ N+ by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By taking s = 1 + ν in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='56) we obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54), which completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 114 NOAH STEVENSON AND IAN TICE Next, we turn our attention to the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The strategy here is the same as in the first item, but due to the dissimilarities between (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55), we cannot apply precisely the same argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our new proposition to be proved inductively is as follows: if s ∈ N+, then for all N ∋ ν ⩾ s − 1 and N ∋ m ⩾ max{2, 1 + ν}, there exists χ ∈ R+, depending on s, m, and ν, and C1/ρ, depending on s, 1/ρ, ν, and m, such that for all N ∈ N satisfying (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39), ρ ⩽ ρreg,normal,ν,m, and (q, u, η) ∈ Xν m,N we have the estimate ∥q, u, η∥q0,u0,η0 Xν m,N ≲ ρ∥q, u, η∥q0,u0,η0 Xν m,N + � |α|⩽s α·en=0 ∥∂αq, ∂αu, ∂αη∥q0,u0,η0 Xν−s m,N + �� w0,γ0 Am,N(q, u, η) �� Yν + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61) As before, the base case, s = 1, was established already, this time in Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose now that N ∋ s ⩾ 2 is such that the induction proposition holds at level s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will now verify it at level s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that N ∋ ν ⩾ s − 1, N ∋ m ⩾ max{2, 1 + ν}, N ∈ N satisfies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='39), ρ ⩽ ρreg,normal,ν,m, and (q, u, η) ∈ Xν m,N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As before, we first invoke Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13 and obtain the estimate ∥q, u, η∥q0,u0,η0 Xν m,N ≲ 1 � σ=0 n−1 � j=1 ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N + �� w0,γ0 Am,N(q, u, η) �� Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) For σ ∈ {0, 1} and j ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , n − 1}, we invoke the (s − 1)-induction hypotheses while again heeding Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 to bound ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N ≲ ρ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N + � |α|⩽s−1 α·en=0 ∥∂α∂σ j q, ∂α∂σ j u, ∂α∂σ j η∥q0,u0,η0 Xν−s m,N + �� w0,γ0 Am,N(∂σ j q, ∂σ j u, ∂σ j η) �� Yν−1 + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63) Now we make three estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we again have the trivial estimate ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 Xν−1 m,N ≲ ∥q, u, η∥q0,u0,η0 Xν m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64) Second, by invoking the second item of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, the log-convexity of the X-norms from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25, and Young’s inequality, we get the bounds �� w0,γ0 Am,N(∂σ j q, ∂σ j u, ∂σ j η) �� Yν−1 ≲ ρ∥q, u, η∥Xν + �� w0,γ0 Am,N(q, u, η) �� Yν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν, ≲ ρ∥q, u, η∥Xν + �� w0,γ0 A (q, u, η) �� Yν + ρ−1−⌊n/2⌋⟨∥q0, u0, η0∥X1+ν⟩2+⌊n/2⌋∥q, u, η∥X−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='65) COMPRESSIBLE TRAVELING WAVES 115 Third, by the log-convexity of the X-norms again and Young’s inequality, we get the bound C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χ∥∂σ j q, ∂σ j u, ∂σ j η∥q0,u0,η0 X−1 m,N ≲ ρ∥q, u, η∥q0,u0,η0 Xν m,N + ρ−1/νC(1+ν)/ν 1/ρ ⟨∥q0, u0, η0∥Xmax{2m,1+m+⌊n/2⌋}⟩χ(1+ν)/ν∥q, u, η∥q0,u0,η0 X−1 m,N .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66) Now we combine estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='62) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='63) with the trio (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='64), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='65), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='66);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' this shows that the induction proposition holds at the level s, and hence for all s ∈ N+ by induction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='55) from the second item now follows by taking s = 1 + ν in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Estimates and existence for the principal part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we first prove a priori estimates for systems (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' After this, we derive the existence theory for the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 (A priori estimates for the principal part and the regularization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, r ∈ {1 + ⌊n/2⌋, 2 + ⌊n/2⌋}, w0 = (q0, u0, η0) ∈ BXr(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval, ν ∈ N, and (g, f, k) ∈ Yν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Let r = 1 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρest,ν ∈ R+, depending only on ν, the dimension, the various physical parameters, and I, such that if ρ ⩽ ρest,ν, then for (q, u, η) ∈ q0,u0,η0 Xν satisfying w0,γ0 A (q, u, η) = (g, f, k) we have the a priori estimate ∥q, u, η∥q0,u0,η0 Xν ≲ ∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) The implicit constants depend on ν, the dimension, the physical parameters, ρest,ν, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Let r = 2 + ⌊n/2⌋ and N ∋ m ⩾ max{2, 1 + ν}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a ρreg,ν,m ∈ R+, depending only on ν, m, the various physical parameters, and I such that if ρ ⩽ ρreg,ν,m, then for (q, u, η) ∈ Xν m,N satisfying w0,γ0 Am,N(q, u, η) = (g, f, k) we have the a priori estimate ∥q, u, η∥q0,u0,η0 Xν m,N ≲ ∥g, f, k∥Yν + ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 X−1 m,N , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) where χν,m ∈ R+ is from the second item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Here the implicit constant depends on ν, m, the dimension, the physical parameters, ρreg,ν,m, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin by proving the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that ρ ⩽ min{ρweak, ρnormal,ν}, where these smallness parameters are from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By combining the a priori estimates for weak solutions from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 with the tangential derivative estimates from the first item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, we learn that � |α|⩽1+ν α·en=0 ∥∂αq, ∂αu, ∂αη∥q0,u0,η0 X−1 ≲ ρ∥q, u, η∥Xν + ∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) We insert the bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) into conclusion (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='54) of the first item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 to acquire the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ ρ∥q, u, η∥Xν +∥g, f, k∥Yν + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Hence, we may define ρest,ν ∈ R+ to be sufficiently small so that when 0 < ρ ⩽ ρest,ν, we may absorb the right hand side’s ∥q, u, η∥Xν-contribution by the left and obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 116 NOAH STEVENSON AND IAN TICE Next we consider the second item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The argument is basically the same, but with an extra step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume 0 < ρ ⩽ min{ρweak,reg, ρreg,normal,ν,m}, where these smallness parameters are from Propositions 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As in the proof of the first item, we combine the conclusions of Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4, the second item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5, and the second item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' however, the resulting estimate is slightly different: ∥q, u, η∥q0,u0,η0 Xν m,N ≲ ρ∥q, u, η∥q0,u0,η0 Xν m,N + ∥g, f, k∥Yν + C1/ρ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥q, u, η∥q0,u0,η0 X−1 m,N + � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) To reach the estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), we note that the log-convexity of the X-norms from Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='25, paired with Young’s inequality grants the bound � 0 if ν ⩽ ⌊n/2⌋, ∥q0, u0, η0∥X1+ν∥q, u, η∥X⌊n/2⌋ if ⌊n/2⌋ < ν, ≲ ρ∥q, u, η∥q0,u0,η0 Xν m,N + ρ−1−⌊n/2⌋⟨∥q0, u0, η0∥X1+ν⟩2+⌊n/2⌋∥q, u, η∥q0,u0,η0 X−1 m,N , (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) and thus we may combine estimates (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6), and then take 0 < ρ ⩽ ρreg,ν,m sufficiently small to obtain (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now are in a position to give an existence result for the principal part of the linearization, system (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 (Existence for the principal part).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD, where the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, and let w0 = (q0, u0, η0) ∈ BX2+⌊n/2⌋(0, ρ) ∩ X∞, γ0 ∈ I for I ⋐ R+ an interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For each ν ∈ N, there exists a ρexi,ν ∈ R+, depending on ν, the physical parameters, and I, such that if ρ ⩽ ρexi,ν then the map w0,γ0 A : q0,u0,η0 Xν → Yν, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) the action of which is given via (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), is a Banach isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Take ρexi,ν = min{ρest,ν, ρreg,ν,max{2,1+ν}}, where these smallness parameters are from the first and second items of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' That the map (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) is well-defined is a consequence of the first item of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This map is injective as a consequence of a priori estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) from the first item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It remains only to verify surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof of the second item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 shows that ρexi,ν ⩽ ρreg,weak, and hence we are in a position to apply Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, which tells us that the maps w0,γ0 Amax{2,1+ν},N : Xν max{2,1+ν},N → Yν are Banach isomorphisms for N ∈ N sufficiently large, say N ⩾ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, given (g, f, k) ∈ Yν we can define the sequence {(qN, uN, ηN)}∞ N=N ⊂ Xν via (qN, uN, ηN) = � w0,γ0 Amax{2,1+ν},N �−1(g, f, k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this we are tacitly using that Xν m,N �→ Xν for any m, N ∈ N+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' in fact, this embedding is non-expansive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, by applying the a priori estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) from the second item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16, followed by the regularized weak solution a priori estimate from Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (and by also invoking Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24), we obtain the uniform bounds sup N⩾N � ∥qN, uN, ηN∥Xν+∥∇·(vw0qN)∥H1+ν � ≲ ⟨∥q0, u0, η0∥Xmax{2m,1+⌊n/2⌋+m}⟩χν,m∥g, f, k∥Yν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) COMPRESSIBLE TRAVELING WAVES 117 Therefore, we may extract a weak limit (q, u, η) ∈ Xν with ∇ · (vw0q) ∈ H1+ν(Ω) such that along some unlabeled subsequence we have that (qN, uN, ηN) ⇀ (q, u, η) in Xν and ∇ · (vw0qN) ⇀ ∇ · (vw0q) in H1+ν(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) By routine weak convergence arguments, we readily deduce that (q, u, η) ∈ q0,u0,η0 Xν and w0,γ0 A (q, u, η) = (g, f, k), which completes the proof of surjectivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Synthesis of linear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this final subsection of linear analysis, we turn to the study of the full derivative of the nonlinear map Ψ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), which is associated to the PDE (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In other words, we consider the question of existence and tame estimates for the system DΨ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) As usual, the unknowns are q : Ω → R, u : Ω → Rn, and η : Σ → R, while the given data are g : Ω → R, f : Ω → Rn, k : Σ → Rn, G, F : Rn → Rn, T : Rn → Rn×n, and γ ∈ R, as well as the background tuple (θ0, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now state our main linear analysis result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 (Analysis of the linearization).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let 0 < ρ ⩽ ρWD (recall that the latter is defined in Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17), I ⋐ R+ be an interval, and (θ0, γ0) = (q0, u0, η0, T0, G0, F0, γ0) ∈ (BX3+⌊n/2⌋(0, ρ) × BW4+⌊n/2⌋(0, ρ) × I) ∩ (X∞ × W∞ × R+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) For every N ∋ ν ⩾ 3 + ⌊n/2⌋ there exists a ρν ∈ R+, depending on ν, the physical parameters, and I, such that if 0 < ρ ⩽ ρν, then the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) The map DΨ(θ0, γ0) : q0,u0,η0 Xν × W1+ν × R → Yν × W1+ν × R (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) is well-defined and a Banach isomorphism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Assume that Ξ = (g, f, k, T , G, F, γ) ∈ Yν ×W1+ν ×R and Θ = (DΨ(θ0, γ0))−1Ξ ∈ q0,u0,η0 Xν × W1+ν × R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have the tame estimate ∥Θ∥q0,u0,η0 Xν ×W1+ν×R ≲ ∥Ξ∥Yν×W1+ν×R + ⟨∥θ0∥X1+ν×W1+ν⟩∥Ξ∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) where the implicit constant depends on ν, the dimension, the physical parameters, ρν, and I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose initially that N ∋ ν ⩾ 2 + ⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The idea of the proof is to prove a priori estimates, uniform with respect to a parameter, and utilize the method of continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To this end, for τ ∈ [0, 1] we define the convex homotopy of operators Lτ(θ0, γ0) : q0,u0,η0 Xν × W1+ν × R → Yν × W1+ν × R (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) via Lτ(θ0, γ0)(q, u, η, T , G, F, γ) = �w0,γ0 A (q, u, η) + τ �w0,γ0 P + θ0Q � (q, u, η, γ) + q0,u0,η0 R (T , G, F) T , G, F, γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Note that L1(θ0, γ0) = DΨ(θ0, γ0) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and that the mapping properties of A, P, Q, and R established in Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22 ensure well-definedness and continuity of the maps Lτ(θ0, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now aim to prove τ-uniform a priori estimates for {Lτ(θ0, γ0)}τ∈[0,1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that (q, u, η, T , G, F, γ) ∈ q0,u0,η0 Xν × W1+ν × R and (g, f, k) ∈ Yν (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) are related via Lτ(θ0, γ0)(q, u, η, T , G, F, γ) = (g, f, k, T , G, F, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) 118 NOAH STEVENSON AND IAN TICE The first three components of the above equation are equivalent to w0,γ0 A (q, u, η) = (g, f, k) − τ �w0,γ0 P + θ0Q � (q, u, η, γ) − q0,u0,η0 R (T , G, F) = (�g, �f, �k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) Assume that 0 < ρ ⩽ ρest,ν;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' then we may invoke the principal part estimates from the first item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 to see that ∥q, u, η∥q0,u0,η0 Xν ≲ ∥g, f, k∥Yν + �� w0,γ0 P (q, u, η, γ) �� Yν + �� θ0Q(q, u, η) �� Yν + �� q0,u0,η0 R (T , G, F) �� Yν + ⟨∥q0, u0, η0∥X1+ν⟩∥q, u, η∥X⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) According to the second item of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 and the first and second items of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22, we have the bounds �� w0,γ0 P (q, u, η, γ) �� Yν ≲ ρ∥q, u, η∥Xν + ∥q0, u0, η0∥X1+ν∥q, u, η, γ∥X⌊n/2⌋×R, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) �� θ0Q(q, u, η) �� Yν ≲ ρ∥q, u, η∥Xν + ∥θ0∥Xν×W1+ν∥q, u, η∥X2+⌊n/2⌋, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) and �� q0,u0,η0 R (T , G, F) �� Yν ≲ ∥T , G, F∥W1+ν + ⟨∥q0, u0, η0∥Xν⟩∥T , G, F∥W3+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) Inserting (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) into the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10), we obtain the estimate ∥q, u, η∥q0,u0,η0 Xν ≲ ρ∥q, u, η∥Xν + ∥g, f, k, T , G, F∥Yν×W1+ν + ⟨∥θ0∥X1+ν×W1+ν⟩∥q, u, η, T , G, F, γ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) Now we take 0 < ρ ⩽ min{ρest,ν, ρexi,ν}, where the latter smallness parameters are from The- orems 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 and 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17, to be sufficiently small so that we can absorb as usual in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) to see that ∥q, u, η∥q0,u0,η0 Xν ≲ ∥g, f, k, T , G, F∥Yν×W1+ν+⟨∥θ0∥X1+ν×W1+ν⟩∥q, u, η, T , G, F, γ∥X2+⌊n/2⌋×W3+⌊n/2⌋×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) We need to handle the norm of (q, u, η) in X2+⌊n/2⌋ on the right hand side of (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' According to the first item of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), we know that ∥q, u, η∥X2+⌊n/2⌋ ⩽ ∥q, u, η∥ q0,u0,η0 X2+⌊n/2⌋ ≲ ∥�g, �f, �k∥Y2+⌊n/2⌋, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) where the latter is defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By arguing as in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12), and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) again but using (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2), we learn that ∥�g, �f, �k∥Y2+⌊n/2⌋ ≲ ρ∥q, u, η, γ∥X2+⌊n/2⌋×R + ∥g, f, k∥Y2+⌊n/2⌋ + ∥T , G, F∥W3+⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) By combining (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) and taking ρ smaller, if necessary, we gain the estimate ∥q, u, η∥X2+⌊n/2⌋ ≲ ∥g, f, k, T , G, F, γ∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) Finally, we insert (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) into (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) to acquire the estimate ∥q, u, η, T , G, F, γ∥q0,u0,η0 Xν ×W1+ν×R ≲ ∥Lτ(θ0, γ0)(q, u, η, T , G, F, γ)∥Yν×W1+ν×R + ⟨∥θ0∥X1+ν×W1+ν⟩∥Lτ(θ0, γ0)(q, u, η, T , G, F, γ)∥Y2+⌊n/2⌋×W3+⌊n/2⌋×R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) The bound (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) establishes the desired τ-uniform a prior bounds for {Lτ(θ0, γ0)}τ∈[0,1], as long as ρ ⩽ ρν for some ρν ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, by the method of continuity (see, for instance, Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in Gilbarg and Trudinger [35]), the invertibility of L1(θ0, γ0) = DΨ(θ0, γ0) is established as soon COMPRESSIBLE TRAVELING WAVES 119 as we know that L0(θ0, γ0) is invertible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The latter holds thanks to Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 (and the fact that ρν ⩽ ρexi,ν) since (L0(θ0, γ0))−1(g, f, k, T , G, F, γ) = ��w0,γ0 A �−1� (g, f, k) − q0,u0,η0 R (T , G, F) � , T , G, F, γ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) This completes the proof of the first item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second item follows from the first and estimate (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) at τ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Conclusion In this section we prove our main result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We begin with Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1, which is an abstract construction in the Nash-Moser inverse function theorem framework from Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then employ the abstract construction for the PDE-style result in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Abstract construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall the scales of Banach spaces from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11), the nonlinear operator Ψ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1), and the parameter ρWD ∈ R+ from Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To state the next theorem we introduce the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we set β = 9 + 2⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Second, for N ∋ s ⩾ β and ρ ∈ R+ we define the open sets Es(ρ) = BEβ(0, ρ) ∩ Es and Fs(ρ) = BFβ(0, ρ) ∩ Fs, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) where Es and Fs are defined by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, it is convenient to introduce a translation of Ψ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' namely, for Γ ∈ R+ we set ΨΓ(q, u, η, T , G, F, γ) = Ψ(q, u, η, T , G, F, Γ + γ) − (0, 0, 0, 0, 0, 0, Γ), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) defined for (q, u, η) ∈ BXβ(0, ρWD), (T , G, F) ∈ W1+β, and R+ ∋ γ > −Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The utility of this definition is that ΨΓ(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (Traveling waves for free boundary compressible Navier-Stokes, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that the parameters µ, λ, and ς satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and that P satisfies P ′ > 0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let Γ ∈ R+, and let β, Es(ρ), Fs(ρ), and ΨΓ be as defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a nonincreasing sequence {εν(Γ)}∞ ν=0 ⊂ (0, min{ρWD, Γ/2}) and κ(Γ) ∈ R+ such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Existence and uniqueness: Given f ∈ Fβ(ε0(Γ)), there exists a unique e ∈ Eβ(κ(Γ)ε0(Γ)) such that ΨΓ(e) = f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This induces the local inverse map Ψ −1 Γ : Fβ(ε0(Γ)) → Ψ −1 Γ (Fβ(ε0(Γ))) ⊆ Eβ(κ(Γ)ε0(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) (2) Higher regularity, given low norm smallness: If ν ∈ N and f ∈ Fβ+ν(εν(Γ)), then Ψ −1 Γ (f) ∈ Eβ+ν, and we have the tame estimate ∥Ψ −1 Γ (f)∥Eβ+ν ≲ ∥f∥Fβ+ν, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) for an implicit constant depending only on the dimension, the physical parameters, ν, ρWD, and Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Continuous dependence: For every ν ∈ N, the restricted map Ψ −1 Γ : Fβ+ν(εν(Γ)) → Eβ+ν (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) is continuous with respect to the norms on Fβ+ν and Eβ+ν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) Continuous differentiability: For every ν ∈ N, the restricted map Ψ −1 Γ : Fβ+ν(εν+2(Γ)) → Eβ+ν−1 (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) is differentiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Moreover, if we view DΨ −1 Γ as a map DΨ −1 Γ : Fβ+ν(εν+2(Γ)) × Fβ+ν−1 → Eβ+ν−1, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) then DΨ −1 Γ is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 120 NOAH STEVENSON AND IAN TICE Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our aim is to show that the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 are satisfied by ΨΓ with the Banach scales EEE and FFF defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thanks to Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, we have that condition II from the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 is satisfied by EEE and FFF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In verifying the rest of the hypotheses we consider the triple (EEE,FFF, ΨΓ) with parameters (µ, r, R) = (1, 3 + ⌊n/2⌋, 17 + 3⌊n/2⌋), which obey the requisite inequality 2(r + µ) + 1 = β < (r + R)/2 with β = 9 + 2⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We next set δr = min{ρWD, Γ/2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17 and Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 prove that the second item (C2 and µ-tameness) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Invoking Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 with ν = R and interval I = [Γ/2, 2Γ] then shows that the third item (derivative inversion) of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20 holds with δR = min{ρν, δr} (where ρν is given by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) and that the remainder of the LRI mapping hypotheses from the definition are also satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence, the hypotheses of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We thus obtain an ε = ε0(Γ) such that the first item holds, and we have the estimate ∥Ψ −1 Γ (f)∥Eβ ≲ ∥f∥Fβ (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) for all f ∈ Fβ(ε0(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, by the second item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, we have that for all s ∈ [β, R + r − β] ∩ N = [9 + ⌊n/2⌋, 11 + 3⌊n/2⌋] ∩ N it holds that if f ∈ Fβ(ε0(Γ)) ∩ Fs, then Ψ −1 Γ (f) ∈ Es and estimate (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) holds with β replaced by s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Therefore, we may set εν(Γ) = ε0 for ν ∈ {0, 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , 2 + 2⌊n/2⌋ = R + r − 2β}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now, given N ∋ ν > 2+2⌊n/2⌋, we define εν(Γ) ∈ (0, ε0] to be the smallness parameter ε provided by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21 with parameter triple (µ, r, �R), where �R = R + ν − (2 + 2⌊n/2⌋).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that the LRI mapping hypotheses are satisfied in this case if we take δ �R = min{ρR+ν−(2+2⌊n/2⌋), δr}, where the former parameter is from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18 (with ν = R + ν − (2 + 2⌊n/2⌋) and I = [Γ/2, 2Γ]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second item now follows from this definition of {εν(Γ)}∞ ν=0, the second item of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21, and uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, the third and fourth items now follow directly from Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As a consequence of the third conclusion of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='24, the local inverse map Ψ −1 Γ produced in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 satisfies certain higher-order differentiability assertions beyond the basic continuous differentiability result in the third item of the previous theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' However, the precise statements become cumbersome to enumerate due to a compounding of the derivative loss in the formulas for higher-order derivatives of the inverse map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' As such, we have chosen not to state these precisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' PDE construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We utilize Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 to prove our main result about the free boundary compressible Navier-Stokes equations, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Traveling wave for free boundary compressible Navier-Stokes, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that the parameters µ, λ, and ς satisfy (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) and that P satisfies P ′ > 0 and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let β = 9 + 2⌊n/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exist a family {V(γ)}γ∈R+ of open sets of Xβ and a nonincreasing sequence {Us}∞ s=β of open sets of W1+β × R+ such that the following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) Nondegeneracy: We have that {0} × R+ ⊆ �∞ s=β Us and 0 ∈ � γ∈R+ V(γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Existence and uniqueness: For all (T , G, F, γ) ∈ Uβ there exists a unique (q, u, η) ∈ V(γ) such that the traveling wave formulation for the free boundary compressible Navier-Stokes equations, system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), is classically satisfied with wave speed γ, data (T , G, F), and solution (q, u, η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (3) Higher regularity, given low norm smallness: If N ∋ s ⩾ β and (T , G, F, γ) ∈ Us∩(W1+s×R), then the corresponding solution satisfies (q, u, η) ∈ V(γ) ∩ X1+s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (4) Continuous dependence: For any N ∋ s ⩾ β, the solution map Us ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (q, u, η) ∈ X1+s (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) is continuous with respect to the W1+s × R and X1+s norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 121 (5) No vacuum formation: There exists positive constants c, C ∈ R+ such that for all (q, u, η) ∈ � γ∈R+ V(γ) we have that c ⩽ σq,η ⩽ C, where σq,η is defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (6) Flattening map diffeomorphism: For any N ∋ s ⩾ β and (q, u, η) ∈ Xs ∩ � γ∈R+ V(γ) we have that the flattening map Fη from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is a smooth diffeomorphism from Ω to Ω[η] that extends to a Cs+2−⌊n/2⌋ diffeomorphism from Ω to Ω[η].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For each Γ ∈ R+, we many invoke Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and acquire a nonincreasing sequence {εν(Γ)}∞ ν=0 ⊂ (0, min{ρWD, Γ/2}) and a κ(Γ) ∈ R+ such that the various conclusions of the theorem hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then define the open sets Us = � Γ∈R+ BW1+β×R((0, Γ), εs−β+1(Γ)/C) for N ∋ s ⩾ β (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) for some constant C ⩾ 1 (independent of s) to be determined, and V(Γ) = BXβ(0, κ(Γ)ε0(Γ)) for Γ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) The first item is now clear by inspection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For the second item, we apply the first conclusion of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, for (T , G, F, γ) ∈ Uβ we have existence by setting (q, u, η) to be the first three components of the tuple Ψ −1 γ (0, 0, 0, T , G, F, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Uniqueness in V(γ) follows from the fact that if Ψ(q, u, η, T , G, F, γ) = Ψ(�q, �u, �η, T , G, F, γ), then Ψγ(q, u, η, T , G, F, 0) = Ψγ(�q, �u, �η, T , G, F, 0) and hence (q, u, η) = (�q, �u, �η) by the uniqueness assertions of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now prove the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' An immediate consequence of the second conclusion of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 is that if N ∋ s ⩾ β and (T , G, F, γ) ∈ Us ∩ (W1+s × R), then the corresponding solution satisfies (q, u, η) ∈ V(γ) ∩ Xs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, the solution is one degree more regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' To see this, we recall the map Φ from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) and note that the equation Ψ(q, u, η, T , G, F, γ) = (0, 0, 0, T , G, F, γ) is equivalent to Ψ(q, u, η, 0, 0, 0, γ) = (−Φ(q, u, η, T , G, F), 0, 0, 0, γ), which in turn is equivalent to Ψγ(q, u, η, 0, 0, 0, 0) = (−Φ(q, u, η, T , G, F), 0, 0, 0, 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Note that Φ(q, u, η, T , G, F) is linear in (T , G, F);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' consequently, Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 (with m = 0) show that ∥Φ(q, u, η, T , G, F)∥Yβ ⩽ c∥T , G, F∥Wβ and ∥Φ(q, u, η, T , G, F)∥Y1+s ≲∥q,u,η∥Xs ∥T , G, F∥W1+s (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) for a constant c depending only on the dimension, the physical parameters, and ρWD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume the constant C ⩾ 1 from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) satisfies C ≳ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then use the inclusion (T , G, F, γ) ∈ Us∩(W1+s×R) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) to see that (−Φ(q, u, η, T , G, F), 0, 0, 0, γ) ∈ F1+s ∩ � Γ∈R+ BFβ×R((0, Γ), εs−β+1(Γ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) Hence, we can apply the first and second conclusions of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 to deduce from (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) that the third item holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' It remains to justify the fourth item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Identity (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and the third conclusion of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 show that for N ∋ s ⩾ β, the solution map Us−1 ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (q, u, η) ∈ Xs (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, we can do one derivative better for the target space topology by arguing as in the proof of the third item above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Indeed, by Propositions 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15 again, along with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7), we find that the map Us ∩ (W1+s × R) ∋ (T , G, F, γ) �→ (−Φ(q, u, η, T , G, F), 0, 0, 0, γ) ∈ F1+s (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In light of the identity Ψ(q, u, η, 0, 0, 0, γ) = (−Φ(q, u, η, T , G, F), 0, 0, 0, γ), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) 122 NOAH STEVENSON AND IAN TICE the third conclusion of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (which implies that Ψ −1 is continuous), the inclusion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) (which implies that the right side of (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is in the domain of Ψ −1), the continuity of the map (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8), and the continuity of compositions, we then complete the verification of the fourth item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The fifth and sixth items are immediate consequences of the first conclusions of Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We conclude this section by recording some simple consequences of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (Open set of data for fixed wave speed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For each γ ∈ R+ there exists a nonempty open set (0, 0, 0) ∈ W(γ) ⊂ W1+β with the property that for all stress-force data tuples (T , G, F) ∈ W(γ) there exists a unique (q, u, η) ∈ V(γ) such that system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) is satisfied with solution (q, u, η), wave speed γ, and data (T , G, F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given γ ∈ R+ we take W(γ) = {(T , G, F) : (T , G, F, γ) ∈ Uβ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Corollary 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (On the Eulerian formulation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Each solution to the flattened, perturbative enthalpy formulation of the traveling wave problem for free boundary compressible Navier-Stokes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e sys- tem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), produced by Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, gives rise to a classical solution to the traveling Eulerian formulation of the problem given by system (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The sixth item of Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 verifies that the flattening map Fη : Ω → Ω[η] from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) is a smooth diffeomorphism that is sufficiently smooth up to the boundary as to preserve the notion of classical solutions upon undoing the flattening.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This same theorem also gives us classical solutions to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) as an easy consequence of various supercritical Sobolev embeddings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By combining these facts and undoing the nonlinear changes of unknowns that took us from (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) to (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9), we obtain the stated result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Standard Sobolev space tools A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Extension operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall the Poisson extension operator E0 and the variant E introduced in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) and (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following lemma records some simple mapping properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (Mapping properties of the Poisson extension operator variants).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) E0 : Hs+1/2(Σ) → 0H1(Ω) ∩ H1+s(Ω) is a bounded linear map for each for s ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) E ◦ Π1 H = E0, and E ◦ Π1 L : H0(Σ) → W ∞,∞((0, b);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H0 (1)(Rn−1)) is a continuous linear map, where H0 (1) is defined by (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first item is clear by standard elliptic theory for the Dirichlet problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second item is immediate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Throughout the paper is is also frequently useful to consider extension operators mapping functions defined on domains to functions defined on the entire Euclidean space that are regularity preserving for the entire Sobolev scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For these we use the celebrated extension operators of Stein;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' the following definition sets our notation for these.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Definition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (Stein-Extension operators and domains).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The notion of a Stein extension operator is given in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 in Chapter VI of Stein [96].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A Stein extension domain is an open set U ⊂ Rd for which there exists a Stein extension operator, which we will denote by EU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will also employ this notation more generally for open subsets of finite dimensional real vector spaces of dimension d via the standard identification with Rd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 123 A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Korn’s inequalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our first result here, which is classical, involves the (not normalized) symmetric gradient, which we recall is defined for differentiable vector fields f by Df = ∇f + ∇ft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 (Korn’s inequality for the symmetric gradient, n ⩾ 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let n ⩾ 2 be the ambient dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then there exists a constant c ∈ R+, depending only Ω and n, such that for all f ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) we have the inequality ∥f∥H1 ⩽ c ∥Df∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The proof can be found in Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 in Beale [8], and is based on Theorem 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='II of Fichera [34], which gives a Korn inequality in cubes of general dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We note that the result in [8] is only stated for n = 3, but the same argument works in general for n ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We are also interested in inequalities involving the trace-free part of the symmetric gradient, also known as the deviatoric gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We denote this for differentiable vector fields f by D0f = ∇f + ∇ft − 2 n(∇ · f)I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In dimensions n ⩾ 3, the analog of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3 for D0 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (Deviatoric Korn’s inequality, n ⩾ 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let n ⩾ 3 be the ambient dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' There exists a constant c ∈ R+, depending only on Ω and n, such that for all f ∈ 0H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) we have the inequality ∥f∥H1 ⩽ c∥D0f∥L2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See, for instance, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 in Dain [22] for a proof of the bound ∥f∥H1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Rn) ≲ ∥D0f∥L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Rn×n) + ∥f∥L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Rn) for all f ∈ H1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) whenever Q is a cube.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By a standard compactness argument, we can then show that ∥f∥H1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Rn) ≲ ∥D0f∥L2(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='Rn×n) for all f ∈ H1(Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rn) vanishing on one side of the cube, where the implicit constant depends on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We then utilize this inequality and the symmetries of Ω, as was done in Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3, to obtain the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 (Failure of deviatoric Korn in dimension n = 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For dimension n = 2, however, the analog of Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 is false.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' See, for instance, Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 of Bauer, Neff, Pauly, and Starke [6], where it is proved that there exists a Lipschitz domain ∅ ̸= U ⊂ R2 and a proper segment ∅ ̸= S ⊂ ∂U with the property that ∥·∥H1 ̸≲ ∥D0(·)∥L2 on the subspace of H1(U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R2) consisting of vector fields vanishing on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Refined interpolation of Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In this subsection we derive refined interpolation inequalities for linear mappings between Sobolev spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' in particular, this discussion considers improvements for the K-method of real interpolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For more information on the basic K-method of interpolation, we refer to Chapter 3 in Bergh and L¨ofstr¨om [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we recall the definition of the K-functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given a pair of Banach spaces X0 and X1 that embed into a Hausdorff topological vector space Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' X0, X1 �→ Z, we define the map K(·, ·;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' X0, X1) : R+ × (X0 + X1) → R, via K(t, x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' X0, X1) = inf{∥x0∥X0 + t∥x1∥X1 : x = x0 + x1, (x0, x1) ∈ X0 × X1} (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) for (t, x) ∈ R+ × (X0 + X1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Our first lemma realizes the K-functional on Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 (Sobolev space K-functional).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let s0, s1, s ∈ R, be such that s > 0 and s0 + s = s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For t ∈ R+ define the bounded linear map P s t : Hs0(Rn) → Hs0+2s(Rn) via P s t = (1 + t2⟨∇⟩2s)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For all f ∈ Hs0(Rn) we have that 2−1K(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs1)2 ⩽ ∥(I − P s t )f∥2 Hs0 + t2∥P s t f∥2 Hs1 ⩽ K(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Define ˆK(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) = infg∈Hs1 � ∥f − g∥2 Hs0 + t2∥g∥2 Hs1 � and note that elementary estimates show that 2−1K(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1)2 ⩽ ˆK(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) ⩽ K(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The benefit of switching to ˆK is that we may readily employ the direct method in the calculus of variations to see that the infimum in the definition of ˆK(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) is actually a minimum, achieved by f⋆ ∈ Hs1 satisfying (by virtue of the Euler-Lagrange equations) − ⟨⟨∇⟩s0(f − f⋆), ⟨∇⟩s0g⟩L2 + t2⟨⟨∇⟩s0+sf⋆, ⟨∇⟩s0+sg⟩L2 = 0 for every g ∈ Hs1(Rn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) 124 NOAH STEVENSON AND IAN TICE This is equivalent to saying that f⋆ is a weak solution to the elliptic pseudo-differential equation (I + t2⟨∇⟩2s)⟨∇⟩s0f∗ = ⟨∇⟩s0f, or alternatively, f⋆ = (I +t2⟨∇⟩2s)−1f = P s t f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Thus, ˆK(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) = ∥(I − P s t )f∥2 Hs0 + t2∥P s t f∥2 Hs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Our next lemma expresses the norm in Sobolev spaces in terms of the K-functional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The point of the following computation is to divide the K-functional norm by a suitable constant to remove the degeneracies near the end-points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7 (A norm computation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let s0, s1, s ∈ R+ be such that s > 0 and s0 + s = s1, and let σ ∈ (0, 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Define c(σ) = � ∞ 0 τ 1−2σ 1+τ 2 dτ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have the equivalence of norms ∥f∥H(1−σ)s0+σs1 ⩽ � 1 c(σ) � ∞ 0 t−2σ−1K(t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1)2 dt �1/2 ⩽ √ 2∥f∥H(1−σ)s0+σs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given t ∈ R+, we may equate ∥(I − P s t )f∥2 Hs0 + t2∥P s t f∥2 Hs1 = � Rn⟨2πξ⟩2s0� t2⟨2πξ⟩2s 1 + t2⟨2πξ⟩2s �2 |F[f](ξ)|2 dξ + � Rn⟨2πξ⟩2(s0+s)� t 1 + t2⟨2πξ⟩2s �2 |F[f](ξ)|2 dξ = � Rn⟨2πξ⟩2s0 t2⟨2πξ⟩2s 1 + t2⟨2πξ⟩2s |F[f](ξ)|2 dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Hence, by Tonelli’s theorem and a change of variables, we have that � ∞ 0 t−2σ−1(∥(I − P s t )f∥2 Hs0 + t2∥P s t f∥2 Hs1) dt = � Rn⟨2πξ⟩2s0 � R+ t−2σ−1 t2⟨2πξ⟩2s 1 + t2⟨2πξ⟩2s dt |F[f](ξ)|2 dξ = c(σ) � Rn⟨2πξ⟩2(s0+σs)|F[f](ξ)|2 dξ = c(σ)∥f∥2 H(1−σ)s0+σs1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) The result then follows by combining this with Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ We now come to the main result of this subsection of the appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8 (Refined interpolation of Sobolev spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Let s0, s1, s, r0, r1, r ∈ R with s, r > 0, s0 + s = s1, and r0 + r = r1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Assume that T ∈ L(Hr0(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0(Rn)) ∩ L(Hr1(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs1(Rn)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) is such that for some constants C0, C1, A ∈ R+ we have the bounds ∥Tf∥s0 ⩽ C0∥f∥r0 and ∥Tf∥s1 ⩽ C1∥f∥r1 + A∥f∥r0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) for all appropriate f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Set r−1 = r0 − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then for all σ ∈ [0, 1] we have the inclusion T ∈ L(H(1−σ)r0+σr1(Rn);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' H(1−σ)s0+σs1(Rn)) (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) and the bound ∥Tf∥H(1−σ)s0+σs1 4C1−σ 0 Cσ 1 ⩽ ∥f∥H(1−σ)r0+σr1 + σ1/2 A C1 ∥f∥H(1−σ)r−1+σr0 (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) for all appropriate f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' COMPRESSIBLE TRAVELING WAVES 125 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given t ∈ R+ and f ∈ Hr0, we decompose f = (I − P r C−1 0 C1t)f + P r C−1 0 C1tf, with (I − P r C−1 0 C1t)f ∈ Hr0 and P r C−1 0 C1tf ∈ Hr1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By the definition of K and the boundedness hypotheses, we see may then estimate K(t, Tf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) ⩽ ∥T(I − P r C−1 0 C1t)f∥Hs0 + t∥TP r C−1 0 C1tf∥Hs1 ⩽ C0 � ∥(I − P r C−1 0 C1t)f∥Hr0 + C−1 0 C1t∥P r C−1 0 C1tf∥Hr1 � + At∥P r C−1 0 C1tf∥Hr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) We apply Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6 to see that K(t, Tf;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hs0, Hs1) ⩽ √ 2C0K(C−1 0 C1t, f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hr0, Hr1) + At∥P r C−1 0 C1tf∥Hr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) Upon squaring, multiplying by t−2σ−1, integrating over R+, and employing Lemma A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7, we acquire the bound 4−1c(σ)∥Tf∥2 H(1−σ)s0+σs1 ⩽ c(σ)C2(1−σ) 0 C2σ 1 ∥f∥2 H(1−σ)r0+σr1 + A2(C−1 0 C1)2(σ−1) � ∞ 0 τ −2σ+1∥P r τ f∥2 Hr0 dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) The final integral above can then be computed explicitly by using Tonelli’s theorem: � ∞ 0 τ −2σ+1∥P r τ f∥2 Hr0 dτ = � Rn⟨2πξ⟩2r0 � ∞ 0 τ −2σ+1 (1 + τ 2⟨2πξ⟩2r)2 dτ |F[f](ξ)|2 dξ = d(σ) � Rn⟨2πξ⟩2(r0+(σ−1)r)|F[f](ξ)|2 dξ = d(σ)∥x∥2 H(1−σ)r−1+σr0, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) where we define d(σ) = � ∞ 0 τ −2σ+1 (1+τ 2)2 dτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Together, (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) imply that ∥Tf∥H(1−σ)s0+σs1 2C1−σ 0 Cσ 1 ⩽ ∥f∥H(1−σ)r0+σr1 + A C1 � d(σ) c(σ) ∥f∥H(1−σ)r−1+σr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) The proof is complete upon noting the bounds c(σ) ⩾ � 1 0 τ 1−2σ 2 dτ + � ∞ 1 τ 1−2σ 2τ 2 dτ = 1 4( 1 1−σ + 1 σ) and d(σ) ⩽ � 1 0 τ −2σ+1 dτ + � ∞ 1 τ −2σ−3 dτ = 1 2( 1 1−σ + 1 1+σ), which imply that d(σ) c(σ) ⩽ 4σ 1+σ ⩽ 4σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Appendix B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Some nonstandard function spaces B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The anisotropic Sobolev spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For R ∋ s ⩾ 0 and d ∈ N+ we define the anisotropic Sobolev space Hs(Rd) = {f ∈ S ∗(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) : F[f] ∈ L1 loc(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' C), ∥f∥Hs < ∞}, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) equipped with the norm ∥f∥Hs = � � Rd � |ξ|−2(ξ2 1 + |ξ|4)1B(0,1)(ξ) + ⟨ξ⟩2s1Rd\\B(0,1)(ξ) � |F[f](ξ)|2 dξ �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2) These spaces were introduced in Leoni and Tice [60], where it was shown, in Proposition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6, that Hs(Rd) is a Hilbert space and Hs(Rd) �→ Hs(Rd) �→ Hs(Rd) + C∞ 0 (Rd), with equality in the first embedding if and only if d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The ‘anisotropic’ descriptor is justified by the low frequency multiplier in (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1) as well as Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in [60], which shows that Hs(Rd) is not closed under composition with rotations when d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We make the following notation for band-limited subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Given κ ∈ R+, we define the space H0 (κ)(Rd) ⊆ � s⩾0 Hs(Rd) via H0 (κ)(Rd) = {f ∈ H0(Rd) : suppF[f] ⊆ B(0, κ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3) We will also sometimes write Hs (κ)(Rd) = H0 (κ)(Rd) for any 0 ⩽ s ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' 126 NOAH STEVENSON AND IAN TICE Now let us enumerate the properties of theses spaces pertinent to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' First, we discuss a high-low decomposition for which the following notational convention is set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' For κ ∈ R+, we define the linear operators Πκ L and Πκ H on the subspace of f ∈ S ∗(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' C) such that F[f] is locally integrable via Πκ Lf = F −1[1B(0,κ)F[f]] and Πκ Hf = (I − Πκ L)f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4) Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 (Frequency splitting for anisotropic Sobolev spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold for 0 ⩽ s ∈ R, f ∈ Hs(Rd), and κ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) We have the equivalence ∥f∥Hs ≍s,κ � ∥Πκ Lf∥2 H0 + ∥Πκ Hf∥2 Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) We have that Πκ Lf ∈ C∞ 0 (Rd) with the estimates ∥Πκ Lf∥W k,∞ ≲k,κ ∥Πκ Lf∥H0 for every k ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5 in Leoni and Tice [60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ The following algebra properties of the anisotropic Sobolev spaces are extremely important in our nonlinear analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 (Algebra properties of anisotropic Sobolev spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Suppose that f1, f2 ∈ H0 (κ)(Rd) for some κ ∈ R+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) The pointwise product f1f2 belongs to H0 (2κ)(Rd) and satisfies the estimate ∥f1f2∥H0 ≲κ ∥f1∥H0∥f2∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Set rd = � 1 + � d+1 d−1 � , if d > 1, 1 if d = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='5) Assume additionally that f3, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' , frd ∈ H0 (κ)(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then the pointwise product �rd j=1 fj belongs to (L2 ∩ H0 (rd·κ))(Rd) and satisfies ��� rd � j=1 fj ��� L2 ≲κ,d rd � j=1 ∥fj∥H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='6) (3) If g ∈ Hs(Rd), then the pointwise product f1g belongs to Hs(Rd) and satisfies ∥f1g∥Hs ≲κ,s ∥f1∥H0∥g∥Hs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The first item is proved in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2 in Koganemaru and Tice [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now prove the second item in the case that κ = 1 and d ⩾ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The general case for d ⩾ 2 can be handled similarly, and the case d = 1 is trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We define the function µ : B(0, 1) → R via µ(ξ) = |ξ|−2(ξ2 1 + |ξ|4) and note that µ is the multiplier that encodes the low frequency control in H0(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We will calculate for which q ∈ [1, ∞) we have 1/µ ∈ Lq(B(0, 1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Consider the decomposition B(0, 1) = E0 ⊔ E1 E0 = {ξ = (ξ1, ξ∗) ∈ B(0, 1) : |ξ1 ± 1/2|2 + |ξ∗|2 < 1/4}, E1 = B(0, 1) \\ E0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='8) An elementary calculation shows that for ξ ∈ B(0, 1) we have that ξ ∈ E0 ⇔ |ξ|4 < |ξ1|2 and ξ ∈ E1 ⇔ |ξ1|2 ⩽ |ξ|4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='9) In turn, these show that � ξ2 1/ |ξ|2 ⩽ µ(ξ) ⩽ 2ξ2 1/ |ξ|2 for ξ ∈ E0 |ξ|2 ⩽ µ(ξ) ⩽ 2 |ξ|2 for ξ ∈ E1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='10) For 1 ⩽ q < ∞, we then have the equivalence � B(0,1) µ−q ≍ � E0 ξ−2q 1 |ξ|2q dξ + � E1 |ξ|−2q dξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) COMPRESSIBLE TRAVELING WAVES 127 We may use spherical coordinates in the ξ∗ ∈ Rd−1 variable to see that � E1 |ξ|−2q dξ < ∞ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) if and only if � 1/2 0 rd−2 � 1/2−√ 1/4−r2 0 dx1 (x2 1 + r2)q dr < ∞, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='13) but for the latter we can bound � 1/2 0 rd−2 � 1/2−√ 1/4−r2 0 dx1 (x2 1 + r2)q dr ⩽ � 1/2 0 rd−2−2q�1 2 − � 1 4 − r2 � dr, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='14) and since the term in parentheses behaves like r2 for r ∼ 0 this will be finite if and only if d − 2 − 2q + 2 > −1 ⇔ q < (d + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' By putting this together, we see that if q < (d + 1)/2 then the integral (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='12) is indeed finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Next we consider the E0 integral, again using spherical coordinates in ξ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We have that � E0 ξ−2q 1 |ξ|2q dξ < ∞ (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) if and only if � 1/2 0 rd−2 � 1/2 1/2−√ 1/4−r2 � 1 + r2 x2 1 �q dx1 dr < ∞, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='16) and we know that this integral is finite if we know that � 1/2 0 rd−2 � 1/2 1/2−√ 1/4−r2 � r2 x2 1 �q dx1 dr < ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='17) For this latter integral we compute � 1/2 0 rd−2 � 1/2 1/2−√ 1/4−r2 � r2 x2 1 �q dx1 dr ⩽ 1 2q − 1 � 1/2 0 rd−2+2q�1 2 − � 1 4 − r2 �1−2q dr, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='18) and this is finite if and only if d − 2 + 2q + 2(1 − 2q) > −1 ⇔ q < d+1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' So once more we guarantee the integral (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='15) is finite if q < (d + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Hence (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='11) is finite for the same range of q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Now suppose that f ∈ H0 (1)(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Since ∥f∥H0 ≍ ∥√µ(∇/2πi)f∥L2, it follows from H¨older’s inequality that for 1 ⩽ p < 2 we have ∥F[f]∥Lp ≲ ∥f∥H0 � � B(0,1) (1/µ)p/(2−p)�1/p−1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='19) The above analysis shows that the coefficient given by the integral on the right hand side is finite if and only if p/(2 − p) < (d + 1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' This is equivalent to p < 2(d+1) d+3 , and thus F[f] ∈ Lp(Rd) for 1 ⩽ p < 2(d+1) d+3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In turn, thanks to the Hausdorff-Young inequality, we have that ∥f∥Lq ≲q ∥f∥H0 for 2(d + 1) d − 1 < q ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20) Hence, by H¨older’s inequality, for any k ∈ N with k ⩽ 2(d+1) d−1 we have that ∥fk∥Lq ≲q,k ∥f∥k H0 for 2(d + 1) k(d − 1) < q ⩽ ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='21) 128 NOAH STEVENSON AND IAN TICE Define rd = min � k ∈ N : 2(d+1) k(d−1) < 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Note that the minimum exists since 2 < 2(d+1) d−1 ⩽ 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, it is easy to see that rd = 1 + �d + 1 d − 1 � = � � � � � 4 if d = 2, 3 if d = 3, 2 if d > 3, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='22) and that ∥frd∥L2 ≲ ∥f∥rd H0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The second item then follows by one more application of H¨older’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Finally, we prove the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 and the first item allow us to estimate ∥f1g∥Hs ⩽ ∥f1Πκ Lg∥Hs + ∥f1Πκ Hg∥Hs ≲ ∥f1∥H0∥Πκ Lg∥H0 + ∥f1Πκ Hg∥Hs ≲ ∥f1∥H0∥g∥Hs + ∥f1∥W s,∞∥Πκ Hg∥Hs ≲ ∥f1∥H0∥g∥Hs, (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='23) which completes the proof of the third item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' □ Remark B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' From the proof of Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='2, specifically (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='20), and Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='1 we deduce that for R ∋ s ⩾ 0, 2(d + 1)/(d − 1) < q ⩽ ∞ and f ∈ Hs(Rd), the low mode part Π1 Lf belongs to Lq(Rd) and the high mode part Π1 Hf belongs to L2(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' In fact, the induced embedding Hs(Rd) �→ (Lq + L2)(Rd) is continuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' We now develop a spatial characterization of the specialized Sobolev spaces that will be useful when employing these spaces in a priori estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Recall that the homogeneous spaces ˙H−1 are defined in (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Proposition B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content='4 (Characterizations of the anisotropic Sobolev spaces).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' The following hold for R ∋ s ⩾ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (1) If f ∈ S ∗(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' R) satisfies ∇f ∈ Hs−1(Rd;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Rd) and ∂1f ∈ ˙H−1(Rd), then there exists a constant c ∈ R such that f − c ∈ Hs(Rd).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' (2) Assume d ⩾ 2 and write pd = 2(d+1) d−1 > 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_NAyT4oBgHgl3EQf3vlI/content/2301.00773v1.pdf'} +page_content=' Then we have the equality Hs(Rd) = � f ∈ L2(Rd) + � pd